On Global Existence and Regularity of Solutions for a Transport Problem Related to Charged Particles

Abstract

The paper considers a class of linear Boltzmann transport equations which models charged particle transport, for example in dose calculation of radiation therapy. The equation is an approximation of the exact transport equation containing hyper-singular integrals in its collision terms. The paper confines to the global case where the spatial domain G is the whole space ${\mathbb{R}}^3.$ Existence results of solutions for the due initial value problem are formulated by applying variational methods. In addition, some regularity results of solutions are verified in scales of relevant anisotropic mixed-norm Sobolev spaces.

Keywords:

• Linear Boltzmann transport equation
• continuous slowing down approximation
• variational formulation
• regularity of solutions
• charged particle transport

AMS CLASSIFICATION:

• 35Q20
• 35R09
• 45E99

1. Introduction

This paper deals with the existence and regularity of solutions of the linear Boltzmann transport equation (BTE) (1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) in ${\mathbb{R}}^3\times S\times I.$ Here S is the unit sphere (velocity direction domain) and $I=[E_0,E_{\mathrm{m}}]$ is the energy interval. ${\nabla }_x$ is the gradient with respect x-variable and ${\nabla }_S$ and Δ_S are the gradient and Laplacian on S, respectively. The solution satisfies the initial condition (2) $$\psi (.,.,E_{\mathrm{m}})=0$$(2) where $E_{\mathrm{m}}$ is the so-called cutoff energy. This condition guarantees that the overall problem is well-posed. The restricted collision operator K_r is a partial integral type operator (see section 3.1 below) (3) $$\begin{array}{l}{K}_r\psi ={\int }_{S\prime }{\int }_{I\prime }{\sigma }^1(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )dE\prime d\omega \prime +{\int }_{S\prime }{\sigma }^2(x,\omega \prime ,\omega ,E)\psi (x,\omega \prime ,E)d\omega \prime \\ +{\int }_{I\prime }{\int }_0^{2\pi }{\sigma }^3(x,E\prime ,E)\psi (x,\gamma (E\prime ,E,\omega )(s),E\prime )\mathit{dsdE}\prime .\end{array}$$(3)

The term $\psi \to \omega ·{\nabla }_x\psi$ on the left in (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) is called a convection (or advection) operator, the term $\psi \to \Sigma \psi$ is a scattering operator and the term $\psi \to K_r\psi$ is a restricted collision operator. The term $\psi \to d·{\nabla }_S\psi$ is needed when external forces (electro-magnetic fields, for example) are present. The term $\psi \to a\frac{\partial \psi }{\partial E}$ represents the energy attenuation and $c{\Delta }_S\psi$ is linked to the diffusion of the angular variable on the sphere. On the right, the function $f=f(x,\omega ,E)$ represent the internal source. The solution ψ of the problem (1, 2) describes the fluence of the considered particle. We remark that in the case where the spatial domain $G\cancel{=}{\mathbb{R}}^3$ one must impose an additional inflow boundary condition (4) $${\psi }_{|{\Gamma }_{-}}=g$$(4) where ${\Gamma }_{-}$ is the “inflow boundary” $\{(y,\omega ,E)\in (\partial G)\times S\times I|\omega ·\nu (y)<0\}.$ Generally speaking the existence and regularity analysis in this case is more sophisticated than in the global case $G={\mathbb{R}}^3.$

In Tervo et al. (2018), Tervo (2019, section 6) and Tervo and Herty (2020) one has given reasons to use the equations like (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) for charged particle transport, for example for electrons and positrons in radiation therapy dose calculation. The starting point is that differential cross sections for charged particles may contain hyper-singularities with respect to energy variable and hence the corresponding exact (original) collision operator is a partial hyper-singular integral operator. This operator can be reasonably approximated which leads to an approximative equation of the form (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) .

We find that the operator in (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) is the sum of the second order partial differential operator and a partial integral operator. Let (5) $$A(x,\omega ,E,D)\psi :=\frac{1}{a}({\Delta }_S\psi +d·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma \psi )$$(5)

Then the equation in (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) is for a > 0 equivalent to (6) $$\frac{\partial \psi }{\partial E}+A(x,\omega ,E,D)\psi +\frac{1}{a}{K}_r\psi =\frac{1}{a}f.$$(6)

In the case where K_r = 0 the problem (1, 2(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) ) is an initial value problem for the partial differential equation (7) $$\frac{\partial \psi }{\partial E}+A(x,\omega ,E,D)\psi =\frac{1}{a}f,\psi (.,.,E_{\mathrm{m}})=0.$$(7)

The existence and regularity results of this reduced problem mirror those of the complete problem (1, 2(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) ).

The operator $\frac{\partial }{\partial E}+A(x,\omega ,E,D)$ is hyperbolic in nature in its wide sense (e.g. Rauch 2012, 47, Pazy 1983, 134). The literature contains numerous contributions for existence and regularity analysis of general partial differential initial boundary value problems which are hyperbolic in nature or which are formally dissipative beginning from Lax and Phillips (1960, Theorem 3.2), Friedrichs (1958) and Phillips and Sarason (1966). More recent results can be found e.g. in Hörmander (1985), Rauch and Massey (1974), Rauch (1985, Chapter XXIII), Morando et al. (2009), Nishitani and Takayama (1998), Nishitani and Takayama (2000).

Some specific results concerning for existence and regularity of solutions of transport problems can also be found in the literature. We mention some of them. In the case where $a=c=d=0$ existence of solutions for transport problems like (1, 4) has been studied for single equations e.g. in Agoshkov (1998), Dautray and Lions (1999), Egger and Schlottbom (2014) and for coupled systems in (Tervo and Kokkonen 2017). In Agoshkov (1998, Chapter 4) a systematic study of the regularity of solutions is exposed. Therein a single monokinetic BTE is considered and the spatial domain G is a bounded subset of ${\mathbb{R}}^n,n=1,2,3$ with sufficiently regular boundary. In the above references it is assumed that the restricted collision operator K_r satisfies a (partial) Schur criterion for boundedness (Halmos and Sunder 1978, 22). In Tervo et al. (2018a, 2018b), we studied existence of solutions for the case $c=d=0.$ In Frank et al. (2010) additionally it has been assumed that $a=a(E).$ In Tervo and Herty (2020) we proved existence results for the problem (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) , (4)(4) $${\psi }_{|{\Gamma }_{-}}=g$$(4) , (2)(2) $$\psi (.,.,E_{\mathrm{m}})=0$$(2) when the spatial domain G was bounded.

Related results for (deterministic and linear) Fokker-Planck type equations can be found in Degond (1986, Appendix A) and Tian (2014). Methods in Degond (1986) are closely related to our techniques in section 3 below. In Le Bris and Lions (2008), existence results of solutions are shown for a class of time-dependent Fokker-Planck equations with irregular (having only Sobolev regularity) coefficients. Results in Chupin (2010) consider existence results for a special form of stationary Fokker-Planck equation in weighted Sobolev spaces. In Herty et al. (2012) and Herty and Sandjo (2011) existence results are obtained in the context of dose calculation for optimal radiation treatment planning.

In Mokhtar-Kharroubi (1991), Sobolev regularity results up to order 1 are proved when $a=c=d=0$ by applying singular integral methods (see also Zeghal 2012). In Bouchut (2002) one has shown that a time-dependent transport problem satisfies a kind of “gain in x-regularity with the help of ω-regularity.” Note that the time-dependent equation is of the form (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) when we replace time t with energy E. In Alonso and Sun (2014, especially Theorem 5.3) and in some of its references one has considered regularity results for the mono-kinetic equation (here $S_{n-1}$ denotes the unit sphere in ${\mathbb{R}}^n$) $$\frac{\partial u}{\partial t}+\omega ·{\nabla }_{x}u-Ku=0\text{in}{\mathbb{R}}^n\times S_{n-1},u(.,.,0)=u_0$$ where the collision operator is of the form $$(Ku)(x,\omega ):={\int }_{S_{n-1}}(u(x,\omega \prime )-u(x,\omega )\sigma (\omega ,\omega \prime )d\omega \prime .$$

Chen and He (2012) have investigated the regularity of solutions of time-dependent nonlinear equation (for general foundations of these equations see e.g. Ukai and Yang 2010) (8) $$\frac{\partial u}{\partial t}+\omega ·{\nabla }_{x}u-Q(u,u)=0,u(.,.,.,0)=u_0.$$(8)

The paper confines to the case n = 3 and to the periodic solutions in spatial variable. $Q(.,.)$ is an appropriate bilinear form.

In this paper we restrict ourselves to the case where the spatial dimension n = 3 and the Lebesgue index p = 2. The regularity results are formulated utilizing the anisotropic Sobolev spaces $H^{(m_1,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°}).$ For the first instance, we in section 2 introduce these spaces for integer indexes $(m_1,m_2,m_3)$ and bring up some of their properties. These spaces are subspaces of $L^2({\mathbb{R}}^3\times S\times I).$

In Section 3 we consider the existence and uniqueness of solutions for the problem (1, 2) in global case $G={\mathbb{R}}^3.$ The solution spaces are certain subspaces of $L^2({\mathbb{R}}^3\times S\times I).$ The proofs are modifications of proofs given in Tervo and Herty (2020) for the case of a bounded spatial domain G. The results are based on the Lions-Lax-Milgram Theorem. In Tervo et al. (2018), section 6.2 we applied same kind of techniques in the case where $c=d=0.$ In Tervo et al. (2018a, 2018b), we considered related results rested on the m-dissipativity of the smallest closed extension of the partial differential part of the transport operator, the methods of which offer an alternative approach for existence analysis.

The focus of this paper is in Sections 4 and 5 where we study the regularity of solutions in the global case $G={\mathbb{R}}^3.$ We restrict ourselves in the main to the spatial regularity (x-regularity) but some outlines for the regularity with respect to all variables are stated as well. We shall find that the main principle operating in the global case is roughly speaking that the “regularity of solutions increases according to the regularity of data.” We proceed in the increasing order of complexity. For the first instance in section 4 we deal with merely the slowing down-convection-scattering equation (that is, $K_r=0,c=d=0$) and therein the proofs are founded on the known explicit formulae of solutions. These treatments suggest relevant anisotropic Sobolev spaces within which regularity results can be formulated. After that the total equation (1)(1) $$a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi +\Sigma (x,\omega ,E)\psi -K_r\psi =f$$(1) is handled in section 5 Therein the proofs are based on the use of partial differences and on the obtained a priori estimates.

We finally expose some notes for the case where $G\cancel{=}{\mathbb{R}}^3.$ In this case we must impose additionally an inflow boundary condition (4)(4) $${\psi }_{|{\Gamma }_{-}}=g$$(4) . This inflow boundary condition causes some problems because the corresponding initial inflow boundary value problem (1, 2, 4(4) $${\psi }_{|{\Gamma }_{-}}=g$$(4) ) has the so called variable multiplicity (Morando et al. 2009; Nishitani and Takayama 1998; Nishitani and Takayama 2000). We also remark that to retrieve smoothness properties of solutions (with respect to evolution variable) in the case of hyperbolic problems, one always must assume that the relevant compatibility conditions are valid for the data. Actually, due to the needed inflow boundary condition (4)(4) $${\psi }_{|{\Gamma }_{-}}=g$$(4) the Sobolev regularity of solutions is strongly dependent on the properties of the so called escape time mapping $t(x,\omega )$ contrary to the global case $G={\mathbb{R}}^n.$ This can be seen transparently e.g. from explicit solution formulas (cf. e.g. Tervo and Kokkonen 2017, section 4.2 and Tervo et al. 2018a, section 10.2). The inflow boundary condition causes that regularity of solutions in the case $G\cancel{=}{\mathbb{R}}^3$ is not necessarily increasing along the data with respect to $(x,\omega )$-variable. In fact, it is known that the regularity of the solution of the transport equation is limited (with respect to x, for example) up to the fractional space $H^{(s_1,0,0)}(G\times S\times I^{°}),s_1<3/2$ regardless of the smoothness of the data (see counterexample given in Tervo and Kokkonen 2017, section 7.1).

2. Preliminaries

2.1. Basic notations and concepts

We restrict ourselves to the spatial dimension n = 3 and we consider the transport problem where the spatial (position) domain $G={\mathbb{R}}^3.$ In this case the related problems are often called global ones. Let S = S₂ be the unit sphere in ${\mathbb{R}}^3$ (S is the velocity direction domain). Furthermore, let $I=[E_0,E_{\mathrm{m}}]$ (I is the energy interval) where $0\le E_0<E_{\mathrm{m}}<\infty .$ We shall denote by $I^{°}$ the interior of I. Define the (Sobolev) space $W^2({\mathbb{R}}^3\times S\times I)$ by $$W^2({\mathbb{R}}^3\times S\times I):=\{\psi \in L^2({\mathbb{R}}^3\times S\times I)|\omega ·{\nabla }_x\psi \in L^2({\mathbb{R}}^3\times S\times I)\}$$ equipped with the inner product $${〈\psi ,v〉}_{W^2({\mathbb{R}}^3\times S\times I)}:={〈\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈\omega ·{\nabla }_x\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$

The space $W^2({\mathbb{R}}^3\times S\times I)$ is a Hilbert space and the space $C_0^1({\mathbb{R}}^3,C^1(S\times I))$ is a dense subspace of $W^2({\mathbb{R}}^3\times S\times I)$ (e.g. Friedrichs 1944).

We recall that the following Green’s formulas are valid (9) $${\int }_{{\mathbb{R}}^3\times S\times I}(\omega ·{\nabla }_x\psi )v\mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}\psi (\omega ·{\nabla }_{x}v)\mathit{dxd}\omega dE$$(9) for every $\psi ,v\in W^2({\mathbb{R}}^3\times S\times I)$ and (10) $${\int }_{{\mathbb{R}}^3\times S\times I}〈{\Delta }_S\psi ,v〉\mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}〈{\nabla }_S\psi ,{\nabla }_{S}v〉\mathit{dxd}\omega dE$$(10) for every $\psi ,v\in L^2({\mathbb{R}}^3\times I,H^1(S))$ for which ${\Delta }_S\psi \in L^2({\mathrm{ℝ}}^3\times S\times I).$ Here ${\nabla }_S$ is the gradient on S and Δ_S is the Laplace-Beltrami operator (so called Laplacian) on S. $H^1(S)$ is the standard Sobolev space on the compact manifold S.

2.2. Function spaces needed in regularity analysis

Let $m=(m_1,m_2,m_3)\in {\mathbb{N}}_0^3$ be a multi-index. Define anisotropic Sobolev spaces $H^m({\mathbb{R}}^3\times S\times I^{°})$ by (11) $$\begin{array}{c}{H}^m({\mathbb{R}}^3\times S\times I^{°}):=\{\psi \in L^2({\mathbb{R}}^3\times S\times I^{°})|{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^l\psi \in L^2({\mathbb{R}}^3\times S\times I^{°}),\\ \text{for all}\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3\},\end{array}$$(11) where the derivatives are taken in the distributional sense. Above $\left\{{\partial }_{{\omega }_j}\right|j=1,2\}$ is a local basis of the tangent space T(S). We recall that for sufficiently smooth functions $f:S\to \mathbb{R}$ $${\frac{\partial f}{\partial {\omega }_j}}_{|\omega }=\frac{\partial }{\partial u_j}{(f°h)}_{|u=h^{-1}(\omega )},j=1,2.$$

Here $h:U\to S\setminus S_0$ is a parametrization of $S\setminus S_0$ where S₀ has the surface measure zero. Moreover, recall that $f\in L^2(S)$ if and only if $f°h\in L^2(U,|A_h|du)$ where $\left|A_h\right|:=\Vert {\partial }_{1}h\wedge {\partial }_{2}h\Vert .$ In $L^2(S)$ we use the inner product (12) $${〈f_1,f_2〉}_{L^2(S)}={\int }_S{f}_1{f}_{2}d\omega :={〈f_1°h,f_2°h〉}_{L^2(U,|A_h|du)}.$$(12)

The space $H^m({\mathbb{R}}^3\times S\times I^{°})$ is a Hilbert space when equipped with the inner product (13) $${〈\psi ,v〉}_{H^m({\mathbb{R}}^3\times S\times I^{°})}:=\sum _{\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3}{〈{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^l\psi ,{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^{l}v〉}_{L^2({\mathbb{R}}^3\times S\times I^{°})}.$$(13)

The corresponding norm is $${\Vert \psi \Vert }_{H^m({\mathbb{R}}^3\times S\times I^{°})}={\left(\sum _{\left|\alpha \right|\le m_1}\sum _{\left|\beta \right|\le m_2}\sum _{l\le m_3}{\Vert {\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^l\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\right)}^{\frac{1}{2}}.$$

Note that for $m\prime \ge m$ (that is, $m_j^{\prime }\ge m_j,j=1,2,3$) $$H^{m\prime }({\mathbb{R}}^3\times S\times I^{°})\subset H^m({\mathbb{R}}^3\times S\times I^{°}).$$

The tensor product $H^{m_1}({\mathbb{R}}^3)\otimes H^{m_2}(S)\otimes H^{m_3}(I^{°})$ is a dense subspace of $H^m({\mathbb{R}}^3\times S\times I^{°})$ but generally the spaces $H^{m_1}({\mathbb{R}}^3)\otimes H^{m_2}(S)\otimes H^{m_3}(I^{°})$ and $H^m({\mathbb{R}}^3\times S\times I^{°})$ are not equal (principles of these kind of results are found e.g. in Aubin 1979, Chapter 12). Moreover, we have

Theorem 2.1.

The space (14) $$\mathcal{D}({\mathbb{R}}^3\times S\times I):=\left\{{\psi }_{|{\mathbb{R}}^3\times S\times I^{°}}\right|\psi \in C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R})\}$$(14) is dense in $H^m({\mathbb{R}}^3\times S\times I^{°}).$

Proof.

The proof follows by using the standard cutting and Friedrichs mollifier smoothing techniques. □

Remark 2.2.

For multi-indexes of the form $m=(m_1,0,m_3)$ the spaces $H^m({\mathbb{R}}^3\times S\times I^{°})$ can be characterized by using Fourier transforms. The (partial) Fourier transform with respect to (x, E) of $\psi \in L^2({\mathbb{R}}^3\times S\times \mathbb{R})$ (in the sense of tempered distributions) is given by

$$({\mathcal{F}}_{(x,E)}\psi )(\xi ,\omega ,\eta ):={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_{\mathbb{R}}\psi (x,\omega ,E)e^{-\mathrm{i}〈(\xi ,\eta ),(x,E)〉}d\mathit{xdE},(\xi ,\eta )\in {\mathbb{R}}^3\times \mathbb{R},\omega \in S.$$

The inverse partial Fourier transform is then $$\psi (x,\omega ,E)=({\mathcal{F}}_{(x,E)}^{-1}\psi )(x,\omega ,E):={(2\pi )}^{-4}{\int }_{{\mathbb{R}}^3}{\int }_{\mathbb{R}}({\mathcal{F}}_{(x,E)}\psi )(\xi ,\omega ,\eta )e^{\mathrm{i}〈(\xi ,\eta ),(x,E)〉}\mathit{\text{dξdη}}.$$

For multi-indexes $m=(m_1,0,m_3)$ we define (15) $${\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R}):=\{\psi \in L^2({\mathbb{R}}^3\times S\times \mathbb{R})|{\Vert \psi \Vert }_{{\widehat{H}}^m({\mathbb{R}}^n\times S\times \mathbb{R})}<\infty \}$$(15) where $${\Vert \psi \Vert }_{{\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})}^2:={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_{\mathbb{R}}{(1+|\xi {|}^2)}^{m_1}{(1+|\eta {|}^2)}^{m_3}|({\mathcal{F}}_{(x,E)}\psi )(\xi ,\omega ,\eta ){|}^2\mathit{\text{dξdωdη}}.$$

The space ${\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})$ is a Hilbert space when equipped with the inner product (16) $$\begin{array}{c}{〈\psi ,v〉}_{{\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})}:={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_{\mathbb{R}}({\mathcal{F}}_{(x,E)}\psi )(\xi ,\omega ,\eta )\overline{({\mathcal{F}}_{(x,E)}v)}(\xi ,\omega ,\eta )\\ ·{(1+|\xi {|}^2)}^{m_1}{(1+|\eta {|}^2)}^{m_3}\mathit{\text{dξdωdη}}.\end{array}$$(16)

Due to Plancherel’s formula (17) $$H^m({\mathbb{R}}^3\times S\times \mathbb{R})={\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})$$(17) and the inner products (13)(13) $${〈\psi ,v〉}_{H^m({\mathbb{R}}^3\times S\times I^{°})}:=\sum _{\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3}{〈{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^l\psi ,{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^{l}v〉}_{L^2({\mathbb{R}}^3\times S\times I^{°})}.$$(13) and (16)(16) $$\begin{array}{c}{〈\psi ,v〉}_{{\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})}:={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_{\mathbb{R}}({\mathcal{F}}_{(x,E)}\psi )(\xi ,\omega ,\eta )\overline{({\mathcal{F}}_{(x,E)}v)}(\xi ,\omega ,\eta )\\ ·{(1+|\xi {|}^2)}^{m_1}{(1+|\eta {|}^2)}^{m_3}\mathit{\text{dξdωdη}}.\end{array}$$(16) are equivalent. The spaces $H^m({\mathbb{R}}^3\times S\times I^{°}$) are isomorphic to the factor spaces $H^m({\mathbb{R}}^3\times S\times \mathbb{R})/H_0^m({\mathbb{R}}^3\times S\times (\mathbb{R}\backslash I))$ where $H_0^m({\mathbb{R}}^3\times S\times (\mathbb{R}\backslash I))$ is the completion of $C_0^{\infty }({\mathbb{R}}^3\times S\times (\mathbb{R}\backslash I))$ with respect to the inner product (13)(13) $${〈\psi ,v〉}_{H^m({\mathbb{R}}^3\times S\times I^{°})}:=\sum _{\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3}{〈{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^l\psi ,{\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^{l}v〉}_{L^2({\mathbb{R}}^3\times S\times I^{°})}.$$(13) .

We finally define the space (18) $$\begin{array}{l}{W}^{\infty ,m}({\mathbb{R}}^3\times S\times I^{°}):=\{f\in L^{\infty }({\mathbb{R}}^3\times S\times I)|{\Vert {\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^{l}f\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}<\infty \\ \text{for}\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3\}\end{array}$$(18) which is equipped with the norm $${\Vert f\Vert }_{W^{\infty ,m}({\mathbb{R}}^3\times S\times I^{°})}:={\max }_{\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3}{\Vert {\partial }_x^{\alpha }{\partial }_{\tilde{\omega }}^{\beta }{\partial }_E^{l}f\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}.$$

3. Existence of solutions when the spatial domain is R³

We assume that the transport operator T is of the form (19) $$T\psi =a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi$$(19) where $b(x,\omega ,E,{\partial }_{\omega })\psi$ is given by (20) $$b(x,\omega ,E,{\partial }_{\omega })\psi =c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi .$$(20)

Here Δ_S is the Laplace-Beltrami operator on sphere and $$d=(d_1,d_2)\sim d_1\frac{\partial }{\partial {\omega }_1}+d_2\frac{\partial }{\partial {\omega }_2}.$$ $d·{\nabla }_S$ is the chosen Riemannian inner product on the tangent space (bundle) T(S). Denote (21) $$d(x,\omega ,E,{\partial }_{\omega })\psi :=d(x,\omega ,E)·{\nabla }_S\psi .$$(21)

We consider an initial value transport problem (22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) in the global case $G={\mathbb{R}}^3$ where $f\in L^2({\mathbb{R}}^3\times S\times I).$ The existence of solutions for the problem (22)(22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) can be studied, by applying e.g. the generalized Lax-Milgram Theorem, the so called Lions-Lax-Milgram Theorem. We will use the following statement that can be found e.g. in Treves (1975, 403) or Grisvard (1985, 234).

Theorem 3.1.

Let X and Y be Hilbert spaces, with Y continuously embedded into X. Assume that $B(·,·):X\times Y\to \mathbb{R}$ is a bilinear form satisfying the following properties with $M\ge 0,c\prime >0,$ (23) $$|B(u,v)|\le M{\Vert u\Vert }_X{\Vert v\Vert }_Y\hspace{1em}\forall u\in X,v\in Y\hspace{1em}(\text{boundedness})$$(23) and (24) $$B(v,v)\ge c\prime {\Vert v\Vert }_X^2\hspace{1em}\forall v\in Y\hspace{1em}(\text{coercivity}).$$(24)

Suppose that $F:X\to \mathbb{R}$ is a bounded linear form. Then there exists $u\in X$ (possibly non-unique) such that (25) $$B(u,v)=F(v)\hspace{1em}\forall v\in Y.$$(25)

3.1. Assumptions for the restricted collision operator

We assume that the restricted collision operator is the sum (for some more details see Tervo et al. 2018, section 5) (26) $$K_r=K_r^1+K_r^2+K_r^3.$$(26)

Here $K_r^1$ is of the form $$(K_r^1\psi )(x,\omega ,E)={\displaystyle {\int }_{S\prime \times I\prime }{\sigma }^1}(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime ,$$ where ${\sigma }^1:{\mathbb{R}}^3\times S^2\times I^2\to \mathbb{R}$ is a non-negative measurable function such that (27) $$\begin{array}{c}{\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \le M_1,\\ {\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \le M_2,\end{array}$$(27) for a.e. $(x,\omega ,E)\in {\mathbb{R}}^3\times S\times I.$

The operator $K_r^2$ is of the form $$(K_r^2\psi )(x,\omega ,E)={\int }_{S\prime }{\sigma }^2(x,\omega \prime ,\omega ,E)\psi (x,\omega \prime ,E)d\omega \prime ,$$ where ${\sigma }^2:{\mathbb{R}}^3\times S^2\times I\to \mathbb{R}$ is a non-negative measurable function such that (28) $$\begin{array}{c}{\int }_{S\prime }{\sigma }^2(x,\omega \prime ,\omega ,E)d\omega \prime \le M_1,\\ {\int }_{S\prime }{\sigma }^2(x,\omega ,\omega \prime ,E)d\omega \prime \le M_2,\end{array}$$(28) for a.e. $(x,\omega ,E)\in {\mathbb{R}}^3\times S\times I.$

Finally, $K_r^3$ is of the form $$(K_r^3\psi )(x,\omega ,E)={\int }_{I\prime }{\int }_0^{2\pi }{\sigma }^3(x,E\prime ,E)\psi (x,\gamma (E\prime ,E,\omega )(s),E\prime )\mathit{dsdE}\prime$$ where $\gamma =\gamma (E\prime ,E,\omega ):[0,2\pi ]\to S$ is a parametrization of the curve (an example for the choice of $\gamma =\gamma (E\prime ,E,\omega )$ is given in Tervo et al. 2018) $$\Gamma (E\prime ,E,\omega )=\{\omega \prime \in S|\omega \prime ·\omega -\mu (E\prime ,E)=0\}.$$

Moreover, ${\sigma }^3:{\mathbb{R}}^3\times I^2\to \mathbb{R}$ is a non-negative measurable function such that (29) $$\begin{array}{c}{\int }_{I\prime }{\sigma }^3(x,E\prime ,E)dE\prime \le M_1,\\ {\int }_{I\prime }{\sigma }^3(x,E,E\prime )dE\prime \le M_2,\end{array}$$(29) for a.e. $(x,E)\in {\mathbb{R}}^3\times I.$ The following result is shown analogously to Theorem 5.13 in Tervo et al. (2018) (in the reference in question we have assumed that the spatial domain G is bounded).

Theorem 3.2.

The operators $K_r^j,j=1,2,3$ are bounded operators $L^2({\mathbb{R}}^3\times S\times I)\to L^2({\mathbb{R}}^3\times S\times I)$ and (30) $$\Vert K_r^1\Vert \le \sqrt{M_1{M}_2},$$(30) (31) $$\Vert K_r^2\Vert \le \sqrt{M_1{M}_2},$$(31) (32) $$\Vert K_r^3\Vert \le 2\pi \sqrt{M_1{M}_2},.$$(32)

In order to render the operator $\Sigma -K_r$ coercive (accretive), we shall assume that (33) $$\begin{array}{l}\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \\ -{\int }_{S\prime }{\sigma }^2(x,\omega ,\omega \prime ,E)d\omega \prime -2\pi {\int }_{I\prime }{\sigma }^3(x,E,E\prime )dE\prime \ge c\prime ,\end{array}$$(33) and (34) $$\begin{array}{l}\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \\ -{\int }_{S\prime }{\sigma }^2(x,\omega \prime ,\omega ,E)d\omega \prime -2\pi {\int }_{I\prime }{\sigma }^3(x,E\prime ,E)dE\prime \ge c\prime ,\end{array}$$(34) for a.e. $(x,\omega ,E)\in {\mathbb{R}}^3\times S\times I.$ In the sequel we assume that $c\prime >0.$

Next result addresses coercivity (accretivity) of $K_r-\Sigma$ under the above assumptions. The result is proven analogously to Theorem 5.14 in Tervo et al. (2018a) (or Tervo et al. 2018a, section 4).

Theorem 3.3.

Suppose that the assumptions (27)(17) $$H^m({\mathbb{R}}^3\times S\times \mathbb{R})={\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})$$(17) , (28, 29, 33(18) $$\begin{array}{l}{W}^{\infty ,m}({\mathbb{R}}^3\times S\times I^{°}):=\{f\in L^{\infty }({\mathbb{R}}^3\times S\times I)|{\Vert {\partial }_x^{\alpha }{\partial }_{\omega }^{\beta }{\partial }_E^{l}f\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}<\infty \\ \text{for}\left|\alpha \right|\le m_1,\left|\beta \right|\le m_2,l\le m_3\}\end{array}$$(18) ) and (34)(34) $$\begin{array}{l}\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \\ -{\int }_{S\prime }{\sigma }^2(x,\omega \prime ,\omega ,E)d\omega \prime -2\pi {\int }_{I\prime }{\sigma }^3(x,E\prime ,E)dE\prime \ge c\prime ,\end{array}$$(34) are valid. Then (35) $${〈(\Sigma -K_r)\psi ,\psi 〉}_{L^2({\mathbb{R}}^3\times S\times I)}\ge c\prime {\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\hspace{1em}\forall \psi \in L^2({\mathbb{R}}^3\times S\times I).$$(35)

3.2. Existence of weak solutions

At first we verify (formally) the corresponding variational equation. Assume that ψ is a solution of (22)(22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) in the classical sense and let $v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S))).$ By the Green’s formula (10)(10) $${\int }_{{\mathbb{R}}^3\times S\times I}〈{\Delta }_S\psi ,v〉\mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}〈{\nabla }_S\psi ,{\nabla }_{S}v〉\mathit{dxd}\omega dE$$(10) we have $${\displaystyle {\int }_S({\Delta }_S\psi )(x,\omega ,E)v(x,\omega ,E)}d\omega =-{\displaystyle {\int }_S〈({\nabla }_S\psi )(x,\omega ,E),({\nabla }_{S}v)(x,\omega ,E)〉}d\omega$$ where $〈{\nabla }_S\psi ,{\nabla }_{S}v〉$ is the chosen Riemannian inner product on S. Hence, (36) $$\begin{array}{c}{〈c(x,E){\Delta }_S\psi ,v〉}_{L^2({\mathrm{ℝ}}^3\times S\times I)}=-{〈{\nabla }_S\psi ,c(x,E){\nabla }_{S}v〉}_{L^2({\mathrm{ℝ}}^3\times S\times I)}\\ :=-{\int }_{{\mathrm{ℝ}}^3\times S\times I}〈{\nabla }_S\psi ,c(x,E){\nabla }_{S}v〉\mathit{\text{dxdωdE}}.\end{array}$$(36)

Moreover, $${〈d(x,\omega ,E,{\partial }_{\omega })\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈\psi ,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}$$ where $d^{*}(x,\omega ,E,{\partial }_{\omega })$ is the formal adjoint of $d(x,\omega ,E,{\partial }_{\omega })$ that is, (37) $$d^{*}(x,\omega ,E,{\partial }_{\omega })v=-d(x,\omega ,E,{\partial }_{\omega })v-\left({\text{div}}_{S}d\right)v=-d(x,\omega ,E)·{\nabla }_{S}v-\left({\text{div}}_{S}d\right)v$$(37) where ${\text{div}}_S$ is the divergence on S. By integration by parts over I (38) $$\begin{array}{l}{〈a\frac{\partial \psi }{\partial E},v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={\int }_{{\mathbb{R}}^3}{\int }_{S}a(x,E_m)\psi (x,\omega ,E_m)v(x,\omega ,E_m)\mathit{\text{dωdx}}\\ -{\int }_{{\mathbb{R}}^3}{\int }_{S}a(x,E_0)\psi (x,\omega ,E_0)v(x,\omega ,E_0)\mathit{\text{dωdx}}-{〈\psi ,\frac{\partial (av)}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =-{〈\psi ,a\frac{\partial v}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}-{〈\psi ,\frac{\partial a)}{\partial E}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈a(.,E_m)\psi (.,.,E_m),v(.,.,E_m)〉}_{L^2({\mathbb{R}}^3\times S)}\\ -{〈a(.,E_0)\psi (.,.,E_0),v(.,.,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}.\end{array}$$(38)

Using Green’s formula (9)(9) $${\int }_{{\mathbb{R}}^3\times S\times I}(\omega ·{\nabla }_x\psi )v\mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}\psi (\omega ·{\nabla }_{x}v)\mathit{dxd}\omega dE$$(9) we obtain (39) $${〈\omega ·{\nabla }_x\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}=-{〈\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$(39)

Finally, (40) $${〈K_r\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈\psi ,K_r^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}{〈\Sigma \psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈\psi ,{\Sigma }^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}$$(40) where ${\Sigma }^{*}=\Sigma$ and $K_r^{*}$ is the adjoint of K_r which can be computed (but we omit computations here).

As a conclusion we see that if ψ is a classical solution of problem (22)(22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) then the following weak formulation is fulfilled (41) $$\begin{array}{l}B(\psi ,v):=-{〈\psi ,a\frac{\partial v}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}-{〈\psi ,\frac{\partial a)}{\partial E}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈a(.,E_0)\psi (.,.,E_0),v(.,.,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}\\ -{〈{\nabla }_S\psi ,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈\psi ,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ +{〈\psi ,{\Sigma }^{*}v-K_r^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈f,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.\end{array}$$(41)

Define in $C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$ inner products (42) $$\begin{array}{l}{〈\psi ,v〉}_{\mathcal{H}}:={〈\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ +{〈\psi (.,.,E_0),v(.,.,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}+{〈\psi (.,.,E_m),v(.,.,E_m)〉}_{L^2({\mathbb{R}}^3\times S)}\\ +{〈\psi ,v〉}_{L^2({\mathbb{R}}^3\times I,H^1(S))}\end{array}$$(42) and (43) $${〈\psi ,v〉}_{\widehat{\mathcal{H}}}={〈\psi ,v〉}_{\mathcal{H}}+{〈\omega ·{\nabla }_x\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈\frac{\partial \psi }{\partial E},\frac{\partial v}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$(43)

Let $\mathcal{H}$ and $\widehat{\mathcal{H}}$ be the completions of $C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$ with respect to the inner products ${〈.,.〉}_{\mathcal{H}}$ and ${〈.,.〉}_{\widehat{\mathcal{H}}},$ respectively.

We assume for the coefficients: (44) $$a\in L^{\infty }({\mathbb{R}}^3,W^{\infty ,1}(I)),c\in L^{\infty }({\mathbb{R}}^3\times I),d_j\in L^{\infty }({\mathbb{R}}^3\times I,W^{\infty ,1}(S))$$(44) (45) $$-\frac{(\partial a)}{\partial E}(x,E)+({\text{div}}_{S}d)(x,\omega ,E))\ge q_1>0,a.e.,$$(45) (46) $$-c(x,E)\ge q_2>0,a.e.,$$(46) (47) $$-a(x,E_0)\ge q_3>0,-a(x,E_m)\ge q_3>0,a.e..$$(47)

We proceed analogously to Tervo and Herty (2020). At first, we show that the bilinear form $B:C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))\times C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))\to \mathbb{R}$ obeys the following boundedness and coercivity conditions:

Theorem 3.4.

Suppose that the assumptions (27)(17) $$H^m({\mathbb{R}}^3\times S\times \mathbb{R})={\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})$$(17) , (28, 29, 33, 34, 44–47(17) $$H^m({\mathbb{R}}^3\times S\times \mathbb{R})={\widehat{H}}^m({\mathbb{R}}^3\times S\times \mathbb{R})$$(17) ) are valid. Then there exists a constant M > 0 such that (48) $$|B(\psi ,v)|\le M{\Vert \psi \Vert }_{\mathcal{H}}{\Vert v\Vert }_{\widehat{\mathcal{H}}}\hspace{1em}\forall \psi ,v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$$(48) and (49) $$B(v,v)\ge c″{\Vert v\Vert }_{\mathcal{H}}^2\hspace{1em}\forall v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$$(49) where (50) $$c″:=\min \{\frac{q_1}{2},\frac{q_3}{2},q_2,\frac{1}{2},c\prime \}.$$(50)

Proof.

A. The boundedness can be seen as in Tervo et al. (2018a, Theorem 6.4), Tervo and Herty (2020, Theorem 6.6) and so we omit details.

B. Secondly, we verify the coercivity (49)(49) $$B(v,v)\ge c″{\Vert v\Vert }_{\mathcal{H}}^2\hspace{1em}\forall v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$$(49) . By partial integration we have for $v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$ (51) $$\begin{array}{l}-{〈v,a\frac{\partial v)}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈v,\frac{\partial (av)}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈v(·,·,E_m),a(·,E_m)v(·,·,E_m)〉}_{L^2({\mathbb{R}}^3\times S)}+{〈v(·,·,E_0),a(·,E_0)v(·,·,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}\end{array}$$(51) and then (52) $$\begin{array}{l}-{〈v,a\frac{\partial v)}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}=\frac{1}{2}({〈v,\frac{\partial a}{\partial E}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈v(·,·,E_m),a(·,E_m)v(·,·,E_m)〉}_{L^2({\mathbb{R}}^3\times S)}+{〈v(·,·,E_0),a(·,E_0)v(·,·,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}).\end{array}$$(52)

Using the Green’s formula we have (53) $$-{〈v,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈\omega ·{\nabla }_{x}v,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}$$(53) which implies (54) $${〈\omega ·{\nabla }_{x}v,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}=0.$$(54)

Furthermore, we have $${〈d(x,\omega ,E,{\partial }_{\omega })v,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈v,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}$$ which implies (by recalling (37)(37) $$d^{*}(x,\omega ,E,{\partial }_{\omega })v=-d(x,\omega ,E,{\partial }_{\omega })v-\left({\text{div}}_{S}d\right)v=-d(x,\omega ,E)·{\nabla }_{S}v-\left({\text{div}}_{S}d\right)v$$(37) that (55) $${〈v,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}=-\frac{1}{2}{〈({\text{div}}_{S}d)v,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$(55)

Finally, we have (56) $$\begin{array}{l}-{〈{\nabla }_{S}v,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =-{\int }_{{\mathbb{R}}^3\times S\times I}c(x,E)〈({\nabla }_{S}v)(x,\omega ,E),({\nabla }_{S}v)(x,\omega ,E)〉\mathit{\text{dxdωdE}}.\end{array}$$(56)

Inserting (52, 54, 55(56) $$\begin{array}{l}-{〈{\nabla }_{S}v,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =-{\int }_{{\mathbb{R}}^3\times S\times I}c(x,E)〈({\nabla }_{S}v)(x,\omega ,E),({\nabla }_{S}v)(x,\omega ,E)〉\mathit{\text{dxdωdE}}.\end{array}$$(56) ) and (56)(56) $$\begin{array}{l}-{〈{\nabla }_{S}v,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =-{\int }_{{\mathbb{R}}^3\times S\times I}c(x,E)〈({\nabla }_{S}v)(x,\omega ,E),({\nabla }_{S}v)(x,\omega ,E)〉\mathit{\text{dxdωdE}}.\end{array}$$(56) in the expression of $B(.,.,)$ (given in (41)(41) $$\begin{array}{l}B(\psi ,v):=-{〈\psi ,a\frac{\partial v}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}-{〈\psi ,\frac{\partial a)}{\partial E}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈a(.,E_0)\psi (.,.,E_0),v(.,.,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}\\ -{〈{\nabla }_S\psi ,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈\psi ,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ +{〈\psi ,{\Sigma }^{*}v-K_r^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈f,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.\end{array}$$(41) ) with $\psi =v$ we get in virtue of the assumptions (45), (46, 47)(56) $$\begin{array}{l}-{〈{\nabla }_{S}v,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =-{\int }_{{\mathbb{R}}^3\times S\times I}c(x,E)〈({\nabla }_{S}v)(x,\omega ,E),({\nabla }_{S}v)(x,\omega ,E)〉\mathit{\text{dxdωdE}}.\end{array}$$(56) and by Theorem 3.3 the required estimate (49)(41) $$\begin{array}{l}B(\psi ,v):=-{〈\psi ,a\frac{\partial v}{\partial E}〉}_{L^2({\mathbb{R}}^3\times S\times I)}-{〈\psi ,\frac{\partial a)}{\partial E}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈a(.,E_0)\psi (.,.,E_0),v(.,.,E_0)〉}_{L^2({\mathbb{R}}^3\times S)}\\ -{〈{\nabla }_S\psi ,c(x,E){\nabla }_{S}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈\psi ,d^{*}(x,\omega ,E,{\partial }_{\omega })v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ -{〈\psi ,\omega ·{\nabla }_{x}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ +{〈\psi ,{\Sigma }^{*}v-K_r^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈f,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.\end{array}$$(41) as in Tervo and Herty (2020, Theorem 6.6).

This completes the proof. □

Because $C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))\times C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$ is dense in $\mathcal{H}\times \widehat{\mathcal{H}}$ and since (48)(48) $$|B(\psi ,v)|\le M{\Vert \psi \Vert }_{\mathcal{H}}{\Vert v\Vert }_{\widehat{\mathcal{H}}}\hspace{1em}\forall \psi ,v\in C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))$$(48) holds, the bilinear form $B(·,·):C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))\times C_0^1({\mathbb{R}}^3,C^1(I,C^2(S)))\to \mathbb{R}$ has a unique extension $\tilde{B}(·,·):\mathcal{H}\times \widehat{\mathcal{H}}\to \mathbb{R}$ which satisfies (57) $$|\tilde{B}(\psi ,v)|\le M{\Vert \psi \Vert }_{\mathcal{H}}{\Vert v\Vert }_{\widehat{\mathcal{H}}}\forall \psi \in \mathcal{H},v\in \widehat{\mathcal{H}}$$(57) and (58) $$\tilde{B}(v,v)\ge c″{\Vert v\Vert }_{\mathcal{H}}^2\hspace{1em}\forall v\in \widehat{\mathcal{H}}.$$(58)

Furthermore, define a bounded linear form (59) $$F:\mathcal{H}\to \mathbb{R};\hspace{1em}F(\psi ):={〈f,\psi 〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$(59)

Note also that the embedding $\widehat{\mathcal{H}}\subset \mathcal{H}$ is continuous.

Let $$P(x,\omega ,E,D)\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi$$ be the differential part of T. The space (60) $$\begin{array}{c}{\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°}):=\{\psi \in L^2({\mathbb{R}}^3\times S\times I)|\\ P(x,\omega ,E,D)\psi \in L^2({\mathbb{R}}^3\times S\times I)\text{in the weak sense}\}\end{array}$$(60) is a Hilbert space when equipped with the inner product $${〈\varphi ,v〉}_{{\mathcal{H}}_P(G\times S\times I^{°})}={〈\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}+{〈P(x,\omega ,E,D)\psi ,P(x,\omega ,E,D)v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$

Using this notation, the equation (22)(22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) can be written shortly as $$P(x,\omega ,E,D)\psi +\Sigma \psi -K_r\psi =f.$$

We formulate the existence of weak solutions without the initial condition $\psi (.,.,E_m)=0.$

Theorem 3.5.

Suppose that the assumptions of Theorem 3.4 that is, (27–29, 33, 34, 44–47(22) $$T\psi =f,\psi (.,.,E_m)=0$$(22) ) are valid. Let $f\in L^2({\mathbb{R}}^3\times S\times I)$. Then the variational equation (61) $$\tilde{B}(\psi ,v)=F(v)\hspace{1em}\forall v\in \widehat{\mathcal{H}}$$(61) has a solution $\psi \in \mathcal{H}$. Furthermore, $\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$ and it is a weak (distributional) solution of the equation (62) $$T\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f.$$(62)

Proof.

The proof is similar as in Tervo and Herty (2020, Theorem 6.8).□

Remark 3.6.

For $f\in L^2({\mathbb{R}}^3\times I,H^{-1}(S),v\in L^2({\mathbb{R}}^3\times I,H^1(S)$ we define $$〈f,v〉:={\int }_{{\mathbb{R}}^3\times I}(f(x,.,E),v(x,.,E))\mathit{dxdE}$$ where $(f(x,.,E),v(x,.,E))$ is the canonical duality between $H^{-1}(S)$ and $H^1(S)$ Since (63) $$〈f,v〉\le {\Vert f\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}{\Vert v\Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S))}$$(63) we find that the above Theorem 3.5 is valid more generally for $f\in L^2({\mathbb{R}}^3\times I,H^{-1}(S)).$

3.3. Existence of solutions for the initial value problem

Let $$Q(x,\omega ,E,D)\psi :=a(x,E)\frac{\partial \psi }{\partial E}+\omega ·{\nabla }_x\psi .$$

Define the space ${\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°})$ $${\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°}):=\{\psi \in L^2({\mathbb{R}}^3\times I,H^1(S))|Q(x,\omega ,E,D)\psi \in L^2({\mathbb{R}}^3\times I,H^{-1}(S))\}$$ equipped with the inner product $${〈\psi ,v〉}_{{\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°})}={〈\psi ,v〉}_{L^2({\mathbb{R}}^3\times I,H^1(S))}+{〈Q(x,\omega ,E,D)\psi ,Q(x,\omega ,E,D)v〉}_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}.$$

Furthermore, define the traces ${\gamma }_{\mathrm{m}}(\psi ):=\psi (·,·,E_{\mathrm{m}}),{\gamma }_0(\psi ):=\psi (·,·,E_0).$ We verify the following trace theorem

Theorem 3.7.

Suppose that (64) $$a\in W^{\infty ,1}({\mathbb{R}}^3\times I^{°})\text{such that}|a(x,E)|\ge q_4>0\mathrm{a}.\mathrm{e}.\text{in}{\mathbb{R}}^3\times I.$$(64)

Then the trace operators $$\begin{array}{c}{\gamma }_{\mathrm{m}}:{\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°})\to L_{\text{loc}}^2({\mathbb{R}}^3\times S),\\ {\gamma }_0:{\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°})\to L_{\text{loc}}^2({\mathbb{R}}^3\times S),\end{array}$$ are well-defined and continuous.

Proof.

In virtue of density results like Friedrichs (Friedrichs 1944; Rauch 1985) the space $C_0^1({\mathbb{R}}^3,C^1(S\times I))$ is dense in ${\mathbf{H}}_Q({\mathbb{R}}^3\times S\times I^{°})$ and so it suffices to show the below boundedness estimate only for $\psi \in C_0^1({\mathbb{R}}^3,C^1(S\times I)).$

By the Green’s formula for $\psi \in C_0^1({\mathbb{R}}^3,C^1(S\times I))$ (65) $$2{\int }_{{\mathbb{R}}^3\times S\times I}\frac{1}{a}(\omega ·{\nabla }_x\psi )\psi \mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){\psi }^2\mathit{\text{dxdωdE}}$$(65) where we used that $$\omega ·{\nabla }_x\left(\frac{1}{a}\psi \right)=\frac{1}{a}\omega ·{\nabla }_x\psi +\omega ·{\nabla }_x\left(\frac{1}{a}\right)\psi .$$

Note that $$\frac{\partial \psi }{\partial E}+\frac{1}{a}\omega ·{\nabla }_x\psi =\frac{1}{a}Q(x,\omega ,E)\psi .$$

Consider the operator ${\gamma }_{\mathrm{m}}.$ Let $\eta \in C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R})$ such that $\eta (.,.,E_0)=0.$ Then we get by (65)(65) $$2{\int }_{{\mathbb{R}}^3\times S\times I}\frac{1}{a}(\omega ·{\nabla }_x\psi )\psi \mathit{dxd}\omega dE=-{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){\psi }^2\mathit{\text{dxdωdE}}$$(65) and (63)(63) $$〈f,v〉\le {\Vert f\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}{\Vert v\Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S))}$$(63) (66) $$\begin{array}{l}{\Vert (\eta \psi )(.,.,E_m)\Vert }_{L^2({\mathbb{R}}^3\times S)}^2={\int }_{{\mathbb{R}}^3\times S}{(\eta \psi )}^2(x,\omega ,E_m)\mathit{\text{dωdx}}\\ ={\int }_{{\mathbb{R}}^3\times S}{\int }_{E_0}^{E_m}\frac{\partial }{\partial E}({(\eta \psi )}^2(x,\omega ,E))\mathit{dEd}\omega dx\\ ={\int }_{{\mathbb{R}}^3\times S\times I}2(\eta \psi )(x,\omega ,E)\frac{\partial (\eta \psi )}{\partial E}(x,\omega ,E)\mathit{dEd}\omega dx\\ =2{\int }_{{\mathbb{R}}^3\times S\times I}(\eta \psi )(x,\omega ,E)(\frac{\partial (\eta \psi )}{\partial E}(x,\omega ,E)+\frac{1}{a}\omega ·{\nabla }_x(\eta \psi ))(x,\omega ,E)\mathit{dEd}\omega dx\\ +{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){(\eta \psi )}^2(x,\omega ,E)\mathit{dxd}\omega dE\\ =2{\int }_{{\mathbb{R}}^3\times S\times I}(\eta \psi )(x,\omega ,E)\frac{1}{a}(Q(x,\omega ,E)(\eta \psi ))(x,\omega ,E)\mathit{dEd}\omega dx\\ +{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){(\eta \psi )}^2(x,\omega ,E)\mathit{dxd}\omega dE\\ \le ({\Vert \eta \psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S)}^2+{\Vert \frac{1}{a}Q(x,\omega ,E,D)(\eta \psi )\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}^2)\\ +{\Vert \omega ·{\nabla }_x\left(\frac{1}{a}\right)\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert \eta \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\end{array}$$(66)

Since $$Q(x,\omega ,E)(\eta \psi )=\eta Q(x,\omega ,E)\psi +(Q(x,\omega ,E)\eta )\psi$$ and for $q\in L^{\infty }({\mathbb{R}}^3\times I,W^{\infty ,1}(S))$ $${\Vert q\psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S))}\le {\Vert q\Vert }_{L^{\infty }({\mathbb{R}}^3\times I,W^{\infty ,1}(S))}{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S))}$$ and $${\Vert qU\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}\le {\Vert q\Vert }_{L^{\infty }({\mathbb{R}}^3\times I,W^{\infty ,1}(S))}{\Vert U\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}$$ we conclude by (66)(66) $$\begin{array}{l}{\Vert (\eta \psi )(.,.,E_m)\Vert }_{L^2({\mathbb{R}}^3\times S)}^2={\int }_{{\mathbb{R}}^3\times S}{(\eta \psi )}^2(x,\omega ,E_m)\mathit{\text{dωdx}}\\ ={\int }_{{\mathbb{R}}^3\times S}{\int }_{E_0}^{E_m}\frac{\partial }{\partial E}({(\eta \psi )}^2(x,\omega ,E))\mathit{dEd}\omega dx\\ ={\int }_{{\mathbb{R}}^3\times S\times I}2(\eta \psi )(x,\omega ,E)\frac{\partial (\eta \psi )}{\partial E}(x,\omega ,E)\mathit{dEd}\omega dx\\ =2{\int }_{{\mathbb{R}}^3\times S\times I}(\eta \psi )(x,\omega ,E)(\frac{\partial (\eta \psi )}{\partial E}(x,\omega ,E)+\frac{1}{a}\omega ·{\nabla }_x(\eta \psi ))(x,\omega ,E)\mathit{dEd}\omega dx\\ +{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){(\eta \psi )}^2(x,\omega ,E)\mathit{dxd}\omega dE\\ =2{\int }_{{\mathbb{R}}^3\times S\times I}(\eta \psi )(x,\omega ,E)\frac{1}{a}(Q(x,\omega ,E)(\eta \psi ))(x,\omega ,E)\mathit{dEd}\omega dx\\ +{\int }_{{\mathbb{R}}^3\times S\times I}\omega ·{\nabla }_x\left(\frac{1}{a}\right){(\eta \psi )}^2(x,\omega ,E)\mathit{dxd}\omega dE\\ \le ({\Vert \eta \psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S)}^2+{\Vert \frac{1}{a}Q(x,\omega ,E,D)(\eta \psi )\Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}^2)\\ +{\Vert \omega ·{\nabla }_x\left(\frac{1}{a}\right)\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert \eta \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\end{array}$$(66) (67) $${\Vert (\eta \psi )(.,.,E_m)\Vert }_{L^2({\mathbb{R}}^3\times S)}^2\le C\prime ({\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^1(S))}^2+{\Vert Q(x,\omega ,E,D)\psi \Vert }_{L^2({\mathbb{R}}^3\times I,H^{-1}(S))}^2)$$(67) and so the assertion holds for γ_m (by choosing e.g. $\eta ={\theta }_1{\theta }_2$ where ${\theta }_1\in C_0^{\infty }({\mathbb{R}}^3\times S)$ and ${\theta }_2\in C_0^{\infty }\left(\mathrm{ℝ}\right)$ such that ${\theta }_2(E_0)=0,{\theta }_2(E_m)=1$). The assertion for γ₀ is similarly proved which completes the proof. □

Let $P^{*}$ be the formal adjoint operator of $P(x,\omega ,E,D)$ that is, $$P^{*}(x,\omega ,E,D)v=-a\frac{\partial v}{\partial E}-\frac{\partial a}{\partial E}v+b^{*}(x,\omega ,E,{\partial }_{\omega })v-\omega ·{\nabla }_{x}v$$ where the formal adjoint of b(x, ω, E, ∂_ω) is $$b^{*}(x,\omega ,E,{\partial }_{\omega })v=c(x,E){\Delta }_{S}v+d^{*}(x,\omega ,E,{\partial }_{\omega })v.$$

We have the next generalized Green’s formula

Lemma 3.8.

Suppose that (44, 70) hold and that $\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})\cap L^2({\mathbb{R}}^3\times I,H^1(S))$ and $v\in \widehat{\mathcal{H}}$ for which $(\text{supp}(v))\cap \partial ({\mathbb{R}}^3\times S\times I)$ is a compact subset of $\partial ({\mathbb{R}}^3\times S\times I)=({\mathbb{R}}^3\times S\times \{E_m\})\cup ({\mathbb{R}}^3\times S\times \{E_0\}).$ Then (here $〈.,.〉$ is the duality defined in Remark 3.6 above) (68) $$\begin{array}{l}〈P(x,\omega ,E,D)\psi ,v〉-〈P^{*}(x,\omega ,E,D)v,\psi 〉\\ ={\int }_{{\mathbb{R}}^3\times S}(a(·,E_m)\psi (·,·,E_m)v(·,·,E_m)-a(·,E_0)\psi (·,·,E_0)v(·,·,E_0))\mathit{dxd}\omega .\end{array}$$(68)

The proof follows by applying the standard Green’s formula and density arguments.

Remark 3.9.

The Green formula has some additional generalizations. Especially, (68)(68) $$\begin{array}{l}〈P(x,\omega ,E,D)\psi ,v〉-〈P^{*}(x,\omega ,E,D)v,\psi 〉\\ ={\int }_{{\mathbb{R}}^3\times S}(a(·,E_m)\psi (·,·,E_m)v(·,·,E_m)-a(·,E_0)\psi (·,·,E_0)v(·,·,E_0))\mathit{dxd}\omega .\end{array}$$(68) holds for $\psi =v$ in the case when $\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})\cap L^2({\mathbb{R}}^3\times I,H^1(S))$ such that ${\gamma }_{\mathrm{m}}(\psi ),{\gamma }_0(\psi )\in L^2({\mathbb{R}}^3\times S)$ (cf. Dautray and Lions 1999, 225).

Under the assumption (66)(62) $$T\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f.$$(62) the weak solution ψ of the equation (62)(62) $$T\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f.$$(62) obtained in Theorem 3.5 can be shown to be a solution of the initial value problem. We have

Theorem 3.10.

Suppose that the assumptions of Theorem 3.5 and (64)(62) $$T\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f.$$(62) are valid. Let $f\in L^2({\mathbb{R}}^3\times S\times I)$. Then the initial value transport problem (69) $$\begin{array}{l}a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f\\ \psi (.,.,E_m)=0\end{array}$$(69) has a unique solution $\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$. In addition, the solution ψ obeys the apriori estimate (70) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(70)

Proof.

The proof is analogous to the proof of Theorem 5.7 (Tervo et al. 2018a) (items (ii)-(iii) of the proof) and we omit the detailed treatments. □

The estimate (70)(70) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(70) implies that (71) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}$$(71) for all $\psi \in D:=\{\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})|\psi (.,.,E_m)=0\}$ where, as above (72) $$T\psi =a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi .$$(72)

Actually, instead of (71)(71) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}$$(71) more can be said

Theorem 3.11.

Suppose that the assumptions of Theorem 3.10 are valid. Then the apriori estimate (73) $$c″{\Vert \psi \Vert }_{\mathcal{H}}^2\le {〈T\psi ,\psi 〉}_{L^2({\mathbb{R}}^3\times S\times I)}\text{for all}\psi \in D$$(73) holds.

Proof.

The proof follows by the generalized Green’s formula using the same kind of estimates as utilized in the proof of the above Theorem 3.4. We omit the detailed proof. □

4. Regularity results of solutions emerging from explicit formulas of solutions

In some cases the transport equation can be solved explicitly. The obtained solution formulas will imply regularity of solutions. The results contain typical (anisotropic) regularity properties of solutions that can be said in the global case $G={\mathbb{R}}^3.$ Moreover, the below case studies expose relevant scales within which the regularity results can be formulated. Hence it is reasonable to consider the subsequent computational methods. To keep the computations limited we focus on the spatial regularity (x-regularity) but some outlooks for regularity with respect to all variables are exposed as well.

4.1. On regularity with respect to x-variable

4.1.1. Convection-scattering equation

We begin by considering a convection-scattering equation (74) $$T\psi :=\omega ·{\nabla }_x\psi +\Sigma \psi =f.$$(74) where Σ obeys (75) $$\Sigma =\Sigma (x,\omega ,E)\ge c\prime >0$$(75) and where $f\in L^2({\mathbb{R}}^3\times S\times I).$ The solution ψ is obtained explicitly (Tervo and Kokkonen 2017, sections 4 and 5 or Dautray and Lions 1999, 244) and it is given by (76) $$\psi ={\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt.$$(76)

Moreover, the solution obeys an estimate (77) $${\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le \frac{1}{c\prime }{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(77)

We utilize repeatedly (without any mention) the following result from analysis.

Theorem 4.1.

Let $G\subset {\mathbb{R}}^3.$ be open and let $(Y,\mu )$ be a measure space. Suppose that $f:G\times Y\to \mathbb{R}$ is a measurable mapping such that the partial derivatives $\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x,y)$ exist for $\left|\alpha \right|\le r$ and that there exist $g_{\alpha }\in L^1(Y)$ such that $$|\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x,y)|\le g_{\alpha }(y),x\in G,y\in Y.$$

Then (78) $$\frac{{\partial }^{\alpha }}{\partial x^{\alpha }}({\int }_{Y}f(x,y)dy)={\int }_Y\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}(x,y)dy$$(78) for $\left|\alpha \right|\le r.$

Proof.

For the proof we refer e.g. to Folland (1999, 56). □

We begin with

Theorem 4.2.

Suppose that $\Sigma \in W^{\infty ,(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ such that (75)(75) $$\Sigma =\Sigma (x,\omega ,E)\ge c\prime >0$$(75) holds and that $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$. Then the solution $\psi \in L^2({\mathbb{R}}^3\times S\times I)$ of the equation (74)(74) $$T\psi :=\omega ·{\nabla }_x\psi +\Sigma \psi =f.$$(74) belongs to $H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

Proof.

A. Constant Σ. To illustrate actual computations we, at first, consider the assertion in the case where $\Sigma ={\Sigma }_0$ is constant. In this case the solution is (by (76)(76) $$\psi ={\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt.$$(76) ) (79) $$\psi ={\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}f(x-t\omega ,\omega ,E)dt$$(79) and by (77)(77) $${\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le \frac{1}{c\prime }{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(77) (80) $${\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le \frac{1}{{\Sigma }_0}{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(80)

Assume that $f\in C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R}).$ For a general $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ the claim is obtained by a limiting process as exposed below. For all .. (by Lemma 4.1) (81) $$\left(\frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}\right)(x,\omega ,E)={\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}\left(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\right)(x-t\omega ,\omega ,E)dt.$$(81)

Furthermore, we find by the Cauchy-Schwartz inequality that (82) $$\left|\right(\frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }})(x,\omega ,E)|\le {({\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}dt)}^{1/2}{({\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}|\left(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\right)(x-t\omega ,\omega ,E){|}^{2}dt)}^{1/2}$$(82) and then for $\left|\alpha \right|\le m_1$ (83) $$\begin{array}{l}{\Vert \frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I|\left(\frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}\right)(x,\omega ,E){|}^2\mathit{\text{dxdωdE}}\\ \le \frac{1}{{\Sigma }_0}{\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I({\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}|(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }})(x-t\omega ,\omega ,E){|}^{2}dt)\mathit{dxd}\omega dE\\ =\frac{1}{{\Sigma }_0}{\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}({\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I|(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }})(x-t\omega ,\omega ,E){|}^2\mathit{\text{dxdωdE}})dt\\ =\frac{1}{{\Sigma }_0^2}{\Vert \frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\end{array}$$(83) where we noticed by using the change of variables $x\prime =x-t\omega$ that $${\int }_{{\mathbb{R}}^3}|\left(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\right)(x-t\omega ,\omega ,E){|}^{2}dx={\int }_{{\mathbb{R}}^3}|\left(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\right)(x,\omega ,E){|}^{2}dx.$$

Due to (83)(83) $$\begin{array}{l}{\Vert \frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2={\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I|\left(\frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}\right)(x,\omega ,E){|}^2\mathit{\text{dxdωdE}}\\ \le \frac{1}{{\Sigma }_0}{\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I({\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}|(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }})(x-t\omega ,\omega ,E){|}^{2}dt)\mathit{dxd}\omega dE\\ =\frac{1}{{\Sigma }_0}{\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}({\int }_{{\mathbb{R}}^3}{\int }_S{\int }_I|(\frac{{\partial }^{\alpha }f}{\partial x^{\alpha }})(x-t\omega ,\omega ,E){|}^2\mathit{\text{dxdωdE}})dt\\ =\frac{1}{{\Sigma }_0^2}{\Vert \frac{{\partial }^{\alpha }f}{\partial x^{\alpha }}\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\end{array}$$(83) (84) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le \frac{1}{{\Sigma }_0}{\Vert f\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R}).$$(84)

Suppose, more generally that $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$ Then there exists a sequence $\left\{f_n\right\}\subset C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R})$ such that ${\Vert f_n-f\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\to 0$ for $n\to \infty$ (recall Theorem 2.1). Let (85) $${\psi }_n(x,\omega ,E):={\int }_0^{\infty }{e}^{-{\Sigma }_{0}t}{f}_n(x-t\omega ,\omega ,E)dt.$$(85)

By the inequality (86)(84) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le \frac{1}{{\Sigma }_0}{\Vert f\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times \mathbb{R}).$$(84) $${\Vert {\psi }_n-{\psi }_m\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le \frac{1}{{\Sigma }_0}{\Vert f_n-f_m\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}$$ and so $\left\{{\psi }_n\right\}$ is a Cauchy sequence in $H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$ Let $\psi \prime \in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ such that ${\Vert {\psi }_n-\psi \prime \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\to 0$ for $n\to \infty .$ Since on the other hand, by (80)(80) $${\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le \frac{1}{{\Sigma }_0}{\Vert f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(80) $${\Vert {\psi }_n-\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}\le \frac{1}{{\Sigma }_0}{\Vert f_n-f\Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}\to 0$$ we conclude that $\psi =\psi \prime \in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}),$ as desired.

B. Variable Σ. Secondly, we consider the claim more generally for $\Sigma \in W^{\infty ,(1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

B.1. At first, suppose that $m_1=1.$ Let $f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$ By routine computations we get (by Lemma 4.1) (86) $$\begin{array}{l}\frac{\partial (\psi )}{\partial x_j}(x,\omega ,E)={\int }_0^{\infty }(-{\int }_0^t\frac{\partial \Sigma }{\partial x_j}(x-s\omega ,\omega ,E)ds)e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\frac{\partial f}{\partial x_j}(x-t\omega ,\omega ,E)dt,\end{array}$$(86)

Furthermore, we find that by (75)(75) $$\Sigma =\Sigma (x,\omega ,E)\ge c\prime >0$$(75) (87) $$e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\le e^{-c\prime t}.$$(87)

Moreover, (88) $$|{\int }_0^t\frac{\partial \Sigma }{\partial x_j}(x-s\omega ,\omega ,E)ds|\le {\Vert \frac{\partial \Sigma }{\partial x_j}\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}t.$$(88)

Using same kind of techniques as in Part A (of the present proof), these estimates imply that there exists C > 0 such that (89) $${\Vert \psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$$(89)

Hence by similar limiting arguments which we applied in Part A show that for $f\in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ the solution $\psi \in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

B.2. Let more generally $m_1\in \mathbb{N}$ and $f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$ The analogous computations as in Parts A and B.1. for higher derivatives show that (90) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$$(90)

For example, (91) $$\begin{array}{l}\frac{{\partial }^2\psi }{\partial x_k\partial x_j}(x,\omega ,E)={\int }_0^{\infty }(-{\int }_0^t\frac{{\partial }^2\Sigma }{\partial x_k\partial x_j}(x-s\omega ,\omega ,E)ds)e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }(-{\int }_0^t\frac{\partial \Sigma }{\partial x_j}(x-s\omega ,\omega ,E)ds)(-{\int }_0^t\frac{\partial \Sigma }{\partial x_k}(x-s\omega ,\omega ,E)ds)\\ ·e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }(-{\int }_0^t\frac{\partial \Sigma }{\partial x_j}(x-s\omega ,\omega ,E)ds)e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\frac{\partial f}{\partial x_k}(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }(-{\int }_0^t\frac{\partial \Sigma }{\partial x_k}(x-s\omega ,\omega ,E)ds)e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\frac{\partial f}{\partial x_j}(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\frac{{\partial }^{2}f}{\partial x_k\partial x_j}(x-t\omega ,\omega ,E)dt\end{array}$$(91) and so by using similar kind of arguments as above we get (90)(90) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$$(90) . We omit further details of this technically more complex part but notice that the Leibniz’s rule $$\frac{{\partial }^{\alpha }\psi }{\partial x^{\alpha }}(x,\omega ,E)=\sum _{\beta \le \alpha }\left(\begin{array}{c}\alpha \\ \beta \end{array}\right){\int }_0^{\infty }\frac{{\partial }^{\alpha -\beta }}{\partial x^{\alpha -\beta }}(e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds})\frac{{\partial }^{\beta }f}{\partial x^{\beta }}(x-t\omega ,\omega ,E)dt.$$ is useful in computations. That is why, we are able to conclude (by utilizing limiting methods as above) that for $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ the solution $\psi \in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ which completes the proof. □

4.1.2. A Continuous slowing down convection-scattering equation

We deal with the following special case of a slowing down convection-scattering equation. Suppose that $a=a(E)$ is independent of x and $\Sigma =\Sigma (x,\omega )$ is independent of E. We assume that $a:I\to \mathbb{R}$ is continuous and that (92) $$\underset{E\in I}{\inf }a(E)=:\kappa >0,\frac{\partial a}{\partial E}\in L^{\infty }(I).$$(92)

Σ is assumed to belong to $L^{\infty }({\mathbb{R}}^3\times S)$ and (93) $$\Sigma (x,\omega )\ge c\prime >0\mathrm{a}.\mathrm{e}.(x,\omega )\in {\mathbb{R}}^3\times S.$$(93)

In addition, for simplicity, we assume that $E_0=0.$ Consider the problem of the form (94) $$T\psi :=-\frac{\partial (a\psi )}{\partial E}+\omega ·{\nabla }_x\psi +\Sigma (x,\omega )\psi =f(x,\omega ,E),\psi (.,.,E_{\mathrm{m}})=0.$$(94)

One can show that the solution of the problem (cf. Example 10.1 of Tervo et al. 2018) (95) $$T\psi =f,\psi (.,.,E_{\mathrm{m}})=0$$(95) is (96) $$\psi (x,\omega ,E)=\frac{1}{a(E)}({\int }_0^{R(E_{\mathrm{m}})-R(E)}{e}^{-{\int }_0^s\Sigma (x-\tau \omega ,\omega )d\tau }\tilde{f}(x-s\omega ,\omega ,R(E)+s)ds)$$(96) where $$R(E)={\int }_0^E\frac{1}{a(\tau )}d\tau$$ and $$\tilde{f}(x,\omega ,\eta )=a(R^{-1}(\eta ))f(x,\omega ,R^{-1}(\eta )).$$

The solution (96)(96) $$\psi (x,\omega ,E)=\frac{1}{a(E)}({\int }_0^{R(E_{\mathrm{m}})-R(E)}{e}^{-{\int }_0^s\Sigma (x-\tau \omega ,\omega )d\tau }\tilde{f}(x-s\omega ,\omega ,R(E)+s)ds)$$(96) possesses the following x-regularity

Theorem 4.3.

Suppose that $a\in W^{\infty ,m_1}(I^{°})$ and $\Sigma \in W^{\infty ,(m_1,0)}({\mathbb{R}}^3\times S)$ such that (92)(92) $$\underset{E\in I}{\inf }a(E)=:\kappa >0,\frac{\partial a}{\partial E}\in L^{\infty }(I).$$(92) , (93)(93) $$\Sigma (x,\omega )\ge c\prime >0\mathrm{a}.\mathrm{e}.(x,\omega )\in {\mathbb{R}}^3\times S.$$(93) hold. Furthermore, suppose that $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$. Then the solution $\psi \in L^2({\mathbb{R}}^3\times S\times I)$ of the problem (94)(94) $$T\psi :=-\frac{\partial (a\psi )}{\partial E}+\omega ·{\nabla }_x\psi +\Sigma (x,\omega )\psi =f(x,\omega ,E),\psi (.,.,E_{\mathrm{m}})=0.$$(94) belongs to $H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

Proof.

We omit the proof but the below computations in Example 4.4 for a more simple case will shed light to the assertion. □

To illustrate the claim of the above theorem we elaborate the following example.

Example 4.4.

Let a = 1 and let $\Sigma ={\Sigma }_0>0$ be constant. Then by (96)(96) $$\psi (x,\omega ,E)=\frac{1}{a(E)}({\int }_0^{R(E_{\mathrm{m}})-R(E)}{e}^{-{\int }_0^s\Sigma (x-\tau \omega ,\omega )d\tau }\tilde{f}(x-s\omega ,\omega ,R(E)+s)ds)$$(96) the solution of the transport problem (97) $$-\frac{\partial \psi }{\partial E}+\omega ·{\nabla }_x\psi +{\Sigma }_0\psi =f,\psi (.,.,E_{\mathrm{m}})=0$$(97) is (98) $$\psi (x,\omega ,E)={\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}f(x-s\omega ,\omega ,E+s)ds.$$(98)

Let $m_1\in \mathbb{N}$ For $\left|\alpha \right|\le m_1$ (99) $$({\partial }_x^{\alpha }\psi )(x,\omega ,E)={\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}({\partial }_x^{\alpha }f)(x-s\omega ,\omega ,E+s)ds.$$(99)

Assuming that $f\in H^{(m_{1}0,,0)}({\mathbb{R}}^3\times S\times I^{°})$ we see by applying similar type of computations as above in section 4.1.1 that by (99)(99) $$({\partial }_x^{\alpha }\psi )(x,\omega ,E)={\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}({\partial }_x^{\alpha }f)(x-s\omega ,\omega ,E+s)ds.$$(99) $\psi \in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

4.2. Some outlines to regularity results with respect to all variables

We give some depictions for regularity with respect to all variables $(x,\omega ,E)$ emerged from explicit formulas. Consider the equation (100) $$\omega ·{\nabla }_x\psi +\Sigma \psi =f(x,\omega ,E)$$(100) where (101) $$\Sigma =\Sigma (x,\omega ,E)\ge c\prime >0\mathrm{a}.\mathrm{e}..$$(101)

Recall that (102) $$\psi ={\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt.$$(102)

We have

Theorem 4.5.

Suppose that $\Sigma \in W^{\infty ,(m_1+m_2,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°})$ such that (101)(101) $$\Sigma =\Sigma (x,\omega ,E)\ge c\prime >0\mathrm{a}.\mathrm{e}..$$(101) holds and that $f\in H^{(m_1+m_2,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°})$. Then the solution $\psi \in L^2({\mathbb{R}}^3\times S\times I)$ of the equation (100)(100) $$\omega ·{\nabla }_x\psi +\Sigma \psi =f(x,\omega ,E)$$(100) belongs to $H^{(m_1,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°}).$

Proof.

We omit the proof but the next Example 4.6 illustrates the assertion; especially the fact that (to guarantee the stated ω-regularity) we need the regularity up to order $m_1+m_2$ for f and Σ with respect to x-variable. □

Example 4.6.

In this example we compute some special cases.

A. Let $m_1=0,m_2=1,m_3=0.$ Assume that $\Sigma \in W^{\infty ,(1,1,0)}({\mathbb{R}}^3\times S\times I^{°})$ and $f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$ Let I_S be the identity mapping of S (that is, $I_S(\omega )=\omega$). We notice that I_S is a $C^{\infty }$-mapping since S is a $C^{\infty }$-manifold. By the chain rule we find that (103) $$\frac{\partial }{\partial {\omega }_j}(f(x-t\omega ,\omega ,E))=-t〈{\nabla }_{x}f(x-t\omega ,\omega ,E),\frac{\partial I_S}{\partial {\omega }_j}(\omega )〉+\frac{\partial f}{\partial {\omega }_j}(x-t\omega ,\omega ,E)$$(103) and similarly for $\frac{\partial }{\partial {\omega }_j}(\Sigma (x-s\omega ,\omega ,E)).$ Moreover, $${\frac{\partial I_S}{\partial {\omega }_j}}_{|\omega }={\overline{\Omega }}_j(\omega ),j=1,2$$ where ${\overline{\Omega }}_j(\omega )$ are tangent vectors on S at ω that is, $${\overline{\Omega }}_1(\omega )=(-{\omega }_2,{\omega }_1,0),{\overline{\Omega }}_2(\omega )=(\frac{{\omega }_1{\omega }_3}{\sqrt{{\omega }_1^2+{\omega }_2^2}},\frac{{\omega }_2{\omega }_3}{\sqrt{{\omega }_1^2+{\omega }_2^2}},-\sqrt{{\omega }_1^2+{\omega }_2^2}).$$

Hence by (102)(102) $$\psi ={\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt.$$(102) we have for j = 1, 2 (104) $$\begin{array}{l}\frac{\partial \psi }{\partial {\omega }_j}(x,\omega ,E)={\int }_0^{\infty }(-{\int }_0^t((-s〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})\Sigma )(x-s\omega ,\omega ,E)ds)\\ ·e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}((-t〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})f)(x-t\omega ,\omega ,E)dt.\end{array}$$(104)

Furthermore, (105) $$\begin{array}{l}{\int }_0^t|((-s〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})\Sigma )(x-s\omega ,\omega ,E)|\\ \le {\int }_0^t({\Vert {\overline{\Omega }}_j(\omega )\Vert }_{L^{\infty }(S)}{\Vert {\nabla }_x\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}s+{\Vert \frac{\partial \Sigma }{\partial {\omega }_j}\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)})\\ =:C_1{t}^2+C_{2}t\end{array}$$(105) and (106) $$\begin{array}{l}|((-t〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})f)(x-t\omega ,\omega ,E)|\\ \le {\Vert {\overline{\Omega }}_j(\omega )\Vert }_{L^{\infty }(S)}t|{\nabla }_{x}f(x-t\omega ,\omega ,E)|+|\frac{\partial f}{\partial {\omega }_j}(x-t\omega ,\omega ,E)|.\end{array}$$(106)

Hence by (104)(104) $$\begin{array}{l}\frac{\partial \psi }{\partial {\omega }_j}(x,\omega ,E)={\int }_0^{\infty }(-{\int }_0^t((-s〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})\Sigma )(x-s\omega ,\omega ,E)ds)\\ ·e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}((-t〈{\overline{\Omega }}_j(\omega ),{\nabla }_x〉+\frac{\partial }{\partial {\omega }_j})f)(x-t\omega ,\omega ,E)dt.\end{array}$$(104) , (87)(87) $$e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\le e^{-c\prime t}.$$(87) (107) $$\begin{array}{l}|\frac{\partial \psi }{\partial {\omega }_j}(x,\omega ,E)|\le {\int }_0^{\infty }{e}^{-c\prime t}(C_1{t}^2+C_{2}t)|f(x-t\omega ,\omega ,E)|dt\\ +{\int }_0^{\infty }{e}^{-c\prime t}(t{\Vert {\overline{\Omega }}_j(\omega )\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}|{\nabla }_{x}f(x-t\omega ,\omega ,E)|+|\frac{\partial f}{\partial {\omega }_j}(x-t\omega ,\omega ,E)|)\end{array}$$(107) which implies as in section 4.1.1 that (108) $${\Vert \psi \Vert }_{H^{(0,1,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(1,1,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$$(108)

From the estimate (108)(108) $${\Vert \psi \Vert }_{H^{(0,1,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(1,1,0)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$$(108) it follows as above that for $f\in H^{(1,1,0)}({\mathbb{R}}^3\times S\times I^{°})$ the solution $\psi \in H^{(0,1,0)}({\mathbb{R}}^3\times S\times I^{°}).$

B. Let $m_1=0,m_2=0,m_3=1.$ Suppose that $\Sigma \in W^{\infty ,(0,0,1)}({\mathbb{R}}^3\times S\times I^{°})$ and $f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°}).$ Then we have (109) $$\begin{array}{l}\frac{\partial \psi }{\partial E}(x,\omega ,E)={\int }_0^{\infty }(-{\int }_0^t\frac{\partial \Sigma }{\partial E}(x-s\omega ,\omega ,E)ds)e^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}f(x-t\omega ,\omega ,E)dt\\ +{\int }_0^{\infty }{e}^{-{\int }_0^t\Sigma (x-s\omega ,\omega ,E)ds}\frac{\partial f}{\partial E}(x-t\omega ,\omega ,E)dt\end{array}$$(109) where $$|{\int }_0^t\frac{\partial \Sigma }{\partial E}(x-t\omega ,\omega ,E)ds|\le {\Vert \frac{\partial \Sigma }{\partial E}\Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}t.$$

Thus again similar computations as in section 4.1.1 show that (110) $${\Vert \psi \Vert }_{H^{(0,0,1)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert f\Vert }_{H^{(0,0,1)}({\mathbb{R}}^3\times S\times I^{°})}\text{for all}f\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$$(110) from which it follows that for $f\in H^{(0,0,1)}({\mathbb{R}}^3\times S\times I^{°})$ the solution $\psi \in H^{(0,0,1)}({\mathbb{R}}^3\times S\times I^{°}).$

For a special case of continuous slowing down equation brought up in section 4.1.2 we have

Theorem 4.7.

Suppose that $a\in W^{\infty ,m_3}(I^{°})$ and $\Sigma \in W^{\infty ,(m_1+m_2,m_2)}({\mathbb{R}}^3\times S)$ such that (92)(92) $$\underset{E\in I}{\inf }a(E)=:\kappa >0,\frac{\partial a}{\partial E}\in L^{\infty }(I).$$(92) , (93)(93) $$\Sigma (x,\omega )\ge c\prime >0\mathrm{a}.\mathrm{e}.(x,\omega )\in {\mathbb{R}}^3\times S.$$(93) hold. Furthermore, suppose that $f\in H^{(m_1+m_2+m_3,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°})$. Then the solution $\psi \in L^2({\mathbb{R}}^3\times S\times I)$ of the problem (111) $$-\frac{\partial (a\psi )}{\partial E}+\omega ·{\nabla }_x\psi +\Sigma (x,\omega )\psi =f(x,\omega ,E),\psi (.,.,E_{\mathrm{m}})=0.$$(111) belongs to $H^{(m_1,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°}).$

Proof.

We omit the proof. The Example 4.8 below gives some insight for the claim. □

Example 4.8.

Consider the case of Example 4.4 for $m_1=m_2=0,m_3=2.$ Recall that ψ is given by (98)(98) $$\psi (x,\omega ,E)={\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}f(x-s\omega ,\omega ,E+s)ds.$$(98) . Suppose that (112) $$f\in H^{(2,0,2)}({\mathbb{R}}^3\times S\times I^{°}).$$(112)

Then we see that (113) $$\begin{array}{l}({\partial }_E\psi )(x,\omega ,E)=-e^{-{\Sigma }_0(E_{\mathrm{m}}-E)}f(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ +{\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}({\partial }_{E}f)(x-s\omega ,\omega ,E+s)ds\end{array}$$(113) and furthermore (114) $$\begin{array}{l}({\partial }_E^2\psi )(x,\omega ,E)=-{\Sigma }_0{e}^{-{\Sigma }_0(E_{\mathrm{m}}-E)}f(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ -e^{-{\Sigma }_0(E_{\mathrm{m}}-E)}(\omega ·{\nabla }_{x}f)(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ +e^{-{\Sigma }_0(E_{\mathrm{m}}-E)}({\partial }_{E}f)(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ +{\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}({\partial }_E^{2}f)(x-s\omega ,\omega ,E+s)ds.\end{array}$$(114)

Due to the Sobolev imbedding Theorem (e.g. Friedman 1976, 22) (115) $${\Vert u\Vert }_{W^{\infty ,k}(I^{°})}\le C{\Vert u\Vert }_{H^{k+1}(I^{°})},u\in H^{k+1}(I^{°}),k=0,1$$(115) and so by (114)(114) $$\begin{array}{l}({\partial }_E^2\psi )(x,\omega ,E)=-{\Sigma }_0{e}^{-{\Sigma }_0(E_{\mathrm{m}}-E)}f(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ -e^{-{\Sigma }_0(E_{\mathrm{m}}-E)}(\omega ·{\nabla }_{x}f)(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ +e^{-{\Sigma }_0(E_{\mathrm{m}}-E)}({\partial }_{E}f)(x-(E_{\mathrm{m}}-E)\omega ,\omega ,E_{\mathrm{m}})\\ +{\int }_0^{E_{\mathrm{m}}-E}{e}^{-{\Sigma }_{0}s}({\partial }_E^{2}f)(x-s\omega ,\omega ,E+s)ds.\end{array}$$(114) $${\Vert {\partial }_{E}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le C{\Vert f\Vert }_{H^{(1,0,2)}({\mathbb{R}}^3\times S\times I)}$$ which implies the claim for $m=(0,0,2).$

Remark 4.9.

The assumptions in Theorems 4.5 and 4.7 are not strict ones. In fact, the above computations show that in Theorem 4.5 it suffices only to assume that

$$f\in {\cap }_{k=0}^{m_2}{H}^{(m_1+k,m_2-k,m_3)}({\mathbb{R}}^3\times S\times I^{°}),\Sigma \in {\cap }_{k=0}^{m_2}{W}^{\infty ,(m_1+k,m_2-k,m_3)}({\mathbb{R}}^3\times S\times I^{°}).$$

Similarly, in Theorem 4.7 it suffices only to assume that

$$f\in {\cap }_{k=0}^{m_2}{\cap }_{j=0}^{m_3}{H}^{(m_1+k+j,m_2-k,m_3-j)}({\mathbb{R}}^3\times S\times I^{°})$$

$$\Sigma \in {\cap }_{k=0}^{m_2}{\cap }_{j=0}^{m_3}{W}^{\infty ,(m_1+k+j,m_2-k,m_3-j)}({\mathbb{R}}^3\times S\times I^{°}).$$

The claim of Theorem 4.7 runs along general regularity results for evolution type equations. The above computations show that the use of explicit formulas in retrieving regularity results for transport equations is very limited.

5. On spatial regularity of solutions for the total transport equation by applying partial differences

In this section we consider the regularity of solutions of the complete transport equation (116) $$T\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi +\Sigma \psi -K_r\psi =f$$(116) where $b(x,\omega ,E,{\partial }_{\omega })\psi =c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi .$ We assume that the assumptions of Theorem 3.10 are valid. Then for any $f\in L^2({\mathbb{R}}^3\times S\times I)$ the solution $\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$ of the initial value problem (117) $$T\psi =f,\psi (.,.,E_{\mathrm{m}})=0$$(117) exists. In addition, there exists a constant $C\ge 0$ such that (118) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(118)

Recall that $$P\psi =P(x,\omega ,E,D)\psi :=a(x,E)\frac{\partial \psi }{\partial E}+b(x,\omega ,E,{\partial }_{\omega })\psi +\omega ·{\nabla }_x\psi$$

Our basic tools to prove regularity are the application of pertinent a priori estimates and partial differences. For simplicity, we restrict ourselves to the case $K_r=K_r^1$ that is, K_r is of the form (119) $$(K_r\psi )(x,\omega ,E)={\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime ,$$(119) where $\sigma :{\mathbb{R}}^3\times S^2\times I^2\to \mathbb{R}$ is a non-negative measurable function such that (27)(27) $$\begin{array}{c}{\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \le M_1,\\ {\int }_{S\prime \times I\prime }{\sigma }^1(x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \le M_2,\end{array}$$(27) holds.

In the following we, for simplicity, assume that a, c and d are independent of x. In more general cases it seems that the techniques based on the use of partial convolutions are more appropriate than the partial differences (see the Discussion section below). We recall the assumptions of Theorem 3.10 for this case: (120) $$\Sigma \in L^{\infty }({\mathbb{R}}^3\times S\times I),\Sigma \ge 0\mathrm{a}.\mathrm{e}.\text{in}{\mathbb{R}}^3\times S\times I,$$(120) (121) $$a\in W^{\infty ,1}(I),c\in L^{\infty }(I),d_j\in L^{\infty }(I,W^{\infty ,1}(S))$$(121) (122) $$-(\frac{\partial a}{\partial E}(E)+({\text{div}}_{S}d)(\omega ,E))\ge q_1>0,a.e.,$$(122) (123) $$-c(E)\ge q_2>0,a.e.,$$(123) (124) $$a(E_0)<0,a(E_m)<0,$$(124) (125) $$|a(E)|\ge q_4>0\mathrm{a},\mathrm{e},.$$(125) (126) $$\begin{array}{c}{\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \le M_1,\\ {\int }_{S\prime \times I\prime }\sigma (x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \le M_2,\end{array}$$(126) (127) $$\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }\sigma (x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \ge c\prime ,$$(127) (128) $$\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \ge c\prime$$(128) where $c\prime$ is a strictly positive constant.

Key techniques based on difference approaches are contained in the proof of the next theorem.

Theorem 5.1.

Let $m_1\in \mathbb{N}$. Assume that a, c and d are independent of x and that the assumptions (120)(120) $$\Sigma \in L^{\infty }({\mathbb{R}}^3\times S\times I),\Sigma \ge 0\mathrm{a}.\mathrm{e}.\text{in}{\mathbb{R}}^3\times S\times I,$$(120) , (121)(121) $$a\in W^{\infty ,1}(I),c\in L^{\infty }(I),d_j\in L^{\infty }(I,W^{\infty ,1}(S))$$(121) , (122)(122) $$-(\frac{\partial a}{\partial E}(E)+({\text{div}}_{S}d)(\omega ,E))\ge q_1>0,a.e.,$$(122) , (123)(122) $$-(\frac{\partial a}{\partial E}(E)+({\text{div}}_{S}d)(\omega ,E))\ge q_1>0,a.e.,$$(122) , (124)(123) $$-c(E)\ge q_2>0,a.e.,$$(123) , (125)(124) $$a(E_0)<0,a(E_m)<0,$$(124) , (126)(125) $$|a(E)|\ge q_4>0\mathrm{a},\mathrm{e},.$$(125) , (127)(126) $$\begin{array}{c}{\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)d\omega \prime dE\prime \le M_1,\\ {\int }_{S\prime \times I\prime }\sigma (x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \le M_2,\end{array}$$(126) , and (128)(127) $$\Sigma (x,\omega ,E)-{\int }_{S\prime \times I\prime }\sigma (x,\omega ,\omega \prime ,E,E\prime )d\omega \prime dE\prime \ge c\prime ,$$(127) are valid. Furthermore, suppose that (129) $$\Sigma \in L^{\infty }(S\times I,W^{\infty ,1}({\mathbb{R}}^3),$$(129) and that (130) $$\begin{array}{c}\text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_{S\prime }{\int }_{I\prime }{\Vert \sigma (.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}<\infty ,\\ \text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_S{\int }_I{\Vert \sigma (.,\omega ,\omega \prime ,E,E\prime )\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}\prime <\infty .\end{array}$$(130)

Let $f\in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ and let $\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$ be a solution of the problem (131) $$T\psi =f,\psi (.,.,E_{\mathrm{m}})=0.$$(131)

Then $\psi \in H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$ Furthermore, there exists a constant $C\ge 0$ such that (132) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert T\psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}.$$(132)

Proof.

A. At first, we assume that $m_1=1.$ Let $\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$ and define (the difference quotient with respect to x_j variable) (133) $$({\delta }_{h,j}\psi )(x,\omega ,E):=\frac{\psi (x+h{e}_j,\omega ,E)-\psi (x,\omega ,E)}{h}.$$(133)

In the sequel we denote shortly ${\delta }_h={\delta }_{h,j}$ (for a fixed $j\in \{1,2,3\}$). We find that (134) $$\begin{array}{l}(\Sigma ({\delta }_h\psi ))(x,\omega ,E)=\frac{1}{h}[\Sigma (x,\omega ,E)(\psi (x+h{e}_j,\omega ,E)-\psi (x,\omega ,E))]\\ =\frac{1}{h}[\Sigma (x,\omega ,E)\psi (x+h{e}_j,\omega ,E)+\Sigma (x+h{e}_j,\omega ,E)\psi (x+h{e}_j,\omega ,E)\\ -\Sigma (x+h{e}_j,\omega ,E)\psi (x+h{e}_j,\omega ,E)-\Sigma (x,\omega ,E)\psi (x,\omega ,E)]\\ ={\delta }_h(\Sigma \psi )(x,\omega ,E)-({\delta }_h\Sigma )(x,\omega ,E)\psi (x+h{e}_j,\omega ,E)\end{array}$$(134) and similarly (135) $$\begin{array}{l}(K_r({\delta }_h\psi ))(x,\omega ,E)={\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)({\delta }_h\psi )(x,\omega \prime ,E\prime )d\omega \prime dE\prime \\ ={\delta }_h(K_r\psi )(x,\omega ,E)-{\int }_{S\prime \times I\prime }({\delta }_h\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x+h{e}_j,\omega \prime ,E\prime )d\omega \prime dE\prime .\end{array}$$(135)

Denote $$(({\delta }_h{K}_r)\psi )(x,\omega ,E):={\int }_{S\prime \times I\prime }({\delta }_h\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime .$$

We see that $${\int }_{S\prime \times I\prime }({\delta }_h\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x+h{e}_j,\omega \prime ,E\prime )d\omega \prime dE\prime =(({\delta }_h{K}_r)({\tau }_{h,j}\psi ))(x,\omega ,E)$$ where $({\tau }_{h,j}\psi )(x,\omega ,E):=\psi (x+h{e}_j,\omega ,E)$ (the translation with respect to x_j variable). We denote shortly ${\tau }_{h,j}={\tau }_h.$ Hence (136) $$(K_r({\delta }_h\psi ))(x,\omega ,E)={\delta }_h(K_r\psi )(x,\omega ,E)-(({\delta }_h{K}_r)({\tau }_{h,j}\psi ))(x,\omega ,E).$$(136)

Since a, c and d are independent of x we see that ${\delta }_h\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I)$ and (137) $$P({\delta }_h\psi )={\delta }_h(P\psi ).$$(137)

In fact, denote $({\overline{\delta }}_{h}v)(x,\omega ,E):=v(x-h{e}_j,\omega ,E)-v(x,\omega ,E).$ Recalling that the formal adjoint of $P=P(x,\omega ,E,D)$ is $$P^{*}v=-a\frac{\partial v}{\partial E}-\frac{\partial a}{\partial E}v+c{\Delta }_{S}v-d·{\nabla }_{S}v-{\text{div}}_S(d)v-\omega ·{\nabla }_{x}v,v\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$$ we find that (138) $${\overline{\delta }}_h(P^{*}v)=P^{*}({\overline{\delta }}_{h}v).$$(138)

Furthermore, we have for all $f,v\in L^2({\mathbb{R}}^3\times S\times I)$ (139) $${〈{\delta }_{h}f,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈f,{\overline{\delta }}_{h}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}$$(139) and so we obtain by (138)(138) $${\overline{\delta }}_h(P^{*}v)=P^{*}({\overline{\delta }}_{h}v).$$(138) for $v\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$ (140) $$\begin{array}{l}{〈{\delta }_h(P\psi ),v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈P\psi ,{\overline{\delta }}_{h}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈\psi ,P^{*}({\overline{\delta }}_{h}v)〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ ={〈\psi ,{\overline{\delta }}_h(P^{*}v)〉}_{L^2({\mathbb{R}}^3\times S\times I)}={〈{\delta }_h\psi ,P^{*}v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\end{array}$$(140) that is, (137)(137) $$P({\delta }_h\psi )={\delta }_h(P\psi ).$$(137) holds. Moreover, (141) $${\delta }_h\psi \in \mathcal{H}$$(141) since if $\left\{{\psi }_n\right\}\subset C^2(S,C^1(\overline{G}\times I))$ is a sequence such that ${\psi }_n\to \psi$ in $\mathcal{H}$ with $n\to \infty$ we see that ${\delta }_h{\psi }_n\to {\delta }_h\psi$ in $\mathcal{H}$ and so (141)(141) $${\delta }_h\psi \in \mathcal{H}$$(141) holds (we omit details).

Applying (137)(137) $$P({\delta }_h\psi )={\delta }_h(P\psi ).$$(137) , (134)(134) $$\begin{array}{l}(\Sigma ({\delta }_h\psi ))(x,\omega ,E)=\frac{1}{h}[\Sigma (x,\omega ,E)(\psi (x+h{e}_j,\omega ,E)-\psi (x,\omega ,E))]\\ =\frac{1}{h}[\Sigma (x,\omega ,E)\psi (x+h{e}_j,\omega ,E)+\Sigma (x+h{e}_j,\omega ,E)\psi (x+h{e}_j,\omega ,E)\\ -\Sigma (x+h{e}_j,\omega ,E)\psi (x+h{e}_j,\omega ,E)-\Sigma (x,\omega ,E)\psi (x,\omega ,E)]\\ ={\delta }_h(\Sigma \psi )(x,\omega ,E)-({\delta }_h\Sigma )(x,\omega ,E)\psi (x+h{e}_j,\omega ,E)\end{array}$$(134) and (135)(135) $$\begin{array}{l}(K_r({\delta }_h\psi ))(x,\omega ,E)={\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)({\delta }_h\psi )(x,\omega \prime ,E\prime )d\omega \prime dE\prime \\ ={\delta }_h(K_r\psi )(x,\omega ,E)-{\int }_{S\prime \times I\prime }({\delta }_h\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x+h{e}_j,\omega \prime ,E\prime )d\omega \prime dE\prime .\end{array}$$(135) we get (140) $$\begin{array}{l}T({\delta }_h\psi )=P({\delta }_h\psi )+\Sigma ({\delta }_h\psi )-K_r({\delta }_h\psi )\\ ={\delta }_h(P\psi )+{\delta }_h(\Sigma \psi )-({\delta }_h\Sigma )({\tau }_h\psi )-{\delta }_h(K_r\psi )+({\delta }_h{K}_r)({\tau }_h\psi )\\ ={\delta }_h(T\psi )-({\delta }_h\Sigma )({\tau }_h\psi )+({\delta }_h{K}_r)({\tau }_h\psi ).\end{array}$$(140)

Since $T\psi =f,{\delta }_h\psi \in \mathcal{H}\cap {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I)$ (by (137)(137) $$P({\delta }_h\psi )={\delta }_h(P\psi ).$$(137) , (141)(141) $${\delta }_h\psi \in \mathcal{H}$$(141) ) and ${\delta }_h\psi (.,.,E_m)=\frac{1}{h}(\psi (.+h{e}_j,.,E_m)-\psi (.,.,E_m))=0$ we obtain in virtue of (118)(118) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(118) , (142)(140) $$\begin{array}{l}T({\delta }_h\psi )=P({\delta }_h\psi )+\Sigma ({\delta }_h\psi )-K_r({\delta }_h\psi )\\ ={\delta }_h(P\psi )+{\delta }_h(\Sigma \psi )-({\delta }_h\Sigma )({\tau }_h\psi )-{\delta }_h(K_r\psi )+({\delta }_h{K}_r)({\tau }_h\psi )\\ ={\delta }_h(T\psi )-({\delta }_h\Sigma )({\tau }_h\psi )+({\delta }_h{K}_r)({\tau }_h\psi ).\end{array}$$(140) (143) $$\begin{array}{l}{\Vert {\delta }_h\psi \Vert }_{\mathcal{H}}\le C{\Vert T({\delta }_h\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ =C\Vert {\delta }_{h}f-({\delta }_h\Sigma )({\tau }_h\psi )+({\delta }_h{K}_r)({\tau }_h\psi )\Vert \\ \le C{\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C{\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert {\tau }_h\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C\Vert {\delta }_h{K}_r\Vert {\Vert {\tau }_h\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ =C{\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C{\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C\Vert {\delta }_h{K}_r\Vert {\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.\end{array}$$(143)

Since by (129)(129) $$\Sigma \in L^{\infty }(S\times I,W^{\infty ,1}({\mathbb{R}}^3),$$(129) $\Sigma (.,\omega ,E)\in W^{\infty ,1}({\mathbb{R}}^3)$ we obtain by the Morrey’s inequality (in Sobolev spaces) that a.e. $(x,\omega ,E)$ $$|\Sigma (x+h{e}_j,\omega ,E)-\Sigma (x,\omega ,E)|\le C_{1}h{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}$$ and so (144) $${\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}\le C_1{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}.$$(144)

In the similar way we see by (136)(130) $$\begin{array}{c}\text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_{S\prime }{\int }_{I\prime }{\Vert \sigma (.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}<\infty ,\\ \text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_S{\int }_I{\Vert \sigma (.,\omega ,\omega \prime ,E,E\prime )\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}\prime <\infty .\end{array}$$(130) that a.e. $$\begin{array}{l}{\int }_{S\prime \times I\prime }|\sigma (x+h{e}_j,\omega \prime ,\omega ,E\prime ,E)-\sigma (x,\omega \prime ,\omega ,E\prime ,E)|d\omega \prime dE\prime \\ \le C_{2}h{\int }_{S\prime \times I\prime }{\Vert ({\partial }_{x_j}\sigma )(.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{L^{\infty }({\mathbb{R}}^3)}\end{array}$$ and $$\begin{array}{l}{\int }_{S\prime \times I\prime }|\sigma (x+h{e}_j,\omega ,\omega \prime ,E,E\prime )-\sigma (x,\omega ,\omega \prime ,E,E\prime )|d\omega \prime dE\prime \\ \le C_{2}h{\int }_{S\prime \times I\prime }{\Vert ({\partial }_{x_j}\sigma )(.,\omega ,\omega \prime ,E,E\prime )\Vert }_{L^{\infty }({\mathbb{R}}^3)}\end{array}$$ and and so by Theorem 3.2 (145) $$\Vert {\delta }_h{K}_r\Vert \le C_2\sqrt{M_1{M}_1^{\prime }}$$(145) where M₁ and $M_1^{\prime }$ are as in (130)(130) $$\begin{array}{c}\text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_{S\prime }{\int }_{I\prime }{\Vert \sigma (.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}<\infty ,\\ \text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_S{\int }_I{\Vert \sigma (.,\omega ,\omega \prime ,E,E\prime )\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}\prime <\infty .\end{array}$$(130) (with $m_1=1$).

B. For $v\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$ $$\frac{v(x+h{e}_j,\omega ,E)-v(x,\omega ,E)}{h}={\displaystyle {\int }_0^1\frac{\partial v}{\partial x_j}}(x+t(h{e}_j),\omega ,E)dt$$ and so (146) $$\begin{array}{l}{\Vert {\delta }_{h}v\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2={\int }_{{\mathbb{R}}^3\times S\times I}|{\int }_0^1\frac{\partial v}{\partial x_j}(x+t(h{e}_j),\omega ,E)dt{|}^2\mathit{\text{dxdωdE}}\\ \le {\int }_0^1{\int }_{{\mathbb{R}}^3\times S\times I}|\frac{\partial v}{\partial x_j}(x+t(h{e}_j),\omega ,E){|}^2\mathit{\text{dxdω}}\mathit{dEdt}={\int }_{{\mathbb{R}}^3\times S\times I}|\frac{\partial v}{\partial x_j}(x,\omega ,E){|}^2\mathit{\text{dxdωdE}}\\ \le {\Vert v\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2.\end{array}$$(146)

Since $C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$ is dense in $H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ we obtain by (146)(146) $$\begin{array}{l}{\Vert {\delta }_{h}v\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2={\int }_{{\mathbb{R}}^3\times S\times I}|{\int }_0^1\frac{\partial v}{\partial x_j}(x+t(h{e}_j),\omega ,E)dt{|}^2\mathit{\text{dxdωdE}}\\ \le {\int }_0^1{\int }_{{\mathbb{R}}^3\times S\times I}|\frac{\partial v}{\partial x_j}(x+t(h{e}_j),\omega ,E){|}^2\mathit{\text{dxdω}}\mathit{dEdt}={\int }_{{\mathbb{R}}^3\times S\times I}|\frac{\partial v}{\partial x_j}(x,\omega ,E){|}^2\mathit{\text{dxdωdE}}\\ \le {\Vert v\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2.\end{array}$$(146) (147) $${\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le {\Vert f\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}.$$(147)

Let $\left\{h_k\right\}$ be a sequence such that $h_k\to 0$ with $k\to \infty .$ By virtue of (143)(143) $$\begin{array}{l}{\Vert {\delta }_h\psi \Vert }_{\mathcal{H}}\le C{\Vert T({\delta }_h\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ =C\Vert {\delta }_{h}f-({\delta }_h\Sigma )({\tau }_h\psi )+({\delta }_h{K}_r)({\tau }_h\psi )\Vert \\ \le C{\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C{\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert {\tau }_h\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C\Vert {\delta }_h{K}_r\Vert {\Vert {\tau }_h\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ =C{\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C{\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C\Vert {\delta }_h{K}_r\Vert {\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.\end{array}$$(143) , (144)(144) $${\Vert {\delta }_h\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}\le C_1{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}.$$(144) , (145)(145) $$\Vert {\delta }_h{K}_r\Vert \le C_2\sqrt{M_1{M}_1^{\prime }}$$(145) , (147)(147) $${\Vert {\delta }_{h}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\le {\Vert f\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}.$$(147) the sequence $\left\{{\delta }_{h_k}\psi \right\}$ is bounded in $\mathcal{H}$ and hence in $L^2({\mathbb{R}}^3\times S\times I).$ That is why there exists a subsequence ${\left\{{\delta }_{h_{k_i}}\psi \right\}}_{i\in \mathbb{N}}$ which convergences weakly to an element $U\in L^2({\mathbb{R}}^3\times S\times I)$ (e.g. Yosida 1980, 126) and so for any $v\in C_0^{\infty }({\mathbb{R}}^3\times S\times I^{°})$ (148) $${〈{\delta }_{h_{k_i}}\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\to {〈U,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}.$$(148)

On the other hand, (149) $$\begin{array}{l}{〈{\delta }_{h_{k_i}}\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =\frac{1}{h_{k_i}}({\int }_{{\mathbb{R}}^3\times S\times I}\psi (x+h_{k_i}{e}_j,\omega ,E)v(x,\omega ,E)\mathit{dxd}\omega dE\\ -{\int }_{{\mathbb{R}}^3\times S\times I}\psi (x,\omega ,E)v(x,\omega ,E)\mathit{dxd}\omega dE)\\ =\frac{1}{h_{k_i}}({\int }_{{\mathbb{R}}^3\times S\times I}\psi (z,\omega ,E)v(z-h_{k_i}{e}_j,\omega ,E)\mathit{dzd}\omega dE\\ -{\int }_{{\mathbb{R}}^3\times S\times I}\psi (z,\omega ,E)v(z,\omega ,E)\mathit{dzd}\omega dE)\\ \to -{〈\psi ,\frac{\partial v}{\partial x_j}〉}_{L^2({\mathbb{R}}^3\times S\times I)}=(\frac{\partial (\psi )}{\partial x_j})(v),i\to \infty .\end{array}$$(149)

That is why $\frac{\partial (\psi )}{\partial x_j}=U\in L^2({\mathbb{R}}^3\times S\times I),j=1,2,3.$ As a conclusion we get that $\psi \in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$

C. We show the a priori estimate (132)(132) $${\Vert \psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C{\Vert T\psi \Vert }_{H^{(m_1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}.$$(132) for $m_1=1.$ Let for $\left|\alpha \right|=1$ $$(({\partial }_x^{\alpha }{K}_r)\psi )(x,\omega ,E):={\int }_{S\prime \times I\prime }({\partial }_x^{\alpha }\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime .$$

The assumption (130) implies that for any $\psi \in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ the partial derivative ${\partial }_x^{\alpha }(K_r\psi ),\left|\alpha \right|=1$ exists (in the weak sense) and (150) $$\begin{array}{l}({\partial }_x^{\alpha }(K_r\psi )(x,\omega ,E)={\int }_{S\prime \times I\prime }{\partial }_x^{\alpha }(\sigma (x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime ))d\omega \prime dE\prime \\ ={\int }_{S\prime \times I\prime }({\partial }_x^{\alpha }\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime \\ +{\int }_{S\prime \times I\prime }\sigma (x,\omega \prime ,\omega ,E\prime ,E)({\partial }_x^{\alpha }\psi )(x,\omega \prime ,E\prime )d\omega \prime dE\prime \\ =(({\partial }_x^{\alpha }{K}_r)\psi )(x,\omega ,E)+(K_r({\partial }_x^{\alpha }\psi ))(x,\omega ,E).\end{array}$$(150)

Since a, c and d do not depend on x we have (similarly to (137)(137) $$P({\delta }_h\psi )={\delta }_h(P\psi ).$$(137) ) (151) $$\begin{array}{l}P({\partial }_{x_j}\psi )={\partial }_{x_j}(P\psi )={\partial }_{x_j}(f-\Sigma \psi +K_r\psi )\\ ={\partial }_{x_j}f-({\partial }_{x_j}\Sigma )\psi -\Sigma ({\partial }_{x_j}\psi )+({\partial }_{x_j}{K}_r)\psi +K_r({\partial }_{x_j}\psi )\end{array}$$(151) and then (by the assumptions (129)(130) $$\begin{array}{c}\text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_{S\prime }{\int }_{I\prime }{\Vert \sigma (.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}<\infty ,\\ \text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_S{\int }_I{\Vert \sigma (.,\omega ,\omega \prime ,E,E\prime )\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}\prime <\infty .\end{array}$$(130) , (130)(130) $$\begin{array}{c}\text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_{S\prime }{\int }_{I\prime }{\Vert \sigma (.,\omega \prime ,\omega ,E\prime ,E)\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}<\infty ,\\ \text{ess}{\text{sup}}_{(\omega ,E)\in S\times I}{\int }_S{\int }_I{\Vert \sigma (.,\omega ,\omega \prime ,E,E\prime )\Vert }_{W^{\infty ,m_1}({\mathbb{R}}^3)}d\omega \prime dE\prime =:M_{m_1}\prime <\infty .\end{array}$$(130) (with $m_1=1$) $P({\partial }_{x_j}\psi )\in L^2({\mathbb{R}}^3\times S\times I)$ because $\psi \in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ and $f\in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$ Hence $${\partial }_{x_j}\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°}).$$

In addition, (152) $${\partial }_{x_j}\psi \in \mathcal{H}.$$(152)

This can be shown as follows. Due to Part B the sequence $\left\{{\delta }_{h_k}\psi \right\}$ therein is bounded in $\mathcal{H}.$ Hence by the Banach-Saks’s Theorem there exists a subsequence $\{{\delta }_{h_{k_i}}\psi \}$ such that the running average sequence $\frac{1}{N}{\sum }_{i=1}^N{\delta }_{h_{k_i}}\psi$ convergences, say to an element $U\prime$ in $\mathcal{H}.$ On the other hand, by (149)(149) $$\begin{array}{l}{〈{\delta }_{h_{k_i}}\psi ,v〉}_{L^2({\mathbb{R}}^3\times S\times I)}\\ =\frac{1}{h_{k_i}}({\int }_{{\mathbb{R}}^3\times S\times I}\psi (x+h_{k_i}{e}_j,\omega ,E)v(x,\omega ,E)\mathit{dxd}\omega dE\\ -{\int }_{{\mathbb{R}}^3\times S\times I}\psi (x,\omega ,E)v(x,\omega ,E)\mathit{dxd}\omega dE)\\ =\frac{1}{h_{k_i}}({\int }_{{\mathbb{R}}^3\times S\times I}\psi (z,\omega ,E)v(z-h_{k_i}{e}_j,\omega ,E)\mathit{dzd}\omega dE\\ -{\int }_{{\mathbb{R}}^3\times S\times I}\psi (z,\omega ,E)v(z,\omega ,E)\mathit{dzd}\omega dE)\\ \to -{〈\psi ,\frac{\partial v}{\partial x_j}〉}_{L^2({\mathbb{R}}^3\times S\times I)}=(\frac{\partial (\psi )}{\partial x_j})(v),i\to \infty .\end{array}$$(149) $\frac{1}{N}{\sum }_{i=1}^N{\delta }_{h_{k_i}}\psi$ convergences weakly to ${\partial }_{x_j}\psi$ in $L^2({\mathbb{R}}^3\times S\times I).$ Hence $U\prime ={\partial }_{x_j}\psi$ and so (152)(152) $${\partial }_{x_j}\psi \in \mathcal{H}.$$(152) holds. Similarly, $\frac{1}{N}{\sum }_{i=1}^N({\delta }_{h_{k_i}}\psi )(.,.,E_m)\to ({\partial }_{x_j}\psi )(.,.,E_m)$ in $L^2({\mathbb{R}}^3\times S)$ and since $\frac{1}{N}{\sum }_{i=1}^N({\delta }_{h_{k_i}}\psi )(.,.,E_m)=0$ we have $({\partial }_{x_j}\psi )(.,.,E_m)=0.$

We find that (153) $$\begin{array}{c}{\partial }_{x_j}(T\psi )-T({\partial }_{x_j}\psi )={\partial }_{x_j}(P\psi )+{\partial }_{x_j}(\Sigma \psi )-{\partial }_{x_j}(K_r\psi )\\ -P({\partial }_{x_j}\psi )-\Sigma ({\partial }_{x_j}\psi )+K_r({\partial }_{x_j}\psi )=({\partial }_{x_j}\Sigma )\psi -({\partial }_{x_j}{K}_r)\psi .\end{array}$$(153)

Hence, due to (118)(118) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(118) (154) $$\begin{array}{l}\frac{1}{C}{\Vert {\partial }_{x_j}\psi \Vert }_{\mathcal{H}}\le {\Vert (T({\partial }_{x_j}\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ ={\Vert {\partial }_{x_j}(T\psi )-({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ \le {\Vert {\partial }_{x_j}(T\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ +\Vert {\partial }_{x_j}{K}_r\Vert {\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\end{array}$$(154) and then again by (118)(118) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(118) (155) $$\begin{array}{l}\frac{1}{C}{\Vert {\partial }_{x_j}\psi \Vert }_{\mathcal{H}}\le {\Vert {\partial }_{x_j}(T\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\\ +C{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}+C\Vert {\partial }_{x_j}{K}_r\Vert {\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}\end{array}$$(155) which implies that (156) $${\Vert \psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C\prime {\Vert T\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})},$$(156) with $C\prime :=C\backslash \sqrt{3}{\max }_{j=1,2,3}\{1+C{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathrm{ℝ}}^3\times S\times I)}+C\Vert {\partial }_{x_j}{K}_r\Vert \}$ as desired.

D. Next we verify that the claim holds for $m_1=2.$ Let $U_j:={\partial }_{x_j}\psi .$ By virtue of Part C, $U_j\in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})\cap \mathcal{H}$ and by (159)(153) $$\begin{array}{c}{\partial }_{x_j}(T\psi )-T({\partial }_{x_j}\psi )={\partial }_{x_j}(P\psi )+{\partial }_{x_j}(\Sigma \psi )-{\partial }_{x_j}(K_r\psi )\\ -P({\partial }_{x_j}\psi )-\Sigma ({\partial }_{x_j}\psi )+K_r({\partial }_{x_j}\psi )=({\partial }_{x_j}\Sigma )\psi -({\partial }_{x_j}{K}_r)\psi .\end{array}$$(153) (recall that $f=T\psi$) (157) $$T{U}_j={\partial }_{x_j}f-({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi =:F_j\in L^2({\mathbb{R}}^3\times S\times I).$$(157)

In addition, $U_j(.,.,E_{\mathrm{m}})=0.$ Hence we are able to apply the results of Parts A-C of the present proof by replacing ψ with U_j which gives that $U_j\in H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ and by (156)(156) $${\Vert \psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C\prime {\Vert T\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})},$$(156) (158) $${\Vert U_j\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C\prime {\Vert T{U}_j\Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}.$$(158)

Hence (159) $$\begin{array}{l}{\Vert {\partial }_{x_j}\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\le C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\\ =C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +C{\prime }^2{\displaystyle \sum _{k=1}^3{\Vert {\partial }_{x_k}({\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2}\\ \le C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +3C{\prime }^2{\displaystyle \sum _{k=1}^3(}{\Vert ({\partial }_{x_k}{\partial }_{x_j})f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}(({\partial }_{x_j}\Sigma )\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_k}(({\partial }_{x_j}{K}_r)\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2).\end{array}$$(159)

Furthermore, we have (160) $${\partial }_{x_k}(({\partial }_{x_j}\Sigma )\psi )=({\partial }_{x_k}{\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}\Sigma ){\partial }_{x_k}\psi$$(160) (161) $${\partial }_{x_k}(({\partial }_{x_j}{K}_r)\psi )=({\partial }_{x_k}{\partial }_{x_j}{K}_r)\psi +({\partial }_{x_j}{K}_r){\partial }_{x_k}\psi$$(161) where $$(({\partial }_{x_k}{\partial }_{x_j}{K}_r)\psi )(x,\omega ,E):={\int }_{S\prime \times I\prime }({\partial }_{x_k}{\partial }_{x_j}\sigma )(x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime .$$

In virtue of the assumption (130) (with $m_1=2$) and Theorem 3.2 the operators $({\partial }_{x_k}{\partial }_{x_j}{K}_r)$ are bounded operators $L^2({\mathbb{R}}^3\times S\times I)\to L^2({\mathbb{R}}^3\times S\times I).$ Hence we finally get by (159)(159) $$\begin{array}{l}{\Vert {\partial }_{x_j}\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\le C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\\ =C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +C{\prime }^2{\displaystyle \sum _{k=1}^3{\Vert {\partial }_{x_k}({\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2}\\ \le C{\prime }^2{\Vert {\partial }_{x_j}f+({\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}{K}_r)\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +3C{\prime }^2{\displaystyle \sum _{k=1}^3(}{\Vert ({\partial }_{x_k}{\partial }_{x_j})f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}(({\partial }_{x_j}\Sigma )\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_k}(({\partial }_{x_j}{K}_r)\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2).\end{array}$$(159) (162) $$\begin{array}{l}{\Vert {\partial }_{x_j}\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\le 3C{\prime }^2({\Vert {\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{W^{\infty ,1}({\mathbb{R}}^3\times S\times I^{°})}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2)+3C{\prime }^2\sum _{k=1}^3({\Vert {\partial }_{x_k}{\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert {\partial }_{x_k}\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert {\partial }_{x_k}\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2).\end{array}$$(162)

Noting that $f=T\psi$ we conclude (as above) from (162)(162) $$\begin{array}{l}{\Vert {\partial }_{x_j}\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\le 3C{\prime }^2({\Vert {\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{W^{\infty ,1}({\mathbb{R}}^3\times S\times I^{°})}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2)+3C{\prime }^2\sum _{k=1}^3({\Vert {\partial }_{x_k}{\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert {\partial }_{x_k}\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert {\partial }_{x_k}\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2).\end{array}$$(162) and (118)(118) $${\Vert \psi \Vert }_{\mathcal{H}}\le C{\Vert T\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}.$$(118) , (156)(156) $${\Vert \psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}\le C\prime {\Vert T\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})},$$(156) that $\psi \in H^{(2,0,0)}({\mathbb{R}}^3\times S\times I^{°})$ and that there exists $C″\ge 0$ such that (163) $${\Vert \psi \Vert }_{H^{(2,0,0)}({\mathbb{R}}^3\times S\times I^{°}}C″{\Vert T\psi \Vert }_{H^{(2,0,0)}({\mathbb{R}}^3\times S\times I^{°})}$$(163) which is the claim for $m_1=2.$

E. For the general $m_1\in {\mathbb{N}}_0$ the assertion is obtained by the induction. The induction step is similar to Part D and we neglect these technicalities. However, we notice that this step needs the generalization of derivation rules (160)(160) $${\partial }_{x_k}(({\partial }_{x_j}\Sigma )\psi )=({\partial }_{x_k}{\partial }_{x_j}\Sigma )\psi +({\partial }_{x_j}\Sigma ){\partial }_{x_k}\psi$$(160) , (161)(161) $${\partial }_{x_k}(({\partial }_{x_j}{K}_r)\psi )=({\partial }_{x_k}{\partial }_{x_j}{K}_r)\psi +({\partial }_{x_j}{K}_r){\partial }_{x_k}\psi$$(161) which is obtained by the Leibniz’s rules (164) $${\partial }_x^{\alpha }(\Sigma \psi )=\sum _{\beta \le \alpha }\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)({\partial }_x^{\alpha -\beta }\Sigma ){\partial }_x^{\beta }\psi =\Sigma ({\partial }_x^{\alpha }\psi )+R_{1,\alpha }\psi ,\left|\alpha \right|\le m_1$$(164) where $$R_{1,\alpha }\psi :=\sum _{\beta <\alpha }\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)({\partial }_x^{\alpha -\beta }\Sigma ){\partial }_x^{\beta }\psi$$ and (165) $$\begin{array}{l}({\partial }_x^{\alpha }(K_r\psi ))(x,\omega ,E)={\int }_{S\prime \times I\prime }{\partial }_x^{\alpha }[\sigma (x,\omega \prime ,\omega ,E\prime ,E)\psi (x,\omega \prime ,E\prime )]d\omega \prime dE\prime \\ ={\int }_{S\prime \times I\prime }\sum _{\beta \le \alpha }\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)({\partial }_x^{\alpha -\beta }\sigma )(x,\omega \prime ,\omega ,E\prime ,E){\partial }_x^{\beta }\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime \\ =:(K_r({\partial }_x^{\alpha }\psi ))(x,\omega ,E)+(R_{2,\alpha }\psi )(x,\omega ,E),\left|\alpha \right|\le m_1\end{array}$$(165) where $$(R_{2,\alpha }\psi )(x,\omega ,E):={\int }_{S\prime }{\int }_{I\prime }\sum _{\beta <\alpha }\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)({\partial }_x^{\alpha -\beta }\sigma )(x,\omega \prime ,\omega ,E\prime ,E){\partial }_x^{\beta }\psi (x,\omega \prime ,E\prime )d\omega \prime dE\prime .$$

This finishes the proof. □

Remark 5.2.

A. Note that the assumptions (130) are equivalent to (166) $$\begin{array}{l}\sigma =\sigma (x,\omega \prime ,\omega ,E\prime ,E)\in L^{\infty }(S\times I,L^1(S\prime \times I\prime ,W^{\infty ,m_1}({\mathbb{R}}^3)))\\ \cap L^{\infty }(S\prime \times I\prime ,L^1(S\times I,W^{\infty ,m_1}({\mathbb{R}}^3))).\end{array}$$(166)

B. Suppose that the assumptions of Theorem 5.1 are valid for every $m_1\in {\mathbb{N}}_0$ and let $f\in W^{(\infty ,0,0)}({\mathbb{R}}^3\times S\times I^{°}).$ Then the solution ψ of the problem (131)(131) $$T\psi =f,\psi (.,.,E_{\mathrm{m}})=0.$$(131) belongs to $W^{(\infty ,0,0)}({\mathbb{R}}^3\times S\times I^{°})\subset C^{\infty }({\mathbb{R}}^3,L^2(S\times I)).$

6. Discussion

We propose that the above regularity result of Theorem 5.1 can be generalized for x-dependent $a,c$ and d and as in section 4 results with respect to all variables $x,\omega ,E$ can be achieved, at least when $K_r=K_r^1.$ We conjecture the following result:

Let $m=(m_1,m_2,m_3)\in {\mathbb{N}}_0^3.$ Suppose that the assumptions of Theorem 3.10 are valid for $K_r=K_r^1.$ Furthermore, suppose that (167) $$\Sigma ,a,c,d\in W^{\infty ,(m_1+m_2+m_3,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°})),$$(167)

Finally, we suppose that (168) $$\sigma \in W^{\infty ,(m_1+m_2+m_3,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°},L^1(S\prime \times I\prime )),$$(168) (169) $$\sigma \in W^{\infty ,(m_1+m_2+m_3,m_2,m_3)}({\mathbb{R}}^3\times S\prime \times I{\prime }^{°},L^1(S\times I)).$$(169)

Let $f\in H^{(m_1+m_2+m_3,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°})$ and let $\psi \in {\mathcal{H}}_P({\mathbb{R}}^3\times S\times I^{°})$ be a solution of the problem (170) $$\begin{array}{l}T\psi =a(x,E)\frac{\partial \psi }{\partial E}+c(x,E){\Delta }_S\psi +d(x,\omega ,E)·{\nabla }_S\psi +\omega ·{\nabla }_x\psi \\ +\Sigma \psi -K_r\psi =f(x,\omega ,E),\\ \psi (.,.,E_{\mathrm{m}})=0.\end{array}$$(170)

Then $\psi \in H^{(m_1,m_2,m_3)}({\mathbb{R}}^3\times S\times I^{°}).$

The assumptions (167)(161) $${\partial }_{x_k}(({\partial }_{x_j}{K}_r)\psi )=({\partial }_{x_k}{\partial }_{x_j}{K}_r)\psi +({\partial }_{x_j}{K}_r){\partial }_{x_k}\psi$$(161) , (168)(162) $$\begin{array}{l}{\Vert {\partial }_{x_j}\psi \Vert }_{H^{(1,0,0)}({\mathbb{R}}^3\times S\times I^{°})}^2\le 3C{\prime }^2({\Vert {\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{W^{\infty ,1}({\mathbb{R}}^3\times S\times I^{°})}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2\\ +{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I^{°})}^2)+3C{\prime }^2\sum _{k=1}^3({\Vert {\partial }_{x_k}{\partial }_{x_j}f\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}\Sigma \Vert }_{L^{\infty }({\mathbb{R}}^3\times S\times I)}^2{\Vert {\partial }_{x_k}\psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2\\ +{\Vert {\partial }_{x_k}{\partial }_{x_j}{K}_r\Vert }^2{\Vert \psi \Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2+{\Vert {\partial }_{x_j}{K}_r\Vert }^2{\Vert {\partial }_{x_k}\psi )\Vert }_{L^2({\mathbb{R}}^3\times S\times I)}^2).\end{array}$$(162) , (169)(163) $${\Vert \psi \Vert }_{H^{(2,0,0)}({\mathbb{R}}^3\times S\times I^{°}}C″{\Vert T\psi \Vert }_{H^{(2,0,0)}({\mathbb{R}}^3\times S\times I^{°})}$$(163) can be weakened along the above Remark 4.9.

In addition to the difference techniques applied here we propose that the proofs can be based on the application of the Friedrich’s mollifier smoothing $J_{ϵ}\psi$ for $\psi \in L^2({\mathbb{R}}^3\times S\times \mathbb{R}).$ The convolution (in mollifier) can be taken simultaneously with respect to all variables. We notice that the mollifier can also be defined on S (see e.g. Fukuoka 2006). For the mollifier it holds that $J_{ϵ}\psi \in H^{(\infty ,\infty ,\infty )}({\mathrm{ℝ}}^3\times S\times \mathrm{ℝ}).$ The relevant a priori estimates together with the (consequences of) well-known Friedrich’s Lemma enable us to deduce the same kind of conclusions as in the proof of the above Theorem 5.1. One additional method would be the application of the known regularity theory of evolution equations.

We finally notice that in the case where $G\cancel{=}{\mathbb{R}}^3$ the relevant problem is an initial inflow boundary value problem of the form (171) $$T\psi =f,{\psi }_{|{\Gamma }_{-}}=g,\psi (.,.,E_m)=0.$$(171)

The regularity of the solution ψ is limited, for example in x-variable, to the scale $H^{(s,0,0)}(G\times S\times I),s<3/2$ (Tervo and Kokkonen 2017, Example 7.4). This is due to the fact that the initial inflow boundary value problem has the so called variable multiplicity (e.g. Nishitani and Takayama 1996; Nishitani and Takayama 2000), which causes irregularity on the inflow boundary.

Acknowledgments

The author thanks an anonymous referee for his/her improvements of the manuscript.