Particle approximation of the 2 - d parabolic-elliptic Keller-Segel system in the subcritical regime
aa r X i v : . [ m a t h . P R ] A p r Particle approximation of the - d parabolic-ellipticKeller-Segel system in the subcritical regime Christian Olivera ∗ Alexandre Richard † Milica Tomaˇsevi´c ‡ April 8, 2020
Abstract
The parabolic-elliptic Keller-Segel partial differential equation is a two-dimensional modelfor chemotaxis. In this work we introduce a stochastic system of moderately interactingparticles which converges, globally in time, to the solution to the Keller-Segel model in 2-d.The advantage of our approach is that we show the convergence in a strong sense for all thesubcritical values of the total mass,
M < π . Key words and phrases:
Stochastic differential equations, Keller-Segel partial differential equa-tion, Interacting particle systems, Analytic semigroups.
MSC2010 subject classification:
In this paper we study a stochastic particle approximation of the two-dimensional parabolic-ellipticKeller-Segel (KS) model that reads ( ∂ t ρ ( t, x ) = ∆ ρ ( t, x ) − ∇ ( ρ ( t, x ) ∇ ( G ∗ ρ ) ( t, x )) , t > , x ∈ R ,ρ (0 , x ) = ρ ( x ) , (1.1)where the convolution is with respect to the x -variable only and G is the fundamental solution ofPoisson’s equation in R , i.e. G ( x ) = − π log | x | , x ∈ R \ { } . This is a closed formulation of the following problem ∂ t ρ ( t, x ) = ∆ ρ ( t, x ) − ∇ ( ρ ( t, x ) ∇ c ( t, x )) , t > , x ∈ R , ∆ c ( t, x ) + ρ ( t, x ) = 0 , t > , x ∈ R ,ρ (0 , x ) = ρ ( x ) , c (0 , x ) = 0 , which describes the time evolution of the density ρ of a cell population whose motion is guidedby the gradient of the concentration c of a chemical stimulus (chemo-attractant). Note that theequation for the chemo-attractant concentration is in the steady state, which justifies to call (1.1)the parabolic-elliptic KS model.System (1.1) is a special case of the general Keller-Segel model for chemotaxis [14, 15] and it hasbeen widely studied: see for instance Horstmann [12, 13] and Perthame [20] for a thorough reviewof the literature up to 2000’s. For a recent review, see Biler [2] and the references therein. Noticingthat (1.1) admits mass conservation, we denote M := R R ρ ( x ) dx = R R ρ ( t, x ) dx. Interestingly, ∗ Departamento de Matem´atica, Universidade Estadual de Campinas, Brazil. [email protected] . † Universit´e Paris-Saclay, CentraleSup´elec, MICS and CNRS FR-3487. [email protected] . ‡ CMAP, Ecole polytechnique, CNRS, I.P. Paris, 91128 Palaiseau, France. [email protected] . M > π , the solutions blow-up in finite time. On the other hand, when M < π , global (in time) existence holds. For these results, see e.g. Blanchet, Dolbeault andPerthame [3], Nagai [17] and Nagai and Ogawa [18]. For more details on the blow-up phenomenon,see Herrero and Velazquez [11].In this work, we are interested in the stochastic particle approximation of (1.1). This is aproblem of noticeable difficulty due to the singularity of the interactions, which has attracted a lotof attention lately. First, Fournier and Jourdain [10] studied the following singularly interactingparticle system associated to (1.1): dX i,Nt = 1 N N X j =1 − ( X i,Nt − X j,Nt )2 π | X i,Nt − X j,Nt | dt + √ dW it , (1.2)where { W it , i ∈ N } is a family of independent standard two-dimensional Brownian motions definedon a filtered probability space (Ω , F , F t , P ). Due to the singular interaction kernel, it is not obviousthat this particle system is well-defined and that the propagation of chaos holds. Nevertheless, theauthors proved its well-posedness when M < π NN − . In addition, when the mass M is smaller than2 π , they proved that any weak limit point of the empirical measure of N particles is a.s. the law ofthe associated non-linear process of McKean-Vlasov type, whose one-dimensional time marginalssatisfy (1.1). They also described complex behaviors of the particle trajectories and proved, usinggeneralized Bessel processes, that (1.2) is well-defined until a time in which a 3-particle collisionoccurs. This time is infinite when M < π N − N − . The existence of solutions to (1.2) was also studiedby Cattiaux and P´ed`eches [6] using Dirichlet forms, and it was proved that (1.2) is well-posed for M ≤ π NN − .Another result regarding the convergence of (1.2) has been established in Bresch, Jabin andWang [4]. Namely, the authors were interested in the convergence, when N → ∞ , of the joint lawof k fixed particles at a time t towards ρ ⊗ kt , where ρ solves (1.1). They worked on a periodic domainΠ ⊂ R . Under the constraint that M ≤ π and the assumption that the particles are well-definedand that ρ ∈ L ∞ ((0 , T ); W , ∞ (Π)), they proved using new techniques of relative entropy the aboveconvergence in L ∞ ((0 , T ); L (Π k )). Moreover, their result is quantitative, in the sense that therate of convergence is explicit as a function of N .Unlike [4, 6, 10], we present a moderately interacting system of stochastic particles (in the senseof Oelschl¨ager [19] and M´el´eard and Roelly [16]). Our objective is to prove the uniform convergenceof its mollified empirical measure towards the solution of (1.1) when the number of particles goesto infinity, for all the subcritical values of the total mass ( M < π ). For that purpose, we followthe new approach presented in Flandoli, Leimbach and Olivera [8], based on semigroup theory anddeveloped first with application to the FKPP equation. This technique permits to approximatenonlinear PDEs by smoothed empirical measures in strong functional topologies. It has alreadyfound many applications: Flandoli and Leocata [7] for a PDE-ODE system related to aggregationphenomena; Olivera and Simon [21] for non-local conservation laws; and Flandoli, Olivera andSimon [9] for the 2d Navier-Stokes equation. The main difficulty here will be the singular natureof the Keller-Segel equation and finding a suitable functional framework.Thus, we consider the following particle system: dX i,Nt = F A N N X k =1 ( ∇ G ∗ V N )( X i,Nt − X k,Nt ) ! dt + √ dW it , t ≤ T, ≤ i ≤ N, (1.3)where V N is a mollifier, F A is a smooth cut-off function that ensures that the drift driving eachparticle remains uniformly bounded in N , and A > N → ∞ , of the mollified empirical measure { g Nt := V N ∗ S Nt } t ∈ [0 ,T ] , where S N is the empirical measure of (1.3), towards the unique mildsolution to (1.1) for M < π . Suitable conditions on the initial law ρ are required among whichwe emphasize that we work with ρ ∈ L ∩ H β ( R ), for some β >
1, where H β is a fractional Sobolevspace. We prove convergence in probability in the following topologies: in the strong topology of C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) , for some γ ∈ (1 , β ) and in the weak topology of L (cid:0) [0 , T ]; H β ( R ) (cid:1) .Compared to the results of [4, 6, 10], the main difference is that we start from a smoothedversion of (1.2) and that we obtain the convergence for the whole range of subcritical parameter M . In addition, we will compare in Subsection 2.2 the modes of convergence of the empiricalmeasures in those works and ours.Finally, let us briefly describe our aproach and point out the main difficulties arising in thiswork. In the definition of the particle system (1.3) and its convergence, it is very convenient to havea bounded drift term, which is ensured by the smooth cut-off function F A ( x ) ≈ sign( x ) × ( | x | ∧ A ).However this implies that the particle system will not converge to the true Keller-Segel PDE (1.1)but rather to a PDE with a modified reaction term involving F A ( ∇ G ∗ ρ ) (see precisely Equation(2.2)). We recall here from [3] that assuming mild conditions on the initial data yields weaksolutions of (1.1) in L ∞ loc (( ε, ∞ ); L p ( R )) for any p ∈ (1 , ∞ ), or from [17] that assuming an initialcondition ρ ∈ L ∩ H ( R ) implies that there exists a unique local (in time) mild solution to(1.1) that belongs to C b ([0 , T ); L ∩ H ( R )). In both cases, it is not clear that ∇ G ∗ ρ remainsbounded on [0 , T ] × R , for an arbitrary T >
0. Hence, we proved that if the initial condition ρ ∈ L ∩ H ( R ), the solution to the original PDE (1.1) satisfies k∇ G ∗ ρ k L ∞ ([0 ,T ] × R ) < ∞ .Thanks to this new estimate, one can choose A larger than k∇ G ∗ ρ k L ∞ ([0 ,T ] × R ) , and it followsthat the solution to (1.1) is a solution to the PDE with cut-off.As for the convergence of g N (the mollified empirical measure of the particle system), weemphasize the closeness of our computations with those that are used in studying the vorticityformulation of the 2d Navier-Stokes equation in [9] (and indeed, this equation has an interactionkernel very close to KS), which rely on semigroup techniques. A new ingredient here comparedto the previous literature on probabilistic KS seems to be a functional inequality of Calder´on-Zygmund type for ∇ G . The convergence of g N is obtained by tightness in C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) ∩ L w (cid:0) [0 , T ]; H β ( R ) (cid:1) , where L w denotes the weak topology of L . Any limit point is then identifiedas a mild solution of the Keller-Segel PDE with cut-off F A . Therefore we provide a uniquenessresult for the Keller-Segel PDE with cut-off F A . Hence in view of the discussion of the previousparagraph, it follows that for A large enough, any limit point of g N is a mild solution to the originalPDE (1.1). Related works and perspectives.
Probabilistic interpretation of the Keller-Segel system inits parabolic-parabolic version yields a non-linear McKean-Vlasov stochastic process proposed byTalay and Tomaˇsevi´c [24] and studied in the 2- d case in Tomaˇsevi´c [25]. The fact that the equationfor the chemo-attractant concentration is not in steady state introduces a memory component inthe non-linear term and the process interacts with all its past time marginal densities in a singularway. The well-posedness of the non-linear process (and the KS system) is proved under an explicitconstraint on M and for ρ ∈ L ( R ) and c ∈ H ( R ).Using the methods developed in this work in the parabolic-elliptic framework, we are currentlyinvestigating the moderately interacting particle system related to the non-linear process in [25].Finally, we point out that Stevens [23] studied the convergence of a moderately interactingstochastic particle system towards a generalized version of the parabolic-parabolic Keller-Segelequation in R d . Her particle system is slightly different than the one proposed in [24], as she con-siders 2 sub-populations of particles, one for the cell population and one for the chemo-attractant.Assuming, among other conditions, that the solution to the parabolic-parabolic KS system is suchthat ρ, c ∈ C , b ([0 , T ] × R d ) ∩ C ([0 , T ]; L ( R d )), Stevens proves the convergence in probability ofthe regularized empirical measure of the particle system towards the solution of the Keller-Segelmodel in the strong topology of C ([0 , T ]; L ( R d )) ∩ L ([0 , T ]; H ( R d )).3 lan of the paper. In Section 2, we present our framework in more details and state our mainresult,Theorem 2.3. Then we compare it to previously known results on the particle approxima-tion of the 2d parabolic-elliptic Keller-Segel PDE. The rest of Section 2 is dedicated to statingimportant intermediate results and exhibiting the organisation of the proof of Theorem 2.3. InSection 3, we treat the well-posedness of (1.1) and its cut-off version: that is, we prove Theorem2.7 about the existence of a mild solution to (1.1) in C b (cid:0) [0 , T ] , L ∩ H ( R ) (cid:1) , its Corollary 3.4that gives an explicit bound on k∇ G ∗ ρ k L ∞ ([0 ,T ] × R ) and finally the proof of Theorem 2.9 thatestablishes uniqueness for the cut-off PDE. In Section 4, we develop the computations that yieldthe mild formulation of g N and its tightness, thus establishing Proposition 2.4. Finally, we gatherin an Appendix some technical computations related to the boundedness of g N in a space thatis compactly embedded in C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) ∩ L w (cid:0) [0 , T ]; H β ( R ) (cid:1) , as well as a couple of usefulinequalities reflated to G , including the Calder´on-Zygmund inequality for ∇ G . Notations and definitions.
For any α ∈ R and p.. , we denote by H α ( R d ) the Bessel potentialspace H α ( R d ) := n u ∈ L ( R d ) ; F − (cid:16)(cid:0) | · | (cid:1) α F u ( · ) (cid:17) ∈ L ( R d ) o , where F u denotes the Fourier transform of u . These spaces are endowed with their norm k u k α, := (cid:13)(cid:13)(cid:13) F − (cid:16) (1 + | · | ) α F u ( · ) (cid:17)(cid:13)(cid:13)(cid:13) L ( R d ) < ∞ . Note that k u k , = k u k L ( R d ) and, for any α ≤ β, k u k α, ≤ k u k β, . For positive α and any ball B (0 , R ) ⊂ R d , the space H α ( B (0 , R )) is defined in Triebel [26, p.310],and corresponds roughly to distributions f on B (0 , R ) which are restrictions of g ∈ H α ( R d ). Then H α loc ( R d ) is the space of distributions f on R d such that f ∈ H α ( B (0 , R )) for any R > e t ∆ ) t ≥ is the heat semigroup. That is, for f ∈ L ( R ), (cid:0) e t ∆ f (cid:1) ( x ) = Z R πt e −| x − y | / (4 t ) f ( y ) dy. Obviously, ∇ e t ∆ f = e t ∆ ∇ f . Applying the convolution inequality [5, Th. 4.15] for p = 2 and usingthe equality (cid:13)(cid:13) ∇ πt e − |·| t (cid:13)(cid:13) L ( R ) = C √ t , it comes that (cid:13)(cid:13) ∇ e t ∆ (cid:13)(cid:13) L → L ≤ C √ t − s . (1.4)The space C ( I ; L ∩ H ( R )) of continuous functions from the time interval I with values in L ∩ H ( R ) is endowed with the norm k f k I,L ∩ H = sup s ∈ I (cid:0) k f s k L ( R ) + k f s k H ( R ) (cid:1) . For any t >
0, we will also need the norm k f k t,L ∩ H = sup s ∈ [0 ,t ] (cid:0) k f s k L ( R ) + k f s k H ( R ) (cid:1) . Finally, if u is a function or stochastic process defined on [0 , T ] × R , we will most of the timeuse the notation u t to denote the mapping x u ( t, x ). The aim of this section is to present our main result and the organisation of its proof, whosetechnical details are presented in separate sections.4 .1 Statement of the theorem
Let us introduce a cut-off in the reaction term of Equation (1.1). Namely, for any
A >
0, let F A be defined as follows: let f A : R → R be a C b ( R ) function such that:(i) f A ( x ) = x , for x ∈ [ − A, A ],(ii) f A ( x ) = A , for | x | > A + 1,(iii) k f ′ A k ∞ ≤ k f ′′ A k ∞ < ∞ .As a consequence, k f A k ∞ ≤ A + 1. Now F A is given by F A : (cid:18) x x (cid:19) (cid:18) f A ( x ) f A ( x ) (cid:19) . (2.1)The modified Keller-Segel PDE with cut-off now reads, in closed form: ( ∂ t e ρ ( t, x ) = ∆ e ρ ( t, x ) − ∇ · ( e ρ ( t, x ) F A ( ∇ G ∗ e ρ ( t, x ))) , t > , x ∈ R e ρ (0 , x ) = ρ ( x ) . (2.2)Although this is implicit, e ρ actually depends on A . Note that if F A is replaced by the identityfunction, one recovers (1.1). Solutions to (2.2) will be understood in the following sense: Definition 2.1.
Given u ∈ L ∩ H ( R ) and A > , a function u on [0 , T ) × R is said to be amild solution to (2.2) on [0 , T ) ifi) u ∈ C b ([0 , T ); L ∩ H ( R )) ;ii) u satisfies the integral equation u t = e t ∆ ρ − Z t ∇ · e ( t − s )∆ ( u s F A ( ∇ G ∗ u s )) ds, < t < T. (2.3) A function u on [0 , ∞ ) × R is said to be a global mild solution to (2.2) if it is a mild solution to (2.2) on [0 , T ) for all < T < ∞ . Remark 2.2.
Similarly, a mild solution to the original PDE (1.1) satisfies Definition 2.1 i ) andsolves u t = e t ∆ ρ − Z t ∇ · e ( t − s )∆ ( u s ∇ G ∗ u s ) ds, < t < T. (2.4)Compared to the singular particle system (1.2), we introduce a mollifier that will be usedboth to regularise the particle system and its empirical measure. Let V : R → R + be a smoothprobability density function. For any x ∈ R , define V N ( x ) := N α V ( N α x ) , for some α ∈ [0 , . (2.5)To cancel out the self-interaction term of a particle, we further assume that V is even (hence ∇ G ∗ V N is odd, so that ∇ G ∗ V N (0) = 0 and the self-interaction does vanish, see below).For each N ∈ N , we consider the following interacting particle system: ( dX i,Nt = F A (cid:0) N P Nk =1 ( ∇ G ∗ V N )( X i,Nt − X k,Nt ) (cid:1) dt + √ dW it , t ≤ T, ≤ i ≤ N,X i,N , ≤ i ≤ N, i.i.d. and independent of { W i } , (2.6)where { W it , i ∈ N } is a family of independent standard two-dimensional Brownian motions definedon a filtered probability space (Ω , F , F t , P ).Let us denote the empirical measure of N particles by S N. = 1 N N X i =1 δ X i,N. , g N · := V N ∗ S N · . The following hypotheses on the initial conditions of the system will be assumed:( C0 ): ( C0 i ) There exists β > p ≥
2, sup N ∈ N E h(cid:13)(cid:13) g N (cid:13)(cid:13) pβ, i < ∞ . ( C0 ii )Let ρ ∈ L ∩ H β ( R ) such that ρ ≥
0. Then h g N , ϕ i → h ρ , ϕ i in probability, for any ϕ ∈ C b ( R ).( C0 iii ) The initial total mass M = k ρ k L ( R ) satisfies M < π .( C0 iv ) For the parameters α and β (which appear respectively in (2.5) and ( C0 i )),assume that 0 < α <
12 + 2 β .
The main result of this paper is the following theorem. It involves a value of the cut-off A which depends only on M and is given precisely in Equation (2.10), and the notion of mild solutionof the PDE (1.1) which is given in Definition 2.1. Theorem 2.3.
Assume that the initial conditions { S N } N ∈ N satisfy ( C0 ) and that the dynamicsof the particle system is given by (2.6) with A greater than A .Then for any γ ∈ (1 , β ) , the sequence of mollified empirical measures { g Nt , t ∈ [0 , T ] } N ∈ N convergesin probability, as N → ∞ , towards the unique mild solution ρ of the parabolic-elliptic Keller-SegelPDE (1.1) , in the following senses: • ∀ ϕ ∈ L (cid:0) [0 , T ]; H β ( R ) (cid:1) , R T h g Nt , ϕ t i H β dt P −→ R T h ρ t , ϕ t i H β dt ; • in the strong topology of C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) . We now proceed to the proof of Theorem 2.3. We will state the most important intermediateresults that are used to prove the main theorem, and refer to subsequent sections for the proof ofthese results.
Fournier and Jourdain [10] proved a tightness – consistency result, not only on the level of timemarginal laws, but on the level of the laws on the space of trajectories. In addition, they analysefine properties of the particle trajectories and obtain existence results for the particle system.However, to prove convergence, they remain in the very subcritical case
M < π . On the otherhand, Bresch et al. [4] are able to improve the constraint on M by passing from 2 π to 4 π and havea quantitative result, but the convergence is only on the level of the time marginal laws (althoughit is in a stronger sense). In addition, it relies on the assumptions that the particles are well definedfor all M < π and all N ≥ π, π ) forthe values of the critical parameter. However, we do analyse a smoothed version of the system andthe empirical measure, which explains the stronger notion of convergence.When comparing our result with the one in [4], notice first that their result is implies theconvergence in law of the time marginals of the empirical measure of N particles, uniformly intime. In addition, it is quantitative. We do not have a quantitative estimate for the conver-gence and we do work with the mollified empirical measure. However, they work on periodicdomains in R , we rather work on the whole domain. To prove the convergence they suppose6 ∈ L ∞ ((0 , T ); W , ∞ (Π)). Our procedure shows that it is enough to find the solution of (1.1) thatbelongs to ρ ∈ C b ([0 , T ); L ∩ W , ( R )). Once again, we fill in the gap for the mass constraintthat is this time [4 π, π ). { g N } First, it will be established in Section 4.2 that { g N } is tight in the space Y := L w (cid:0) [0 , T ]; H β ( R ) (cid:1) ∩ C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) , (2.7)where L w ( R ) denotes the L ( R ) space endowed with the weak topology. It suffices for that toprove the boundedness of the sequence in a space that is compactly embedded in Y . By Prokhorov’stheorem, the tightness of { g N } implies that it is relatively compact in a sense that we precise now(because Prokhorov’s theorem applies only in Polish spaces, and L w is not metrizable). Indeed, wewill make a slight abuse of language in the following when we say that g N converges in law (resp.in probability, or almost surely) in Y : it will be understood that for any ϕ ∈ L (cid:0) [0 , T ]; H β ( R ) (cid:1) , h g N , ϕ i converges in law (resp. in probability or a.s.), and of course that g N converges in law (resp.in probability or a.s.) in C (cid:0) [0 , T ]; H γ loc ( R ) (cid:1) .Hence there is a subsequence of g N which converges in law in Y , and we still denote this subse-quence g N by a slight abuse of notation. We deduce from the previous discussion and Skorokhod’srepresentation theorem the following proposition. Proposition 2.4.
There exists a probability space (Ω , F , P ) rich enough to support { g N } N ∈ N andthere exists a Y -valued random variable ξ defined on (Ω , F , P ) such that g N Y −→ ξ a.s. Remark 2.5.
For each N and t ∈ [0 , T ] , the definition of g Nt yields that g Nt ∈ L ∞ ( R ) , andsince g Nt is a probability density function, it is also in L ( R ) . Hence by interpolation, g Nt ∈ T ∞ p =1 L p ( R ) .Now by Fatou’s lemma, one gets that ξ t ∈ L ( R ) . Moreover, by Sobolev embedding in dimension , ξ t ∈ H β with β > implies that ξ t ∈ L ∞ ( R ) (see e.g. [1, Thm 1.66]). Hence by interpolation, ξ t ∈ T ∞ p =1 L p ( R ) . In Subsection 4.1, we will then prove that for any test function ϕ , g N satisfies the followingequation h g Nt , ϕ i = h g N , ϕ i + Z t h S Ns , ∇ ( V N ∗ ϕ ) · F A (cid:0) ∇ G ∗ g Ns (cid:1) i ds + 1 N N X i =1 Z t ∇ ( V N ∗ ϕ )( X i,Ns ) · dW is + Z t h g Ns , ∆ ϕ i ds. (2.8)In Subsection 4.3, using (2.8) and the convergence result of Proposition 2.4, we will prove that thefollowing equality holds for all t > ϕ ∈ C ∞ c ( R ), h ξ ( t, · ) , ϕ i = h ρ , ϕ i + Z t Z R ξ ( s, x ) ∇ ϕ ( x ) · F A ( ∇ G ∗ ξ ( s, · ))( x ) dx ds + Z t h ξ ( s, · ) , ∆ ϕ i ds. (2.9)Observing that ξ ∈ Y , one deduces that R t ∇ ( ξ ( s, · ) F A ( ∇ G ∗ ξ ( s, · ))) ds ∈ L ( R ), hence thefollowing mild formulation in distribution holds: for any ϕ ∈ C ∞ c ( R ), h ξ ( t, · ) , ϕ i = h e t ∆ ρ , ϕ i − h Z t ∇ e ( t − s )∆ ( ξ ( s, · ) F A ( ∇ G ∗ ξ ( s, · ))) ds, ϕ i . ξ is non-random and that, for any t ∈ (0 , T ), it satisfiesalmost surely in R the following equation: ξ ( t, · ) = e t ∆ ρ − Z t ∇ e ( t − s )∆ ( ξ ( s, · ) F A ( ∇ G ∗ ξ ( s, · ))) ds. The next proposition will be useful in identifying ξ as a mild solution to (1.1). Its proof is givenin Subsection 4.4. Proposition 2.6.
Let ξ be as in Proposition 2.4. Then, ξ ∈ C b ([0 , T ); L ∩ H ( R )) . In the next subsection, we will state the existence and uniqueness of a function ξ ∈ C (cid:0) [0 , T ] , L ∩ H (cid:1) that satisfies the previous equation. Thus ξ will be called a mild solution of the cut-off PDE (2.2). In Section 3, we will study the mild solutions of the PDEs (1.1) and (2.2) with L ∩ H ( R ) spaceregularity. Although our results are close to the work of Nagai [17], it seems that they do notappear as such in the previous literature. Hence in Subsection 3.1, we will prove the followingtheorem for the Keller-Segel PDE (1.1). Theorem 2.7.
Let ρ ∈ L ∩ H ( R ) be a non-negative initial data. Then, there exists a uniquenon-negative mild solution to (1.1) locally in time.Assuming further that M < π , the non-negative mild solution of (1.1) exists globally in time. Remark 2.8.
In Corollary 3.4, we will obtain the following useful bound to compare the solutionsof (1.1) and a solution of (2.2) for a given
A > : There exists a universal constant C > suchthat the unique mild solution ρ of (1.1) satisfies ∀ t > , k∇ G ∗ ρ t k L ∞ ( R ) ≤ C ( M k ρ k L ∩ H ( R ) ) ∨ ( k ρ k L ∩ H ( R ) ) =: A . (2.10)As for the Keller-Segel PDE with cut-off, we will obtain in Subsection 3.2 the following unique-ness result. Theorem 2.9.
Let ρ ∈ L ∩ H ( R ) . Then for any A > and F defined in (2.1) , there is atmost one mild solutionto the cut-off PDE (2.2) . In Section 2.3, we have obtained that on the probability space (Ω , F , P ), g N converges almostsurely in Y to ξ , which satisfies the mild formulation (2.4) of the Keller-Segel equation (Proposition2.4 and the discussion below it). Thanks to Propostion 2.6, ξ is the unique mild solution to thePDE (2.2).Observe now that when M < π and A ≥ A , it follows from Equation (2.10) that a mildsolution ρ to (1.1) is also a mild solution to (2.2). Hence by uniqueness, ξ = ρ and now, ξ is a mildsolution to (1.1), as claimed in Theorem 2.3.Let us now come back to the original probability space (Ω , F , P ). We have obtained that everysubsequence of { g N } has a further subsequence that converges in law to ρ , the unique mild solutionof (1.1), in Y . Hence g N converges in law to ρ , and since ρ is non-random, the convergence alsohappens in probability for the topology of Y , which concludes the proof of Theorem 2.3. In this section we first prove Theorem 2.7 combining the results obtained in Nagai [17]. We chooseto work in the functional space C b ([0 , T ]; L ∩ H ( R )) for ρ . The latter will imply that ∇ c belongs8o the space C b ([0 , T ]; L ∞ ( R )) as seen below. This choice enables us to adapt all the techniques(described in the introduction) in order to obtain the uniform convergence of the mollified empiricalmeasure towards the solution of the KS model.Then, we focus on the cut-off equation (2.2) for some A >
We recall that the definition of a mild solution to (1.1) is given in Remark 2.2.
Lemma 3.1.
Let u be a mild solution to (1.1) on [0 , T ) . Then sup t ∈ [0 ,T ) k∇ G ∗ u t k L ∞ ( R ) < ∞ . Proof.
For any t > p ∈ [1 , ∞ ], we apply Lemma 2.5 of [17] with q = 3 to get k∇ G ∗ u t k L ∞ ( R ) ≤ C k u t k L ( R ) k u t k L ( R ) . (3.1)By applying Inequality (20) of [5, p.280] (with m = in the notation of [5]) , one obtains k u t k L ( R ) ≤ k u t k L ( R ) k∇ u t k L ( R ) . (3.2)Thus k∇ G ∗ u t k L ∞ ( R ) ≤ C k u t k L ( R ) k∇ u t k L ( R ) ≤ c k u k T,L ∩ H . Having in mind the fact that u is a mild solution to (1.1) and as such it belongs to C b ([0 , T ); L ∩ H ( R )), the proof is finished. Remark 3.2.
Let u a mild solution on (0 , T ) to (1.1) . Repeat the arguments of [17, Prop. 2.4]with the following modification. Everytime one needs to control k u ( ∇ G ∗ u ) k L ( R ) , it is possible touse the previous lemma and the fact that u satisfies Definition 2.1- i ) . Then, one obtains that Z R u t ( x ) dx = Z R u ( x ) dx. Moreover, when the initial data is such that u ≥ and u , then by repeating the argumentsof [17, Prop. 2.7], u is such that u ( t, x ) > , for ( t, x ) ∈ (0 , T ) × R . Remark 3.3.
In [17], mild solutions are considered in the space C − p ,T ( L p ( R )) of functions suchthat sup t ∈ [0 ,T ) t − p k u t k L p ( R ) < ∞ , with p = . Observe that if u ∈ C b ([0 , T ) , L ∩ H ( R )) , then by (3.2) , sup t ∈ [0 ,T ) k u t k L ( R ) is finite, and by an interpolation inequality, so is sup t ∈ [0 ,T ) k u t k L ( R ) .Hence u ∈ C − p ,T ( L p ( R )) for p = , therefore a mild solution in the sense of Remark 2.2 is alsoa solution in the sense of Nagai [17, Def. 2.1].It thus follows from the uniqueness result of Nagai ([17, Prop. 2.1]) that there can be at mostone mild solution in the sense of Definition 2.1 (see also the Proof of Proposition 3.6). Now we are in a position to prove the existence result given in Theorem 2.7.
Proof of Theorem 2.7.
In view of the above remarks it only remains to discuss the existence.The existence of a solution ρ to (2.4) in the sense of [17], i.e. such that ρ ∈ C − p ,T ( L p ( R ))for p = (see Remark 3.3) on some [0 , T ], is given by [17, Prop. 2.6]. We need to prove that ρ ∈ C ([0 , T ] , L ∩ H ( R )). We rely on the explicit bounds given by Inequality (2.24) of [17], inorder to get k ρ k t,H ∩ L ≤ k ρ k H ∩ L ( R ) + C ( t + √ t ) k ρ k t,H ∩ L .
9y the standard arguments of [17, Lemma 2.3] choosing T such that4 C ( T + p T ) k ρ k H ∩ L ( R ) < , we have that k ρ k T ,H ∩ L ≤ − q − C ( T + √ T ) k ρ k H ∩ L ( R ) C ( T + √ T ) . Choosing T such that, for example, 4 C ( T + √ T ) k ρ k H ∩ L ( R ) = , one has k ρ k T ,H ∩ L < k ρ k H ∩ L ( R ) . (3.3)Now, [17, Thm. 5.2] implies the global existence in time of the solution to (2.4) that satisfies, forany 1 ≤ p ≤ ∞ , k ρ t k L p ( R ) ≤ C p t − p , t > . (3.4)Now, noticing that k ρ t + T k L ( R ) ≤ C √ T for t ≥
0, it remains to control k∇ ρ t + T k L ( R ) for t > ρ t + T = e t ∆ ρ T − Z t ∇ · e ( t − s )∆ ( ρ s + T ( ∇ G ∗ ρ s + T )) ds. As ∇ ( ∇ G ∗ ρ ) = − ρ , we have, in view of (1.4), that k∇ ρ t + T k L ( R ) ≤k∇ ρ T k L ( R ) + Z t C √ t − s k∇ ρ s + T k L ( R ) k∇ G ∗ ρ s + T k ∞ ds + Z t C √ t − s k ρ s + T k L ( R ) ds. In view of (3.1) and (3.4), we have that k∇ G ∗ ρ s + T k ∞ ≤ C M √ T and k ρ s + T k L ( R ) ≤ CT k ρ s + T k L ( R ) ≤ CT √ s . Plugging this in the above inequality we obtain k∇ ρ t + T k L ( R ) ≤ k ρ k H ∩ L ( R ) + CT β ( , ) + C M √ T Z t k∇ ρ s + T k L ( R ) √ t − s ds, where β denotes the usual beta function.Singular Gronwall’s lemma allows us to conclude that ρ ∈ C b ([0 , T ); L ∩ H ( R )), for any T > Corollary 3.4.
Let
T > . Then, for any t ∈ [0 , T ] , one has k∇ G ∗ ρ t k L ∞ ( R ) ≤ C ( M k ρ k L ∩ H ( R ) ) ∨ ( k ρ k L ∩ H ( R ) ) . Proof.
Fix T as in the proof of Theorem 2.7 and let t < T . In view of Lemma 3.1 and (3.3), onehas k∇ G ∗ ρ t k L ∞ ( R ) ≤ C k ρ k H ∩ L ( R ) . Now, let t ∈ [ T , T ]. Combine (3.1) and (3.4). It comes k∇ G ∗ ρ t k L ∞ ( R ) ≤ CM / T / . Given the choice of T , one obtains the desired estimate.10 .2 Mild solutions of the modified Keller-Segel PDE In this section, we consider the cut-off system (2.2) and its mild solution from Definition 2.1. Here, F A is given in (2.1), but we denote it simply by F for the sake of readability. Remark 3.5.
Let ρ ∈ L ∩ H ( R ) such that ρ ≥ and ρ . Then, same arguments as inthe Remark 3.2 enable us to conclude that a solution to (2.3) is non-negative and that it admitsthe mass conservation. Proposition 3.6.
Let ρ ∈ L ∩ H ( R ) . Then there is at most one mild solution to (2.2) . Remark 3.7.
In view of Theorem 2.7 and Corollary 3.4, if one chooses
A > A , it follows thatthe mild solution ρ to (1.1) is a mild solution to (2.2) . We are now ready to prove Theorem 2.9 about the uniqueness of mild solutions to (2.2).
Proof of Theorem 2.9.
Assume there are two mild solutions ρ and ρ to (2.2). Then, ρ t − ρ t = − Z t ∇ · e ( t − s )∆ (cid:8) ρ s F ( ∇ ( G ∗ ρ s )) − ρ s F ( ∇ ( G ∗ ρ s )) (cid:9) ds = − Z t ∇ · e ( t − s )∆ (cid:8) ( ρ s − ρ s ) F ( ∇ ( G ∗ ρ s )) + ρ s ( F ( ∇ ( G ∗ ρ s )) − F ( ∇ ( G ∗ ρ s ))) (cid:9) ds. Hence k ρ t − ρ t k L ( R ) + k ρ t − ρ t k L ( R ) ≤ C A Z t √ t − s ( k ρ s − ρ s k L ( R ) + k ρ s ∇ G ∗ ( ρ s − ρ s ) k L ( R ) ) ds + C A Z t √ t − s k ρ s − ρ s k L ( R ) ds + Z t C √ t − s k ρ s ∇ G ∗ (cid:0) ρ s − ρ s (cid:1) k L ( R ) ds ≤ C √ t k ρ − ρ k τ,L ∩ H + C Z t k ρ s k L ( R ) + k ρ s k L ( R ) √ t − s k∇ G ∗ (cid:0) ρ s − ρ s (cid:1) k L ∞ ( R ) ds. (3.5)As in (3.2), k ρ s − ρ s k L ( R ) ≤ k ρ s − ρ s k L ( R ) k∇ ( ρ s − ρ s ) k L ( R ) and by using [17, Lemma 2.5] with q = 3, k∇ G ∗ (cid:0) ρ s − ρ s (cid:1) k L ∞ ( R ) ≤ C k ρ s − ρ s k L ( R ) k ρ s − ρ s k L ( R ) ≤ C k ρ s − ρ s k L ( R ) k∇ ( ρ s − ρ s ) k L ( R ) ≤ C k ρ − ρ k τ,L ∩ H . (3.6)Plugging this upper bound in (3.5) gives k ρ t − ρ t k L ( R ) + k ρ t − ρ t k L ( R ) ≤ k ρ − ρ k τ,L ∩ H C √ t (cid:0) k ρ k τ,L ∩ H (cid:1) . (3.7)Consider now ∇ ( ρ t − ρ t ) = − Z t ∇ · e ( t − s )∆ (cid:8) ∇ · (cid:8) ( ρ s − ρ s ) F ( ∇ ( G ∗ ρ s )) + ρ s ( F ( ∇ ( G ∗ ρ s )) − F ( ∇ ( G ∗ ρ s ))) (cid:9)(cid:9) ds = − Z t ∇ · e ( t − s )∆ ( ∇ ( ρ s − ρ s ) · F ( ∇ ( G ∗ ρ s ))) ds − Z t ∇ · e ( t − s )∆ (( ρ s − ρ s ) ∇ · F ( ∇ ( G ∗ ρ s )) ds − Z t ∇ · e ( t − s )∆ ( ∇ ρ s · ( F ( ∇ ( G ∗ ρ s )) − F ( ∇ ( G ∗ ρ s ))) ds − Z t ∇ · e ( t − s )∆ ( ρ s ( ∇ · F ( ∇ ( G ∗ ρ s )) − ∇ · F ( ∇ ( G ∗ ρ s )))) . k∇ ( ρ t − ρ t ) k L ( R ) ≤ C A Z t (cid:13)(cid:13) ∇ e ( t − s )∆ (cid:13)(cid:13) L → L ( k∇ ( ρ s − ρ s ) k L ( R ) + k ( ρ s − ρ s ) ∇ · F ( ∇ ( G ∗ ρ s ) k L ( R ) ++ k∇ ρ s ∇ G ∗ (cid:0) ρ s − ρ s (cid:1) k L ( R ) + k ρ s ( ∇ · F ( ∇ ( G ∗ ρ s )) − ∇ · F ( ∇ ( G ∗ ρ s ))) k L ( R ) ) ds ≤ C A Z t C ( t − s ) ( k ρ − ρ k τ,L ∩ H + k ρ s − ρ s k L ( R ) k∇ · F ( ∇ ( G ∗ ρ s ) k L ( R ) + k∇ ρ s k L ( R ) k∇ G ∗ (cid:0) ρ s − ρ s (cid:1) k L ∞ ( R ) + k ρ s k L ( R ) k∇ · F ( ∇ ( G ∗ ρ s )) − ∇ · F ( ∇ ( G ∗ ρ s )) k L ( R ) ) ds =: C A Z t C ( t − s ) ( k ρ − ρ k τ,L ∩ H + I ( s ) + I ( s ) + I ( s )) ds. (3.8)In view of (3.6), one has I ( s ) ≤ C k ρ k τ,L ∩ H k ρ − ρ k τ,L ∩ H . (3.9)To treat the other terms we first notice that, for i = 1 , ∇ · F ( ∇ G ∗ ρ is ) = f ′ A ( ∂ G ∗ ρ is ) ∂ ( G ∗ ρ is ) + f ′ A ( ∂ G ∗ ρ is ) ∂ ( G ∗ ρ is ) . Now, we need the following general result. For a function u ∈ C ([0 , T ]; L ∩ H ( R )), [5, Cor. 9.11](more precisely the first inequality in the proof, with m = N = 2) implies that k u t k L ( R ) ≤ k u t k L ( R ) k∇ u t k L ( R ) ≤ C k u k τ,L ∩ H . (3.10)Apply (3.10) for u = ρ − ρ , (A.20) for p = 4 and again (3.10) for u = ρ . It comes I ( s ) ≤ C k ρ − ρ k τ,L ∩ H k f ′ A k L ∞ ( R ) ( k∇ ( G ∗ ρ s ) k L ( R ) + k∇ ( G ∗ ρ s ) k L ( R ) ) ≤ C k ρ − ρ k τ,L ∩ H k∇ ( ∇ ( G ∗ ρ s )) k L ( R ) ≤ C k ρ − ρ k τ,L ∩ H k ρ s k L ( R ) ≤ C k ρ − ρ k τ,L ∩ H k ρ k τ,L ∩ H . (3.11)It remains to treat I ( s ). Similarly as above, k∇ · F ( ∇ ( G ∗ ρ s )) − ∇ · F ( ∇ ( G ∗ ρ s )) k L ( R ) (3.12) ≤ k f ′′ A k L ∞ ( R ) ( k∇ ( G ∗ ( ρ s − ρ s )) k L ( R ) + k∇ ( G ∗ ( ρ s − ρ s )) k L ( R ) ) ≤ C k∇ ( ∇ ( G ∗ ( ρ s − ρ s ))) k L ( R ) ≤ C k ρ s − ρ s k τ,L ∩ H . Thus, in view of (3.10) for u = ρ , one has I ( s ) ≤ C k ρ k τ,L ∩ H k ρ s − ρ s k τ,L ∩ H (3.13)Thus in view of (3.11), (3.9) and (3.13), we obtain k∇ ( ρ t − ρ t ) k L ( R ) ≤ C √ t k ρ − ρ k τ,L ∩ H (cid:0) k ρ k τ,L ∩ H + k ρ k τ,L ∩ H (cid:1) . (3.14)Therefore (3.7) and (3.14) yield k ρ − ρ k τ,L ∩ H ≤ C √ τ k ρ − ρ k τ,L ∩ H (cid:0) k ρ k τ,L ∩ H + k ρ k τ,L ∩ H (cid:1) . Hence for τ small enough, we deduce that k ρ − ρ k τ,L ∩ H = 0. Therefore the uniqueness holdsfor mild solutions on [0 , τ ]. Then by restarting the equation and using the same arguments asabove combined with similar arguments at the end of the proof of Theorem 2.7, one gets globaluniqueness. Recall that V : R → R + is an even smooth probability density function and that V N is defined byEquation (2.5), that { X i,N } is the particle system defined by (2.6) with cutoff F A given in (2.1).In this section, we use again the notation F instead of F A , for the sake of readability.12 .1 Equation satisfied by the regularised empirical measure: Proof ofEquality (2.8) Consider the mollified empirical measure g Nt := V N ∗ S Nt : x ∈ R Z R V N ( x − y ) dS Nt ( y ) = 1 N N X k =1 V N ( x − X k,Nt ) . Using this definition, we rewrite the particle system in (2.6) as dX i,Nt = F (cid:0) ( ∇ G ∗ g Nt )( X i,Nt ) (cid:1) dt + √ dW it , t ∈ [0 , T ] , ≤ i ≤ N. (4.1)Fix x ∈ R and 1 ≤ i ≤ N . Apply Itˆo’s formula to the function V N ( x − · ) and the particle X i,N .Then, sum for all 1 ≤ i ≤ N and divide by N . It comes g Nt ( x ) = g N ( x ) − N N X i =1 Z t ∇ V N ( x − X i,Ns ) · F (cid:0) ( ∇ G ∗ g Ns )( X i,Ns ) (cid:1) ds − N N X i =1 Z t ∇ V N ( x − X i,Ns ) · dW is + 1 N N X i =1 Z t ∆ V N ( x − X i,Ns ) ds. (4.2)Notice that1 N N X i =1 Z t ∇ V N ( x − X i,Ns ) · F (cid:0) ( ∇ G ∗ g Ns )( X i,Ns ) (cid:1) ds = Z t h S Ns , ∇ V N ( x − · ) · F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i ds and 1 N N X i =1 Z t ∆ V N ( x − X i,Ns ) ds = Z t ∆ g Ns ( x ) ds. The preceding equalities combined with (4.2) and the fact that ∇ V N ( − x ) = −∇ V N ( x ) (because V N is even) lead to g Nt ( x ) = g N ( x ) + Z t h S Ns , ∇ V N ( · − x ) · F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i ds + 1 N N X i =1 Z t ∇ V N ( X i,Ns − x ) · dW is + Z t ∆ g Ns ( x ) ds. (4.3)and for ϕ ∈ D ( R ), then (4.3) implies (2.8).For further use in Section 4.2, we also get the following mild form g Nt ( x ) = e t ∆ g N ( x ) + Z t e ( t − s )∆ h S Ns , ∇ V N ( · − x ) · F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i ds + 1 N N X i =1 Z t e ( t − s )∆ ∇ V N ( X i,Ns − x ) · dW is . (4.4)Finally, developing the scalar product, one has h S Ns , ∇ V N ( · − x ) · F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i = ∇ x · h S Ns , V N ( · − x ) F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i . Combining the latter with the fact that e t ∆ ∇ f = ∇ e t ∆ f , (4.4) reads g Nt ( x ) = e t ∆ g N ( x ) + Z t ∇ e ( t − s )∆ h S Ns , V N ( · − x ) · F (cid:0) ( ∇ G ∗ g Ns )( · ) (cid:1) i ds + 1 N N X i =1 Z t e ( t − s )∆ ∇ V N ( X i,Ns − x ) · dW is . (4.5)13 .2 Tightness of g N : Proof of Proposition 2.4 First, we point to [26, p.169, p.310] for the definition of the Sobolev spaces W η,q . Here, we willneed more particularly the space W η,q (cid:0) [0 , T ] ; H − ( R ) (cid:1) , for η > p >
1, with norm given in[26, p.323]: k f k pW η,q ([0 ,T ]; H − ( R )) ∼ k f k pL p ([0 ,T ]; H − ( R )) + Z T Z T k f t − f s k p − , | t − s | qη dsdt. Let us now prove the tightness of { g N } in the space Y defined in (2.7). This will be achievedby proving boundedness in the following space Y = L p (cid:0) [0 , T ] ; H β ( R ) (cid:1) ∩ W η,q (cid:0) [0 , T ] ; H − ( R ) (cid:1) , which is compactly embedded in Y , as proven in [7] (see also Section 2.3 of [9]), if ηϑ ≥ − ϑp + ϑq where ϑ := β − λ β .In the next two propositions, we compute the moments of g N in Y . Proposition 4.1.
Let the Assumption ( C0 ) hold. Let p ≥ . Then there exists a constant C β,T,A,p > such that, for all t ∈ (0 , T ] and N ∈ N , it holds: E (cid:20)(cid:13)(cid:13)(cid:13) (I − ∆) β/ g Nt (cid:13)(cid:13)(cid:13) pL ( R ) (cid:21) ≤ C β,T,A,p . Proposition 4.2.
Let the Assumption ( C0 ) hold. Let η ∈ (0 , ) and q ≥ . There exists aconstant C β,T,A,q > such that, for any N ∈ N , it holds: E " Z T Z T (cid:13)(cid:13) g Nt − g Ns (cid:13)(cid:13) q − , | t − s | qη ds dt ≤ C η,T,A,q . The proofs of these two results are similar to the proofs of Propositions 6 and 7 in [9] (thekernel plays no role here), but we reproduce them in Appendix for the sake of completeness.
Remark 4.3.
We recall the following classical inequality for β > , based on the isometry propertyof F − , k ( I − ∆) f k L ( R ) = k (cid:0) | · | (cid:1) F f k L ( R ) ≤ Z R (cid:0) | x | (cid:1) β |F f ( x ) | dx = k ( I − ∆) β f k L ( R ) . Hence, Proposition 4.1 implies that E (cid:20)(cid:13)(cid:13)(cid:13) (I − ∆) g Nt (cid:13)(cid:13)(cid:13) pL ( R ) (cid:21) ≤ C β,T,A,p . Observing that k∇ ξ t k L ( R ) ≤ C k ( I − ∆) ξ t k L ( R ) and by applying Fatou’s lemma, it follows that ∀ t ∈ [0 , T ] , k∇ ξ t k L ( R ) ≤ C lim inf n →∞ E (cid:20)(cid:13)(cid:13)(cid:13) (I − ∆) g Nt (cid:13)(cid:13)(cid:13) L ( R ) (cid:21) ≤ C β,T,A, , (4.6) since ξ is deterministic. Similarly, one gets that ∀ t ∈ [0 , T ] , k ξ t k L ( R ) ≤ C β,T,A, . The Chebyshev inequality ensures that P (cid:0) k g N · k Y > R (cid:1) ≤ E (cid:2)(cid:13)(cid:13) g N · (cid:13)(cid:13) Y (cid:3) R , for any
R > . P (cid:0)(cid:13)(cid:13) g N · (cid:13)(cid:13) Y > R (cid:1) ≤ CR , for any
R > , N ∈ N . Let P N be the law of g N in Y . The last inequality implies that there exists a bounded set B ǫ ∈ Y such that P N ( B ǫ ) < − ǫ for all N , and therefore there exists a compact set K ǫ ∈ Y such that P N ( K ǫ ) < − ǫ . That is, the sequence of random variables { g N } is tight in Y . Therefore wededuce that Proposition 2.4 holds. (2.9) First it comes from Assumption ( C0 ) on the initial condition that h g N , ϕ i → h ρ , ϕ i . In view of Proposition 2.4, recall that g N → ξ in the space Y which was defined in Equation (2.7).First, it is clear that this result implies that we can to pass to the limit in (2.8): Z t h g Ns , ∆ ϕ i ds → Z t h ξ s , ∆ ϕ i ds, and E N N X i =1 Z t ∇ ( V N ∗ ϕ )( X i,Ns ) · dW is ! = 1 N N X i =1 Z t E h(cid:0) ∇ ( V N ∗ ϕ )( X i,Ns ) (cid:1) i ds → . To conclude that ξ satisfies Equation (2.9), it remains to prove Lemma 4.4.
For any t ∈ [0 , T ] , the following convergence happens a.s. Z t h S Ns , ∇ ( V N ∗ ϕ ) · F (cid:0) ∇ G ∗ g Ns (cid:1) i ds → Z t Z R ξ s ( x ) ∇ ϕ ( x ) · F ( ∇ G ∗ ξ s )( x ) dx ds. (4.7) Proof.
First, let ǫ > B ǫ be a ball centred in 0, with a sufficiently large radius to ensurethat Z R B cǫ ( y ) |∇ G ( y ) | dy ≤ ǫ . (4.8)In view of Proposition 2.4, one has that for all x ∈ R , there is N large enough such thatsup t ∈ [0 ,T ] ,y ∈ B ǫ | g Nt ( x − y ) − ξ t ( x − y ) | ≤ ǫ . It follows, using the Cauchy-Schwarz inequality inthe second inequality and the bound (4.8) in the third, that |∇ G ∗ ( g Ns − ξ s )( x ) | = | Z R B ǫ ( y ) ∇ G ( y )( g Ns − ξ s )( x − y ) dy + Z R B cǫ ( y ) ∇ G ( y )( g Ns − ξ s )( x − y ) dy |≤ ǫ Z R B ǫ ( y ) |∇ G ( y ) | dy + (cid:18)Z R B cǫ ( y ) |∇ G ( y ) | dy Z R | ( g Ns − ξ s )( x − y ) | dy (cid:19) ≤ ǫ (cid:18)Z B ǫ |∇ G ( y ) | dy + k g Ns − ξ s k L ( R ) (cid:19) . Hence it follows that for any s ∈ [0 , T ] and any x ∈ R , (cid:0) ∇ G ∗ g Ns (cid:1) ( x ) → ( ∇ G ∗ ξ s ) ( x ) a.s. Next, observe that (cid:12)(cid:12)(cid:12)D S Ns , ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) E − D g Ns , ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) E(cid:12)(cid:12)(cid:12) ≤ sup x ∈ R (cid:12)(cid:12)(cid:12) ∇ ( V N ∗ ϕ )( x ) · F ( ∇ G ∗ g Ns ) − (cid:0) ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) (cid:1) ∗ V N ( x ) (cid:12)(cid:12)(cid:12) . R R V = 1 and V ≥
0, one first gets that (cid:12)(cid:12)(cid:12) ∇ ( V N ∗ φ )( x ) · F ( ∇ G ∗ g Ns )( x ) − (cid:0) ∇ ( V N ∗ φ ) · F ( ∇ G ∗ g Ns ) (cid:1) ∗ V N ( x ) (cid:12)(cid:12)(cid:12) ≤ Z R V ( y ) (cid:12)(cid:12) ∇ ( V N ∗ φ )( x ) (cid:12)(cid:12) (cid:12)(cid:12) F ( ∇ G ∗ g Ns )( x ) − F ( ∇ G ∗ g Ns ) (cid:0) x − yN α (cid:1)(cid:12)(cid:12) dy + Z R V ( y ) (cid:12)(cid:12) ∇ ( V N ∗ φ )( x ) − ∇ ( V N ∗ φ ) (cid:0) x − yN α (cid:1)(cid:12)(cid:12) (cid:12)(cid:12) F ( ∇ G ∗ g Ns ) (cid:0) x − yN α (cid:1)(cid:12)(cid:12) dy ≤ C Z R V ( y ) (cid:12)(cid:12) ∇ ( V N ∗ φ )( x ) (cid:12)(cid:12) (cid:12)(cid:12) ∇ G ∗ g Ns ( x ) − ∇ G ∗ g Ns (cid:0) x − yN α (cid:1)(cid:12)(cid:12) dy + CN α Z R V ( y ) | y | dy where the second inequality comes using the Lipschitz continuity and boundedness of F . Now inview of (A.21), for some p ∈ (2 , ∞ ) and η = 1 − p , one has (cid:12)(cid:12) ∇ G ∗ g Ns ( x ) − ∇ G ∗ g Ns (cid:0) x − yN α (cid:1)(cid:12)(cid:12) ≤ (cid:13)(cid:13) ∇ G ∗ g Ns (cid:13)(cid:13) C η (cid:16) | y | N α (cid:17) η ≤ C p k g Ns k L p ( R ) | y | η N ηα . Therefore, (cid:12)(cid:12)(cid:12) ∇ ( V N ∗ ϕ )( x ) · F ( ∇ G ∗ g Ns )( x ) − (cid:0) ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) (cid:1) ∗ V N ( x ) (cid:12)(cid:12)(cid:12) ≤ C Z R V ( y ) (cid:12)(cid:12) ∇ ( V N ∗ ϕ )( x ) (cid:12)(cid:12) C p k g Ns k L p ( R ) | y | η N ηα dy + CN α Z R V ( y ) | y | dy. Thus we have obtained (cid:12)(cid:12) ∇ ( V N ∗ ϕ )( x ) · F ( ∇ G ∗ g Ns )( x ) − (cid:0) ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) (cid:1) ∗ V N ( x ) (cid:12)(cid:12) ≤ C N α + k g Ns k L p ( R ) N ηα ! . Recall that { g N } N ∈ N converges almost surely in L ([0 , T ] , H β ) for the weak topology, hence itis bounded in this space (by the uniform boundedness principle). Thus, sup N R T k g Ns k H β ds < ∞ ,and by interpolation inequality and Sobolev embedding, sup N R T k g Ns k θL p ( R ) ds < ∞ for θ = 1 − p .It follows that sup N R T k g Ns k L p ( R ) ds < ∞ and thereforelim N →∞ Z t D S Ns , ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) E ds = lim N →∞ Z t D g Ns , ∇ ( V N ∗ ϕ ) · F ( ∇ G ∗ g Ns ) E ds = lim N →∞ Z t Z R g Ns ( x ) ∇ ( V N ∗ ϕ )( x ) · F ( ∇ G ∗ g Ns )( x ) dxds = Z t Z R ξ s ( x ) ∇ ϕ ( x ) · F ( ∇ G ∗ ξ s )( x ) dxds where in the last equality we used that g N a.s. −→ ξ strongly in L ([0 , T ]; C ( D )) for D the compactsupport of ϕ (recall that g N converges a.s. in C ([0 , T ] , H γ loc ), hence by Sobolev embedding anddominated convergence, the convergence in L ([0 , T ]; C ( D )) holds). ξ : Proof of Proposition 2.6 As ξ ∈ Y , we know that for any t ∈ [0 , T ), ξ t ∈ L ∩ H ( R ). Observe that for p = 1 , t ∈ [0 , T ), we have k g Nt k L p ( R ) ≤ C T,p : indeed, for p = 1, this is because g Nt is a probability densityfunction; for p = 2, this follows from Proposition 4.1 and is explained in Remark 4.3. Hence withFatou’s lemma, this implies thatsup t ≤ T (cid:0) k ξ t k L ( R ) + k ξ t k L ( R ) (cid:1) ≤ C T . (4.9)16n addition, in view of Remark 4.3, one hassup t ≤ T k∇ ξ t k L ( R ) ≤ C T . (4.10)It only remains to prove that for any t ∈ [0 , T ), one haslim s → t k ξ t − ξ s k L ∩ H ( R ) = 0 . (4.11)This follows from the above properties of ξ and the mild form satisfied by ξ . Namely, almosteverywhere in R , one has ξ t = e ( t − s )∆ ξ s + Z ts ∇ e ( t − r )∆ ( ξ r F ( ∇ G ∗ ξ r )) dr. To check (4.11), we need to ensure thatlim s → t Z ts k∇ e ( t − r )∆ ( ξ r F ( ∇ G ∗ ξ r )) k L ∩ H ( R ) dr = 0 . (4.12)This will follow from the continuity of the integral if the integral is well-defined. For p = 1 ,
2, wehave that Z ts k∇ e ( t − r )∆ ( ξ r F ( ∇ G ∗ ξ r )) k L p ( R ) dr ≤ Z ts C A √ t − s k ξ r k L p ( R ) dr. In view of (4.9), the integral is well-defined. Now we turn to the H -norm. Notice that Z ts k∇ e ( t − r )∆ ( ξ r F ( ∇ G ∗ ξ r )) k H ( R ) dr ≤ Z ts k∇ e ( t − r )∆ ( ∇ ξ r F ( ∇ G ∗ ξ r )) k L ( R ) dr + Z ts k∇ e ( t − r )∆ ( ξ r f ′ ( ∇ G ∗ ξ r )) ∇ G ∗ ξ r k L ( R ) dr =: I + II.
Now let us use the boundedness of F , a convolution inequality and (4.10). Then it comes I ≤ C A Z ts √ t − s k∇ ξ r k dr ≤ C T,A √ t − s. For II , we use the properties of f A , the property (A.20) of G and (4.9). It comes II ≤ C A Z ts √ t − s k∇ G ∗ ξ r k L ( R ) dr ≤ C Z ts √ t − s k ξ r k L ( R ) dr ≤ C √ t − s. Hence the proof is complete.
Appendix
A.1 Proofs of technical results
Proof of Proposition 4.1.
Step 1 . Let H = L ( R ) and let F stand for the function F A defined in(2.1). From (4.5) after applying (I − ∆) β/ and by the triangular inequality we have (cid:13)(cid:13)(cid:13) (I − ∆) β/ g Nt (cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ (cid:13)(cid:13)(cid:13) (I − ∆) β/ e t ∆ g N (cid:13)(cid:13)(cid:13) L p (Ω; H ) (A.13)+ Z t (cid:13)(cid:13)(cid:13) (I − ∆) β/ ∇ e ( t − s )∆ (cid:0) V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) L p (Ω; H ) ds (A.14)+ (cid:13)(cid:13)(cid:13)(cid:13) N N X i =1 Z t (I − ∆) β/ ∇ e ( t − s )∆ (cid:0) V N (cid:0) · − X i,Ns (cid:1)(cid:1) dW is (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω; H ) . (A.15)17 tep 2 . Noticing that by a convolution inequality k (I − ∆) β/ e t ∆ g N k L ( R ) ≤ k e t ∆ k L → L k (I − ∆) β/ g N k L ( R ) ,one gets that the first term (A.13) can be estimated by (cid:13)(cid:13)(cid:13) (I − ∆) β/ e t ∆ g N (cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ (cid:13)(cid:13)(cid:13) (I − ∆) β/ g N (cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ C β , with C β >
0. The boundedness of the norm of g N follows from Assumption ( C0 i ). Step 3 . Let us come to the second term (A.14): Z t (cid:13)(cid:13) (I − ∆) β/ ∇ e ( t − s )∆ (cid:0) V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1)(cid:1) (cid:13)(cid:13) L p (Ω; H ) ds ≤ C Z t (cid:13)(cid:13) ∇ e t − s ∆ (cid:13)(cid:13) L → L (cid:13)(cid:13) (I − ∆) β/ e t − s ∆ (cid:0) V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1)(cid:1) (cid:13)(cid:13) L p (Ω; H ) ds. We have (cid:13)(cid:13) (I − ∆) / e (( t − s ) / (cid:13)(cid:13) L → L ≤ C ( t − s ) / . On the other hand, for any x ∈ R , (cid:12)(cid:12) (cid:0) V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1)(cid:1) ( x ) (cid:12)(cid:12) ≤ (cid:13)(cid:13) F ( ∇ G ∗ g Ns ) (cid:13)(cid:13) ∞ (cid:12)(cid:12) V N ∗ S Ns ( x ) (cid:12)(cid:12) ≤ A (cid:12)(cid:12) g Ns ( x ) (cid:12)(cid:12) . By Lemma 16 in [8] we have (cid:13)(cid:13)(cid:13) (I − ∆) β/ e (( t − s ) / h V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1) i(cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ C A (cid:13)(cid:13) e (( t − s ) / (I − ∆) β/ g Ns (cid:13)(cid:13) L p (Ω; H ) ≤ C A (cid:13)(cid:13) (I − ∆) β/ g Ns (cid:13)(cid:13) L p (Ω; H ) . To summarize, we have Z t (cid:13)(cid:13)(cid:13) (I − ∆) β/ ∇ e ( t − s )∆ (cid:0) V N ∗ (cid:0) F ( ∇ G ∗ g Ns ) S Ns (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) L p (Ω; H ) ds ≤ C β,A Z t ( t − s ) − (cid:13)(cid:13)(cid:13) (I − ∆) β/ g Ns (cid:13)(cid:13)(cid:13) L p (Ω; H ) ds. This bounds the second term.
Step 4 . For the third term (A.15), we have by Lemma 10 in [9] that for any δ >
0, there exists C β,T,p,δ > (cid:13)(cid:13)(cid:13)(cid:13) N N X i =1 Z t (I − ∆) β/ ∇ e ( t − s )∆ (cid:0) V N (cid:0) · − X i,Ns (cid:1)(cid:1) dW is (cid:13)(cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ C β,T,p,δ N ( α (2+2 δ +2 β ) − . Therefore, taking α < β and δ small enough, the last quantity is bounded by some C β,T,p .Collecting the three bounds together, we get (cid:13)(cid:13)(cid:13) (I − ∆) β/ g Nt (cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ C β,T,p + C β,A Z t ( t − s ) − (cid:13)(cid:13)(cid:13) (I − ∆) β/ g Ns (cid:13)(cid:13)(cid:13) L p (Ω; H ) ds. We can now apply Gronwall’s Lemma to deduce (cid:13)(cid:13)(cid:13) (I − ∆) β/ g Nt (cid:13)(cid:13)(cid:13) L p (Ω; H ) ≤ C β,T,A,p . roof of Proposition 4.2 . Let us now prove the second estimate on g N given in Proposition 4.2.In this proof we use the fact that L ( R ) ⊂ H − with continuous embedding, and that the linearoperator ∆ is bounded from L ( R ) to H − .We first observe that g Nt ( x ) − g Ns ( x ) = Z ts (cid:10) S Nr , ( ∇ G ∗ F ( g Nr )) ∇ V N ( x − · ) (cid:11) dr + ν Z ts ∆ g Nr ( x ) dr + 1 N N X i =1 Z ts ∇ (cid:0) V N (cid:1) (cid:0) x − X i,Nr (cid:1) dW ir . Thus we obtain E h (cid:13)(cid:13) g Nt ( x ) − g Ns ( x ) (cid:13)(cid:13) q − , i ≤ ( t − s ) q − Z ts E h (cid:13)(cid:13)(cid:10) S Nr , F ( ∇ G ∗ g Nr ) ∇ V N ( x − · ) (cid:11)(cid:13)(cid:13) q − , i dr (A.16)+ ( t − s ) q − Z ts E h (cid:13)(cid:13) ∆ g Nr ( x ) (cid:13)(cid:13) q − , i dr (A.17)+ E (cid:20)(cid:13)(cid:13)(cid:13)(cid:13) N N X i =1 Z ts ∇ (cid:0) V N (cid:1) (cid:0) x − X i,Nr (cid:1) dW ir (cid:13)(cid:13)(cid:13)(cid:13) q − , (cid:21) . (A.18)To estimate the first term (A.16) we observe first that E h(cid:13)(cid:13) (cid:10) S Nr , F ( ∇ G ∗ g Nr ) ∇ V N ( x − · ) (cid:11) (cid:13)(cid:13) q − , i = E h (cid:13)(cid:13) ∇ ( S Nr F ( ∇ G ∗ g Nr ) ∗ V N ) (cid:13)(cid:13) q − , i ≤ E h (cid:13)(cid:13) ( S Nr F ( ∇ G ∗ g Nr ) ∗ V N (cid:13)(cid:13) q − , i . ≤ C A E h (cid:13)(cid:13) g Nt (cid:13)(cid:13) qL ( R ) i ≤ C A . Moreover, for the second term (A.17) we have E h (cid:13)(cid:13) ∆ g Nr (cid:13)(cid:13) q − , i ≤ C E h (cid:13)(cid:13) g Nr (cid:13)(cid:13) qL ( R ) i ≤ C. (A.19)Now, we bound the last term (A.18): E (cid:20)(cid:13)(cid:13)(cid:13)(cid:13) N N X i =1 Z ts ∇ (cid:0) V N (cid:1) (cid:0) x − X i,Nr (cid:1) dW ir (cid:13)(cid:13)(cid:13)(cid:13) q − , (cid:21) ≤ C q E (cid:20) N N X i =1 Z ts (cid:13)(cid:13)(cid:13)(cid:13) ∇ (cid:0) V N (cid:1) (cid:0) x − X i,Nr (cid:1) (cid:13)(cid:13)(cid:13)(cid:13) q − , dr (cid:21) q/ Then we have1 N Z R N X i =1 Z ts (cid:12)(cid:12)(cid:12) (I − ∆) − ∇ (cid:0) V N (cid:1) (cid:0) x − X i,Nr (cid:1) (cid:12)(cid:12)(cid:12) drdx = ( t − s ) 1 N (cid:13)(cid:13) V N (cid:13)(cid:13) − , ≤ ( t − s ) 1 N (cid:13)(cid:13) V N (cid:13)(cid:13) , ≤ CN α − ( t − s ) ≤ C ( t − s ) . In order to conclude the lemma, we need to divide (A.16)–(A.18) by | t − s | qη . From the previousestimates, we always get a term of the form | t − s | ρ with ρ < η < ).19 .2 Properties of the Kernel G Recall that ∇ G ( x ) = − x / k x k , x / k x k ). Then for any ball D ⊂ R , it follows from a polarcoordinate change of variables that ∇ G ∈ L p ( D ) if and only if p ∈ [1 , B denote the unitball of R . Let p ∈ [1 , ∞ ). Then it follows from Young’s inequality that for any q ∈ ( pp +2 , p ], any q ∈ ( pp +1 , pp +2 ) and any f ∈ L q ∩ L q ( R ), k∇ G ∗ f k L p ( R ) ≤ k ( B ∇ G ) ∗ f k L p ( R ) + k ( B c ∇ G ) ∗ f k L p ( R ) ≤ k B ∇ G k L r ( R ) k f k L q ( R ) + k B c ∇ G k L r ( R ) k f k L q ( R ) ≤ C G (cid:0) k f k L q ( R ) + k f k L q ( R ) (cid:1) where r − = 1 + p − q ∈ ( ,
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