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Abstract
A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.
1. Introduction
The aim of this article is to derive and analyze the following nonlocal porous-medium equation:
(1)
(1)
where
and
is a nondecreasing function satisfying
This model describes a particle system that evolves according to a continuity equation for the density
with velocity
The velocity is assumed to be the gradient of a potential, which expresses Darcy’s law. The pressure P is related to the density in a nonlinear and nonlocal way through
The nonlocal operator
can be written as a convolution operator with a singular kernel,
(2)
(2)
where
and Γ denotes the Gamma function [Citation1, theorem 5].
If s = 0, we recover the porous-medium equation (for nonnegative solutions), while the case s = 1 was investigated in [Citation2, Citation3] with f(u) = u for the evolution of the vortex density in a superconductor. Related models (with f(u) = u) appear in the dynamics of dislocations (line defects) in crystals [Citation4, (1.5)]. Other applications include particle systems with long-range interactions [Citation5, subsection 6.2]. The case corresponds to long-range repulsive interactions. This model, still with f(u) = u, was investigated in [Citation4], but a mathematical justification is missing. In this article, we provide a rigorous derivation from an interacting particle system for general functions f(u). In this way, we aim to contribute to the understanding of mean-field limits involving nonquadratic nonlinearities.
EquationEquation (1)(1)
(1) was first analyzed in [Citation6] with f(u) = u for nonnegative solutions and in [Citation7] with
(m > 1) for sign-changing solutions. The nonnegative solutions have the interesting property that they propagate with finite speed, which is not common in other fractional diffusion models [Citation6, Citation8]. EquationEquation (1)
(1)
(1) was probabilistically interpreted in [Citation9], and it was shown that the probability density of a so-called random flight process is given by a Barenblatt-type profile. Previous mean-field limits leading to (Equation1
(1)
(1) ) were concerned with the linear case f(u) = u only; see [Citation10] (using the technique of [Citation11]) and [Citation12] (including additional diffusion as in (Equation7
(7)
(7) ) below). In [Citation13], EquationEq. (1)
(1)
(1) (with f(u) = u) was derived in the high-force regime from the Euler–Riesz equations, which can be derived in the mean-field limit from interacting particle systems [Citation14]. A direct derivation from particle systems with Lévy noise was proved in [Citation15] for cross-diffusion systems, but still with f(u) = u. Up to our knowledge, a rigorous derivation of (Equation1
(1)
(1) ) from stochastic interacting particle systems for general nonlinearities f(u) like power functions is missing in the literature. In this article, we fill this gap.
1.1. Problem setting
EquationEquation (1)(1)
(1) is derived from an interacting particle system with N particles, moving in the whole space
Because of the singularity of the integral kernel and the degeneracy of the nonlinearity, we approximate (Equation1
(1)
(1) ) using three levels. First, we introduce a parabolic regularization adding a Brownian motion to the particle system with diffusivity
and replacing f by a smooth approximation
Second, we replace the interaction kernel
by a smooth kernel
with compact support, where
Third, we consider interaction functions
with
which approximate the delta distribution. (We refer to Subsection 1.3 for the precise definitions.)
The particle positions are represented on the microscopic level by the stochastic processes evolving according to
(3)
(3)
where the convolution has to be understood with respect to xi,
are independent d-dimensional Brownian motions defined on a filtered probability space
and ξi are independent identically distributed random variables in
with the same probability density function
(defined in (Equation12
(12)
(12) ) below).
The mean-field-type limit is performed in three steps. First, for fixed system (Equation3
(3)
(3) ) is approximated for
on the intermediate level by
(4)
(4)
where
is the probability density function of
and a strong solution to
(5)
(5)
System (Equation4
(4)
(4) ) is uncoupled, since
depends on N only through the initial datum.
Second, passing to the limit in the intermediate system leads on the macroscopic level to
(6)
(6)
where
is the density function of
and a weak solution to
(7)
(7)
We perform the limits and
simultaneously. In this limit, we use the logarithmic scaling
for some
between the strength of interaction β and the number of particles N. This can be viewed as a moderately interacting particle system. For the smoothing parameter ζ of the singularity from
we can even allow an algebraic dependence on N, i.e.,
for some
see Theorem 2 for details. Our approach also implies the two-step limit but leading to weak convergence only, compared to the convergence in expectation obtained in Theorem 3.
Third, the limit is performed on the level of the diffusion equation, based on a priori estimates uniform in σ and the div-curl lemma.
The main result of this article is that the particles of system (Equation3(3)
(3) ) become independent in the limit with a common density function that is a weak solution to (Equation1
(1)
(1) )–(Equation2
(2)
(2) ).
1.2. State of the art
We already mentioned that the existence of weak solutions to (Equation1(1)
(1) ) with f(u) = u was proved first in [Citation6]. The convergence of the weak solution to a self-similar profile was shown by the same authors in [Citation16]. The convergence becomes exponential, at least in one space dimension, when adding a confinement potential [Citation17]. EquationEquation (1)
(1)
(1) with f(u) = u was identified as the Wasserstein gradient flow of a square fractional Sobolev norm [Citation18], implying time decay as well as energy and entropy estimates. The Hölder regularity of solutions to (Equation1
(1)
(1) ) was proved in [Citation19] for f(u) = u and in [Citation20] for
and
In the literature, related equations have been analyzed too. EquationEquation (1)(1)
(1) for f(u) = u and the limit case s = 1 was shown in [Citation21] to be the Wasserstein gradient flow on the space of probability measures, leading to the well-posedness of the equation and energy-dissipation inequalities. The existence of local smooth solutions to the regularized EquationEq. (Equation7
(7)
(7) )
(7)
(7) are proved in [Citation22]. The solutions
with
in
propagate with finite speed if and only if
[Citation8]. The existence of weak solutions to this equation with
and n > 0 is proved in [Citation23] (in bounded domains). While (Equation1
(1)
(1) ) has a parabolic-elliptic structure, parabolic-parabolic systems have been also investigated. For instance, the global existence of weak solutions to
and
where
was shown in [Citation24]. In [Citation25], the algebraic decay toward the steady state was proved in the case β = 2. We also mention that fractional porous-medium equations of the type
in
have been studied in the literature; see, e.g., [Citation26]. Compared to (Equation1
(1)
(1) ), this problem has infinite speed of propagation. For a review and comparison of this model and (Equation1
(1)
(1) ), we refer to [Citation27].
There is a huge literature concerning mean-field limits leading to diffusion equations, and the research started already in the 1980s; we refer to the reviews [Citation28, Citation29] and the classical works of Sznitman [Citation30, Citation31]. Oelschläger proved the mean-field limit in weakly interacting particle systems [Citation32], leading to deterministic nonlinear processes, and moderately interacting particle systems [Citation33], giving porous-medium-type equations with quadratic diffusion. First, investigations of moderate interactions in stochastic particle systems with nonlinear diffusion coefficients were performed in [Citation34]. The approach of moderate interactions was extended in [Citation35, Citation36] to multispecies systems, deriving population cross-diffusion systems. Reaction-diffusion equations with nonlocal terms were derived in the mean-field limit in [Citation37]. The large population limit of point measure-valued Markov processes leads to nonlocal Lotka–Volterra systems with cross diffusion [Citation38]. Further references can be found in [Citation12, subsection 1.3].
Compared to previous works, we consider a singular kernel and derive a partial differential equation without Laplace diffusion by taking the limit
The authors of [Citation39] derived the viscous porous-medium equation by starting from a stochastic particle system with a double convolution structure in the drift term, similar to (Equation4
(4)
(4) ). The main difference to our work is that (besides different techniques for the existence and regularity of solutions to the parabolic problems) we consider a singular kernel in one part of the convolution and a different scaling for the approximating regularized kernel
where
is a
cutoff function (see Subsection 1.3 for the exact definition), in comparison to the interaction scaling
The two different scalings β and ζ allow us to establish a result, for which the kernel regularization on the particle level does not need to be of logarithmic type but of power-law type only.
1.3. Main results and key ideas
We impose the following hypotheses:
(H1) Data: Let
(H2) satisfies
in
and
(H3) Nonlinearity: is nondecreasing,
and
for u > 0 is strictly convex.
Let us discuss these assumptions. We assume that the case d = 1 can be treated if s < 1∕2; see [Citation6]. Extending the range of s to s < 0 leads to the fractional (higher-order) thin-film equation, which is studied in [Citation40]. The case
may be considered too, since it yields better regularity results; we leave the details to the reader. Conversely, the case
is more delicate since the multiplier in the definition of
using Fourier transforms does not define a tempered distribution. The case
for
(with a logarithmic Riesz kernel) was analyzed in [Citation10]. We need the moment bound for the initial datum
to prove the same moment bound for
which in turn is used several times, for instance to show the entropy balance and the convergence
as
in the sense of
The monotonicity of f and the strict convexity of
are needed to prove the strong convergence of
which then allows us to identify the limit of
An example of a function satisfying Hypothesis (H3) is
with
Our first result is concerned with the existence analysis of (Equation1(1)
(1) ). This result is needed to prove the main theorem below. We write
for the
norm and define the so-called entropy density
by
Theorem 1
(Existence of weak solutions to (Equation1(1)
(1) )). Let Hypotheses (H1)–(H3) hold. Then, there exists a weak solution
to (Equation1
(1)
(1) ) satisfying (i) the regularity
(ii) the weak formulation
(8)
(8) for all
and T > 0, (iii) the initial datum
in the sense of
, and (iv) the following properties for t > 0:
Mass conservation:
Dissipation of the
norm:
Moment estimate:
Entropy inequality:
Note that the Hardy–Littlewood–Sobolev-type inequality (B3) (see Appendix B) implies that
such that
and the weak formulation (Equation8
(8)
(8) ) is well defined.
The main ideas of the proof of Theorem 1 are as follows. A priori estimates for strong solutions to the regularized EquationEq. (Equation7
(7)
(7) )
(7)
(7) are derived from mass conservation, the entropy inequality, and energy-type bounds. The energy-type bound allows us to show, for sufficiently small
that the
norm of
is bounded by the
norm of
up to some factor depending on the moment bound for
The existence of a strong solution
is proved by regularizing (Equation7
(7)
(7) ) in a careful way to deal with the singular kernel. The regularized equation is solved locally in time by Banach’s fixed-point theorem. Entropy estimates allow us to extend this solution globally in time and to pass to the de-regularization limit. The second step is the limit
in (Equation7
(7)
(7) ). Since the bounds only provide weak convergence of (a subsequence of)
the main difficulty is the identification of the nonlinear limit
This is done by applying the div-curl lemma and exploiting the monotonicity of f and the strict convexity of
[Citation41].
We already mentioned that the existence of local smooth solutions to (Equation7
(7)
(7) ) has been proven in [Citation13]. However, we provide an independent proof that allows for global strong solutions and that yields a priori estimates needed in the mean-field limit.
Our second and main result is the propagation of chaos, which shows a mean-field-type convergence of the particle system (Equation3(3)
(3) ) to a solution of (Equation1
(1)
(1) ). To define our particle system properly, we need some definitions. Introduce the smooth approximation
(9)
(9)
where the mollifier
for
is given by
and
satisfies
while the cutoff function
satisfies
in
and
for
Then, thanks to
we have
The cutoff function guarantees that the derivatives
are bounded and compactly supported for all
Furthermore, it holds that
In a similar way, we introduce the mollifier function for
and
by
(10)
(10)
Let us define the cutoff version of the singular kernel
by
(11)
(11)
Then, the regularized kernel
is given by
where
Let the cutoff function
satisfy
in
and
for
Then, we define the regularized initial datum for
by
(12)
(12)
This definition guarantees the mass conservation since
see Subsection 2.1.
Note that our particle system (Equation3(3)
(3) ) depends on 4 parameters:
denotes the number of particles,
models the strength of interaction between the particles,
describes the regularization of the singular kernel
and
is a measure of the additional diffusion. The quantities
are regularization parameters needed to overcome the singularity of the kernel
and the (possible) degeneracy of the nonlinearity f.
In the limit we prove the following propagation-of-chaos result.
Theorem 2
(Propagation of chaos). Let and
for some constants C1,
, and let
be the joint distribution of
for
and
. Then, there exists a subsequence in σ such that
where the limit is locally uniform in t, and the measure
is absolutely continuous with respect to the Lebesgue measure with the probability density function
that is a weak solution to (Equation1
(1)
(1) ).
It is well known (see, e.g., [Citation31]) that the result of Theorem 2 implies the weak convergence of the empirical measure associated to the particle system (Equation3(3)
(3) ) toward the deterministic measure
i.e.,
for a subsequence in σ. Furthermore, Theorem 2 shows that at any time t > 0, in the limit
any finite selection of k particles in (Equation3
(3)
(3) ) becomes independent with limiting distribution
If EquationEq. (1)
(1)
(1) was uniquely solvable, we would obtain the convergence of the whole sequence in σ. Unfortunately, the regularity of the solution ρ to (Equation1
(1)
(1) ) is too weak to conclude the uniqueness of weak solutions. Up to our knowledge, none of the known methods, such as [Citation42, Citation43], seem to be applicable to EquationEq. (Equation1
(1)
(1) )
(1)
(1) .
Theorem 2 is proved in two steps: First, we show (strong) error estimates between particles of systems (Equation3(3)
(3) ) and (Equation6
(6)
(6) ), respectively; see Proposition 3 below. Second, we show the weak convergence of (a subsequence of)
to a solution ρ to (Equation1
(1)
(1) ); see Corollary 13.
Proposition 3
(Error estimate for the stochastic system). Let and
be the solutions to (Equation3
(3)
(3) ) and (Equation6
(6)
(6) ), respectively. We assume that
for some constant
. Let
and
. Then, there exist constants
, depending on σ and δ, and
, depending on σ and T, such that if
then,
The proposition is proved by estimating the differences
and applying the triangle inequality. For the first difference, we estimate expressions like
for appropriate functions u and
for
in terms of negative powers of β (here,
denotes the kth-order partial derivatives). Using properties of Riesz potentials, in particular Hardy–Littlewood–Sobolev-type inequalites (see Lemmas 22 and 23), we show that for some
(i = 1, 2, 3),
By applying the Gronwall lemma and choosing a logarithmic scaling for β and an algebraic scaling for ζ with respect to N, we infer that
for some
For the second difference E2, we need the estimates
(Lemma 21), and
(Proposition 14), recalling that
The proof of these estimates is very technical. The idea is to apply several times fractional Gagliardo–Nirenberg inequalities that are proved in Appendix B and Hardy–Littlewood–Sobolev inequalities that are recalled in Lemmas 22 and 23. Then, after suitable computations,
and we conclude with Gronwall’s lemma that
The article is organized as follows. The existence of global nonnegative weak solutions to (Equation1(1)
(1) ) is proved in Section 2 by establishing an existence analysis for (Equation7
(7)
(7) ) and performing the limit
Some uniform estimates for the solution
to (Equation5
(5)
(5) ) and for the difference
are shown in Section 3. Section 4 is devoted to the proof of the error estimate in Theorem 3 and the propagation of chaos in Theorem 2. In Appendices A–C we recall some auxiliary results and Hardy–Littlewood–Sobolev-type inequalities, prove new variants of fractional Gagliardo–Nirenberg inequalities, and formulate a result on parabolic regularity.
1.3.1. Notation
We write for the
or
norm with
The ball around the origin with radius R > 0 is denoted by BR. The partial derivative
is abbreviated as
for
and
denotes a partial derivative of order
where
is a multi-index. The notation
refers to the kth-order tensor of partial derivatives of order
In this situation, the norm
is the sum of all Lp norms of partial derivatives of u of order k. Finally, C > 0,
etc. denote generic constants with values changing from line to line.
2. Analysis of EquationEq. (Equation1
(1)
(1) )
(1)
(1)
In this section, we prove the existence of global nonnegative weak solutions to (Equation1(1)
(1) ). We first prove the existence of a solution
to (Equation7
(7)
(7) ) by a fixed-point argument, and then, perform the limit
In Subsection 2.1, we prove some basic estimates for a strong solution
to (Equation7
(7)
(7) ). Entropy and moment estimates as well as higher-order estimates are derived in Subsections 2.2 and 2.3, respectively. The existence of a unique strong solution to (Equation7
(7)
(7) ) is proved in Subsection 2.4 using a regularized version of (Equation7
(7)
(7) ) and Banach’s fixed-point theorem. The strong
limit
is performed in Subsection 2.5 using the div-curl lemma. Finally, Subsection 2.6 is concerned with the proof of a time-uniform weak
limit of
which is needed in the proof of Proposition 3. Recall definition (Equation12
(12)
(12) ) of the number
which is stated in (iv) below.
Proposition 4.
Let Hypotheses (H1)–(H3) hold. Then, for all , there exists a unique weak solution
to (Equation7
(7)
(7) ) satisfying (i) the regularity
(ii) the weak formulation of (Equation7
(7)
(7) ) with test functions
, (iii) the inital datum
in
, and (vi) the following properties for t > 0, which are uniform in σ for sufficiently small
:
Mass conservation:
Dissipation of the
norm:
Moment estimate:
Entropy inequality:
Additionally, for any T > 0, , and
, there exists C > 0, depending on T, σ, p, and q, such that
i.e.,
is even a strong solution to (Equation7
(7)
(7) ) and
for
2.1. Basic estimates for ![](//:0)
![](//:0)
We prove a priori estimates in Lp spaces and an energy-type estimate. Let and let
be a nonnegative strong solution to (Equation7
(7)
(7) ). Integration of (Equation7
(7)
(7) ) in
and the definition of
immediately yield the mass conservation
(13)
(13)
Lemma 5
(Energy-type estimate). Let be convex and let
. Then,
(14)
(14) where
for
and
is defined after (Equation2
(2)
(2) ).
Proof.
First, we assume that is additionally bounded. Then,
is an admissible test function in the weak formulation of (Equation7
(7)
(7) ), since
It follows from definition (B1) of the fractional Laplacian and integration by parts that
A symmetrization of the last integral yields (Equation14
(14)
(14) ).
In the general case, we introduce for k > 0. Then,
is bounded and (Equation14
(14)
(14) ) follows for F replaced by Fk. The result follows after taking the limit
using monotone convergence. □
We need a bound on defined in (Equation12
(12)
(12) ), to derive uniform
bounds for
Lemma 6
(Bound for ). There exists C > 0 such that, for sufficiently small
Proof.
By Young’s convolution inequality (Lemma 19), we have
which shows that
To prove the upper bound, we use the triangle inequality
:
Using the property
for the second term on the right-hand side and
for the third term, we find that
Because of
for
we obtain
which proves the lemma. □
Lemma 7
(Bounds for ). The following bounds hold:
(15)
(15)
(16)
(16) where (Equation15
(15)
(15) ) holds for sufficiently small
Lemma 7 and mass conservation imply that is bounded for all t > 0 and
Observe that
as
So, if
a.e., the dissipation of the
norm follows, as stated in Theorem 1 (iv).
Proof.
The convexity of F shows that G, defined in Lemma 5, is nondecreasing. Therefore, and
We choose a convex function
such that F(u) = 0 for
F(u) > 0 for
and satisfying
for
Then,
Consequently,
for t > 0, showing the
bound. Finally, choosing
in Lemma 5, the
estimate follows. □
2.2. Entropy and moment estimates
We need a fractional derivative estimate for which is not an immediate consequence of Lemma 5. To this end, we define the entropy density
Lemma 8
(Entropy balance). It holds for all t > 0 thatIn particular, for all T > 0, there exists C > 0 such that
(17)
(17)
Proof.
The idea is to apply Lemma 5. Since we cannot use the lemma directly. Instead, we apply it to the regularized function
where
Choosing
in Lemma 5 gives
(18)
(18)
where
for
Step 1: Estimate of The pointwise limit
holds a.e. in
as
We observe that for all
while for all u > 1, since
in
The last inequality follows after integration of
in
Therefore, since
a.e. in
we find that
where
is arbitrary, and consequently, because of mass conservation,
(19)
(19)
Step 2: Estimate of Let
and
Then, by Young’s inequality,
since the choice of θ guarantees that
so
To control the right-hand side, we need to bound a suitable moment of
For this, we use the test function in the weak formulation of (Equation7
(7)
(7) ), where
is a cutoff function with the properties
and k > 1 is arbitrary. We find that
Since
and
in
the terms involving
and
are bounded in
It follows from the choice of ξk that
Thus, taking into account the assumption on
and mass conservation,
Next, we apply the Hardy–Littlewood–Sobolev inequality (see Lemma 22) and the Hölder inequality and use the fact that
is bounded in any
:
for all
Since
for
k > 1, and
a.e. in
as
we deduce from monotone convergence that in the limit
for all
This proves that
Step 3: A priori estimate. Inserting the previous estimate into (Equation19(19)
(19) ) leads to
We integrate (Equation18
(18)
(18) ) in time and use the previous estimate:
We wish to pass to the limit
in the previous inequality. We deduce from dominated convergence that
a.e. in
The integrand of the second term on the left-hand side is nonnegative, and we obtain from Fatou’s lemma that
(20)
(20)
By the integral representation of the fractional Laplacian,
the last term in (Equation20
(20)
(20) ) can be absorbed for sufficiently small
by the second term on the left-hand side. This leads to the estimate
Thus, we can pass to the limit
in (Equation18
(18)
(18) ) giving the desired entropy balance. Finally, bound (Equation17
(17)
(17) ) follows from the definition of the
norm and the facts that
since
is locally Lipschitz continuous,
and
is bounded both in
and
independently of σ. □
Lemma 9
(Moment estimate). It holds thatwhere C > 0 depends on T and the
norms of
and
Proof.
For the following computations, we would need to use cutoff functions to make the calculations rigorous. We leave the details to the reader, as we wish to simplify the presentation. Let Since
by assumption, we can compute
(21)
(21)
By Young’s inequality and mass conservation, we have
It follows from (Equation17
(17)
(17) ) that
is bounded in
In particular, because of the Sobolev embedding
Furthermore, using
Young’s inequality, and the property
(recall that
)
Thus, we infer from (Equation21
(21)
(21) ) and the identity
that
and Gronwall’s lemma concludes the proof. □
2.3. Higher-order estimate
We need some estimates in higher-order Sobolev spaces.
Proposition 10
(Higher-order regularity). Let T > 0, and
. Then, there exists C > 0, depending on T, σ, p, and q, such that
Proof.
Step 1: Case s > 1∕2. If s > 1∕2, then, does not involve any derivative of
Thus,
for
and Lemma 26 in Appendix C implies that
Iterating the argument leads to the conclusion. Thus, in the following, we can assume that
Step 2: Estimate of in
We claim that w can be estimated in
for any
Then, by Lemma 26,
We use the
bound for
the fractional Gagliardo–Nirenberg inequality (Lemma 24), and Young’s inequality to find that
where
is arbitrary. By estimate (C2) in Lemma 26,
Choosing
sufficiently small shows the claim.
Step 3: Estimate of in
We use Hölder’s inequality with
to obtain
By the fractional Gagliardo–Nirenberg inequality (Lemma 25 with
and Lemma 24 with s replaced by
) and Young’s inequality, it follows that
where
is arbitrary. Taking the
norm of the previous inequality and observing that
(because of
), it follows that
Lemma 26 and Step 2 (
) show that
Choosing
sufficiently small, this yields
and
We deduce from Lemma 20, applied to
that
for any
(At this point, we need the restriction
)
Step 4: Higher-order regularity. To improve the regularity of we differentiate (Equation7
(7)
(7) ) in space. Recall that
Then,
(22)
(22)
We estimate the right-hand side term by term. Let
First, by Hölder’s inequality with
and the fractional Gagliardo–Nirenberg inequality (Lemma 24),
The second and third term on the right-hand side of (Equation22
(22)
(22) ) can be treated in a similar way, observing that
The last term is estimated according to
and the last expression can be absorbed by the corresponding estimate of
from the left-hand side of (Equation22
(22)
(22) ). Then, we deduce from Lemma 26 that
for all p > 1 and Lemma 20, applied to
yields
for all
Next, if we use the second inequality in Lemma 24 and argue similarly as before. This finishes the proof. □
Lemma 11.
Under the assumptions of Proposition 10, for every , there exists a constant
, depending on σ, such that
The embedding for q > d yields a bound for
in
Proof.
We first prove the bound in By differentiating (Equation7
(7)
(7) ) twice in space, estimating similarly as in Step 4 of the previous proof, and using the regularity results of Proposition 10, we can show that
is bounded in
for any
It remains to show the bound for
In view of mass conservation and Gagliardo–Nirenberg–Sobolev’s inequality, it suffices to show a bound for
in
To this end, we define the weights
for
and test EquationEq. (Equation7
(7)
(7) )
(7)
(7) for
with
Then,
Arguing as in Step 4 of the previous proof, we can find a bound in
for vn. Indeed, we can proceed by induction over n, since the additional terms in In can be controlled by Sobolev norms of v0,…,vn−1. The definition of
implies that
for every
Then, choosing n > d yields, for
that
We conclude from
that
This proves the desired bound. □
2.4. Existence of solutions to (Equation7
(7)
(7) )
We show that the regularized EquationEq. (Equation7(7)
(7) )
(7)
(7) possesses a unique strong solution
Step 1: Existence for an approximated system. Let T > 0 arbitrary, define the spaces
and consider the mapping
(23)
(23)
where
is a regularized version of
defined by
and
is given by
The regularization with parameter η is needed for the entropy estimates.
Lemma 12.
For any and a.e.
, the function
is nonincreasing for
Proof.
Let We can write
with
Then,
and its derivative equals
We show that
is nonincreasing in
for
We have for
Furthermore, we have
while
for
so it holds that
At this point, the above representation formula together with elementary monotonicity considerations show that
is nonincreasing in
for
It remains to show that
is nonincreasing in
for
It holds that
Once again, we conclude from the above representation formula together with elementary monotonicity considerations that
is nonincreasing in
for
This finishes the proof. □
We derive some estimates for First, we have
for
since
Furthermore,
which implies that
for
This shows that there exists
such that for any ρ1,
It follows that
for
Since a standard argument shows that (Equation23
(23)
(23) ) has a unique solution
Therefore, the mapping S is well-defined. Additionally, the nonnegativity of u follows immediately after by testing (Equation23
(23)
(23) ) with
We show now that S is a contraction on for sufficiently small T > 0. We start with a preparation. By testing (Equation23
(23)
(23) ) with u and taking into account the
bound for
we deduce from Young’s inequality for products and convolutions that
since
for
Then, if
we infer from Gronwall’s lemma that
(24)
(24)
Let
and set
i = 1, 2. We compute
Therefore, using (Equation24
(24)
(24) ), for
(25)
(25)
Next, we write (Equation23
(23)
(23) ) for (ui, vi) in place of (u, v), i = 1, 2, take the difference between the two equations, and test the resulting equation with
:
It follows from (Equation25
(25)
(25) ) that
and we conclude from Gronwall’s lemma that
This inequality implies that S is a contraction in
provided that T is sufficiently small. Therefore, by Banach’s theorem, S admits a unique fixed point
for T > 0 sufficiently small.
It remains to show that the local solution can be extended to a global one. To this end, we note that the function satisfies (Equation23
(23)
(23) ) with v = u:
(26)
(26)
Then, defining the truncated entropy density
and testing (Equation26
(26)
(26) ) with
yields, in view of the definition of
that
(27)
(27)
for
This inequality and the definitions of
and
yield a
-uniform bound for u in
which in turn (together with (Equation26
(26)
(26) )) implies a
-uniform bound for u in XT, and a fortiori in YT. This means that the solution u can be prolonged to the whole time interval
and exists for all times.
Finally, we point out that, since then,
and so
This fact yields the conservation of mass for u, i.e.,
for t > 0. Indeed, it is sufficient to test (Equation26
(26)
(26) ) with a cutoff
satisfying
for
for
for
and then, to take the limit
Step 2: Limit Let
be the solution to (Equation26
(26)
(26) ). An adaption of the proof of [Citation24, lemma 1] shows that the embedding
is compact. Thus, because of the δ-uniform bounds for
the Aubin–Lions Lemma implies that (up to a subsequence)
strongly in
for every T > 0. We wish now to study the convergence of the nonlinear and nonlocal terms in (Equation26
(26)
(26) )–(Equation27
(27)
(27) ) as
It follows from (Equation27(27)
(27) ) that (up to a subsequence)
(28)
(28)
To identify the limit U, we first notice that, by construction,
a.e. in
Furthermore, the Hardy–Littlewood–Sobolev inequality, the bound for
and then, the Gagliardo–Nirenberg–Sobolev inequality yield that
Therefore, since
meaning that
Taking into account that
and that
is nonincreasing (see Lemma 12), we deduce from monotone convergence that
(29)
(29)
Furthermore, arguing as before and using the estimates for
leads to
Since
is bounded in
it follows that (up to a subsequence)
converges weakly to some limit in
However, Hölder’s inequality and the fact that
strongly in
for every
which follows from
imply that
We conclude that
(30)
(30)
We deduce from (Equation29(29)
(29) ) to (Equation30
(30)
(30) ) that
which, together with (Equation28
(28)
(28) ), implies that
that is,
(31)
(31)
Let
Because of
we find that
Our goal is to show that
strongly in
as
We can assume without loss of generality that
a.e. in
Indeed, for general functions ψ, we may write
where
and
and we have
Once again, since
a.e. in
it is sufficient to show that
The Hardy–Littlewood–Sobolev inequality (see Appendix B) yields
It follows from (Equation31
(31)
(31) ), the previous argument, and the fact that
that
for every
which means that
(32)
(32)
Since
strongly in
and
is bounded in
(via mass conservation), it also holds that
strongly in
Therefore, the convergence (Equation32
(32)
(32) ) is sufficient to pass to the limit
in (Equation26
(26)
(26) ).
Step 3: Limit and conclusion. The limit
in (Equation26
(26)
(26) ) shows that the limit u solves
(33)
(33)
Fatou’s Lemma and the weakly lower semicontinuity of the L2 norm allow us to infer from (Equation27
(27)
(27) ) that for t > 0,
(34)
(34)
At this point, all the bounds for u, derived in the previous subsections, and the moment estimate, contained in Lemma 9, can be proved like in Subsections 2.1–2.2. All these estimates are uniform in η. It is rather straightforward to perform the limit
in (Equation33
(33)
(33) )–(Equation34
(34)
(34) ) to obtain a weak solution to (Equation7
(7)
(7) ). However, the higher regularity bounds obtained in Subsection 2.3 imply that u is actually a strong solution to (Equation7
(7)
(7) ), which in turn yields the uniqueness of u as a weak solution to (Equation7
(7)
(7) ). This finishes the proof of Theorem 4.
2.5. Limit ![](//:0)
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We prove that there exists a subsequence of that converges strongly in
to a weak solution ρ to (Equation1
(1)
(1) ).
The uniform bound for
in Lemma 7 implies that, up to a subsequence,
weakly* in
as
We deduce from the uniform
bound (Equation13
(13)
(13) ) and the moment bound for
in Lemma 9 that
is equi-integrable. Thus, by the Dunford–Pettis theorem, again up to a subsequence,
weakly in
It follows from the
estimate (Equation16
(16)
(16) ) that
strongly in
The estimates in (Equation17
(17)
(17) ) and Lemma 7 show that
is bounded in
and consequently, up to a subsequence,
weakly in
Therefore, the limit
in (Equation7
(7)
(7) ) leads to
(35)
(35)
where the overline denotes the weak limit of the corresponding sequence.
We need to identify the weak limit on the right-hand side. The idea is to use the div-curl lemma [Citation41, theorem 10.21]. For this, we define the vector fields with d + 1 components
Let R > 0 be arbitrary and write BR for the ball around the origin with radius R. The
bound (Equation15
(15)
(15) ) for
and the
bound (Equation17
(17)
(17) ) for
show that
is bounded in
for some p > 1, while
is bounded in
Furthermore, by (Equation17
(17)
(17) ),
where
is the antisymmetric part of the Jacobian matrix of
Hence, by the compact embedding
(since
), the sequence
is relatively compact in
for some r > 1. Therefore, we can apply the div-curl lemma giving
or
By definition (Equation9(9)
(9) ) of
it follows for arbitrary
and sufficiently large L > 0, that
We use the properties that
is uniformly bounded and
in
Then, choosing
sufficiently small,
setting
Hence, using
we find that
Taking into account the fundamental theorem of calculus for the function
we can estimate as follows:
Then, since
is compact,
and
is uniformly bounded, we conclude that
This means that
strongly in
and it shows that
a.e. in
As f is nondecreasing, we can apply [Citation41, theorem 10.19] to infer that
a.e. in
Consequently,
As
is assumed to be strictly convex, we conclude from [Citation41, theorem 10.20] that
converges a.e. in
Since
is bounded in
it follows that
strongly in
for all
Using the moment estimate from Lemma 9, we infer from
that
strongly in
for all
The strong convergences of
and
in
for all
allow us to identify the weak limit in (Equation35
(35)
(35) ), proving the weak formulation (Equation8
(8)
(8) ).
Finally, we deduce from the uniform bound for
and the fact that
strongly in
for any
that
in the sense of
Properties (iv) of Theorem 1 follow from the corresponding expressions satisfied by
in the limit
2.6. Time-uniform convergence of ![](//:0)
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The following lemma is needed in the proof of Proposition 3. It is essentially a consequence of the bound of
and the Ascoli–Arzelà theorem.
Corollary 13.
Under the assumptions of Theorem 1, it holds for all that, possibly for a subsequence,
Proof.
Let and
The uniform
bound of
implies that
Hence, the sequence of functions
is bounded and equicontinuous in
By the Ascoli–Arzelá theorem, up to a
-depending subsequence,
strongly in
as
Since
weakly* in
we can identify the limit,
Since
is separable, a Cantor diagonal argument together with a density argument allows us to find a subsequence (which is not relabeled) such that for all
(36)
(36)
Since
is bounded in
another density argument shows that this limit also holds for all
Now, let Using
it follows from (Equation36
(36)
(36) ) and the moment estimate for
that
This shows that
concluding the proof. □
3. Analysis of EquationEq. (Equation5
(5)
(5) )
(5)
(5)
This section is devoted to the analysis of EquationEq. (Equation5(5)
(5) )
(5)
(5) ,
(37)
(37)
where
and
is defined in (Equation10
(10)
(10) ), as well as to an estimate for the difference
which is needed in the mean-field analysis. The existence and uniqueness of a strong solution to (Equation37
(37)
(37) ) follows from standard parabolic theory, since we regularized the singular kernel and smoothed the nonlinearity.
Proposition 14
(Uniform estimates). Let Hypotheses (H1)–(H3) hold and let T > 0, p > d. Set , let
be the strong solution to (Equation7
(7)
(7) ), and let
be the strong solution to (Equation5
(5)
(5) ). Then, there exist constants
, and
, both depending on σ, p, and T, such that if
then,
(38)
(38)
(39)
(39)
Furthermore, for every there exists
depending on σ and T, such that
(40)
(40)
(41)
(41)
The proof is presented in the following subsections. The most difficult part is the proof of (Equation38
(38)
(38) ) in Subsection 3.1. We first prove an estimate for
that depends on a lower-order estimate of this difference. Second, this lower-order estimate is shown by testing the equation satisfied by the difference
with a suitable nonlinear test function. Based on the arguments of this section, estimates (Equation39
(39)
(39) )–(Equation41
(41)
(41) ) are, then, shown in Subsections 3.2–3.4, respectively.
3.1. Proof of (Equation38
(38)
(38) )
We introduce the difference which satisfies
(42)
(42)
and the initial datum
in
where
We show first an estimate for that depends on a lower-order estimate for u.
Lemma 15
(Conditional estimate for u). For any p > d, there exists a number such that, if
, then,
recalling that
, and where C > 0 is independent of u, β, and ζ, but may depend on σ.
Proof.
Let be such that
We will find a constraint for Γp at the end of the proof. The aim is to derive an estimate for the right-hand side of (Equation42
(42)
(42) ) in
We observe that
for
In the following, we denote by C > 0 a generic constant that may depend on σ, without making this explicit. Furthermore, we denote by μ a generic exponent in (0, 1), whose value may vary from line to line.
Step 1: Estimate of Let
Then, by the Hardy–Littlewood–Sobolev-type inequality (B3),
We use the Young convolution inequality, the Gagliardo–Nirenberg inequality, the smoothness of
the property
and the fact
to estimate the terms on the right-hand side:
where
and
Therefore,
and
(43)
(43)
Next, let
Then, we write
By the Hardy–Littlewood–Sobolev-type inequality (Lemma 22),
Next, we apply the Gagliardo–Nirenberg inequality with
:
which is possible as long as
or equivalently
which is true. Consequently, using
where
is arbitrary. It follows from the Hardy–Littlewood–Sobolev-type inequality and the Gagliardo–Nirenberg inequality
that
Finally, using similar ideas, we obtain
Summarizing the estimates for D1, D2, and D3 and integrating in time leads to
(44)
(44)
Step 2: Estimate of We write
for the three terms in the definition of
below (Equation42
(42)
(42) ).
Step 2a: Estimate of R1. If s > 1∕2, we can argue similarly as in the derivation of (Equation43(43)
(43) ), which gives
If
we write
where
All terms except the last one can be treated by the Hardy–Littlewood–Sobolev and Gagliardo–Nirenberg inequalities as before. For the last term, we use these inequalities and the
bound for
:
We infer that (possibly with a different
than before)
Step 2b: Estimate of R2. Since is bounded on the interval
we obtain for s > 1∕2,
For
we write
where
Similar estimations as before allow us to treat all terms except the third one:
The first term can be estimated similarly as above by
while
It follows from
and the fractional Gagliardo–Nirenberg inequality (Lemma 24) that
This shows that
and we conclude that
Step 2c: Estimate of R3. We write where
We start with the estimate of R31. We use the Hardy–Littlewood–Sobolev inequality (Lemma 22) and Lemma 21 to estimate
:
also taking into account the
bound for
see Proposition 10. With this regularity, we can estimate all other terms except R34 and R36. Since they have similar structures, we only treat R34. This term is delicate since the factor
cannot be bounded in
for any
Therefore, one might obtain via Hardy–Littlewood–Sobolev’s inequality factors like
and
with either
or
However, for such factors, an
bound in time is currently lacking (Proposition 10 provides such a bound only for
). Our idea is to add and subtract the term
since
can be controlled. This leads to
as the first term can be estimated in a standard way. For the estimate of Q341, we need to distinguish two cases.
If we infer from the Hardy–Littlewoord–Sobolev-type inequality (B3) that
Next, let
Then, we apply the Hardy–Littlewoord–Sobolev-type inequality (B2), the standard Gagliardo–Nirenberg inequality for some
and Lemma 21:
We conclude that
and eventually
Summarizing the estimates for R1, R2, and R3 finishes this step:
(45)
(45)
Step 3: Estimate of We formulate this term as
where
The terms S1, S2, and S3 can be treated as the terms in
since they have the same structure and the techniques used to estimate integrals involving
can be applied to those involving
This leads to (for some
)
(46)
(46)
It remains to estimate S4. We write
where
Observe that, because of the definition of
with
(defined in (Equation11
(11)
(11) )), we have
for every function v for which the convolution is defined, and therefore, by the Hardy–Littlewood–Sobolev-type inequality (B2), Young’s convolution inequality, and Lemma 21,
Given the regularity properties of
(see Lemma 11) and the assumptions on
it follows that
(47)
(47)
We conclude from (Equation46(46)
(46) ) and (Equation47
(47)
(47) ) that
(48)
(48)
where
Step 4: End of the proof. Summarizing (44, 45), and (Equation48(48)
(48) ), we infer that the right-hand side of (Equation42
(42)
(42) ) can be bounded (for some
) by
By parabolic regularity (C1),
Choosing
sufficiently small finishes the proof. □
It remains to estimate the norm of u. This is done in the following lemma.
Lemma 16
(Unconditional estimate for u). For any p > d, there exist constants C > 0, and , both depending on σ, p, and T, such that for
recalling that
Proof.
The idea is to test (Equation42(42)
(42) ) with
Integration by parts and some elementary computations lead to
Consequently, we have
(49)
(49)
We infer from Lemmas 20 and 26 that Therefore, since
it holds that
for all
and
for
Let
We have shown in the proof of the previous lemma that
Hence, we can estimate the right-hand side
of (Equation49
(49)
(49) ) as follows:
where
Choosing δ sufficiently small, the last term is absorbed by the corresponding expression on the left-hand side of (Equation49
(49)
(49) ), and we infer from (Equation49
(49)
(49) ) that for
We assume without loss of generality that C(p, t) is nondecreasing in t. Then, Gronwall’s lemma implies that for
Choosing
we find that
for
and
By definition of
it follows that
In particular,
for
which finishes the proof. □
3.2. Proof of (Equation38
(38)
(38) ) and (Equation39
(39)
(39) )
Combining Lemmas 15 and 16 leads to
(50)
(50)
as long as
and p > d. Next, we differentiate (Equation42
(42)
(42) ) with respect to xi (writing
for
):
Taking into account estimate (Equation50
(50)
(50) ) and arguing as in the proof of Lemma 15, we can show that for
We infer from parabolic regularity (Lemma 26) for sufficiently small
that
Then, Lemma 20, applied to
leads to (Equation38
(38)
(38) ), which with Proposition 10 implies (Equation39
(39)
(39) ).
3.3. Proof of (Equation40
(40)
(40) )
Let We use the definitions of
and
to find that
Let
be such that
and
in B1. Then, (since we can assume without loss of generality that
), by arguing like in the derivation of (Equation47
(47)
(47) ), we obtain
A computation shows that for
where we note that
if
Then, in view of the regularity of
in Lemma 11, we find that
3.4. Proof of (Equation41
(41)
(41) )
The bound for
is shown in a similar way as the corresponding bound for
in Lemma 11.
4. Mean-field analysis
This section is devoted to the proof of Proposition 3 and Theorem 2. The existence of solutions to (Equation4(4)
(4) ) and (Equation6
(6)
(6) ) as well as the existence of density functions is shown in Subsection 4.1. In Subsection 4.2, we estimate the difference
of the processes of the original system (Equation3
(3)
(3) ) and the intermediate system (Equation4
(4)
(4) ), while the difference
of the processes of the intermediate system (Equation4
(4)
(4) ) and the macroscopic system (Equation6
(6)
(6) ) is estimated in Subsection 4.3. These estimates are combined in Subsection 4.4 to conclude with the proof of Proposition 3 and Theorem 2.
4.1. Existence of density functions for (Equation4
(4)
(4) ) and (Equation6
(6)
(6) )
First, we show that the coefficients of the stochastic differential EquationEq. (Equation6(6)
(6) )
(6)
(6) , satisfied by
are globally Lipschitz continuous and of at most linear growth. The latter condition follows from
where p > d and
according to the Hardy–Littlewood–Sobolev inequality, and we used the regularity bounds for
from Lemma 26. The global Lipschitz continuity is a consequence of the mean-value theorem, the Hardy–Littlewood–Sobolev inequality, and the
regularity of
from Lemma 11:
These two conditions yield the existence and uniqueness of solutions to the associated particle systems [Citation44, theorems 2.5 and 2.9]. Moreover, by [Citation45, theorem 2.3.1], the law of the process
is absolutely continuous with respect to the Lebesgue measure. By Radon-Nikodym’s theorem, there exists a density function
for all t > 0 on
which is measurable and integrable with respect to the Lebesgue measure. (Since all
are copies of the same process, their density functions are the same almost everywhere.) The processes
have continuous paths, which implies the continuity of the distribution function of
with respect to time, and this implies in turn the Bochner measurability of
Clearly, we have
which shows that
Similar arguments show that has a density function
Next, we show that and
can be identified with the weak solutions
and
respectively, using Itô’s lemma. Indeed, let
We infer from Itô’s formula that
Taking the expectation, the Itô integral vanishes, and we end up with
(51)
(51)
Hence,
is a very weak solution in the space
to the linear equation
(52)
(52)
where
is the unique solution to (Equation7
(7)
(7) ).
It can be shown that (Equation52(52)
(52) ) is uniquely solvable in the class of functions in
This implies that
in
(and similarly
). The proof is technical but standard; see, e.g., [Citation35, theorem 7] for a sketch of a proof.
Another approach is as follows. Because of the linearity of (Equation51(51)
(51) ), it is sufficient to prove that
in
if
First, we verify that
and
for
Then, by density, (Equation51
(51)
(51) ) holds for all
with q > d and
Choosing ψ to be the unique strong solution to the dual problem
in the very weak formulation of (Equation51
(51)
(51) ), we find that
for all
which implies that
4.2. Estimate of ![](//:0)
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We derive an estimate for the expectation of the difference To this end, we need to estimate the difference of the microscopic average
and the macroscopic average
By a careful choice of β and ζ, we show that this estimate is of the order
for
Lemma 17.
Let and
be the solutions to (Equation3
(3)
(3) ) and (Equation4
(4)
(4) ), respectively, and let
. Under the assumptions of Theorem 3 on β and ζ, it holds that
Proof.
To simplify the presentation, we set
and we write
Taking the difference of EquationEqs. (Equation3
(3)
(3) )
(3)
(3) and Equation(Equation4
(4)
(4) )
(4)
(4) in the integral formulation leads to
(53)
(53)
Step 1: Estimate of I1. To estimate I1, we formulate where
We start with the first integral:
We claim that
(54)
(54)
For the proof, we introduce
Then, by definition of
We estimate the right-hand side:
where we used the Hardy–Littlewood–Sobolev inequality for
in the last step. It follows from the Faàdi Bruno formula, after an elementary computation that the last term is estimated according to
since
and
This verifies (Equation54
(54)
(54) ). We infer from (Equation54
(54)
(54) ) with k = 2 that
The term I13 is estimated in a similar way, with
replaced by
:
The estimate of the remaining term I12 is more involved. Since
is assumed to be symmetric, we find that
where
is a random value. We write
and note that
for all
Then,
Observe that we did not use the compact support for
(which depends on ζ), because a negative exponent of ζ at this point would lead to a logarithmic connection between ζ and N in the end, which we wish to avoid.
Furthermore, by the convolution, Sobolev, and Hardy–Littlewood–Sobolev inequalities as well as the fact that
where we recall that
and we choose p > d satisfying
The previous two estimates lead to
We summarize:
(55)
(55)
Step 2: Estimate of I2. We take the expectation of I2 and use the mean-value theorem:
(56)
(56)
where
We deduce from
that
due to the compact support of
and
We claim that
for all
To show the claim, we compute the expectation
We estimate first the terms with
(omitting the argument (y, s) to simplify the notation). Then, an elementary but tedious computation leads to
The diagonal terms contribute in the following way:
since
This shows that
We infer that (Equation56
(56)
(56) ) becomes
(57)
(57)
Step 3: End of the proof. We insert (Equation55(55)
(55) ) and (Equation57
(57)
(57) ) into (Equation53
(53)
(53) ) to infer that
By Gronwall’s lemma,
We choose
for some arbitrary
Then, since by assumption,
and
we find that
proving the result. □
4.3. Estimate of ![](//:0)
![](//:0)
Next, we compute the expectation of by estimating the difference between
and
as well as the difference between
and
The estimate depends on β and ζ.
Lemma 18.
Let and
be the solutions to (Equation4
(4)
(4) ) and (6), respectively. Then, there exists a constant C > 0, depending on σ, such that
where
Proof.
We compute the difference
where
the convolution is taken with respect to xi, and
Step 1: Estimate of J1. We write and add and subtract the expression
:
By the mean-value theorem and using
we obtain for some random variable
(58)
(58)
We need to estimate the last integral. For this, we write for k = 1, 2
where
and
Note that
A similar argument as for the estimate of I12 in the proof of Lemma 17 shows that for
where we used Proposition 14 ((Equation39
(39)
(39) ) and (Equation41
(41)
(41) )) with
in the last inequality. We conclude from (Equation58
(58)
(58) ) that
(59)
(59)
Step 2: Estimate of J2. We treat the two cases s < 1∕2 and separately. Let first
Then,
Recalling the definition of
in (Equation11
(11)
(11) ) and writing
for
we find that
(60)
(60)
We estimate the right-hand side term by term. Because of
we use Sobolev’s embedding
for any p > d, and then, the boundedness of the Riesz operator
[Citation46, chapter IV, §3.1] in case s = 1/2 or the Hardy–Littlewood–Sobolev inequality for
(see Lemma 22) in case s > 1∕2 to control the first norm in (Equation60
(60)
(60) ) by
where r = p in case s = 1/2 and
in case s > 1∕2. Choosing
guarantees that r > d always holds.
For the second norm in (Equation60(60)
(60) ), Hölder’s inequality yields for q > d and
for every t > 0,
which can be bounded by
since
By similar arguments and the fact that
we find that
and hence, using
we conclude for the second and third term in (Equation60
(60)
(60) ) that
The choice
guarantees on the one hand that q > d and on the other hand that the exponent
is strictly positive (which allows us to use the property
).
Using these estimates in (Equation60(60)
(60) ), we arrive (for
) at
where we recall that
These norms can be estimated by
and
The
bound for
from Lemma 11 and the definition of
finally show for
and
that
(61)
(61)
Now, let s < 1∕2. In this case, we cannot estimate
and put the gradient to the second factor of the convolution. Adding and subtracting an appropriate expression as in Step 1, using the embedding
for p > d, the estimate
and the Hardy–Littlewood–Sobolev inequality, we find that
where r > d is such that
(this is needed for the Hardy–Littlewood– Sobolev inequality) and p > d (because of Sobolev’s emebdding). Note that r > d can be only guaranteed if s < 1∕2. Together with the fact that
(choose q > d in (Equation41
(41)
(41) ) and use Sobolev’s embedding), this shows that for s < 1∕2,
(62)
(62)
It follows from estimate (Equation38
(38)
(38) ) and Lemma 21 in Appendix A for p > d that
where we used the
estimate for
in (Equation41
(41)
(41) ). Then, we deduce from estimates (Equation61
(61)
(61) ) and (Equation62
(62)
(62) ) that for all
where we recall that
Step 3: Estimate of J3 and end of the proof. Arguing similarly as in Subsection 3.3, we have
This implies that
(63)
(63)
Taking the expectation, we infer from (Equation59
(59)
(59) ) to (Equation63
(63)
(63) ) that
An application of Gronwall’s lemma gives the result. □
4.4. Proof of Theorem 2 and Proposition 3
Lemmas 17 and 18 show that
and this expression converges to zero as
and
under the conditions stated in Theorem 3. This result implies the convergence in probability of the k-tuple
to
Since convergence in probability implies convergence in distribution, we obtain
where
and
denote the joint distributions of
and
respectively. By Subsection 4.1,
is absolutely continuous with the density function
Using the test function
in Corollary 13, we have, up to a subsequence,
locally uniformly for t > 0. Since the convergence also holds for the initial condition, the result is shown.
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Funding
References
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Appendix A.
Auxiliary results
We recall some known results. The following result is proved in [Citation47, theorem 4.33].
Lemma 19
(Young’s convolution inequality). Let , and
. Then,
and
The following lemma slightly extends [Citation48, lemma 7.3] from the L2 to the Lp setting.
Lemma 20.
Let and T > 0. Then, the following embedding is continuous:
Proof.
Let and
Then,
(A1)
(A1)
where
Direct computations using Young’s inequality lead to
We infer from (A1) and the continuity of the integrals with respect to the time integration boundaries that
is continuous and
(A2)
(A2)
Next, let
be arbitrary and let
as
such that
Estimate (A2) implies that
is bounded in
Thus, there exists a subsequence
of
such that
weakly in
as
for some
We can show, using estimate (A2) and dominated convergence for the integral
that in the limit
which yields
Moreover, since is continuous, we have
Since
is uniformly convex, we deduce from [Citation47, prop. 3.32] that
strongly in
Since the limit is unique, the whole sequence converges. Together with (A2), this concludes the proof. □
Let be nonnegative with
and define
for
and
Lemma 21.
Let and
. Then,
Proof.
We use Hölder’s inequality and the fact that to find that
which shows the lemma. □
Appendix B.
Fractional Laplacian
We recall that the fractional Laplacian for
can be written as the pointwise formula
(B1)
(B1)
and the integral is understood as principal value if
[Citation1, theorem 2]. The inverse fractional Laplacian
is defined in (Equation2
(2)
(2) ). The following lemma can be found in [Citation46, chapter V, subsection 1.2].
Lemma 22
(Hardy–Littlewood–Sobolev inequality). Let and
. Then, there exists a constant C > 0 such that for all
Applying Hölder’s, and then, Hardy–Littlewood–Sobolev’s inequality gives the following result.
Lemma 23.
Let and
. Then, there exists C > 0 such that for all
(B2)
(B2)
(B3)
(B3)
Lemma 24
(Fractional Gagliardo–Nirenberg inequality I). Let and
. Then, there exists C > 0 such that for all
or
, respectively,
Proof.
It follows from the properties of the Riesz and Bessel potentials [Citation46, theorem 3, p. 96] that the operator is bounded for
while the operator
is bounded for
Thus, if
Replacing u by
with
yields
We minimize the right-hand side with respect to λ giving the value
and therefore,
The case
is proved in a similar way. □
Lemma 25
(Fractional Gagliardo–Nirenberg inequality II). Let , and
. If
, we assume additionally that
. Then, there exists C > 0 such that for all
where
Proof.
The statement is true for s = 1/2 since the operator is bounded for any
[Citation46, theorem 3, p. 96]. Then, the inequality follows from the standard Gagliardo–Nirenberg inequality.
Thus, let We claim that it is sufficient to prove that
is bounded. Indeed, assume that
(B4)
(B4)
Replacing, as in the proof of Lemma 24, u by
with
yields
where θ is defined in the statement of the theorem. Minimizing the right-hand side with respect to λ gives the value
and therefore,
It remains to show Equation(B4)
(B4)
(B4) . To this end, we distinguish two cases. First, let
By assumption,
We apply the Hardy–Littlewood–Sobolev inequality (Lemma 22) to find that
Furthermore, by using (in this order) the boundedness of
lemma 2 in [Citation46, p. 133], Equationeq. (Equation40
(40)
(40) )
(40)
(40) in [Citation46, p. 135], and theorem 3 in [Citation46, p. 135f],
(B5)
(B5)
These inequalities hold for any
Now, it is sufficient to interpolate with
:
Second, let
We choose
and apply the Hardy–Littlewoord– Sobolev inequality:
where
Since
we deduce from Equation(B5)
(B5)
(B5) that
Since
as
the result follows. □
Appendix C.
Parabolic regularity
Lemma 26
(Parabolic regularity). Let , T > 0 and let u be the (weak) solution to the heat equation
where
and
. Then, there exists C > 0, depending on T and p, such that
(C1)
(C1)
Furthermore, if for some
then,
(C2)
(C2)
Proof.
We use a known result on the parabolic regularity for the equation
(C3)
(C3)
It holds that [Citation49]
(C4)
(C4)
We apply this result to
where
is the solution to the homogeneous heat equation in
with initial datum u0. Then,
solves Equation(C3)
(C3)
(C3) and satisfies estimate Equation(C4)
(C4)
(C4) . Inserting the definition of
and observing that
we obtain Equation(C1)
(C1)
(C1) .
If for some
the uniqueness of solutions to the heat equation yields
where U solves
Then, we deduce from the regularity result of [Citation49] with
and v = w that
Since
inequality Equation(C2)
(C2)
(C2) follows. □