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Abstract
First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this article we construct global in time weak solutions to a first order mean field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. This is achieved by characterizing solutions through two convex optimization problems in duality. As part of our approach, we develop tools for the analysis of mean field games on a non-compact domain by variational methods. We introduce a notion of ‘reachable set’, built from the initial measure, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions.
1. Introduction
The aim of the theory of mean field games (MFG for short) is to characterize limits of Nash equilibria of stochastic or deterministic differential games when the number of agents tends to infinity. Such models were first proposed about 15 years ago, simultaneously by Lasry–Lions [Citation1–3] and Huang-Malhamé-Caines [Citation4].
This theory turned out to be extremely rich in applications and it provided excellent mathematical questions. Its literature has witnessed a huge increase in the last decade. From the theoretical viewpoint, there are two main approaches to the study of MFG. One is based on analytical and PDE techniques, while the other is a probabilistic approach. The first approach goes back to the original works of Lasry-Lions and has been extended in a great variety of directions in the subsequent years by many authors. If a non-degenerate idiosyncratic noise is present in the models, this typically yields a parabolic structure for the corresponding PDEs and one can expect (strong) classical solutions or a suitable regularity for weak solutions to the corresponding PDE systems, even when the corresponding Lagrangians are local functions of the density variable. For a non-exhaustive list of works in this direction we refer the reader to [Citation5–12]. The probabilistic approach proved to be equally successful for problems involving Lagrangians that are nonlocal functions of the measure variable. This approach seems to be very powerful for handling different kinds of noises in combination with the non-degenerate idiosyncratic one, such as the common noise. For a non-exhaustive collection of works in this direction we refer to [Citation13–16].
When the model lacks a non-degenerate idiosyncratic noise, this clearly poses technical difficulties in the analysis. Typically, it means that additional structural assumptions need to be imposed on the data to be able to hope for (weak) solutions. Such conditions are, for instance, suitable notions of convexity/monotonicity (cf. [Citation17,Citation18]), or the presence of a suitable variational structure, as in the case of potential games ([Citation19–24]). In the case of local couplings, it was pointed out by Lions in [Citation25] that the MFG system (including the planning problem) can be transformed into a degenerate elliptic system in space-time with oblique boundary conditions. Relying on this idea, in a quite general setting, under suitable assumptions on the data (such as strict monotonicity and strong convexity of the Hamiltonians in the measure and momentum variables, respectively; regularity and positivity conditions on the initial data), it has been proven recently in [Citation26,Citation27] that the corresponding first order MFG systems have smooth classical solutions.
For an excellent, relatively complete account on the subject and a summary of results to date we refer the reader to the collection [Citation17].
In this article we study a class of first order kinetic MFG systems, involving Lagrangians that are local functions of the density variable and that possess a variational structure, in the sense of [Citation19–21].
In our setting, the MFG system can be formally written as
(1.1)
(1.1)
Here denotes either that d-dimensional flat torus
or the whole d-dimensional Euclidean space
and is the physical space for the position x of the agents, while the velocity vector v of the agents lies in
T > 0 is an arbitrary time horizon,
is the Hamiltonian function, while
stand for the running and final costs of the agents, respectively.
Under suitable assumptions on the data, we obtain the global in time existence, uniqueness and Sobolev regularity of weak solutions to Equation(1.1)(1.1)
(1.1) , relying on two convex optimization problems in duality. One of these problems can be seen as an optimal control problem for the Hamilton-Jacobi equation, while its dual is an optimal control problem for the continuity equation (cf. [Citation19–21]).
Review of the literature in connection to our work
MFG systems of type Equation(1.1)(1.1)
(1.1) have been introduced in the context of models when agents control their acceleration. It seems that such a model can be traced back to the work [Citation28] (in the engineering community), where the authors proposed a MFG model where agents control their acceleration. In the mathematical community, the first works in this framework seem to be the ones [Citation29–31]. These works consider Hamiltonians (with our notation H – f) and final cost functions that are nonlocal regularizing functions in the measure variable. Moreover, the Hamiltonians need to be either purely quadratic or have quadratic growth in the momentum variable. In addition, in [Citation29,Citation30] further conditions on the initial measure m0 are also imposed. In [Citation29] m0 is taken to be compactly supported and Hölder continuous, while in [Citation30] m0 is taken to be compactly supported. These two works construct weak solutions to the corresponding MFG system in the sense that the Hamilton-Jacobi equation has to be understood in the viscosity sense, while the continuity equation is understood in the sense of distributions. In [Citation31] the initial measure m0 can be quite general and the corresponding Hamiltonian does not need to have the so-called ‘separable structure’ which was assumed in [Citation29,Citation30] and is also assumed in this article. These more general hypotheses come at the price of obtaining a weaker notion of solution to the MFG system: the so-called mild solutions. However, the authors show that, under the additional separability assumption on the Hamiltonian, mild solutions become more standard weak solutions in the sense described above.
Several interesting new works are built on the models introduced in [Citation29–31]. In [Citation32] the authors study the ergodic behavior of MFG systems, for the case of Hamiltonians that are purely quadratic in the momentum variable and nonlocal regularizing coupling functions f, g, with additional growth assumption on f in the v variable. In [Citation33] the authors obtain mild solutions to MFG under acceleration control and state constraints, under assumptions similar to the ones in [Citation29] on the Hamiltonians, with the possibility to consider Hamiltonians that are power-like functions in the momentum variable. Lastly, in [Citation34] the author studies a perturbation problem associated to MFG under acceleration control, where the (Lagrangian) cost associated to the acceleration vanishes.
MFG models with degenerate diffusion share some common features with kinetic type problems. In this context we can mention several works. In [Citation35] and [Citation36] the authors study time independent MFG systems with purely quadratic Hamiltonians and nonlocal regularizing coupling functions, where the diffusion operator is hypoelliptic or satisfies a suitable Hörmander condition. It is also worth mentioning that our system Equation(1.1)(1.1)
(1.1) shares some similarities with MFG models where agents interact also through their velocities. In this direction we refer to the works [Citation37–41].
Finally, a second order MFG system of type Equation(1.1)(1.1)
(1.1) has been recently studied in [Citation42]. In this work the author obtains weak and renormalized solutions (in the spirit of [Citation12]) to a MFG system that involves a non-degenerate diffusion in the v direction. This seems to be the only work in the context of kinetic type MFG models where the coupling functions f and g are taken to be local functions of the density variable m. Here the Hamiltonian H is assumed to depend only on the momentum variable and either to be globally Lipschitz continuous or to have quadratic/sub-quadratic growth. There are several summability properties and moment bounds imposed on the initial density m0. In the case of Lipschitz continuous Hamiltonians, the coupling functions f, g are supposed to fulfill several further assumptions: a strong uniform increasing property in the m variable and their derivatives in the (x, v) variable must have a linear growth condition in the m variable.
In [Citation42] the presence of the diffusion in the v direction allows the author to use suitable De Giorgi type arguments to show that the solution to the Fokker-Planck equation is bounded and has fractional Sobolev regularity. These estimates seem to be instrumental to set up a fixed point scheme and to show that the MFG system has a weak solution. Furthermore, the presence of this diffusion allows to obtain second order Sobolev estimates for the MFG system.
Description of our results
As highlighted above, in this work we are inspired by [Citation19–21] and we obtain existence and uniqueness of weak solutions to Equation(1.1)(1.1)
(1.1) (in the sense of Definition 2.3) via two convex optimization problems in duality (Problem 3.1 and Problem 3.3). Compared to these works, several major differences arise which require new ideas. A first obvious difference is that in our setting (in contrast to the compact setting of the flat torus which is considered in the mentioned references) the velocity variable v lives in the non-compact space
This clearly introduces technical issues in the analysis.
To prove our main results, the general outline of our programme is the same as the one of [Citation19–21]: prove the duality for Problem 3.1 and Problem 3.3; suitably relax Problem 3.1 (this will be Problem 3.8) and show that the value of this is the same as the original one; show existence of optimizers for the relaxed problem and apply the duality result again to obtain existence of solutions in a suitable weak sense. In this article H is supposed to have a superlinear growth in the momentum variable, and f and g are supposed to have polynomial growth in their last variables. The growth of f, g may be taken independently of the growth of the Hamiltonian (we refer to the next section for the precise assumptions).
To show that the value of the relaxed problem is the same as the original one, a standard approach used in [Citation19–21] is to test the Hamilton-Jacobi inequality of any competitor by competitors of the dual problem (i.e. solutions to the continuity equation). To justify this computation a mollification argument was applied for solutions to the continuity equation. In our case, this mollification alone is not enough because of the non-compact setting. Therefore a delicate cutoff argument has to be also implemented.
The most delicate part, however, is to obtain existence of optimizers to the relaxed problem and in particular to obtain proper compactness results for the minimizing sequences. First, in our case the time trace of the solutions to the Hamilton–Jacobi inequality constraint in Problem 3.8 is quite weak: has to be understood as a locally finite signed Radon measure. Since in this work m0 may have non-compact support, it takes additional effort to give a meaning to
(a term that appears in the objective functional present in Problem 3.8). Our construction, although completely different, has some similarities in spirit with the one in [Citation43], to define similar time boundary traces.
In order to obtain suitable estimates for the minimizing sequence of the relaxed problem, in [Citation19–21] a typical trick was to test the Hamilton-Jacobi inequality constraint by the initial measure m0. For this reason, it was necessary to impose enough regularity, and more importantly a uniform positive lower bound of this density everywhere. Because of this, estimates on the quantity would readily yield summability estimates on u0 solely. We emphasize that in this article we assume that m0 is merely a bounded and continuous probability density and so we take a completely different route when obtaining such estimates. We introduce the reachable set
a set of points in time, space and velocity that can be reached from
with arbitrary smooth admissible controls (cf. Definition 2.2). In fact, by the controllability of the underlying ODE system, which satisfies the Kalman rank condition, we have
In order to obtain our crucial estimates on the corresponding minimizing sequence we use well chosen test functions that are supported in
This construction seems to be new in the literature on variational MFG and we believe that it could be instrumental also in other settings, to possibly relax regularity, positivity or compact support assumptions on m0.
As there is no Hopf–Lax type representation formula available for solutions to our Hamilton-Jacobi equations (which was the case in [Citation19,Citation20]), first, we obtain estimates on truncations of the solutions. These are similar in flavor to the corresponding estimates in [Citation21], and such ideas date back to [Citation44]. As our terminal data typically have merely local summability, this will be the source of additional technical issues (in contrast to [Citation21], where the terminal data was taken to be regular enough).
Let us underline that the ideas and constructions that we have described so far allow us to obtain summability estimates on u and using the structure of the problem. This is not sufficient to yield weak precompactness for minimizing sequences due to the lack of regularity estimates in x. To recover the necessary compactness we make use of averaging lemmas available in kinetic theory. Averaging lemmas go back to the works [Citation45,Citation46] and provide improved regularity and compactness properties for velocity averages of solutions of kinetic transport equations (see Subsection Citation6.Citation1 for the precise definitions). For more details and a survey of results we refer the reader to the review [Citation47] and the references cited therein. When regularity with respect to v is additionally available, similar properties can be deduced for the full density function: we refer for instance to [Citation48] for regularity results in the Lp case for
We carefully tailor this approach to our setting, combining our estimates on
with L1 averaging lemmas [Citation49–51] to deduce precompactness for minimizing sequences. In this way we prove Theorem 6.8 on the existence of a minimizer of Problem 3.8. This in turn implies Theorem 2.4, that system Equation(1.1)
(1.1)
(1.1) has a (unique) weak solution. As was similarly obtained in [Citation19–21], we show the uniqueness of m and the uniqueness of u on
A natural question that arises in the context of variational MFG is whether the variational structure and further strong monotonicity and convexity assumptions on the data would yield higher order Sobolev estimates on weak solutions. Such estimates were recently obtained in more classical frameworks in [Citation23,Citation24, Citation39, Citation52–54]. In this article we pursue similar Sobolev estimates, implied by taking stronger assumptions on the data. In comparison with the works [Citation23,Citation24], in our setting we need to work with a considerably weaker notion of time trace of u, which is not stable under perturbations of the initial measure m0. Therefore, our Sobolev estimates remain local in time on Another delicate difference is due to the presence of the kinetic transport term. Because of this, a careful choice of perturbations need to be used, which take into account the kinetic nature of the problem. As a result of this, interestingly, first we obtain estimates on differential operators of the form
applied to m and Dvu. For the precise results in this direction we refer to Theorem 8.2, Corollary 8.4 and Corollary 8.5.
The structure of the article is as follows. In Section 2 we state our standing assumptions and main results. In Section 3 we present the two variational problems in duality along with the relaxed problem of the primal problem. In Section 4 we have collected some preliminary estimates on weak solutions of the Hamilton-Jacobi inequality obtained on the reachable set In Section 5 we show that the relaxed problem has the same value as the primal problem and hence the duality result holds. Section 6 contains the existence result of a solution to the relaxed problem. Here we rely on the combination of the estimates derived in the previous sections and suitably tailored averaging lemmas from kinetic theory, applied in our context for distributional subsolutions to kinetic Hamilton-Jacobi equations. In Section 7 we show that optimizers of the variational problems in duality provide weak solutions to the MFG system and, conversely, weak solutions are also optimizers of the variational problems. Furthermore, strong convexity yields (partial) uniqueness of these solutions. Section 8 is devoted to the derivation of higher order Sobolev estimates for the weak solutions. These require further assumptions on the data.
We end the paper with two appendix sections. In Appendix A we discuss the time regularity of distributional subsolutions to kinetic Hamilton-Jacobi equations which allow us to construct suitable notions of time traces. Finally, in Appendix B we show that truncations and maxima of distributional subsolutions to kinetic Hamilton-Jacobi equations remain distributional subsolutions to suitably modified equations.
2. Standing assumptions and main results
In this section we state our main results on the existence, uniqueness and Sobolev regularity of solutions to the MFG system.
We define and
to be the anti-derivatives of the coupling functions f and g with respect to m:
Throughout, we make the following assumptions on the Hamiltonian and coupling functions.
Assumption 1.
(H1) The Hamiltonian
is continuous in all variables, and convex and differentiable with respect to p. Furthermore, for some r > 1, H satisfies bounds of the form
for all and some constants
and
. Finally, the function
has positive part
, where
denotes the closure of the space
with respect to the uniform norm.
(H2)
is continuous in all variables and strictly convex and differentiable with respect to m for m > 0. Moreover, it satisfies the growth condition
where q > 1 and the function . For m < 0, we set
(H3)
is continuous and strictly convex. Moreover, it satisfies the growth condition
for some and
. For m < 0, we set
(H4) The initial datum
is a probability density.
We note that since m0 is imposed to be a bounded probability density, by interpolation, it is uniformly bounded in for any
We emphasize that here we impose growth conditions on
rather than on f, g.
Example 2.1.
For any q > 1 and continuous bounded function c such that with
a strictly positive constant, the function
satisfies the given assumptions.
Definition 2.2
(Reachable set). It will be useful to define the set to be the set of points potentially reachable by a collection of agents initially distributed according to m0 and evolving according to the control system
for some control
Observe that the previous control system satisfies the classical Kalman rank condition, and so we have
Under these standing assumptions, we define the following notion of weak solution to the MFG system.
Definition 2.3.
We say that (u, m) is a weak solution to Equation(1.1)(1.1)
(1.1) , if the following are fulfilled:
and
and
and
is a locally finite Radon measure supported in
The continuity equation from Equation(1.1)
(1.1)
(1.1) holds in
is finite.
The following energy equality holds:
(2.4)
(2.4)
2.1. Existence and uniqueness
The first of our main results is the existence and uniqueness of these weak solutions.
Theorem 2.4.
Let Assumption 1 hold. Then there exists a weak solution (u, m) of the mean field game system Equation(1.1)(1.1)
(1.1) in the sense of Definition 2.3. This solution is unique, in the sense that if (u1, m1) and (u2, m2) are both weak solutions in the sense of Definition 2.3, then m1 = m2 almost everywhere and u1 = u2 almost everywhere on the set
2.2. Regularity
Our second main result is Sobolev regularity for weak solutions of the mean field games system Equation(1.1)(1.1)
(1.1) . For this result we assume quadratic growth of the Hamiltonian (r = 2) and stronger convexity and regularity hypotheses on the data, as follows.
Assumption 2.
(H5) (Conditions on the coupling functions) There exists C > 0 such that the functions f, g satisfy
Moreover, there exists such that
(2.7)
(2.7)
(2.8)
(2.8)
In the above assumptions, if q < 2 or s < 2 one should interpret and
as
. In this way, when
, for instance, Equation(2.7)
(2.7)
(2.7) reduces to
, as in the more regular case
. Similar comments can be made for Equation(2.8)
(2.8)
(2.8) .
(H6)
In particular, and in light of our restriction Equation(2.1)(2.1)
(2.1) , we assume that j1 and j2 have linear growth.
(H7)
Under these additional assumptions, we prove the following result. The proof is carried out in Section 8.
Theorem 2.5.
Suppose that (u, m) is a weak solution to Equation(1.1)(1.1)
(1.1) in the sense of Definition 2.3 and that (H5), (H6), (H7) hold.
Then, there exists such that
and
Remark 2.6.
The estimates appearing in this statement are informal; we in fact obtain uniform L2-type summability of differential quotients (see estimate Equation(8.8)(8.8)
(8.8) below). The corresponding Sobolev estimates, however, are more delicate to obtain, because these would need to be understood in the sense of weighted Sobolev spaces or more generally in the sense of Sobolev spaces with respect to measures. Their precise versions would need to involve tangent spaces with respect to the measure m, but these are beyond the scope of the current article. We refer to [Citation55] on this topic.
3. Variational problems in duality
We will prove existence of a solution to the MFG system Equation(1.1)(1.1)
(1.1) through a variational characterization. In this section we set up the variational problems used to obtain solutions. We recall that here and throughout the rest of the article, we will work under Assumption 1.
3.1. Optimal control of the Hamilton-Jacobi equation: smooth setting
We define the Fenchel conjugates of and
respectively by
Under our assumptions on we have the bounds
(3.1)
(3.1)
where denotes the Hölder conjugate exponent of q. Note also that
is non-decreasing. Similar observations hold for
Using this, we define the following functional: for let
whenever the integrals are meaningful, and set
otherwise. We define a first variational problem associated to this problem.
Problem 3.1.
Minimize over
where E0 denotes the space
(3.2)
(3.2)
Remark 3.2.
E0 is a Banach space when equipped with the norm
3.2. Optimal control of the continuity equation
To state the dual problem we define the Lagrangian which is the Fenchel conjugate of the Hamiltonian H in the last variable. In other words, for any
we define
Note that L then satisfies upper and lower bounds of the form
where
denotes the Hölder conjugate exponent of r.
For pairs we define the functional
with the convention that
We then define a second variational problem, (formally) dual to the first.
Problem 3.3.
Minimize over the set
of pairs
with
subject to
satisfying the following continuity equation:
(3.3)
(3.3)
and
in the sense of a weak trace.
Remark 3.4.
Let us comment on the weak trace of m with respect to the time variable. Since we are interested in competitors (m, w) for which is finite, there must exist a vector-valued measurable function
that is, for which
such that
(i.e. V is the density of w with respect to m). So, we notice that the previous equation can be written as
Since we have
We are then able to prove that m has a narrowly continuous representative
so that in particular
and
are meaningful. This is essentially a consequence of [Citation56, Lemma 8.1.2], with minor modifications to account for the fact that
is only locally integrable; we sketch this in the appendix in EquationLemma A.12
(A.12)
(A.12) .
3.3. Duality
Lemma 3.5.
We have the following duality:
Proof.
This is an application of the classical Fenchel–Rockafeller duality theorem. Recall that we defined the Banach space E0 above in Equation(3.2)(3.2)
(3.2) . Then let E1 be defined by
we will express elements of E1 as pairs
of continuous bounded functions, where
is real-valued and ψ is vector-valued. E1 is a Banach space with respect to the uniform norm. On these spaces we define the respective functionals
and
Note that these functionals are convex. We also define the bounded linear map by
Then
We wish to apply Fenchel–Rockafeller duality. In order to do this we must verify the existence of such that
and
is continuous at
For example, we may take u to be of the form
where CH denotes the constant from the bounds on the Hamiltonian Equation(2.1)
(2.1)
(2.1) . We then take
non-negative to have sufficiently strong decay at infinity so that
(3.4)
(3.4)
Explicitly, for the case we may take for example
in which case
and therefore
For the case we may take
In this case,
and so
which implies that
Then, in either case,
It follows that the positive part satisfies
and thus by the bounds on
Equation(3.1)
(3.1)
(3.1) we obtain
That is, is finite.
Moreover, and thus
Finally, since and m0 is a probability density,
Thus is finite.
Now we verify that is continuous at
with respect to convergence in E1. Consider the sequence of pairs
such that
where
satisfy
Then
Using the bounds Equation(2.1)(2.1)
(2.1) on the Hamiltonian, we obtain
for some constant C > 0. Therefore, for all n large enough that
for the positive part we have
Then the bounds Equation(3.1)(3.1)
(3.1) on
imply that
(where the constant C > 0 has changed line to line). The right hand side is in L1 because we constructed ζ to satisfy Equation(3.4)(3.4)
(3.4) . We may therefore use it as a dominating function: since
certainly converges to zero pointwise (in fact in uniform norm), and
is continuous with respect to the variable β, by dominated convergence we may conclude that
Thus is indeed continuous at
It remains to check that is bounded below on E0. Let
and set
Then, using the growth assumptions on
and
similarly to the inequality Equation(4.2)
(4.2)
(4.2) below, we have
where c0 was set to be a large positive constant depending only on
Therefore, we are in position to apply the Fenchel–Rockafeller duality theorem (cf. [Citation57, Chapter Citation3, Theorem 4.1]), to conclude
Here denotes the dual space of E1. By [Citation58, IV.6] the dual space of
may be identified with the space of bounded, regular, finitely additive set functions. Thus
is the space of pairs (m, w), where m is a real-valued regular finitely additive set function, and w is a
-valued regular finitely additive set function.
It remains to identify
In what follows, we are going to show that the above maximization problem actually admits solutions in a better space than So, we have
where the set
stands for pairs (m, w) such that m is a finite Radon measure on
and w is a finite vector-valued Radon measure on
taking values in
The proof of this is postponed to Lemma 3.6 below.
Then, by arguing as in [Citation19, Section 3.3], we may identify that
where the maximum is taken over
such that
and
almost everywhere, such that
Thus
□
Lemma 3.6.
Using the notations and assumptions from Lemma 3.5, we have
Proof.
Observe that any pair induces functionals on
and
Therefore, there exist a signed Radon measure
with finite total variation and a finite vector-valued measure
which coincide with, respectively, m and w on (the closure with respect to the uniform norm of)
and
Then
By considering functions of the form for
(note that our assumptions on H imply in particular that
) and any non-negative
and l > 0, and ψ = 0, we find that
unless m is a positive functional. Indeed, note that
and
if
Next, by taking the supremum over the smaller set we have
Let us underline that the assumption on H0 plays a crucial role, otherwise the integral of might not be finite for compactly supported test functions.
Since H is convex, for any such that
Thus
and in particular we can compare the positive parts:
Since is non-decreasing,
Hence, for all such that
by dominated convergence we have
where
for some continuous
converging pointwise to the constant function 1 as R tends to positive infinity. We conclude that, for any
(respectively signed, vector-valued) Radon measures with finite total variation,
where we have used that H0 is also a function in order to relabel
We have thus proved that
Next, note that if is a positive functional with Radon measure part
then
is also a positive functional: given
let
be a sequence of continuous functions, non-decreasing with R and converging pointwise to the constant function 1 as R tends to positive infinity. Then, since
by dominated convergence and the positivity of m,
Since and thus
for all m such that
is finite. Then
We now consider We assume from now on that
is a positive functional, since we only wish to consider (m, w) for which
Then, taking supremum over the smaller set we have
If then
Thus
We show that the right hand side is in fact equal to given
let
be a sequence of cutoff functions such that
We construct χR such that their support is contained in
the closed ball of radius 2 R,
on
the closed ball of radius R, and
for some constant C > 0 independent of R. Thus note in particular that
and
pointwise as
Let
Then
Since we have
Moreover
is bounded and therefore integrable with respect to m0. Finally, note that
Hence, if
we may apply the dominated convergence theorem to find that
This completes the proof that the suprema over E0 and are equal for the Radon measure parts. We conclude that
Now observe that, since the set is contained in
(it is precisely the set of Radon measure parts of elements of
),
All of the above inequalities are therefore equalities. Moreover, since
if (m, w) attains the supremum then the same is true of the Radon measure part
Thus, without loss of generality, the optimizer is given by some
i.e. a finite measure and a finite
-valued measure. □
Remark 3.7.
Let us notice that the minimizer of is unique (by the convexity of
and L in their last variables). Moreover, the growth conditions on
and L imply that
and
Furthermore, by Hölder’s inequality,
with
These arguments are similar to the ones in [Citation20, Theorem 2.1] and [Citation19, Lemma 2]. Furthermore, the equation satisfied by m conserves mass, so that
and in fact
for all
3.4. The relaxed problem
The third problem we define is a relaxation of Problem 3.1. Consider the functional
Problem 3.8.
Minimize over the set
of triples
satisfying
The positive part of u satisfies
The positive part of β satisfies
The positive part of βT satisfies
Definition 3.9.
We say that a triple that belongs to the spaces from Problem 3.8 is a weak distributional solution to
(3.5)
(3.5) if
(3.6)
(3.6) for any
nonnegative.
Remark 3.10.
Let us emphasize that the weak form Equation(3.6)
(3.6)
(3.6) encodes both inequalities from Equation(3.5)
(3.5)
(3.5) , as we show this in Lemma A.6.
u0 is similarly understood as a certain notion of a trace at t = 0 in a weak sense. In particular, the term
4. The Hamilton–Jacobi equation
In this section, we analyze the Equationequation (3.5)(3.5)
(3.5) . We take the assumptions appropriate to the minimization problem we will consider. Therefore, we suppose throughout that
is such that
From the finiteness of the energy we deduce in particular that
4.1. Upper bounds
We prove upper bounds on u. First, we observe that for any constant the function
satisfies (see Lemma B.1)
(4.1)
(4.1)
We use the notation to denote the set of functions
which becomes a Banach space when equipped with the norm
We also use the notation to denote the intersection of L1 and Lq made into a Banach space under the norm
Note that the dual space is given by
Lemma 4.1.
Let be given and let
satisfy Equation(4.1)
(4.1)
(4.1) .
Then
, with the a priori estimate
Suppose in addition that
. Then, there exists
Proof.
First, let us note that, since and
(by Assumption 1),
and thus also
Let
be fixed. Let
and consider
Then ζ is smooth and compactly supported and satisfies
By using ζ as a test function for over
we obtain
Recall that when we write we are always referring to the version of u that is weakly right continuous with respect to time (cf. Appendix A, Lemma A.1).
Since we have
Then
We compute
Similarly
and
Thus
This extends by density to all non-negative and general
by non-negativity of
We conclude by the fact that
The result follows.
By the definition of
and the assumptions on the data one has
(4.2)
(4.2)
where in the last inequality we used the estimate from (i). This further yields the claim in (ii). □
Corollary 4.2.
Let be as in the statement of Lemma 4.1 such that there exists
with
Then, there exists
such that the following hold.
Proof.
We notice that (i) is a simple consequence of Lemma 4.1(i)-(ii), by setting l = 0 and t = 0 (in the sense of weak trace, given in EquationDefinition A.5(A.5)
(A.5) ).
For (ii), we observe
By the bounds Equation(3.1)(3.1)
(3.1) on
and the corresponding estimates for
Hence, using the above bounds and (i), we obtain
which completes the proof. □
4.2. Local L1 bounds
Next, we prove bounds on the negative parts of u and β. We will obtain bounds, by use of a duality argument involving a certain class of test functions which satisfy the continuity equation associated to the control system.
Lemma 4.3.
Let be a bounded control. Let
satisfy
. Let
be the solution of the continuity equation
Then, for any such that
, the following hold:
, that is,
The negative part of β satisfies
The negative part of βT satisfies
The following estimate holds:
Proof.
Note the following properties of
is non-negative,
has compact support contained in
In particular, since for any
then
By Lemma 4.1,
(4.3)
(4.3)
and thus for
we have
where
is understood in the sense of weak trace (cf. Lemma A.1, EquationDefinition A.5
(A.5)
(A.5) ).
For the negative part we make use of the equation. A density argument shows that is admissible as a test function in the weak form of the Hamilton-Jacobi inequality satisfied by u. Thus for
We apply this in the case s = 0, Using the fact that
satisfies the continuity equation in a pointwise sense,
Here, let us notice that we have used the existence of weak traces in the sense of Lemma A.1. In particular the integral is meaningful and finite, since
(EquationDefinition A.10
(A.10)
(A.10) ).
Since and
has compact support contained in
we may integrate by parts to obtain
Then estimate
where, in order to lighten the notation, we have used the shorthand
to denote Lp norms with respect to the measure on with density
with respect to Lebesgue measure. Thus
Using the lower bounds on the Hamiltonian H, rearranging terms and using Young’s inequality for products (with a small parameter), we obtain
Then
(4.4)
(4.4)
Finally, since we use the
bounds on the positive part
(EquationEquation (4.3)
(4.3)
(4.3) ) to conclude that
Notice that by setting t = T, Equation(4.4)(4.4)
(4.4) and the fact that
(together with the bounds that we already have on
) readily yield also that
This completes the proof. □
Corollary 4.4.
Let such that
. Then
and
Proof.
First, consider a compact set
where
denotes the solution of the continuity equation
(4.5)
(4.5)
By compactness of K, there exist finitely many such that
The function is continuous and so
Then
By Lemma 4.3, this leads to the estimate
where
We now claim that
This follows from the controllability of the ODE system
(4.6)
(4.6)
on
That is, for any initial datum
and target
there exists a control function a such that the solution
of the ODE Equation(4.5)
(4.5)
(4.5) with
satisfies
Next, note that (since m0 is continuous) contains a closed ball
for some point (x0, v0) and some r > 0. Thus there exists
such that
on
Consider the solution
of Equation(4.4)
(4.4)
(4.4) for the control a found above and with this choice of
It follows that
Finally, we notice that by the structure of the set we have the bound
□
5. Duality for the relaxed problem
Theorem 5.1.
Problems 3.3 and 3.8 are in duality:
Proof.
For such that
the triple
lies in
Thus
By the duality result of Lemma 3.5,
It therefore remains only to prove the reverse inequality. This follows from Lemma 5.2 below, which states that for all and
Taking the infimum over and supremum over
gives
as required. □
Lemma 5.2.
Let and
such that
. Then
In the proof of this lemma we require the following observation regarding the commutator between the operator and the operator given by convolution with a fixed function.
Lemma 5.3.
Let be a function such that
. Let
for
. Then
Proof.
By direct computation, for all
□
Proof of Lemma 5.2.
The overall idea of the proof is to use m as a test function in the weak form of the inequality
and its terminal condition
To make this valid, we must first introduce an approximation procedure.
First, we introduce a lower cutoff on u and β. Let and define
Similarly, for
let
Then by Lemma B.1 we obtain
(5.1)
(5.1)
in the sense of distributions. By Lemma 4.1,
We emphasize that k and l are taken to be possibly independent at this point.
Next, we approximate m by a function in which is then an admissible test function for the Hamilton-Jacobi EquationEquation (5.1)
(5.1)
(5.1) . We regularize m by convolution with a mollifier. For ease of presentation, it will be convenient to work with the time, space and velocity variables separately. Fix
and define
for
by
For the space variable, consider and for
let
For the time variable, fix and for
let
We then define the full mollifier by
Then define the smooth functions
Notice that for the convolution in time, (m, w) needs to be extended. We choose the following extensions. We set to for t < 0 and t > T. Then, if t < 0, we set
to be the solution to the problem
Similarly, for t > T we set to be the solution to
where mT is the trace of m in time at t = T.
As the final step in the approximation, we localize As localizers we consider smooth functions
such that
(5.2)
(5.2)
We then define and
Then satisfies the equation
where the error term is given by
(5.3)
(5.3)
Here, we use the standard commutator notation where
are some operators acting on the function f.
5.1. Convergence of the error term
We show that the error term defined by Equation(5.3)
(5.3)
(5.3) converges to zero in the space
as
and
under a certain relationship between these parameters.
For the first term, either for p = 1 or p = q, using the explicit formula for the commutator we estimate
where we have used Lemma 5.3 in the third inequality. Thus by choosing
sufficiently small with respect to δ, we may ensure that
For the second term, observe that for all R > 0, and thus
It follows that
For the third term, for either p = 1 or p = q we have
Taking as above, we can then ensure this term converges to zero by choosing
sufficiently large with respect to δ and
Thus, for this choice of
we have
Altogether, we have found that there exists a regime and
such that
5.2. Testing the equation
Using as a test function in the weak form of the equation for ul, one obtains
Using the equation satisfied by
we have
(5.4)
(5.4)
Next, note that By the chain rule for Lipschitz functions composed with Sobolev-regular functions,
Thus, using the definition of distributional derivative we may integrate by parts to obtain
Since is finite, w is absolutely continuous with respect to m. It follows that there exists
such that
Thus
Substituting this, we obtain
(5.5)
(5.5)
We have shown above that there exists a regime and
such that the final term converges to zero uniformly in η as δ tends to zero, since
We now discuss the convergence of the other terms.
5.3. Boundary terms
We consider the boundary terms at Note that Lemma 4.1 and Corollary 4.2 yield
(since
and
is bounded below), while by Lemma 4.1 and Corollary 4.4 we have
We first show that converges to m0 in
and
converges to mT in
in the limit as δ tends to zero for a certain regime
and
according to the regime already found above.
For we write
(5.6)
(5.6)
We first note that by the assumption that it is a bounded probability density, while
since the energy
is finite, and
since the continuity equation conserves mass.
Then, since if
(where
or
) then by dominated convergence
Moreover, by continuity of translations in Lp,
Since for all R > 0
it follows that
Therefore, the latter two terms of Equation(5.6)(5.6)
(5.6) converge to zero as δ tends to zero with
and
as already specified above, in
for t = 0 and in
for t = T.
It remains to estimate the difference For any function
(
to be specified),
We use the notation Writing the time convolution explicitly, we obtain
Next, we use estimates on
Then, since
We estimate
and similarly
Thus
Finally
where
Thus it is possible to choose
depending on
in such a way that
We apply this in the case f = fi for i = 1, 2, where
Consequently, for
and
as
with
chosen to depend on δ in the manner specified.
Finally, we take the limit For the term, t = 0, convergence holds by monotonicity, and the limit is finite since
by finiteness of For the term t = T, we first note that
The second term on the right hand side converges due to the assumption on Equation(2.3)
(2.3)
(2.3) , since the integrand is dominated by
5.4. Term involving βm
Since by standard results on approximation by mollification in Lp spaces we have
and thus the same limit holds with
chosen to depend on δ as described above. Then, since
we deduce that
By the definition of Fenchel conjugate,
We then take the limit Note
and
are both lower bounded by integrable functions (conditions Equation(2.2)
(2.2)
(2.2) and Equation(3.1)
(3.1)
(3.1) ). Then, by monotonicity,
Moreover
Since the lower bound is integrable and has finite integral by finiteness of the energy, both of these limits are finite. Thus
A similar argument shows that
where the right hand side is finite.
Finally, we consider Note that
by assumption (see Equation(3.1)
(3.1)
(3.1) ). Since
is a continuous non-decreasing function of β, as k decreases to negative infinity
is decreasing and converges almost everywhere to
Thus we deduce the convergence
By the bounds Equation(3.1)(3.1)
(3.1) , the right hand side is finite. Moreover, for any
Thus we conclude that
5.5. Lagrangian term
For the term involving the Lagrangian, we use a similar argument as was used in [Citation20]. This argument is based on the joint convexity of as a function of (m, w). In our case we must additionally account for the convergence of the localizer ζR. By convexity, for all (t, x, v), the integrand satisfies the inequality
Then, note that
Then, since
converges to
in
as
tend to zero. Since
It follows that if we take the regime established above, then
We stay with this regime and consider the remaining term
The integrand converges to zero almost everywhere: note then that
The right hand side converges in L1 to as δ tends to zero. Thus by dominated convergence we conclude that
The reverse inequality follows from Fatou’s lemma. Thus
Finally, we take the limit Since
in the limit we obtain
5.6. Conclusion
From the discussion above, we have obtained
where all terms are finite. Rearranging this inequality, we obtain the statement
□
Corollary 5.4.
Let and
be such that
and
Then
Moreover, for almost all
and
(5.7)
(5.7)
In particular,(5.8)
(5.8)
Proof.
This is a consequence of the proof of Lemma 5.2 and in particular the inequality Equation(5.5)(5.5)
(5.5) . First, let us show the first part of the statement, i.e. that
As in the mentioned proof, let us first pass to the limit with in the inequality Equation(5.5)
(5.5)
(5.5) . Then, we pass to the limit as
and
the remaining terms.
All the terms, except the ones involving and
pass to the limit, as in the proof. After rearranging, we also find that both
and
are uniformly bounded, independently of l and k. Therefore, the monotone convergence theorem yields
and
The summability results follow, and so does Equation(5.8)(5.8)
(5.8) .
For the case of general we begin by testing the equation to find that, for example,
We note that we are referring to the version of ul that is weakly right continuous in time (see Appendix A). The only term that requires attention is
For almost all and
Thus the arguments for the boundary terms in Lemma 5.2 show that
The limit is then taken by monotone convergence, noting that
Note that the argument for the case Equation(5.7)
(5.7)
(5.7) shows that limit is not negative infinity, since all other terms have finite limits. □
Corollary 5.5.
Let and
be such that
and
. Then
is uniformly bounded in
, by a constant that depends only on the data and
and
The following estimate holds:
Proof.
This is a consequence of Equation(5.4)(5.4)
(5.4) . Using the same notation as in the proof of Lemma 5.2, we rewrite Equation(5.4)
(5.4)
(5.4) as
Let us observe that for some
parameter that we choose later, Young’s inequality yields
We notice that Thus, by using the growth condition Equation(2.1)
(2.1)
(2.1) on the Hamiltonian and choosing θ appropriately, we can conclude that there exists a constant C > 0 (independent of the parameters
), such that after passing to the limit with
as in the proof of Lemma 5.2, we obtain
Since the right hand side of this inequality is uniformly bounded, independently of l (by Lemma 4.1(ii), Corollary 4.2(ii) and Remark 3.7), the result follows by Fatou’s lemma by sending
Using Equation(5.4)(5.4)
(5.4) , we have
Passing to the limit with as in the proof of Lemma 5.2, by Fatou’s lemma we obtain
Finally, taking the limit as in the proofs of Lemma 5.2 and Corollary 5.4, we obtain
6. Existence of a solution of the relaxed problem
In this section we prove the existence of a solution for the relaxed problem. Consider a minimizing sequence We will extract a convergent subsequence, and show that the limit constitutes a minimizer of the objective functional.
6.1. Compactness of the minimizing sequence
The following proposition enables the extraction of a convergent subsequence.
Proposition 6.1.
Let be a sequence of solutions to the Hamilton-Jacobi equations
Assume that:
The family
is uniformly bounded in
The family
is uniformly bounded in
The family
is uniformly bounded in
Then there exists a subsequence that is strongly convergent in
The proof of this result will be a consequence of some intermediate results that we detail below.
Remark 6.2.
Let us notice that the assumptions of Proposition 6.1 hold true by Corollary 4.4 and Lemma 4.1.
Proposition 6.1
is proved by treating the Hamilton-Jacobi equation as a kinetic transport equation with right hand side bounded in
A form of compactness for the solutions can be obtained by using an averaging lemma. Averaging lemmas are results in kinetic theory showing that, for Lp-bounded families of solutions to the kinetic transport equation, with Lp-bounded source terms, the velocity averages
enjoy additional fractional Sobolev regularity and/or strong Lp-compactness. In our case we are in the setting p = 1, and we use an L1 averaging result from [Citation49]. It is necessary to assume a certain equi-integrability condition on the solution un. This condition is defined below.
Definition 6.3
(Equi-integrability in velocity). Let be a bounded family in
The family is locally equi-integrable in v if, for all
and all compact sets
there exists
such that for all measurable families
of measurable subsets of
for which
The required averaging lemma is quoted below. This result was proved in [Citation49] for the stationary case, i.e. the equation The result can be adapted to the time dependent equation by standard techniques; see [Citation50] or [Citation51] for statements in the time dependent setting.
Theorem 6.4.
Let be a bounded family in
satisfying
(6.1)
(6.1)
where is a bounded family in
. Assume that
is equi-integrable in v. Then
The family
is locally equi-integrable in all variables in
For each
, the family of averages
is relatively (strongly) compact in
In our setting we expect to have local summability estimates in rather than
To deal with this technicality we make use of a localization procedure: given a compact set K, consider a smooth bump function ζ supported in K. If
satisfies the kinetic transport EquationEquation (6.2)
(6.2)
(6.2) , then
satisfies
(6.2)
(6.2)
The right hand side of the above equation is bounded in uniformly in λ, as long as
and
are bounded in
We wish to apply this to the solutions of the Hamilton-Jacobi equation. To do this, we verify the equi-integrability condition. To prove equi-integrability, we make use of the Lr estimates available for the v-derivative
Lemma 6.5.
Let be a bounded family in
. Assume that
is a bounded family in
. Then:
is bounded in
where
if r < d, or any
if
is equi-integrable in v, locally in
Proof.
We obtain higher integrability in the velocity variable by using Sobolev embedding. We first apply a localization procedure. Given a compact set let ζK denote a smooth bump function with compact support contained in
such that ζK takes values contained in
and
on K. Then
Let denote the support of ζK. Then
thus
is bounded in L1, uniformly in λ. Moreover it is compactly supported.
We then apply Sobolev embedding in the v variable. Letting we have
Thus is uniformly bounded in
We now apply a bootstrap argument: it follows from the above that is bounded in
and thus
is bounded in
for
This process can be repeated until we obtain that
is bounded in
for
if r < d; otherwise we may obtain
for any
We now prove local equi-integrability. Let be a measurable family of measurable subsets of
such that
for all t, x. Then
where denotes a Hölder conjugate exponent. From the condition on the measure of
we have
which proves equi-integrability. □
It follows that, under the assumptions of Proposition 6.1, Theorem 6.4 can be applied to for any
We deduce strong
-compactness for the averages
We now use this to prove strong compactness for the full solutions un.
Lemma 6.6.
Assume that the family satisfies the following:
is uniformly bounded in
.
is equi-integrable in all variables.
share the same compact support K.
is uniformly bounded in
For each
, the family of averages
is relatively (strongly) compact in
Then the family is relatively (strongly) compact in L1.
Proof.
First, note that the first two assumptions imply the weak L1 sequential compactness of We pass (without relabelling) to a weakly convergent subsequence un, and let u denote the weak limit. In the remainder of the proof we improve the mode of convergence of un to u to strong convergence in L1.
Step 1: Approximation by smoothing in v. We approximate un by a function that is smooth with respect to the v variable. Fix
and define, for
Let
Step 2: Compactness for the approximations. Fix
For any
with compact support we consider testing the sequence
against the test function
Since
we deduce that
converges weakly in
to
as
Note moreover for each fixed
is a velocity average with respect to the test function
Therefore, by Theorem 6.4 the convergence in fact holds strongly in
for each
Furthermore, for fixed the family
is equi-continuous in v into
indeed
Thus
By an Arzelà-Ascoli argument the convergence therefore holds locally uniformly in v, with respect to the strong topology on that is, for all compact sets
and
Consequently, the convergence holds in in fact, since
is supported for all n in
the convergence holds in L1.
Step 3: Removing the approximation.
The bound on implies that, for any
It follows that
Indeed, by definition of
Thus, for any
That is,
then, one integrates in t, x and takes supremum.
Finally, estimate
Thus
as
which completes the proof. □
Proof of Proposition 6.1.
The proof of this proposition follows by applying Lemma 6.6 to □
It remains to obtain the necessary convergence of and
Lemma 6.7.
Let be a minimizing sequence for Problem 3.8. There exists a modification
of this sequence that is also minimizing such that
is weakly precompact in
and
is weakly precompact in
Proof.
We replace βn with some and
with
such that
is uniformly integrable, and
is still a minimizing sequence. We do this in a similar manner to [Citation20]: since
is bounded in
using a compact exhaustion of
and a diagonal argument, by [Citation59] it is possible to pass to a subsequence such that the following holds for some
We define
by
Then it is possible to choose Jn in such a way that:
For each compact set
the sequence
is uniformly integrable.
The measure of the set
converges to zero as n tends to infinity.
We use the exact same construction for and we can get the same properties (now taking
).
We notice, that by construction the constraints
and
are still satisfied. Finally,
By the estimate Equation(3.1)(3.1)
(3.1) , the integrand on the right hand side is dominated by
and thus the right hand side converges to zero as n tends to infinity.
The exact same arguments apply to and
too. Thus
is a minimizing sequence. Moreover, there exists
such that up to passing to a subsequence,
converges to u strongly in
converges weakly to β in
and
converges to
weakly in
□
6.2. Existence of a minimizer of ![](//:0)
over ![](//:0)
![](//:0)
In this subsection, we prove that there exists a minimizer by passing to the limit in the functional
Theorem 6.8.
Under our standing assumptions, the functional admits a minimizer over
Proof.
Let be a minimizing sequence. Without loss of generality, for example by considering
we may assume equality in the Hamilton-Jacobi equation:
For this minimizing sequence we have, for some constant C > 0,
We have discussed that this implies uniform in n bounds on the following quantities:
To get the uniform integrability on and
we perform the surgery argument as in Lemma 6.7. So, let
be the modification of the minimizing sequence (which will still have uniformly bounded energy). By Proposition 6.1 we know that
is strongly precompact in
while Lemma 6.7 yields that
and
are weakly precompact in
and
respectively. In particular, after passing to a subsequence let us denote by u the strong
limit of
In what follows, to ease the notation, we drop the tilde symbol, but whenever we write
and
we mean the corresponding modified versions.
Passing to further subsequences (that we do not relabel), there exist limit functions so that we may also assume the following weak convergences:
weakly in
as
weakly in
as
weakly in
as
weakly in
as
weakly in
as
With these convergences in hand, we are ready to pass to the limit in the Hamilton-Jacobi inequality constraint and the functional. Note that the weak form of the inequality (Definition 3.9) implies that, for all n and all test functions such that
(6.3)
(6.3)
Note again that is compactly supported in
By the weak convergence of
in
and the convexity of H it follows that
All the other convergences stated above are sufficient to guarantee convergence against So, we obtain that the limit
satisfies Equation(3.6)
(3.6)
(3.6) .
Next, we consider the convergence in the functional. In addition to the previous convergences, along the previously chosen subsequence, we have
weakly-* in
as
The convergence of the sequence requires special attention. The boundedness of this sequence in
lets us conclude that there exists a nonnegative Radon measure ν such that after passing to a subsequence (that we do not relabel)
This means in particular that for all we have
Since the the sequence is supported in the open set
we get that
Now, let us take
arbitrary and define
Since
by assumption, we have that
and so
Thus, this means that as converges weakly-
to the nonnegative Radon measure
i.e.
has density
with respect to ν. We notice that this means that
is absolutely continuous with respect to ν. In fact, we also have that ν is absolutely continuous with respect to
and so we can write
Let us take now and test the inequalities satisfied by
similarly to Equation(6.3)
(6.3)
(6.3) , to obtain
Incorporating also the previously described convergence of we can pass to the limit along the chosen subsequence and obtain
where
We notice that
is a signed Radon measure, supported in
Having in hand this last inequality, we readily check that the assumptions of EquationLemma A.11(A.11)
(A.11) are fulfilled with the choice of
and
as before. This means in particular that u satisfies
where when writing the traces u0 and uT, we are referring to the right continuous version of u in time. Since by construction,
is finite, we have that
is meaningful and finite, with
Lower semicontinuity of the term involving
Claim.
Proof of Claim.
First, notice that since is a positive distribution, it can be represented by a Radon measure. We may therefore write, for some
such that
and
It follows that the Hahn-Jordan decomposition of u0 satisfies
Now consider any compactly supported function such that
Then
Since as measures,
Then take a non-decreasing sequence of functions ζk such that ζk converges pointwise to the indicator function of the set as k tends to infinity: consider for example functions such that
This is always possible since m0 is continuous. Then, by monotone convergence, we indeed have
as desired and the claim follows.
By the weak star convergence of to
in
we also have that, for
the positive part
converges to
strongly in
Since
as signed measures, we deduce that
(6.4)
(6.4)
as required.
The term involving
For the term involving we notice that by convexity
(6.5)
(6.5)
Indeed, by classical results (cf. [Citation57, Proposition I.2.3, Corollary I.2.2]), this is a consequence of the convexity of the integrand in the last variable and Fatou’s lemma that yields the lower semi-continuity with respect to the strong topology on
The term involving
First note that, for functions β such that and
by Equation(3.1)
(3.1)
(3.1) the following inequality holds:
Thus
(6.6)
(6.6)
Indeed, since for all
we have
The integrand is bounded by the L1 function and converges to zero almost everywhere as δ tends to zero. Thus, taking
we obtain Equation(6.6)
(6.6)
(6.6) . A similar equality holds for all βn.
Therefore, by the convexity of (and by arguments similar to the one for
), we conclude that
(6.7)
(6.7)
Thus, collecting all the previous arguments, one deduces that
The thesis of the theorem follows. □
Corollary 6.9.
In the setting and notation of the previous theorem, in fact on
Proof.
Since is a minimizer,
where in the last equality we have used that
is a minimizing sequence.
All the above inequalities are therefore equalities. From the inequalities Equation(6.4)(6.4)
(6.4) , Equation(6.5)
(6.5)
(6.5) and Equation(6.7)
(6.7)
(6.7) for each of the terms, we deduce that
It follows that as signed measures on
Indeed, first note that
as signed measures, or in other words
is a nonnegative measure. For any non-negative test function
we have
on the support of
for some
Thus there exists a constant C such that
Thus
Thus as signed measures on
□
7. Existence and uniqueness of a solution to the MFG system
In this section we prove Theorem 2.4. First, we show that the minimizers of Problems 3.3 and 3.8 that we have obtained in the previous sections provide weak solutions (u, m) of the MFG.
Theorem 7.1.
Let be a minimizer of
over
and let (m, w) be a minimizer of
over
. Then
for a.e.
for a.e.
;
for a.e.
As a consequence, (u, m) is a weak solution to Equation(1.1)(1.1)
(1.1) in the sense of Definition 2.3.
Proof.
By Theorem 5.1,
Substituting the definitions of the functionals, we obtain
Fenchel’s inequality then implies that
By Corollary 5.4, the left hand side is non-negative, and therefore equality holds:
(7.1)
(7.1)
Moreover, equality also holds almost everywhere in the applications of Fenchel’s inequality. Thus the following hold almost everywhere in
(7.2)
(7.2)
By Equation(7.1)(7.1)
(7.1) and Corollary 5.5,
By Fenchel’s inequality, the integrand on the right hand side is non-negative; we deduce that equality holds in the above estimate and thus the integrand is equal almost everywhere to zero. It follows that
almost everywhere on the support of m. Moreover,
(7.3)
(7.3)
The energy equality then follows from substituting Equation(7.2)
(7.2)
(7.2) and Equation(7.3)
(7.3)
(7.3) into Equation(7.1)
(7.1)
(7.1) . □
We show now, conversely, that weak solutions to the MFG system are in fact minimizers in the corresponding variational problems. The proof of this result follows similar ideas as the corresponding ones from [Citation20,Citation21].
Theorem 7.2.
Let (u, m) be a weak solution to Equation(1.1)(1.1)
(1.1) in the sense of Definition 2.3. Then by setting
and
, we find that (m, w) is a solution of Problem 3.3, while
is a solution of Problem 3.8.
Proof.
First let us notice that by Fenchel’s equality one has
We define the Borel set Restricted to this set, we find
where in the first inequality we have used our assumptions (3.1). Since, CF,
and
are summable, this implies in particular that
Using the growth condition on
we find furthermore that
Now, on Bc, i.e. when we find
Again, the summability of the right hand side, we find that Using the exact same arguments for
we find similarly that
and
Moreover, we have that
and
so (m, w) and
are admissible competitors for the two optimization problems.
Now, take as an admissible competitor for the problem involving the functional
By the convexity and differentiability of
and
in their last variable we have
where we have used the fact that
and
(by the arguments at the beginning of this proof). Moreover,
and
(cf. Corollary 5.4) and
Now, using Equation(2.4)(2.4)
(2.4) , one obtains
where in the last line we have used
By Corollary 5.4 we conclude that as desired.
Using the very same ideas and the convexity of and
we can conclude similarly that (m, w) must be a minimizer in Problem 3.3. □
Finally, we show that solutions in the sense of Definition 2.3 are unique, again following similar ideas as the corresponding ones from [Citation21]. One major difference, however, is that we develop a suitable comparison principle for the distributional solutions to the corresponding Hamilton-Jacobi inequalities. This completes the proof of Theorem 2.4.
Proof of Theorem 2.4.
The existence of a weak solution (u, m) follows from combining Theorem 6.8 (existence of a minimizer for ), Theorem 5.1 (duality, and the fact that the infimum for
is attained) and Theorem 7.1 (minimizers are weak solutions in the sense of Definition 2.3).
For the uniqueness, we first apply Theorem 7.2 to obtain that for i = 1, 2, are minimizers of
over
and
are minimizers of
over
Since the minimizer of
is unique by strict convexity,
almost everywhere and
almost everywhere.
To show that u1 = u2 almost everywhere on the set we first define
By Lemma B.2, u also satisfies the Hamilton-Jacobi inequality, with
and
Since
for i = 1, 2, we have
and thus
Since ui is a minimizer, equality holds. By duality, equality then holds in the energy inequalities of Corollary 5.4 for u and m, with
as defined previously. Thus, for almost all
The same is true replacing u by ui, and so
Thus, since also we deduce that ui = u almost everywhere on the set
□
8. Sobolev estimates on the solutions
In this section, we obtain Sobolev estimates on the optimizers of the variational problems, and hence on weak solutions for the MFG system Equation(1.1)(1.1)
(1.1) . The general idea is to “compare” the optimality of the optimizers in the variational problems with their carefully chosen translates. Then using strong convexity of the data one can deduce differential quotient estimates.
These results are inspired by [Citation23,Citation24]. However, because of the kinetic nature of the model we need completely new ideas when we consider perturbations. So, the estimates that we obtain are on suitable kinetic differential operators applied to the solutions. Another crucial difference between our results and the ones in [Citation23,Citation24] is that our Sobolev estimates in the x and v variables are local in time on The main reason behind this is that we have a weaker notion of trace for u0, that we cannot ensure to be stable under perturbations. This imposed further technical complications that require us to work in the case of r = 2.
We emphasize that these estimates are consequences of the stronger convexity and regularity assumptions on the data stated in Assumption 2.
8.1. Local in time Sobolev estimates
Let be a smooth cutoff function such that
and
for all t > 0. We define
as
For competitors (m, w) in Problem 3.3, without loss of generality one might assume the representation w = Vm, for a suitable vector field V. Let with
and define
We use the notation
We notice that by construction, if is a distributional solution to Equation(3.3)
(3.3)
(3.3) , so is
and
Similarly, for competitors in Problem 3.8 we define
Furthermore, we define
When computing the Legendre transforms of these functions in their last variables we obtain
Let us notice that satisfies in particular the hypotheses imposed in Assumptions 1. Correspondingly, we define the functionals
and
and the constraint sets
and
using the shifted versions of the data functions. By construction, as a consequence of a change of variable formula, the proof of the following lemma is immediate.
Lemma 8.1.
(m, w) is an optimizer of over
if and only if
is an optimizer of
over
. Similarly,
is an optimizer of
over
if and only if
is an optimizer of
over
Proof.
We provide the proof of one of the statements only, the other ones follow similar steps. Suppose that is an optimizer of
over
This means in particular the minimality of the quantity
where in the last equality we have used the change of variables
So, this means that the minimality of
after a change of variables, yields the minimality of (m, w). □
Now we are ready to state the main result of this subsection.
Theorem 8.2.
Suppose that (u, m) is a weak solution to Equation(1.1)(1.1)
(1.1) in the sense of Definition 2.3 and that (H5), (H6), (H7) hold.
Then, there exists such that
and
Remark 8.3.
As for Theorem 2.5, this is an informal statement: the result we obtain is on suitable difference quotients as in estimate Equation(8.8)(8.8)
(8.8) below.
Proof of Theorem 8.2.
Let be a minimizing sequence for Problem 3.8 such that
Let us recall that after passing to a subsequence, that we do not relabel, as a consequence of Proposition 6.1, Lemma 6.7 and by Claim 2 in the proof of Theorem 7.1, we have that
weakly in
as
weakly in
as
weakly in
as
weakly in
as
weakly-
in
as
weakly in
as
Notice that the previous arguments imply also that the subsequence can be chosen such that for all M < 0
(8.1)
(8.1)
and
Furthermore, by Theorem 7.1, we have that and
Let
Fix such that
and
as described at the beginning of this subsection.
Now, the main idea is to use as a test function in the weak formulation of the equation satisfied by (m, w) and un as test function in the weak form of the equation satisfied by
Then we combine these inequalities with the energy equality Equation(2.4)
(2.4)
(2.4) written for (m, w) and
respectively, and rely on the strong convexity and regularity properties of the data to deduce a differential quotient estimate.
Following these steps, we obtain
We combine this with the energy equality Equation(2.4)(2.4)
(2.4) for (m, w) to get
(8.2)
(8.2)
Similarly, using un as a test function in the weak form of the equation for and combining with Equation(2.4)
(2.4)
(2.4) for
(8.3)
(8.3)
Adding Equation(8.2)(8.2)
(8.2) and Equation(8.3)
(8.3)
(8.3) , after some changes of variables (translations) and a Taylor expansion of L, we deduce
(8.4)
(8.4)
where we have also used the facts that by the choice of η and ζ, we have
and
Our aim now is to pass to the limit in Equation(8.4)
(8.4)
(8.4) and derive a differential quotient estimate. For this, we consider each of the terms separately.
Step 1. First, we notice that by (H7) and by the fact that
there exists C > 0 such that
Step 2. Second, let us notice that by the fact that
and by Equation(8.1)
(8.1)
(8.1) , for any M < 0 we have
Therefore,
Now, sending we conclude that
(8.5)
(8.5)
where we have used the fact that so that the integrand is upper bounded by an L1 function to allow us to apply the monotone convergence theorem. Since the left hand side of inequality Equation(8.4)
(8.4)
(8.4) is bounded from below by zero, it follows the right hand side of Equation(8.5)
(8.5)
(8.5) is not negative infinity.
By the very same arguments one can conclude that
Step 3. By Young’s inequality, we have
Step 4. By the previous steps we can conclude that
is uniformly bounded above, independently of Let us recall that
and so is
Using the growth condition on H, by choosing c > 0 small enough in our application of Young’s inequality we deduce that
is uniformly bounded in
By a change of variable, one can similarly deduce that
is uniformly bounded in
Claim.
After passing to a subsequence that we do not relabel, we have weakly in
as
Proof of Claim.
By the uniform boundedness of the sequence, we know that there exists a subsequence of it (that we do not relabel) and as weak limit, i.e.
Thus, we aim to show that As
weakly in
as
we can argue similarly as in the proof of Claim 2, in the proof of Theorem 7.1 to deduce the claim.
Step 5. By summarizing, Equation(8.4)
(8.4)
(8.4) implies that
Using the additional assumption (2.10) and the inequality for c > 0 sufficiently small, one can conclude that there exists
depending only on the data, such that
(8.6)
(8.6)
Now, our aim is to pass to the limit with first in Equation(8.6)
(8.6)
(8.6) . For this we take
of the left hand side and
of the right hand side. We notice that the term
needs special attention, since we do not have upper semicontinuity of it. Because of this, we add to both sides of Equation(8.6)
(8.6)
(8.6) the quantity
before passing to the limit. Thus we obtain
All the arguments in the previous steps allow us to pass to the limit. By the fact that is a minimizing sequence, we get that
So, after simplification, one obtains
(8.7)
(8.7)
Now, using Equation(2.5)(2.5)
(2.5) and Equation(2.7)
(2.7)
(2.7) the very same arguments as in [Citation23, computation (4.25)] yield
Similarly, Equation(2.6)(2.6)
(2.6) and Equation(2.8)
(2.8)
(2.8) yield
Combining these estimates with Equation(8.7)(8.7)
(8.7) , we get
(8.8)
(8.8)
Dividing by and letting
we easily obtain the result. □
8.1.1. Recovering estimates on the operator ![](//:0)
applied to solutions
By choosing a specific structure for the cutoff function ζ, we can recover estimates on more particular differential operators. Suppose that for
and
for
for some
(to be chosen to be arbitrary), in such a way that also
Then in Theorem 8.2, the operator
for
becomes
So, one can state the following local in time corollary.
Corollary 8.4.
Suppose that the assumptions of Theorem 8.2 take place. Then, there exists such that
and
8.1.2. Recovering estimates on the operator Dx applied to solutions
Now suppose that for
and
for
(where
can be chosen arbitrarily). We still require that
With this choice of cutoff functions
we can formulate the following result as a corollary of Theorem 8.2.
Corollary 8.5.
Suppose that the assumptions of Theorem 8.2 take place. Then, there exists such that
and
8.1.3. Proof of Theorem 2.5
Finally, the proof of Theorem 2.5 follows from the previous two corollaries and the inequality
(8.1.3) for any Sobolev function h.
Acknowledgements
We thank Mikaela Iacobelli for useful discussions regarding Lemma A.4. We thank the two anonymous referees for carefully reading our manuscript and for their valuable comments.
Disclosure statement
The authors report that there are no competing interests to declare.
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Appendix A.
Time regularity
In this appendix, we collect some facts about the regularity with respect to time of solutions u of
(A.1)
(A.1)
By this we mean that, for any non-negative test function
(A.2)
(A.2)
What we discuss is close to the standard theory of distributional solutions. However, in our case technical difficulties arise since, firstly, Equation(A.1)(A.1)
(A.1) is an inequality and, secondly, we wish to work on the atypical domain
We therefore found it useful to clarify several points. Our main goal is to give a precise sense to the specification of boundary data for this problem at time t = T, and to give a meaning to u0 (the ‘value of u at time t = 0’), which appears in the functional
defining the variational problem.
Throughout this appendix we impose the following summability conditions on the pair and that H satisfies Equation(2.1)
(2.1)
(2.1) .
Assumption 3.
The pair satisfies the following assumptions:
The positive part of β satisfies
;
;
Under Assumption 3, by a density argument the weak form Equation(A.2)(A.2)
(A.2) extends additionally to test functions in
Lemma A.1.
Let be a distributional solution to Equation(A.1)
(A.1)
(A.1) satisfying Assumption 3. Then
for any
the function
is of locally bounded variation and therefore has a right continuous representative with a countable number of jump discontinuities.
There exists a path
which is right continuous with respect to the weak-star topology on
and such that
as elements of
for almost every
Proof.
Since is a positive distribution, it is given by some Radon measure ν on
We have
(A.3)
(A.3)
which we will use to deduce weak time regularity for u.
Consider a test function The function
(A.4)
(A.4)
has distributional derivative
(A.5)
(A.5)
By Assumption 3, u, β and are all locally integrable functions on
and so in particular on
Thus the distributional derivative
defined in Equation(A.5)
(A.5)
(A.5) is a Radon measure on
and so the path Equation(A.4)
(A.4)
(A.4) is of locally bounded variation.
It follows that has a unique right-continuous version. That is, there exists a set
of full measure and a right continuous function
such that
for all
The function
satisfies
for all
Now consider (independent of time). The path
has time derivative
For each compact subset we define the following Radon measure on
for
Borel,
For the right-continuous versions we have, for all and all
with support contained in K,
(A.6)
(A.6)
Using these estimates, it is possible to construct a right continuous version of u: that is, a path that is right continuous with respect to the weak-star topology.
Such a construction is classical, but because of the lack of a precise reference in our context, we sketch the main ideas here. Take a countable dense set there is a full measure set
such that
for all
and all
and moreover
(the latter is true for almost all t since u is
). Then
defines a bounded linear functional on Z, for all
The estimate Equation(A.6)
(A.6)
(A.6) can be used to show that this is in fact true for all
The resulting functional
extends by density to a continuous linear functional on
Then the estimate Equation(A.6)
(A.6)
(A.6) can be used to prove that
is right continuous for all
not just on Z. □
Next, we construct the extension of to the boundaries
Definition A.2
(Transport shift). Let . The operator
is defined by
Remark A.3
(Group property). For any
Lemma A.4.
Let u be a solution to Equation(A.1)(A.1)
(A.1) and let
be its right continuous representative, obtained in Lemma A.1. Let
be non-negative. Consider the function
. Then
As t tends to
either tends to a finite limit or to positive infinity.
As t tends to
either tends to a finite limit or to negative infinity.
Proof.
Observe that
It follows that
Then the negative part of the time derivative satisfies
Thus can be written as the difference of monotone functions, where the decreasing part is absolutely continuous on
and can be extended to finite limits at the endpoints. By monotonicity, the increasing part either has a finite limit at t = T, or tends to positive infinity; similarly, at t = 0 it either has a finite limit or tends to negative infinity. □
Definition A.5
(Weak traces). For any , let
(A.7)
(A.7)
These define linear maps from to
in the case of u0, and
in the case of uT.
We now suppose that, in addition to the weak Hamilton-Jacobi inequality in the interior Equation(A.2)(A.2)
(A.2) , u satisfies the following: for all non-negative test functions
(A.8)
(A.8)
In our setting, we will have that is a given function whose positive part satisfies
In this case, we show below that the time trace uT, enjoys some more properties.
Lemma A.6.
If u satisfies Equation(A.8)(A.8)
(A.8) with
, then uT as defined in Equation(A.7)
(A.7)
(A.7) is a bounded linear functional on
and
in the sense of distributions: that is, for all
non-negative,
In particular, we have that does not occur for any
Remark A.7.
Since then is a positive distribution, if
then we in fact have that uT is represented by a signed Radon measure with absolutely continuous positive part.
Proof of Lemma A.6.
In what follows, we will use the right continuous representative of u constructed in Lemma A.1. By the abuse of notation, we write simply u for Fix
non-negative. For each
small consider a smooth, non-negative test function
chosen such that
for all
and the derivative
satisfies
Note that as a consequence of the fundamental theorem of calculus, one has
We define the following non-negative test function
Substituting this choice of into Equation(A.8)
(A.8)
(A.8) , we obtain
(A.9)
(A.9)
where
Note that since β and
are locally integrable and
are bounded in
uniformly in
We exclude the possibility that Indeed, if this occurs, then for any M > 0, there exists
such that for any
Then by bounding the left hand side of inequality Equation(A.9)(A.9)
(A.9) from below we obtain that for any
Taking the limit gives
Since this holds for any M > 0, we derive a contradiction. Thus—using also Lemma A.4—uT is in fact a linear map from to
We note also that the map
extends to a function that is bounded and continuous (from the left) at t = T.
Next, we show that as functionals on
We have
(A.10)
(A.10)
where
For the second term here we have
which converges to zero as
since the trajectory
is bounded near t = T. Thus
Taking the limit in inequality Equation(A.10)
(A.10)
(A.10) , we conclude that
Since is a positive linear functional on
it is bounded, and therefore uT is also a bounded linear functional on
□
Corollary A.8.
If u satisfies Equation(A.8)(A.8)
(A.8) with
then, in the notation of Equationequation (A.3)
(A.3)
(A.3) , the measure ν extends to a finite Radon measure on
given by
Proof.
We show that, for any non-negative test function
(A.11)
(A.11)
It suffices to prove Equation(A.11)(A.11)
(A.11) for test functions of the form
where
and
Then
It follows that (once again using the right continuous version of u), for t < T,
Taking the limit by definition of uT,
□
We now discuss the trace of u at t = 0: u0 as defined in Definition A.7. is defined for all
Our aim is to give a meaning to the quantity
which appears in the definition of the functional
In the case where
this is straightforward, noting that we allow the possible value
We now consider the more general case where
Lemma A.9.
Assume that, for all . Then u0 is represented by a Radon measure on
. Furthermore, the positive part
has the property that
Proof.
Let be non-negative. Since
we have
The right hand side is linear in and satisfies
here KT denotes the set
where K is the support of and
is the set
Thus S defines a bounded linear functional on In particular it is a distribution; moreover, it is represented by a signed Radon measure.
Observe next that is a positive linear functional on
and thus bounded and a distribution. By positivity it is given by a Radon measure ν0 on
We deduce that
That is, u0 is a signed Radon measure.
Moreover, from the definition of the Hahn-Jordan decomposition we have the following estimate for the positive part:
Let be an increasing sequence of functions such that
converges to m0 as
Since
we conclude that
is finite. □
Based on the previous lemma, we make the following definition.
Definition A.10.
We define as follows:
If there exists
such that
then we define
Otherwise, let
be an increasing sequence of functions such that
converges to m0 as
and define
This is well-defined (allowing for the possible value ) by Lemma A.9.
Lemma A.11.
Suppose that the assumptions of Lemma A.6 hold and suppose in addition that(A.12)
(A.12)
holds for all , where
is also given. Then for the trace u0 of the right continuous version of u we have
and in particular
for any
If in addition we suppose that is such that
is meaningful and finite, then
is finite and
Proof.
The proof of this result follows the same lines as the proof of Lemma A.6, so we point out only the main differences. Let u stand for the right continuous representative constructed in Lemma A.1. Fix non-negative. For each
small consider a smooth, non-negative test function
chosen such that
for all
and the derivative
satisfies
Note that as a consequence of the fundamental theorem of calculus, one has
We define the following non-negative test function
Substituting this choice of into Equation(A.12)
(A.12)
(A.12) , we obtain
where
As before, we note that We exclude the possibility that
For this, we rewrite the previous inequality as
and use the same arguments as when proving
in the proof of Lemma A.6.
Therefore, u0 defines a linear map on Having this, we can show the inequality
in the same way as corresponding inequality in Lemma A.6.
Now, using the Definition A.10, is meaningful, having also the possibility that it is
However, if the additional assumption that
is finite takes place, taking a an increasing sequence of test functions, we find that
so clearly, the latter term cannot be
□
Through similar arguments it is possible to justify the existence of weak time traces for competitors m in Problem 3.3, thereby giving meaning to the initial value problem
Recall that in Remark 3.4 we established that, in the cases of interest to us, there exists a function such that
and so we may assume that m is a distributional solution of the following equation:
with
This setting is much more standard since here the time derivatives will be in
for any
rather than measures, that is, we expect absolutely continuous rather than right continuous trajectories. Moreover we can work on the whole space
rather than only the reachable set
Deducing that m has a narrowly continuous representative is essentially an application of [Citation56, Lemma 8.1.2]. However, since we do not necessarily have
due to the unbounded drift v, we cannot immediately apply this lemma. Below we briefly sketch the adaptation to our case.
Lemma A.12.
(See [Citation56, Lemma 8.1.2]). Let be a non-negative function satisfying
in the sense of distributions on , where V is given, such that
Then there exists a continuous curve such that
for almost all
. Thus
is well-defined as an element of
(or in fact, by positivity, a Radon measure).
Furthermore, if is a probability measure, then
extends uniquely to a narrowly continuous curve in the space of probability measures, i.e.
Proof.
Since for any compact set
we have
It follows that, as in the proof of [Citation56, Lemma 8.1.2], we may select a dense subset of
and take a version
of mt such that
is continuous with respect to t for all n and
for almost all
and a define a unique weak-* continuous extension of
to
Thus
is well-defined as the unique element of
satisfying
Moreover, since is non-negative, in fact
is a Radon measure on
for all
Furthermore, it follows from the continuity of in the weak-* sense of
and the fact that
is locally finite, that, for any (N.B. now time-dependent)
the path
is continuous. Thus we may also use the final argument from [Citation56, Lemma 8.1.2] (similar to our argument for the time traces at the boundary in Lemma A.6) to prove the following equality (c.f. [Citation56, EquationEquation (8.1
(8.1)
(8.1) .4)]): for any
and any
(A.13)
(A.13)
Next we wish to show that, if is a probability measure, then
is a probability measure for all
If this is the case, then we may apply [Citation56, Remark 5.1.6]—if
in the sense of distributions as n tends to infinity, then this convergence also holds in the narrow sense—to deduce that
is a narrowly continuous path in the space of probability measures, as desired.
To do this we use the argument of Lemma A.4 to avoid the need for to be integrable. First, fix a sequence
of smooth, compactly supported functions, approximating the constant function 1 in a monotone limit as R tends to infinity—that is, let ζR satisfy the assumptions given in EquationEquation (5.2)
(5.2)
(5.2) . We note in particular that
for some constant C > 0 independent of R. Then, for each
consider the test function
(recall
from Definition A.2), which satisfies
Using Equation(A.13)(A.13)
(A.13) with
and
we find that
We observe that, for all
Thus (since for almost all t)
The right hand side tends to zero as R tends to infinity, since Moreover, since
is a probability measure and
increases monotonically to 1 pointwise as R tends to infinity, it follows that
Finally, since is a Radon measure and
is a monotone limit,
That is, is a probability measure for all
This completes the proof. □
Appendix B.
Truncations and maxima
Given a distributional solution to the Hamilton-Jacobi inequality
(B.1)
(B.1)
in the sense of Definition 3.9, we show that the truncations of u from below, that is, the functions
for some l < 0, satisfy a similar inequality. In a similar vein, we also show that given u1 and u2 both satisfying Equation(B.1)
(B.1)
(B.1) , their maximum satisfies the same inequality Equation(B.1)
(B.1)
(B.1) .
Lemma B.1.
Let satisfy Equation(B.1)
(B.1)
(B.1) in the sense of distributions. Assume that
and
Then
satisfies
A similar result holds for the truncation Moreover it suffices to consider the case l = 0.
Lemma B.2.
Let u1, u2 satisfy Assumption 3 and Equation(B.1)(B.1)
(B.1) . Then
also satisfies Equation(B.1)
(B.1)
(B.1) .
The result here is in the spirit of renormalization [Citation60]. Bouchut [Citation61, Theorem 1.1] proved a chain rule for the kinetic transport operator, i.e. the identity
(B.2)
(B.2)
that applies when h is a Lipschitz function and
However, since in our case
may only be a measure, we are not able to use this result directly, or indeed prove a chain rule with equality as in EquationEquation (B.2)
(B.2)
(B.2) . Nevertheless, the ideas of the proofs in [Citation24, Citation61] can be used to obtain the inequality that is sufficient for our case.
The argument proceeds in several steps.
B.1. Extension
We define the following time shift and extension of u on the time interval for
Then, defining
satisfies the following inequality in the sense of distributions on
(see the similar construction in [Citation21, Section 6.3]):
B.2. Fenchel’s inequality
Since L is the Fenchel conjugate of H, for any continuously differentiable vector field we have
B.3. Regularisation
Fix non-negative, symmetric, unit mass mollifiers and
Assume that θ is supported on the set
Then define, for
Then define the full mollifier by
Then the regularization satisfies the following inequality, in a pointwise sense on
where
and
while
denotes the commutator
With the choice this error converges to zero in
by Lemma 5.3 and [Citation60].
B.4. The maximum function
We fix a smooth approximation of the functions and
First, for each
we fix a smooth function
approximating
in such a way that
and
for all
and
for all x and
for x > 0. Note in particular that then
for all
and
so that
converges pointwise to the function
as α tends to zero.
We similarly define an approximation of the maximum function by
Observe that satisfies
for i = 1, 2, and
B.5. Inequality for the truncations and maximum
Since is non-negative,
(B.3)
(B.3)
Thus, since and
is smooth in all variables, applying the usual chain rule
Similarly, given two subsolutions u1 and u2,
Thus
B.6. Limits
We now take the limit as the smoothing parameters tend to zero in the previous inequalities. This procedure yields the proofs of the Lemmas B.1 and B.2. We continue to choose to ensure convergence of the commutator. We detail the procedure in the case of the truncation Equation(B.3)
(B.3)
(B.3) ; the case of the maximum function is similar.
Proof of Lemma B.1.
We first test the inequality Equation(B.3)(B.3)
(B.3) with an arbitrary non-negative smooth function
We fix an extension of ζ to a function
and consider integrating over
For all
small enough that the support of ζ is contained in
We have used that Since
we may estimate the boundary term from above to obtain
Since
converge respectively to these strongly in
by standard results on convolutions. We have already noted that
converges to zero in
We therefore also obtain pointwise convergence along a subsequence. Similarly,
converges to ζ pointwise since ζ is smooth. From this we obtain convergence of all terms, by continuity of
and
and applying dominated convergence. Hence we obtain the following inequality:
The convergences as all follow by dominated convergence: for example, since
and ζ has support contained in
we have
A similar argument is used for the term involving βT.
For the remaining term, use that (the bound being uniform in
), and both β and
are in
by assumption. Then
Finally, taking a sequence of vector fields a converging in to
we conclude that
that is, the following holds in the sense of distributions:
□