Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


IMPACT FACTOR 2018: 6.636

CiteScore 2018: 5.03

SCImago Journal Rank (SJR) 2018: 3.215
Source Normalized Impact per Paper (SNIP) 2018: 3.225

Mathematical Citation Quotient (MCQ) 2017: 1.89

Open Access
Online
ISSN
2191-950X
See all formats and pricing
More options …

On the Cauchy problem of a degenerate parabolic-hyperbolic PDE with Lévy noise

Imran H. Biswas
  • Centre for Applicable Mathematics, Tata Institute of Fundamental Research, P.O. Box 6503, GKVK Post Office, Bangalore 560065, India
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ananta K. Majee / Guy Vallet
  • Corresponding author
  • Université de Pau & Pays de l’Adour – LMAP UMR – CNRS 5142, IPRA BP 1155, 64013 Pau Cedex, France
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2017-09-12 | DOI: https://doi.org/10.1515/anona-2017-0113

Abstract

In this article, we deal with the stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasis is on analyzing the effects of a multiplicative Lévy noise on such problems and on establishing a well-posedness theory by developing a suitable weak entropy solution framework. The proof of the existence of a solution is based on the vanishing viscosity technique. The uniqueness of the solution is settled by interpreting Kruzhkov’s doubling technique in the presence of a noise.

Keywords: Stochastic PDEs; Lévy noise; degenerate parabolic equations; entropy solutions; Young measures

MSC 2010: 35K65 - 60H15

1 Introduction

Stochastic partial differential equations (SPDEs) often result from the efforts to model complex physical phenomena where uncertainty/randomness is inherent. A brief survey of relevant literature reveals the use of SPDEs in a wide variety of studies in areas that include physics, biology, engineering and finance. As examples we mention the studies of neural dynamics, the spread of infectious disease (a tracer or a passive population in a flow subject to possibly random external forces), Navier–Stokes-type flow under random forces, ferromagnetism under random influences (stochastic Landau–Lifschitz–Gilbert equation) and the modeling of forward rate curve in finance. A popular way to account for the randomness has been to add some noise to the deterministic models and, most often, the noise is assumed to be Gaussian/Brownian. However, the data collected through surveys/experiments often exhibit properties such as heavy-tailedness, which can not be adequately explained by a Brownian noise. This inadequacy has lately prompted a lot of interest in models with Lévy-type noise, which necessitates the development of a well-rounded mathematical theory for SPDEs driven by such type of noise. In this article, we embark on the well-posedness study of a class of nonlinear and degenerate parabolic SPDEs with a Lévy noise.

Let (Ω,P,,{t}t0) be a filtered probability space satisfying the usual hypothesis, i.e., {t}t0 is a right-continuous filtration such that 0 contains all the P-null subsets of (Ω,). In addition, let (E,,m) be a σ-finite measure space and let N(dt,dz) be a Poisson random measure on (E,) with intensity measure m(dz) with respect to the same stochastic basis. The existence and construction of such a general notion of Poisson random measure with a given intensity measure are detailed in [19]. We are interested in the Cauchy problem for a nonlinear degenerate parabolic stochastic PDE of the following type:

du(t,x)-Δϕ(u(t,x))dt-divxf(u(t,x))dt=Eη(x,u(t,x);z)N~(dz,dt),(t,x)ΠT,(1.1)

with the initial condition

u(0,x)=u0(x),xd,

where ΠT=[0,T)×d with T>0 fixed, u(t,x) is the unknown random scalar-valued function, F:d is a given flux function and N~(dz,dt)=N(dz,dt)-m(dz)dt is the compensated Poisson random measure. Furthermore, (x,u,z)η(x,u;z) is a real-valued function defined on the domain d××E, and ϕ: is a given non-decreasing Lipschitz continuous function. The stochastic integral on the right-hand side of (1.1) is defined in the Lévy–Itô sense.

Remark 1.1.

Since ϕ is a real-valued non-decreasing and Lipschitz continuous function, we know that the set A={r:ϕ(r)=0} is not empty in general and hence the problem is called degenerate. Furthermore, A is not negligible either and the problem is strongly degenerate in the sense of [9].

Remark 1.2.

The analysis of this paper remains valid if the noise in the right-hand side of (1.1) is of jump-diffusion type. In other words, the same analysis holds if we add the term σ(x,u)dWt in the right-hand side of (1.1), where Wt denotes a cylindrical Brownian motion. Moreover, we will carry out our analysis under the structural assumption that E=𝒪×*, where 𝒪 is a subset of the Euclidean space. The measure m on E is defined as the product measure λ×μ, where λ is a Radon measure on 𝒪 and μ is a so-called Lévy measure on *. In such a case, the noise in the right-hand side would be called an impulsive white noise with the jump position intensity λ and the jump size intensity μ. We refer to [19] for more on Lévy sheet and related impulsive white noises.

Remark 1.3.

In so far as the techniques are concerned, one anticipates to follow closely the methodology used in analyzing a version of (1.1) with a Brownian noise in [3]. However, there will be additional difficulties specific to the discontinuous character of the Lévy noise. Note that the solution of problem (1.1) is interpreted in the entropy sense as in [9, 3]. The entropy inequalities will have an Itô–Lévy correction term, and they will have non-localities resulting from the jump nature of the noise term in the right-hand side of (1.1). This non-locality tends to derail the stability analysis, and proving the uniqueness becomes a trickier affair. We are able to manage this added difficulty here and establish the uniqueness, but under a slightly restrictive assumption (iv) on the jump vector η (see Section 2.1).

Equation (1.1) becomes a multi-dimensional deterministic degenerate parabolic-hyperbolic equation if η=0. It is well-documented in the literature that the solution has to be interpreted in the weak sense and one needs an entropy formulation to prove the well-posedness of the problem. We refer to [1, 9, 11, 12, 4, 20] and the references therein for more on the entropy solution theory for deterministic degenerate parabolic-hyperbolic equations.

1.1 Studies on degenerate parabolic-hyperbolic equations with a Brownian noise

The study of stochastic degenerate parabolic-hyperbolic equations has so far been limited to mainly equations with a Brownian noise. In particular, hyperbolic conservation laws with a Brownian noise are examples of such problems that have attracted the attention of many. The first documented development in this direction is [17], where Holden and Risebro established a result of existence of a path-wise weak solution (possibly non-unique) for one-dimensional balance laws via the splitting method. In a separate development, Khanin, Mazel and Sinai [15] published their celebrated work that described some statistical properties of Burgers equations with a noise. Kim [18] extended the Kruzhkov entropy formulation and established the well-posedness of the Cauchy problem for one-dimensional balance laws driven by an additive Brownian noise. The multi-dimensional analogue on bounded domains was studied by Vallet and Wittbold [21]. They established a result of well-posedness of the entropy solution with the theory of Young measures.

This approach is not applicable to the multiplicative noise case. This case was studied by many authors [2, 10, 14, 16]. In [16], Feng and Nualart found a way to recover the necessary information in the form of the strong entropy condition from the parabolic regularization, and they established a result of uniqueness of the strong entropy solution in the Lp-framework for the multi-dimensional case, but the existence of a solution was established only in the one-dimensional case. We also add here that Feng and Nualart [16] used an entropy formulation which is strong in time but weak in space, which in our view may give rise to problems when the solutions are not shown to have continuous sample paths. We refer to [7], where a few technical questions are raised on the strong in time formulation and remedial measures have been proposed. In [14], Debussche and Vovelle obtained the existence of a solution via a kinetic formulation, and Chen, Ding and Karlsen [10] used the BV solution framework. Bauzet, Vallet and Wittbold [2] established a result of well-posedness via the Young measure approach. The well-posedness of the problem to a multi-dimensional degenerate parabolic-hyperbolic stochastic problem has been studied by Debussche, Hofmanová and Vovelle [13] and Bauzet, Vallet and Wittbold [3]. The former adapted the notion of kinetic formulation and developed a well-posedness theory, while the latter revisited [1, 9, 11] and established the well-posedness of the entropy solution via the Young measure theory.

1.2 Relevant studies on problems with a Lévy noise

Over the last decade, there have been many contributions on the larger area of stochastic partial differential equations that are driven by a Lévy noise. A worthy reference on this subject is [19]. However, very little is available on the specific problem of degenerate parabolic problems with a Lévy noise such as (1.1). This article marks an important step in our quest to develop a comprehensive theory of stochastic degenerate parabolic equations that are driven by jump-diffusions. The relevant results in this context are made available recently and they are on conservation laws that are perturbed by a Lévy noise. In recent articles, Biswas, Karlsen and Majee [5] and Biswas, Koley and Majee [6] established the well-posedness of the entropy solution for multi-dimensional conservation laws with a Poisson noise via the Young measure approach. In [6], Biswas et al. developed a continuous dependence theory on nonlinearities within the BV solution setting.

Stochastic degenerate parabolic-hyperbolic equations are one of the most important classes of nonlinear stochastic PDEs. Nonlinearity and degeneracy are two main features on these equations and yield several striking phenomena. Therefore, this requires new mathematical ideas, approaches, and theories. It is well known that due to the presence of nonlinear flux terms, solutions to (1.1) are not smooth even for smooth initial data u0(x). Therefore, a solution must be interpreted in the weak sense. Before introducing the concept of weak solutions, we first recall the notion of predictable σ-field. By a predictable σ-field on [0,T]×Ω, denoted by 𝒫T, we mean the σ-field generated by the sets of the form {0}×A and (s,t]×B for any A0, Bs, 0<s,tT. The notion of a stochastic weak solution is defined as follows.

Definition 1.4 (Stochastic weak solution).

We say that an L2(d)-valued {t:t0}-predictable stochastic process u(t)=u(t,x) is a weak solution to problem (1.1) provided the following conditions are satisfied:

  • (i)

    uL2(Ω×ΠT) and ϕ(u)L2((0,T)×Ω;H1(d)).

  • (ii)

    It holds

    t[u-0tEη(x,u(s,);z)N~(dz,ds)]L2((0,T)×Ω;H-1(d))

    in the sense of distribution.

  • (iii)

    For almost every t[0,T] and P-a.s., the following variational formulation holds:

    0=t[u-0tEη(x,u(s,);z)N~(dz,ds)],vH-1(d),H1(d)+d{ϕ(u(t,x))+f(u(t,x))}.vdx

    for any vH1(d).

However, it is well known that weak solutions may be discontinuous and are not uniquely determined by their initial data. Consequently, an admissibility criterion for the so-called entropy solution (see Section 2 for the definition of an entropy solution) must be imposed to single out the physically correct solution.

1.3 Goal of the study and outline of the paper

The case of a strongly degenerate stochastic problem driven by a Brownian noise is studied by Bauzet et al. [3]. In this article, drawing primary motivation from [3, 5, 9], we propose to establish a result of well-posedness of the entropy solution to a degenerate Cauchy problem (1.1) by using the vanishing viscosity method along with few a priori bounds.

The rest of the paper is organized as follows: We state the assumptions, details of the technical framework and the main results in Section 2. Section 3 is devoted to prove the existence of a weak solution for the viscous problem via an implicit time discretization scheme and to derive some a priori estimates for the sequence of viscous solutions. In Section 4, we establish first the uniqueness of the limit of the viscous solutions when the viscosity parameter goes to zero via the Young measure theory, and then we establish the existence of an entropy solution. The uniqueness of the entropy solution is presented in Section 5.

2 Technical framework and statements of the main results

Here and in the sequel, we denote by Nω2(0,T,L2(d)) the space of predictable L2(d)-valued processes u such that

𝔼[ΠT|u|2𝑑t𝑑x]<+.

Moreover, we use the letter C to denote various generic constants. There are situations where the constant may change from line to line, but the notation is kept unchanged so long as it does not impact the primary implication. We denote by cϕ and cf the Lipschitz constants of ϕ and f, respectively. Also, we use , to denote the pairing between H1(d) and H-1(d).

2.1 Entropy inequalities

We begin this subsection with a formal derivation of the entropy inequalities à la Kruzhkov. Remember that we need to replace the traditional chain rule for deterministic calculus by the Itô–Lévy chain rule.

Definition 2.1 (Entropy flux triple).

A triplet (β,ζ,ν) is called an entropy flux triple if βC2() is Lipschitz continuous and β0, ζ=(ζ1,ζ2,,ζd):d is a vector-valued function and ν: is a scalar-valued function such that

ζ(r)=β(r)f(r)andν(r)=β(r)ϕ(r).

An entropy flux triple (β,ζ,ν) is called convex if β′′(s)0.

For a small positive number ε>0, assume that the parabolic perturbation

du(t,x)-Δϕ(u(t,x))dt=divxf(u(t,x))dt+Eη(x,u(t,x);z)N~(dz,dt)+εΔu(t,x)dt,(t,x)ΠT

of (1.1) has a unique weak solution uε(t,x). Note that this weak solution uε is in L2((0,T)×Ω;H1(d)). Moreover, for the time being, we assume that it satisfies the initial condition in the sense of (A.2). This enables one to derive a weak version of the Itô–Lévy formula for the solutions to (1.1) as detailed in Theorem A.1 in Appendix A.

Let (β,ζ,ν) be an entropy flux triple. Given a non-negative test function ψCc1,2([0,)×d), we apply the generalized version of the Itô–Lévy formula (cf. Appendix A) to have, for almost every T>0,

dβ(uε(T,x))ψ(T,x)𝑑x-dβ(uε(0,x))ψ(0,x)𝑑x=ΠTβ(uε(t,x))tψ(t,x)𝑑x𝑑t-ΠTψ(t,x)ζ(uε(t,x))𝑑x𝑑t   +ΠTE01η(x,uε(t,x);z)β(uε(t,x)+θη(x,uε(t,x);z))ψ(t,x)𝑑θN~(dz,dt)𝑑x   +ΠTE01(1-θ)η2(x,uε(t,x);z)β′′(uε(t,x)+θη(x,uε(t,x);z))ψ(t,x)𝑑θm(dz)𝑑x𝑑t   -ΠT(εxψ(t,x).xβ(uε(t,x))+εβ′′(uε(t,x))|xuε(t,x)|2ψ(t,x))dxdt   -ΠTϕ(uε(t,x))β′′(uε(t,x))|uε(t,x)|2ψ(t,x)𝑑x𝑑t+ΠTν(uε(t,x))Δψ(t,x)𝑑x𝑑t.

Let G be the associated Kirchhoff function of ϕ, given by

G(x)=0xϕ(r)𝑑r.

A simple calculation shows that

|G(uε(t,x))|2=ϕ(uε(t,x))|uε(t,x)|2.

Since β and ψ are non-negative functions, we obtain

0dβ(uε(0,x))ψ(0,x)𝑑x+ΠT{β(uε(t,x))tψ(t,x)-ψ(t,x)ζ(uε(t,x))}𝑑x𝑑t-ΠTβ′′(uε(t,x))|G(uε(t,x))|2ψ(t,x)𝑑x𝑑t+ΠTν(uε(t,x))Δψ(t,x)𝑑x𝑑t+𝒪(ε)+ΠTE01η(x,uε(t,x);z)β(uε(t,x)+θη(x,uε(t,x);z))ψ(t,x)𝑑θN~(dz,dt)𝑑x+ΠTE01(1-θ)η2(x,uε(t,x);z)β′′(uε(t,x)+θη(x,uε(t,x);z))ψ(t,x)𝑑θm(dz)𝑑x𝑑t.

Clearly, the above inequality is stable under the limit ε0 if the family {uε}ε>0 has Llocp-type stability. Just as the deterministic equations, the above inequality provides us with the entropy condition. We now formally define the entropy solution.

Definition 2.2 (Stochastic entropy solution).

A stochastic process uNω2(0,T,L2(d)) is called a stochastic entropy solution of (1.1) if the following conditions hold:

  • (i)

    For each T>0,

    G(u)L2((0,T)×Ω;H1(d))andsup0tT𝔼[u(t)22]<.

  • (ii)

    Given a non-negative test function ψCc1,2([0,)×d) and a convex entropy flux triple (β,ζ,ν), the following inequality holds:

    ΠT{β(u(t,x))tψ(t,x)+ν(u(t,x))Δψ(t,x)-ψ(t,x)ζ(u(t,x))}𝑑x𝑑t   +ΠTE01η(x,u(t,x);z)β(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θN~(dz,dt)𝑑x   +ΠTE01(1-θ)η2(x,u(t,x);z)β′′(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θm(dz)𝑑x𝑑tΠTβ′′(u(t,x))|G(u(t,x))|2ψ(t,x)𝑑x𝑑t-dβ(u0(x))ψ(0,x)𝑑x,P-a.s.(2.1)

Remark 2.3.

We point out that, by a classical separability argument, it is possible to choose a subset of Ω of P-full measure such that (2.1) holds on that subset for every admissible entropy triplet and test function.

The primary aim of this paper is to establish a result of existence and uniqueness of an entropy solution for the Cauchy problem (1.1) in accordance with Definition 2.2, and we do so under the following assumptions:

  • (i)

    ϕ: is a non-decreasing Lipschitz continuous function with ϕ(0)=0. Moreover, if η is not a constant function with respect to the space variable x, then tϕ(t) has a modulus of continuity ωϕ such that

    ωϕ(r)r2/30as r0.

  • (ii)

    f=(f1,f2,,fd):d is a Lipschitz continuous function with fk(0)=0 for all 1kd.

  • (iii)

    The space E is of the form 𝒪×*, and the Borel measure m on E has the form λ×μ, where λ is a Radon measure on 𝒪 and μ is a so-called one-dimensional Lévy measure.

  • (iv)

    There exist positive constants K>0, λ*(0,1) and h1(z)L2(E,m) with 0h1(z)1 such that

    |η(x,u;z)-η(y,v;z)|(λ*|u-v|+K|x-y|)h1(z)for all x,yd,u,v,zE.

  • (v)

    There exist a non-negative function gL(d)L2(d) and h2(z)L2(E,m) such that for all

    (x,u,z)d××E

    one has

    |η(x,u;z)|g(x)(1+|u|)h2(z).

The above definition does not say anything explicitly about the way the entropy solution satisfies the initial condition. However, the initial condition is satisfied in a certain weak sense. Here we state the lemma whose proof follows the same line of argument as the one of [5, Lemma 2.3].

Lemma 2.4.

Any entropy solution u(t,) of (1.1) satisfies the initial condition in the following sense: for every non-negative test function ψCc2(Rd) such that supp(ψ)=K,

limh0𝔼[1h0hK|u(t,x)-u0(x)|ψ(x)𝑑x𝑑t]=0.

Next, we describe a special class of entropy functions that plays an important role in the sequel. Let β: be a C and Lipschitz continuous function satisfying

β(0)=0,β(-r)=β(r),β′′0

and

β(r){=-1if r-1,[-1,1]if |r|<1,=+1if r1.

For any ϑ>0, define βϑ: by

βϑ(r)=ϑβ(rϑ).

Then

|r|-M1ϑβϑ(r)|r|and|βϑ′′(r)|M2ϑ𝟏|r|ϑ,

where

M1=sup|r|1||r|-β(r)|,M2=sup|r|1|β′′(r)|.

By simply dropping ϑ, for β=βϑ we define

ϕβ(a,b)=baβ(σ-b)ϕ(σ)d(σ),Fkβ(a,b)=baβ(σ-b)fk(σ)d(σ),Fk(a,b)=sign(a-b)(fk(a)-fk(b)),F(a,b)=(F1(a,b),F2(a,b),,Fd(a,b)).

We conclude this section by stating the main results of this paper.

Theorem 2.5 (Existence).

Let assumptions (i)(v) be true and that the L2(Rd)-valued F0-measurable random variable u0 satisfies E[u022]<. Then there exists an entropy solution of (1.1) in the sense of Definition 2.2.

Theorem 2.6 (Uniqueness).

Let assumptions (i)(v) be true and that the L2(Rd)-valued F0-measurable random variable u0 satisfies E[u022]<. Then the entropy solution of (1.1) is unique.

Remark 2.7.

In addition, if u0 is in Lp(d) for p[2,), then it can be concluded that

uL(0,T,Lp(Ω×d)).

Furthermore, if u0L and there is M>0 such that η(x,u;z)=0 for |u|>M and

M1=supx,|u|M,z|η(x,u;z)|<,

then |u(t,x)|max{M+M1,u0} for almost every (t,x,ω)ΠT×Ω. We sketch a justification of this claim in Section 4.

3 Existence of a weak solution to the viscous problem

Just as in the case of the deterministic problem, here we also study the problem regularized by adding a small diffusion operator and derive some a priori bounds. Due to the nonlinear function ϕ and related degeneracy, one cannot expect a classical solution and instead seeks a weak solution.

3.1 Existence of a weak solution to the viscous problem

For a small parameter ε>0, we consider the viscous approximation of (1.1) as

du(t,x)-Δϕ(u(t,x))dt=divxf(u(t,x))dt+Eη(x,u(t,x);z)N~(dz,dt)+εΔu(t,x)dt,t>0,xd.(3.1)

In this subsection, we establish the existence of a weak solution for problem (3.1). To do this, we use an implicit time discretization scheme. Let Δt=TN for some positive integer N1. Further, set tn=nΔt for n=0,1,2,,N.

Define

𝒩=L2(Ω;H1(d)),𝒩n={the nΔt measurable elements of 𝒩},=L2(Ω;L2(d)),n={the nΔt measurable elements of }.

Proposition 3.1.

Assume that Δt is small. For any given unHn, there exists a unique un+1Nn+1 with ϕ(un+1)Nn+1 such that P-a.s. for any vH1(Rd) the following variational formula holds:

d((un+1-un)v+Δt{ϕ(un+1)+εun+1+f(un+1)}v)𝑑x=dtntn+1Eη(x,un;z)vN~(dz,ds)𝑑x.(3.2)

Before proving this proposition, we first state a key deterministic lemma, related to the weak solution of parabolic equations. We have the following lemma, a proof of which can be found in [8, p. 19].

Lemma 3.2.

Assume that Δt is small and XL2(Rd). Then, for a fixed parameter ε>0, the following holds:

  • (i)

    There exists a unique uH1(d) with ϕ(u)H1(d) such that, for any vH1(d),

    d(uv+Δt{ϕ(u)+εu+f(u)}v)𝑑x=dXv𝑑x.

  • (ii)

    There exists a constant C=C(Δt)>0 such that the following a priori estimate holds:

    uL2(d)2+ϕ(u)H1(d)2+εuL2(d)2CXL2(d)2.(3.3)

  • (iii)

    The map Θ:XL2(d)(u,ϕ(u))H1(d)2 is continuous.

Proof of Proposition 3.1.

Let un𝒩n. Take

X=un+tntn+1Eη(x,un;z)N~(dz,ds).

Then, by assumption (v), we obtain

𝔼[XL2(d)2]un2+CΔt(gL2(d)2+un2).

This shows that XL2(d) almost surely. Therefore, one can use Lemma 3.2 and conclude that for almost surely all ωΩ there exists a unique u(ω) satisfying the variational equality (3.2). Moreover, by construction Xn+1. Thus, due to the continuity of Θ for the (n+1)Δt measurability and to the a priori estimate (3.3), we conclude that u𝒩n+1 with ϕ(u)𝒩n+1. We denote this solution u by un+1. Hence the assertion of the proposition follows. ∎

3.1.1 A priori estimate

Note that df(v)vdx=0 for any v𝒟(d), and hence it holds true for any vH1(d) by a density argument. We choose the test function v=un+1 in (3.2) to have

d(un+1-un)un+1𝑑x+Δtdϕ(un+1)|un+1|2𝑑x+εΔtd|un+1|2𝑑x=dtntn+1Eη(x,un;z)N~(dz,ds)un+1𝑑xdtntn+1Eη(x,un;z)unN~(dz,ds)𝑑x+α2un+1-unL2(d)2   +12αd(tntn+1Eη(x,un;z)N~(dz,ds))2dfor some α>0.

Since

d|ϕ(u)|2𝑑x=d|ϕ(u)u|2𝑑xcϕdϕ(u)|u|2𝑑x,

we see that

Δtcϕϕ(u)2Δt𝔼[dϕ(u)|u|2𝑑x].(3.4)

In view of assumption (v), inequality (3.4), the Itô–Lévy isometry and the fact that for any a,b one has (a-b)a=12(a2+(a-b)2-b2), we obtain

12[un+12+un+1-un2-un2]+Δtcϕϕ(un+1)2+εΔtun+12α2un+1-un2+CΔt2α(1+un2).

Since α>0 is arbitrary, one can choose α>0 so that

un2+k=0n-1uk+1-uk2+Δtcϕk=0n-1ϕ(uk+1)2+εΔtk=0n-1uk+12C1+C2Δtk=0n-1uk2for some constants C1,C2>0.(3.5)

Thanks to the discrete Gronwall lemma, we can deduce from (3.5) that

un2+k=0n-1uk+1-uk2+Δtcϕk=0n-1ϕ(uk+1)2+εΔtk=0n-1uk+12C(3.6)

For fixed Δt=TN, we define

uΔt(t)=k=1Nuk𝟏[(k-1)Δt,kΔt)(t),u~Δt(t)=k=1N[uk-uk-1Δt(t-(k-1)Δt)+uk-1]𝟏[(k-1)Δt,kΔt)(t)

with uΔt(t)=u0 for t<0. Similarly, we define

B~Δt(t)=k=1N[Bk-Bk-1Δt[t-(k-1)Δt]+Bk-1]𝟏[(k-1)Δt,kΔt)(t),

where

Bn=k=0n-1kΔt(k+1)ΔtEη(x,uk;z)N~(dz,ds)=0nΔtEη(x,uΔt(s-Δt);z)N~(dz,ds).

A straightforward calculation shows that

uΔtL(0,T;)=maxk=1,2,,Nuk,u~ΔtL(0,T;)=maxk=0,1,,Nuk,uΔt-u~ΔtL2(0,T;)2Δtk=0N-1uk+1-uk2.

Since ϕ is a Lipschitz continuous function with ϕ(0)=0, in view of the above definitions and the a priori estimate (3.6), we have the following proposition.

Proposition 3.3.

Assume that Δt is small. Then uΔt and u~Δt are bounded sequences in L(0,T;H), ϕ(uΔt) and ϵuΔt are bounded sequences in L2(0,T;N), and uΔt-u~ΔtL2(0,T;H)2CΔt.

Moreover, uΔt-uΔt(-Δt)0 in L2(Ω×ΠT).

Next, we want to find some upper bounds for B~Δt(t). Regarding this, we have the following proposition.

Proposition 3.4.

B~Δt is a bounded sequence in L2(Ω×ΠT) and

B~Δt()-0Eη(x,uΔt(s-Δt);z)N~(dz,ds)L2(Ω×d)2CΔt.

Proof.

First, we prove the boundedness of B~Δt(t). By using the definition of B~Δt(t), assumption (v) and the boundedness of uΔt in L(0,T;), we obtain

B~ΔtL2(0,T;L2(Ω,L2(d)))2Δtk=0NBkL2(Ω×d)2Δtk=0N𝔼[|d0kΔtEη(x,uΔt(s-Δt);z)N~(dz,ds)𝑑x|2]CΔtk=0N𝔼[d0kΔtg2(x)(1+|uΔt(s-Δt)|2)𝑑x𝑑s]C(1+uΔtL(0,T;L2(Ω×d)))<+.

Thus, B~Δt is a bounded sequence in L2(Ω×ΠT).

To prove the second part of the proposition, we see that for any t[nΔt,(n+1)Δt),

B~Δt(t)-0tEη(x,uΔt(s-Δt);z)N~(dz,ds)=t-nΔtΔtnΔt(n+1)ΔtEη(x,uΔt(s-Δt);z)N~(dz,ds)-nΔttEη(x,uΔt(s-Δt);z)N~(dz,ds)=t-nΔtΔtnΔt(n+1)ΔtEη(x,un;z)N~(dz,ds)-nΔttEη(x,un;z)N~(dz,ds).

Therefore, in view of (3.6) and assumption (v), we have

B~Δt(t)-0tEη(x,uΔt(s-Δt);z)N~(dz,ds)L2(Ω×d)2=𝔼[d|t-nΔtΔtnΔt(n+1)ΔtEη(x,un;z)N~(dz,ds)-nΔttEη(x,un;z)N~(dz,ds)|2𝑑x]2d𝔼[(t-nΔtΔt)2nΔt(n+1)ΔtEη2(x,un;z)m(dz)𝑑s+nΔttEη2(x,un;z)m(dz)𝑑s]𝑑xC(1+un2)[(t-nΔt)2Δt+(t-nΔt)]CΔt.

This completes the proof. ∎

3.1.2 Convergence of uΔt(t,x)

Thanks to Proposition 3.3 and the Lipschitz property of f and ϕ, there exist u, ϕu and fu such that (up to a subsequence)

{uΔt*uin L(0,T;L2(Ω×d)),uΔtuin L2((0,T)×Ω;H1(d)) (for a fixed positive ϵ),ϕ(uΔt)ϕuin L2((0,T)×Ω;H1(d)),F(uΔt)fuin L2((0,T)×Ω×d).(3.7)

Next, we want to identify the weak limits ϕu and fu. Note that, for any vH1(d), we can rewrite (3.2) in terms of uΔt, u~Δt and B~Δt as

d(t(u~Δt-B~Δt)(t)v+{ϕ(uΔt(t))+εuΔt(t)+f(uΔt(t))}v)𝑑x=0.(3.8)

Lemma 3.5.

{uΔt} is a Cauchy sequence in L2(Ω×ΠT).

Proof.

Consider two positive integers N and M and denote Δt=TN and Δs=TM. Then, for any vH1(d), from (3.8) one gets

d(t[(u~Δt-B~Δt)(t)-(u~Δs-B~Δs)(t)]v+{(ϕ(uΔt(t))-ϕ(uΔs(t)))+ε(uΔt(t)-uΔs(t))+(f(uΔt(t))-f(uΔs(t)))}v)dx=0.(3.9)

Let w=(u~Δt-B~Δt)(t)-(u~Δs-B~Δs)(t). Set v=(I-Δ)-1w in (3.9); then one has

12tv(t)H1(d)2+d(ϕ(uΔt(t))-ϕ(uΔs(t)))w𝑑x-d(ϕ(uΔt(t))-ϕ(uΔs(t)))v𝑑x+εd(uΔt-uΔs)w𝑑x-εd(uΔt-uΔs)v𝑑x+d(f(uΔt)-f(uΔs))vdx=0.(3.10)

Note that w=(uΔt-uΔs)-(B~Δt-B~Δs)-(uΔt-u~Δt)+(uΔs-u~Δs).

Therefore,

d(ϕ(uΔt(t))-ϕ(uΔs(t)))w𝑑x=d(ϕ(uΔt(t))-ϕ(uΔs(t)))(uΔt-uΔs)𝑑x-d(ϕ(uΔt(t))-ϕ(uΔs(t)))(B~Δt-B~Δs)𝑑x   -d(ϕ(uΔt(t))-ϕ(uΔs(t))){(uΔt-u~Δt)+(u~Δs-uΔs)}𝑑xd(ϕ(uΔt(t))-ϕ(uΔs(t)))(uΔt-uΔs)𝑑x-12cϕd(ϕ(uΔt(t))-ϕ(uΔs(t)))2𝑑x   -cϕ2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x12d(ϕ(uΔt(t))-ϕ(uΔs(t)))(uΔt-uΔs)𝑑x   -cϕ2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x-cϕ2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x(by (i)).(3.11)

Similarly, we also have

εd(uΔt-uΔs)w𝑑xε2d(uΔt-uΔs)2𝑑x-ε2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x.(3.12)

Combining (3.10), (3.11) and (3.12), we obtain

12tv(t)H1(d)2+ε2d(uΔt-uΔs)2𝑑x(cϕ+ε)2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x   +εd(uΔt-uΔs)v𝑑x-d(F(uΔt)-F(uΔs))vdx+d(ϕ(uΔt(t))-ϕ(uΔs(t)))v𝑑x(cϕ+ε)2d{(B~Δt-B~Δs)+(uΔt-u~Δt)+(u~Δs-uΔs)}2𝑑x+β4d|v|2𝑑x   +ε2αd(uΔt-uΔs)2𝑑x+(β4+εα2)dv2𝑑x+(cf)2+(cϕ)2βd(uΔt-uΔs)2𝑑x

for some α and β>0. Since α,β>0 are arbitrary, there exist some positive constants C1, C2 and C3 such that

𝔼[v(t)H1(d)2]-C10t𝔼[v(r)H1(d)2]𝑑r+C20t𝔼[uΔt-uΔsL2(d)2]𝑑rC3{uΔt-u~ΔtL2(Ω×ΠT)2+uΔs-u~ΔsL2(Ω×ΠT)2+0t𝔼[B~Δt-B~ΔsL2(d)2]𝑑r}.(3.13)

In view of Proposition 3.3, we notice that

uΔt-u~ΔtL2(Ω×ΠT)2+uΔs-u~ΔsL2(Ω×ΠT)2C(Δt+Δs).(3.14)

In addition,

𝔼[B~Δt(r)-B~Δs(r)L2(d)2]3B~Δt(r)-0rEη(x,uΔt(σ-Δt);z)N~(dz,dσ)L2(Ω×d)2+3B~Δs(r)-0rEη(x,uΔs(σ-Δs);z)N~(dz,dσ)L2(Ω×d)2+30rE(η(x,uΔt(σ-Δt);z)-η(x,uΔs(σ-Δs);z))N~(dz,dσ)L2(Ω×d)2C(Δt+Δs)+C0r𝔼[uΔt(σ-Δt)-uΔs(σ-Δs)L2(d)2]𝑑σ,

where we have used Proposition 3.4 and assumption (v). Thus, we get

0t𝔼[B~Δt-B~ΔsL2(d)2]𝑑rC(Δt+Δs)+C0t0r𝔼[uΔt(σ-Δt)-uΔs(σ-Δs)L2(d)2]𝑑σ𝑑r.(3.15)

We combine (3.14) and (3.15) in (3.13) and have

𝔼[v(t)H1(d)2]-C10t𝔼[v(r)H1(d)2]𝑑r+C20t𝔼[uΔt-uΔsL2(d)2]𝑑rC(Δt+Δs)+C0t0r𝔼[uΔt(σ-Δt)-uΔs(σ-Δs)L2(d)2]𝑑σ𝑑r(by Proposition 3.3)C(Δt+Δs)+C0t0r𝔼[uΔt-uΔsL2(d)2]𝑑σ𝑑r.

Hence, an application of Gronwall’s lemma yields

𝔼[v(t)H1(d)2]+0t𝔼[uΔt-uΔsL2(d)2]𝑑rC(Δt+Δs)eCt.

This implies that

uΔt-uΔsL2(Ω×ΠT)2C(Δt+Δs)eCT,

i.e., {uΔt} is a Cauchy sequence in L2(Ω×ΠT). ∎

We are now in position to identify the weak limits ϕu and fu. We have shown that uΔtu and that uΔt is a Cauchy sequence in L2(Ω×ΠT). Thanks to the Lipschitz continuity of ϕ and f, one can easily conclude that ϕu=ϕ(u) and fu=f(u).

In view of the variational formula (3.8), one needs to show the boundedness of

t(u~Δt-B~Δt)

in L2(Ω×(0,T);H-1(d)) and then identify its weak limit. To this end, we have the following lemma.

Lemma 3.6.

The sequence {t(u~Δt-B~Δt)(t)} is bounded in L2(Ω×(0,T);H-1(Rd)), and

t(u~Δt-B~Δt)t(u-0Eη(x,u;z)N~(dz,ds))in L2(Ω×(0,T);H-1(d)),

where u is given by (3.7).

Proof.

To prove the lemma, let Γ=Ω×[0,T]×E, 𝒢=𝒫T×(E) and ς=Ptm, where 𝒫T represents the predictable σ-algebra on Ω×[0,T] and (E) represents the Lebesgue σ- algebra on E.

The space L2((Γ,𝒢,ς);) represents then the space of the square integrable predictable integrands for the Itô–Lévy integral with respect to the compensated compound Poisson random measure N~(dz,dt). Moreover, the Itô–Lévy integral defines a linear operator from L2((Γ,𝒢,ς);) to L2((Ω,T);) and it preserves the norm (see, for example, [19]).

Thanks to Propositions 3.3 and 3.5, uΔt(t-Δt) converges to u in L2(Ω×ΠT). Therefore, in view of Proposition 3.4, the Lipschitz property of η and the above discussion, we conclude that

B~Δt0Eη(x,u;z)N~(dz,ds)in L2(Ω×ΠT).

Again, note that

t(u~Δt-B~Δt)(t)=k=1N(uk-uk-1)-(Bk-Bk-1)Δt𝟏[(k-1)Δt,kΔt).

From (3.2), we see that for any vH1(d),

d(un+1-unΔt-1ΔtnΔt(n+1)ΔtEη(x,un;z)N~(dz,ds))v𝑑x=-dϕ(un+1)vdx-εdun+1vdx-dF(un+1)vdx{ϕ(un+1)L2(d)+εun+1L2(d)+cfun+1L2(d)}vH1(d),

and hence

supvH1(d){0}d(un+1-unΔt-1ΔtnΔt(n+1)ΔtEη(x,un;z)N~(dz,ds))v𝑑xvH1(d)ϕ(un+1)L2(d)+εun+1L2(d)+cfun+1L2(d).

This implies that t(u~Δt-B~Δt)(t) is a bounded sequence in L2(Ω×(0,T);H-1(d)).

To prove the second part of the lemma, we recall that

B~Δt0Eη(x,u;z)N~(dz,ds)andu~Δtu

in L2(Ω×ΠT). In view of the first part of this lemma, one can conclude that, up to a subsequence,

t(u~Δt-B~Δt)t(u-0Eη(x,u;z)N~(dz,ds))in L2(Ω×(0,T);H-1(d)).

This completes the proof. ∎

3.1.3 Existence of a weak solution

As we have emphasized, our aim is to prove the existence of a weak solution to the viscous problem. For this, it is required to pass to the limit as Δt0. To this end, let us choose αL2(0,T) and βL2(Ω). Then, in view of the variational formula (3.8), we obtain

Ω×(0,T)t(u~Δt-B~Δt),vαβ𝑑t𝑑P+εΩ×ΠTuΔtvαβdxdtdP+Ω×ΠTϕ(uΔt)vαβdxdtdP+Ω×ΠTf(uΔt)vαβdxdtdP=0.

We make use of (3.7) and Lemmas 3.5 and 3.6 to pass to the limit as Δt0 in the above variational formulation, and then arrive at

Ω×(0,T)t(u-0tEη(x,u;z)N~(dz,ds)),vαβ𝑑t𝑑P+εΩ×ΠTuvαβdxdtdP+Ω×ΠT{ϕ(u)v+f(u)v}αβ𝑑x𝑑t𝑑P=0.(3.16)

Since H1(d) is a separable Hilbert space, the above formulation (3.16) yields almost surely in Ω,

t(u-0tEη(x,u;z)N~(dz,ds)),v+d(εu+ϕ(u)+f(u))vdx=0

for almost every t[0,T].

This proves that u is a weak solution of (3.1). For every ϕH1(d), it is easily seen that

𝔼|1δ0δd(u(s,x)-u0(x))ψ(x)𝑑x𝑑s|CΔtif δ<Δt.

Therefore,

lim supδ0𝔼|1δ0δd(u(s,x)-u0(x))ψ(x)𝑑x𝑑s|CΔtfor very δ<Δt.

Now, by letting Δt0, we have

limδ01δ0δdu(s,x)ψ(x)𝑑x𝑑s=du0(x)ψ(x)𝑑xP-almost surely.

3.2 A priori bounds for the viscous solutions

Note that for a fixed ε>0 there exists a weak solution, denoted as uεH1(d), which satisfies the following variational formulation: P-almost surely in Ω, and for almost every t(0,T),

t[uε-0tEη(,uε(s,);z)N~(dz,ds)],v+dϕ(uε(t,x))vdx+d{f(uε(t,x))+εuε(t,x)}vdx=0

for any vH1(d). Let β(u)=u2. Applying the Itô–Lévy formula (cf. Theorem A.1) to β(u), one gets that for any t>0,

duε2(t)𝑑x+20td[ε+ϕ(uε(s))]|uε|2𝑑s+2df(uε)uεdx=duε2(0)𝑑x+20tEd01η(x,uε(s,x);z)(uε+θη(x,uε;z))𝑑θ𝑑xN~(dz,ds)   +0tEdη2(x,uε(s,x);z)𝑑xm(dz)𝑑s.

Note that df(uε)uεdx=0 as uεH1(d). Taking the expectation, we obtain

𝔼[uε(t)22]+ε0t𝔼[uε22]𝑑s+0t𝔼[G(uε(s))22]𝑑s𝔼[uε(0)22+CtgL2(d)2]+C0t𝔼[uε(s)22]𝑑s.

An application of Gronwall’s inequality yields

sup0tT𝔼[uε(t)22]+ε0T𝔼[uε(s)22]𝑑s+0T𝔼[G(uε(s))22]𝑑sC.

The achieved result can be summarized by the following theorem.

Theorem 3.7.

For any ε>0, there exists a weak solution uε to problem (3.1). Moreover, it satisfies the following estimate:

sup0tT𝔼[uε(t)22]+ε0T𝔼[uε(s)22]𝑑s+0T𝔼[G(uε(s))22]𝑑sC,(3.17)

where G is an associated Kirchhoff’s function of ϕ, defined by G(x)=0xϕ(r)𝑑r.

Remark 3.8.

Let us remark that since any solution to (3.1) is an entropy solution, the solution uε in unique.

4 Existence of an entropy solution

In this section, we will establish the existence of an entropy solution. In view of the a priori estimates given in (3.17), we can apply [5, Lemmas 4.2 and 4.3] (see also [2]) and show the existence of a Young measure-valued limit process solution u(t,x,α), α(0,1) associated with the sequence {uε(t,x)}ε>0.

The basic strategy in this case is to apply the Young measure technique and to adapt Kruzhkov’s doubling variable method in the presence of a noise to viscous solutions with two different parameters, and then to send the viscosity parameters to zero. One needs a version of the classical L1 contraction principle (for conservation laws) to get the uniqueness of the Young measure-valued limit and to show that the Young measure-valued limit process is independent of the additional (dummy) variable. Hence, it will imply the point-wise convergence of the sequence of viscous solutions.

4.1 Uniqueness of the Young measure-valued limit process

To do this, we follow the same line of argument as in [3] for the degenerate parabolic part and [5] for the Lévy noise. For the convenience of the reader, we have chosen to provide detailed proofs of a few crucial technical lemmas and the rest is referred to appropriate resources. Bauzet et al. [2] and Biswas et al. [5] used the fact that ΔuεL2(Ω×ΠT). Note that, in this case, uεH1(d). Therefore, we need to regularize uε by convolution. Let {τκ} be a sequence of mollifiers in d. Since uε is a solution to problem (3.1), as shown in the proof of Theorem A.1, uετκ is a solution to the problem

(uετκ)-0tΔ(ϕ(uε)τκ)𝑑s=0tdivx(f(uε)τκ)𝑑s+0tE(η(x,uε;z)τκ)N~(dz,ds)+0tεΔ(uετκ(t,x))𝑑sa.e. t>0,xd.(4.1)

Note that Δ(uετκ)L2(Ω×ΠT) for fixed ε>0.

Let ρ and ϱ be standard non-negative mollifiers on and d, respectively, such that supp(ρ)[-1,0] and supp(ϱ)=B¯1(0). We define

ρδ0(r)=1δ0ρ(rδ0)andϱδ(x)=1δdϱ(xδ),

where δ and δ0 are two positive constants. Given a non-negative test function ψCc1,2([0,)×d) and two positive constants δ and δ0, define

φδ,δ0(t,x,s,y)=ρδ0(t-s)ϱδ(x-y)ψ(s,y).

Clearly ρδ0(t-s)0 only if s-δ0ts, and hence φδ,δ0(t,x;s,y)=0 outside s-δ0ts.

Let uθ(t,x) be the weak solution to the viscous problem (3.1) with parameter θ>0 and initial condition uθ(0,x)=v0(x). Moreover, let J be the standard symmetric non-negative mollifier on with support in [-1,1] and Jl(r)=1lJ(rl) for l>0. We now simply write down the Itô–Lévy formula for uθ(t,x) against the convex entropy triple (β(-k),Fβ(,k),ϕβ(,k)), multiply by Jl(uετκ(s,y)-k) for k and then integrate with respect to s, y and k. Taking the expectation of the resulting expressions, we have

θ𝔼[ΠTΠTβ′′(uθ(t,x)-k)|uθ(t,x)|2φδ,δ0(t,x,s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   +𝔼[ΠTΠTβ′′(uθ(t,x)-k)|G(uθ(t,x))|2φδ,δ0(t,x,s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]𝔼[ΠTdβ(v0(x)-k)φδ,δ0(0,x,s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑y𝑑s]   +𝔼[ΠTΠTβ(uθ(t,x)-k)tφδ,δ0(t,x,s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   +𝔼[ΠT0TEd01η(x,uθ(t,x);z)β(uθ(t,x)+τη(x,uθ(t,x);z)-k)      ×φδ,δ0(t,x,s,y)dτdxN~(dz,dt)Jl(uετκ(s,y)-k)dkdyds]   +𝔼[ΠT0TEd01(1-τ)η2(x,uθ(t,x);z)β′′(uθ(t,x)+τη(x,uθ(t,x);z)-k)      ×φδ,δ0(t,x,s,y)Jl(uετκ(s,y)-k)dτdkdxm(dz)dtdyds]   -𝔼[ΠTΠTFβ(uθ(t,x),k)xϱδ(x-y)ψ(s,y)ρδ0(t-s)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   +𝔼[ΠTΠTϕβ(uθ(t,x),k)Δxϱδ(x-y)ψ(s,y)ρδ0(t-s)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   -θ𝔼[ΠTΠTβ(uθ(t,x)-k)xuθ(t,x)xφδ,δ0(t,x,s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s],

i.e.,

I0,1+I0,2I1+I2+I3+I4+I5+I6+I7.(4.2)

We now apply the Itô–Lévy formula to (4.1) and obtain

ε𝔼[ΠTΠTβ′′(uετκ-k)|(uετκ)|2φδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   +𝔼[ΠTΠTβ′′(uετκ-k)(ϕ(uε)τκ)(uετκ)φδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]𝔼[ΠTdβ(uετκ(0,y)-k)φδ,δ0(t,x,0,y)Jl(uθ(t,x)-k)𝑑k𝑑x𝑑y𝑑t]   +𝔼[ΠTΠTβ(uετκ(s,y)-k)sϕδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑y𝑑s𝑑x𝑑t]   +𝔼[ΠT0TEd01(η(y,uε;z)τκ)β(uετκ+θ(η(y,uε;z)τκ)-k)      ×φδ,δ0Jl(uθ(t,x)-k)dθdydkN~(dz,ds)dxdt]   +𝔼[ΠT0TEd01(1-θ)(η(y,uε;z)τκ)2β′′(uετκ+θ(η(y,uε;z)τκ)-k)      ×φδ,δ0Jl(uθ(t,x)-k)dθdydkm(dz)dsdxdt]   -𝔼[ΠTΠTβ(uετκ-k)(ϕ(uε)τκ)yφδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   -𝔼[ΠTΠTβ(uετκ-k)(F(uε)τκ)yφδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   -𝔼[ΠTΠTβ′′(uετκ-k)(F(uε)τκ)y(uετκ)φδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]   -ε𝔼[ΠTΠTβ(uετκ(s,y)-k)y(uετκ)yφδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑y𝑑s𝑑x𝑑t],

i.e.,

J0,1+J0,2J1+J2+J3+J4+J5+J6+J7+J8.(4.3)

We now add (4.2) and (4.3) and look for the passage to the limit with respect to the different approximation parameters. Note that I0,1 and J0,1 are both non-negative terms, they are the left-hand sides of inequalities (4.2) and (4.3), respectively. Hence we can omit these terms. Let us consider the expressions I0,2 and J0,2. Recall that

I0,2=𝔼[ΠTΠTβ′′(uθ(t,x)-k)|G(uθ(t,x))|2φδ,δ0Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]

and

J0,2=𝔼[ΠTΠTβ′′(uετκ-k)(ϕ(uε)τκ)(uετκ)φδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s].

By using the properties of Lebesgue points, convolutions and approximations by mollifications, one is able to pass to the limit in I0,2 and J0,2, and conclude the following lemma.

Lemma 4.1.

It holds that

liml0limκ0limδ00I0,2=𝔼[ΠTdβ′′(uθ(t,x)-uε(t,y))|G(uθ(t,x))|2ψ(t,y)ϱδ(x-y)𝑑x𝑑t𝑑y]

and

liml0limκ0limδ00J0,2=𝔼[ΠTdβ′′(uε(s,y)-uθ(s,x))ϕ(uε(s,y))uε(s,y)×ψ(s,y)ϱδ(x-y)𝑑x𝑑y𝑑s].

Lemma 4.2.

It follows that

lim supθ0lim supε0liml0limκ0limδ00(I0,2+J0,2)2𝔼[ΠTd0101u~(t,x,γ)u(t,y,α)(s=σu~(t,x,γ)β′′(σ-s)ϕ(s)ds)ϕ(σ)dσ×divyx[ψ(t,y)ϱδ(x-y)]dγdαdxdydt].(4.4)

Proof.

In view of Lemma 4.1, we see that

liml0limκ0limδ00(I0,2+J0,2)=𝔼[ΠTdβ′′(uε(t,y)-uθ(t,x))(|yG(uε(t,y))|2+|xG(uθ(t,x))|2)ψ(t,y)ϱδ(x-y)𝑑x𝑑y𝑑t].

Let u(t,y,α) and u~(t,x,γ) be the Young measure-valued narrow limits associated with the sequences {uε(t,y)}ε>0 and {uθ(t,x)}θ>0, respectively. With these at hand, one can use a similar argument as in [3, Lemma 3.4] and arrive at the conclusion that (4.4) holds. ∎

We consider the terms (I1+J1) originating from the initial conditions. Note that I1=0 as suppρδ0[-δ0,0). Under a slight modification of the same line of arguments as in [5], we arrive at the following lemma.

Lemma 4.3.

It holds that

limθ0limε0liml0limκ0limδ00(I1+J1)=𝔼[ddβϑ(v0(x)-u0(y))ψ(0,y)ϱδ(x-y)𝑑x𝑑y]

and

lim(ϑ,δ)(0,0)𝔼[ddβϑ(v0(x)-u0(y))ψ(0,y)ϱδ(x-y)𝑑x𝑑y]=𝔼[d|v0(x)-u0(x)|ψ(0,x)𝑑x].

We now turn our attention to (I2+J2). Note that tρδ0(t-s)=-sρδ0(t-s) and that β and Jl are even functions. A simple calculation gives

I2+J2=𝔼[ΠTΠTβ(uετκ(s,y)-k)sψ(s,y)ρδ0(t-s)ϱδ(x-y)×Jl(uθ(t,x)-k)𝑑k𝑑y𝑑s𝑑x𝑑t].

One can now pass to the limit in (I2+J2) and have the following conclusion.

Lemma 4.4.

It holds that

limθ0limε0liml0limκ0limδ00(I2+J2)=𝔼[ΠTd0101β(u(s,y,α)-u~(s,x,γ))sψ(s,y)ϱδ(x-y)𝑑γ𝑑α𝑑y𝑑x𝑑s]

and

lim(ϑ,δ)(0,0)𝔼[ΠTd0101β(u(s,y,α)-u~(s,x,γ))sψ(s,y)ϱδ(x-y)𝑑γ𝑑α𝑑y𝑑x𝑑s]=𝔼[0Td0101|u(s,y,α)-u~(s,y,γ)|sψ(s,y)𝑑γ𝑑α𝑑y𝑑s].

In regard to I6 and J5, we have the following lemma.

Lemma 4.5.

It holds that

limθ0limε0liml0limκ0limδ00I6=𝔼[ΠTd0101ϕβ(u~(s,x,γ),u(s,y,α))Δxϱδ(x-y)ψ(s,y)𝑑γ𝑑α𝑑x𝑑y𝑑s]

and

limθ0limε0liml0limκ0limδ00J5=𝔼[ΠTd[0,1]2ϕβ(u(s,y,α),u~(s,x,γ))Δy[ψ(s,y)ϱδ(x-y)]𝑑γ𝑑α𝑑x𝑑y𝑑s].

Proof.

Let us consider the passage to the limits in I6. To do this, we define

1=𝔼[ΠTΠTϕβ(uθ(t,x),k)Δxϱδ(x-y)ψ(s,y)ρδ0(t-s)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑t𝑑y𝑑s]-𝔼[ΠTdϕβ(uθ(s,x),k)Δxϱδ(x-y)ψ(s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑y𝑑s].

Note that, for all a,b,c,

|ϕβ(a,b)-ϕβ(c,b)|C|c-a|.(4.5)

By using (4.5), we have

1=𝔼[ΠTΠTϕβ(uθ(t,x),uετκ(s,y)-k)Δxϱδ(x-y)ψ(s,y)ρδ0(t-s)Jl(k)𝑑k𝑑x𝑑t𝑑y𝑑s]-𝔼[ΠTdϕβ(uθ(s,x),uετκ(s,y)-k)Δxϱδ(x-y)ψ(s,y)Jl(k)𝑑k𝑑x𝑑y𝑑s]=𝔼[ΠTΠT(ϕβ(uθ(t,x),uετκ(s,y)-k)-ϕβ(uθ(s,x),uετκ(s,y)-k))Δxϱδ(x-y)   ×ψ(s,y)ρδ0(t-s)Jl(k)dkdxdtdyds]-𝔼[ΠTdϕβ(uθ(s,x),uετκ(s,y)-k)ψ(s,y)(1-0Tρδ0(t-s)dt)   ×Δxϱδ(x-y)Jl(k)dkdxdyds].

Then

|1|C𝔼[s=δ0TΠTd|uθ(t,x)-uθ(s,x)||Δxϱδ(x-y)|ψ(s,y)ρδ0(t-s)Jl(k)𝑑k𝑑x𝑑t𝑑y𝑑s]+𝒪(δ0)C𝔼[s=δ0T0TKδ|uθ(t,x)-uθ(s,x)|ρδ0(t-s)𝑑x𝑑t𝑑s]+𝒪(δ0)C𝔼[r=010TKδ|uθ(t+δ0r,x)-uθ(t,x)|ρ(-r)𝑑x𝑑t𝑑r]+𝒪(δ0),

where Kδd is a compact set depending on ψ and δ.

Note that

limδ000TKδ|uθ(t+δ0r,x)-uθ(t,x)|𝑑x𝑑t0

almost surely for all r[0,1]. Therefore, by the dominated convergence theorem, we have

limδ00I6=𝔼[ΠTdϕβ(uθ(s,x),k)Δxϱδ(x-y)ψ(s,y)Jl(uετκ(s,y)-k)𝑑k𝑑x𝑑y𝑑s].

Moreover, one can use the property of convolution to conclude

limκ0limδ00I6=𝔼[ΠTdϕβ(v(s,x),k)Δxϱδ(x-y)ψ(s,y)Jl(uε(s,y)-k)𝑑k𝑑x𝑑y𝑑s].

Passage to the limit as l0: Let

2:=𝔼[ΠTdϕβ(uθ(s,x),k)Δxϱδ(x-y)ψ(s,y)Jl(uε(s,y)-k)𝑑k𝑑x𝑑y𝑑s]-𝔼[ΠTdϕβ(uθ(s,x),uε(s,y))Δxϱδ(x-y)ψ(s,y)𝑑x𝑑y𝑑s]=𝔼[ΠTd(ϕβ(uθ(s,x),k)-ϕβ(uθ(s,x),uε(s,y)))Δxϱδ(x-y)ψ(s,y)×Jl(uε(s,y)-k)dkdxdyds].

Note that for all a,b,c,

|ϕβ(a,b)-ϕβ(a,c)|C(1+|a-b|)|b-c|.(4.6)

Therefore, by (4.6) we have

|2|C𝔼[ΠTd(1+|uθ(s,x)-k|)|uε(s,y)-k||Δxϱδ(x-y)|ψ(s,y)×Jl(uε(s,y)-k)𝑑k𝑑x𝑑y𝑑s]Cl{1+(supθ>0supt>0𝔼[uθ(t)22])12+(supε>0supt>0𝔼[uε(t)22])12}0as l0.

One can justify the passage to the limit as ε0 and θ0 in the sense of Young measures as in [2, 5] and conclude

limθ0limε0𝔼[ΠTdϕβ(uθ(s,x),uε(s,y))Δxϱδ(x-y)ψ(s,y)𝑑x𝑑y𝑑s]=𝔼[ΠTd0101ϕβ(u~(s,x,γ),u(s,y,α))Δxϱδ(x-y)ψ(s,y)𝑑γ𝑑α𝑑x𝑑y𝑑s].

This proves the first part of the lemma.

To prove the second part, let us recall that

J5:=-𝔼[ΠTΠTβ(uετκ-k)(ϕ(uε)τκ)yφδ,δ0Jl(uθ(t,x)-k)𝑑k𝑑x𝑑t𝑑y𝑑s].

A classical property of Lebesgue points and convolution yields

limκ0limδ00J5=-𝔼[ΠTdβ(uε(s,y)-k)ϕ(uε(s,y))y[ψ(s,y)ϱδ(x-y)]×Jl(uθ(s,x)-k)𝑑k𝑑x𝑑y𝑑s].

Recall that ϕβ(a,b)=baβ(s-b)ϕ(s)𝑑s. Making use of Green’s type formula along with the Young measure theory and keeping in mind that u(s,y,α) and u~(s,x,γ) are Young measure-valued limit processes associated with the sequences {uε(s,y)}ε>0 and {uθ(s,x)}θ>0, respectively, we arrive at the following conclusion:

limθ0limε0liml0limκ0limδ00J5=𝔼[ΠTd0101ϕβ(u(s,y,α),u~(s,x,γ))Δy[ψ(s,y)ϱδ(x-y)]𝑑γ𝑑α𝑑x𝑑y𝑑s].

This completes the proof of the lemma. ∎

Next, we want to pass to the limits in (J6+J7) and I5, respectively. A slight modification of an argument used in [3, 2, 5] yields the following lemma.

Lemma 4.6.

It holds that

liml0limκ0limδ00(J6+J7)=-𝔼[ΠTdFβ(uε(s,y),uθ(s,x))y[ψ(s,y)ϱδ(x-y)]dxdyds]ε0-𝔼[ΠTd01Fβ(u(s,y,α),uθ(s,x))y[ψ(s,y)ϱδ(x-y)]dαdxdyds]θ0-𝔼[ΠTd0101Fβ(u(s,y,α),u~(s,x,γ))y[ψ(s,y)ϱδ(x-y)]dγdαdxdyds]

and

limθ0limε0liml0limκ0limδ00I5=-𝔼[ΠTd0101Fβ(u~(s,x,γ),u(s,y,α))x[ϱδ(x-y)ψ(s,y)]dγdαdxdyds].

In view of the above, we now want to pass to the limit as (ϑ,δ)(0,0). We follow a line of argument similar to the proof of the second part of [5, Lemmas 5.7 and 5.8], and arrive at the following lemma.

Lemma 4.7.

Assume that ϑ0, δ0 and ϑδ0. Then

limϑ0limδ0limϑδ0limθ0limε0liml0limκ0limδ00[(J6+J7)+I5]=-𝔼[ΠT0101F(u(s,y,α),u~(s,y,γ))yψ(s,y)𝑑γ𝑑α𝑑y𝑑s].

We now focus on I7+J8 and establish the following lemma.

Lemma 4.8.

For fixed δ>0 and β, it holds that

lim sup(θ,ε,l,κ,δ0,)0|I7+J8|=0.

Proof.

Note that

|J8|εβ𝔼[ΠTd|y(uετκ(s,y))||y[ψ(s,y)ϱδ(x-y)]|𝑑y𝑑x𝑑s]εβ𝔼[|y|K0Td|y(uετκ(t,y))||y[ψ(t,y)ϱδ(x-y)]|𝑑x𝑑t𝑑y]C(β)ε12(𝔼[ΠTε|y(uετκ(t,y))|2𝑑y𝑑t])12(𝔼[KΠT|y[ψ(t,y)ϱδ(x-y)]|2𝑑x𝑑t𝑑y])12C(β,ψ,δ)ε12(supε>0𝔼[|ε0Td|yuε(t,y)|2dydt|])12C(β,ψ,δ)ε12,

where the second line follows by the Cauchy–Schwarz inequality. Similarly, we have |I7|C(β,ψ,δ)θ1/2. This completes the proof. ∎

Next we consider the stochastic terms I3+J3. To this end, we cite [5] and assert that for two constants T1,T20 with T1<T2,

𝔼[XT1T1T2EJ(t,z)N~(dz,dt)]=0,(4.7)

where J is a predictable integrand and X is an adapted process.

For each βC() with β,β′′Cb() and each non-negative φCc(Π×Π), we define

M[β,φ](s;y,v):=0TEd{β(uθ(r,x)+η(x,uθ(r,x);z)-v)-β(uθ(r,x)-v)}×φ(r,x,s,y)𝑑xN~(dz,dr),

where 0sT, (y,v)d×. Furthermore, we extend the process uετκ(,y) for negative time simply by uετκ(s,y)=uε(0,)τk(y) if s<0. With this convention, it follows from (4.7) that

𝔼[vΠTM[β,φδ,δ0](s;y,v)Jl(uε(s-δ0,y)-v)𝑑y𝑑s𝑑v]=0,

and hence we have J3=0 and

I3=𝔼[ΠTM[β,φδ,δ0](s;y,v)(Jl(uετκ(s,y)-v)-Jl(uετκ(s-δ0,y)-v))𝑑y𝑑s𝑑v].(4.8)

Our aim is to pass to the limit into the stochastic terms I3+J3. This requires the following moment estimate of M[β,φδ,δ0], a proof of which can be found in [5].

Lemma 4.9.

Let γC(R) be a function such that γCc(R) and let p be a positive integer of the form p=2k for some kN. If pd+3, then there exists a constant C=C(γ,ψ,δ) such that

sup0sT(𝔼[M[γ,φδ,δ0](s;,)L(d×)2])C(γ,ψ,δ)δ02(p-1)/p,

and the following identities hold:

vM[γ,φ](s;y,v)=M[-γ,φ](s;y,v),ykM[γ,φ](s;y,v)=M[γ,ykφ](s;y,v).

Lemma 4.10.

It holds that

liml0limκ0limδ00I3=𝔼[ΠTdE{β(uθ(r,x)+η(x,uθ(r,x);z)-uε(r,y)-η(y,uε(r,y);z))-β(uθ(r,x)-uε(r,y)-η(y,uε(r,y);z))+β(uθ(r,x)-uε(r,y))-β(uθ(r,x)+η(x,uθ(r,x);z)-uε(r,y))}ψ(r,y)ϱδ(x-y)m(dz)dxdydr].

Proof.

Note that uετκ(,y) satisfies for all yd,

duετκ(s,y)=div(f(uε)τκ(s,y))ds+Δ(ϕ(uε)τκ(s,y))ds+εΔuετκ(s,y)ds+E(η(,uε;z)τκ(s,y))N~(dz,ds).

Now apply the Itô–Lévy formula on Jl(uετκ(s,y)-v) to get

Jl(uετκ(s,y)-v)-Jl(uετκ(s-δ0,y)-v)=s-δ0sJl(uετκ(σ,y)-v)(div(f(uε)τκ(σ,y))+εΔ(uετκ(σ,y))+Δ(ϕ(uε)τκ(σ,y)))𝑑σ   +s-δ0sE(Jl(uετκ(σ,y)+(η(,uε;z)τκ(σ,y))-v)-Jl(uετκ(σ,y)-v))N~(dz,dσ)   +s-δ0sEλ=01(1-λ)Jl′′(uετκ(σ,y)-v+λ(η(,uε;z)τκ(σ,y)))      ×(η(,uε;z)τκ(σ,y))2dλm(dz)dσ=-vs-δ0s(div(f(uε)τκ(σ,y))+εΔ(uετκ(σ,y))+Δ(ϕ(uε)τκ(σ,y)))Jl(uετκ(σ,y)-v)𝑑σ   +s-δ0sE(Jl(uετκ(σ,y)+(η(,uε;z)τκ(σ,y))-v)-Jl(uετκ(σ,y)-v))N~(dz,dσ)   +s-δ0sEλ=01(1-λ)Jl′′(uετκ(σ,y)-v+λ(η(,uε;z)τκ(σ,y)))      ×(η(,uε;z)τκ(σ,y))2dλm(dz)dσ.

Therefore, from (4.8) and Lemma 4.9 we have

I3=-𝔼[ΠTM[β,φδ,δ0](s;y,v)(s-δ0sJl(uετκ(σ,y)-v)div(f(uε)τκ(σ,y))𝑑σ)𝑑s𝑑y𝑑v]-𝔼[ΠTM[β,φδ,δ0](s;y,v)(s-δ0sJl(uετκ(σ,y)-v)εΔ(uετκ(σ,y))𝑑σ)𝑑s𝑑y𝑑v]-𝔼[ΠTM[β,φδ,δ0](s;y,v)(s-δ0sJl(uετκ(σ,y)-v)Δ(ϕ(uε)τκ(σ,y))𝑑σ)𝑑s𝑑y𝑑v]+𝔼[ΠTs-δ0sdE(β(uθ(r,x)+η(x,uθ(r,x);z)-v)-β(uθ(r,x)-v))   ×(Jl(uετκ(r,y)+η(,uε;z)τκ(r,y)-v)-Jl(uετκ(r,y)-v))   ×ρδ0(r-s)ψ(s,y)ϱδ(x-y)m(dz)dxdrdvdyds]+𝔼[ΠTM[β,ϕδ,δ0](s;y,v){s-δ0sEλ=01Jl′′(uετκ(σ,y)-v+λ(η(,uε;z)τκ(σ,y)))   ×(1-λ)(η(,uε;z)τκ(σ,y))2dλm(dz)dσ}dydsdv]A1κ,l,ε(δ,δ0)+A2κ,l,ε(δ,δ0)+A3κ,l,ε(δ,δ0)+Bε,l,κ+A4κ,l,ε(δ,δ0).

Note that Δ(uετκ)L2(Ω×ΠT) for fixed κ and ε. One can use Young’s inequality for convolution and replace uε by uετκ to adapt the same line of argument as in [5] and conclude

A1κ,l,ε(δ,δ0)0,A2κ,l,ε(δ,δ0)0,A3κ,l,ε(δ,δ0)0,A4κ,l,ε(δ,δ0)0  as δ00.

Again, it is routine to pass to the limit in Bε,l,κ and arrive at the conclusion that

liml0limκ0limδ00Bε,l,κ=𝔼[ΠTdE{β(uθ(r,x)+η(x,uθ(r,x);z)-uε(r,y)-η(y,uε(r,y);z))-β(uθ(r,x)-uε(r,y)-η(y,uε(r,y);z))+β(uθ(r,x)-uε(r,y))-β(uθ(r,x)+η(x,uθ(r,x);z)-uε(r,y))}ψ(r,y)ϱδ(x-y)m(dz)dxdydr].

This completes the proof. ∎

Let us consider the additional terms I4+J4. A line of arguments similar to the ones in [3, 2, 5] and classical properties of convolution yield the following lemma.

Lemma 4.11.

It holds that

liml0limκ0limδ00J4=𝔼[ΠTdEλ=01(1-λ)β′′(uε(s,y)-uθ(s,x)+λη(y,uε(s,y);z))×η2(y,uε(s,y);z)ψ(s,y)ϱδ(x-y)dλm(dz)dxdyds]

and

liml0limκ0limδ00I4=𝔼[ΠTdEλ=01(1-λ)β′′(uθ(s,x)-uε(s,y)+λη(x,uθ(s,x);z))×η2(x,uθ(s,x);z)ψ(s,x)ϱδ(x-y)dλm(dz)dxdyds].

Now we add these terms (cf. Lemma 4.11) with the terms resulting from Lemma 4.10 and have the following lemma.

Lemma 4.12.

Assume that ϑ0+, δ0+ and ϑ-1δ20+. Then

lim supϑ0+,δ0+,ϑ-1δ20+lim supθ,ε0[liml0limκ0limδ00((I3+J3)+(I4+J4))]=0.

Proof.

In view of Lemmas 4.10 and 4.11, we see that

liml0limκ0limδ00((I3+J3)+(I4+J4))=𝔼[ΠTd(E{β(uθ(t,x)-uε(t,y)+η(x,uθ(t,x);z)-η(y,uε(t,y);z))   -(η(x,uθ(t,x);z)-η(y,uε(t,y);z))β(uθ(t,x)-uε(t,y))   -β(uθ(t,x)-uε(t,y))}m(dz))ψ(t,y)ϱδ(x-y)dxdydt]=𝔼[ΠTd(Eτ=01b2(1-τ)β′′(a+τb)𝑑τm(dz))ψ(t,y)ϱδ(x-y)𝑑x𝑑y𝑑t],(4.9)

where

a=uθ(t,x)-uε(t,y)andb=η(x,uθ(t,x);z)-η(y,uε(t,y);z).

By using arguments similar to the ones used in the proof of [5, Lemma 5.11], we arrive at

liml0limκ0limδ00((I3+J3)+(I4+J4))C1(ϑ+ϑ-1δ2)T,

where the constant C1 depends only on ψ and is in particular independent of ε. We now let ϑ0, δ0 and ϑ-1δ20, yielding

lim supϑ0,δ0,ϑ-1δ20lim supθ,ε0[liml0limκ0limδ00((I3+J3)+(I4+J4))]0.

This concludes the proof as

liml0limκ0limδ00((I3+J3)+(I4+J4))0,

thanks to (4.9). ∎

We now turn our attention back to the terms which are originating from Lemmas 4.2 and 4.5. To this end, define

:=-2𝔼[ΠTd0101u~(s,x,γ)u(s,y,α)(r=σu~(s,x,γ)β′′(σ-r)ϕ(r)dr)ϕ(σ)dσ   ×divyx[ψ(s,y)ϱδ(x-y)]dγdαdxdyds]+𝔼[ΠTd0101ϕβ(u~(s,x,γ),u(s,y,α))Δxϱδ(x-y)ψ(s,y)𝑑γ𝑑α𝑑x𝑑y𝑑s]+𝔼[ΠTd0101ϕβ(u(s,y,α),u~(s,x,γ))Δy[ψ(s,y)ϱδ(x-y)]𝑑γ𝑑α𝑑x𝑑y𝑑s]=𝔼[ΠTd0101{2Iβ(u~(s,x,γ),u(s,y,α))+ϕβ(u~(s,x,γ),u(s,y,α))   +ϕβ(u(s,y,α),u~(s,x,γ))}ψ(s,y)Δyϱδ(x-y)dγdαdxdyds]+𝔼[ΠTd0101(2Iβ(u(s,y,α),u~(s,x,γ))+2ϕβ(u~(s,x,γ),u(s,y,α)))   ×yψ(s,y)yϱδ(x-y)dγdαdxdyds]+𝔼[ΠTd0101ϕβ(u(s,y,α),u~(s,x,γ))Δyψ(s,y)ϱδ(x-y)𝑑γ𝑑α𝑑x𝑑y𝑑s]1+2+3,

where

Iβ(a,b)=abμaβ′′(μ-σ)ϕ(σ)𝑑σϕ(μ)𝑑μfor any a,b.

Our aim is to pass to the limit in as (ϑ,δ)(0,0). For this, we need some a priori estimates on Iβ(a,b). Here we state the required lemma whose proof can be found in [3].

Lemma 4.13.

The following holds:

Iβ(a,b)=Iβ(b,a)𝑎𝑛𝑑Iβ(a,b)=-12ababβ′′(σ-μ)ϕ(μ)ϕ(σ)𝑑μ𝑑σ,2Iβ(a,b)+ϕβ(a,b)+ϕβ(b,a)=12ababβ′′(μ-σ)[ϕ(μ)-ϕ(σ)]2𝑑μ𝑑σ.

Moreover, if ϕ has a modulus of continuity ωϕ, then

2Iβ(a,b)+ϕβ(a,b)+ϕβ(b,a)C|b-a||ωϕ(|ϑ|)|2

and

2Iβ(a,b)+ϕβ(b,a)C|b-a||ωϕ(|ϑ|)|2+Cmin{2ϑ,|b-a|}.

We now shift our focus back to the expression and prove the following lemma.

Lemma 4.14.

It holds

lim(ϑ,δ)(0,0)=𝔼[ΠT0101|ϕ(u(s,y,α)-ϕ(u~(s,y,γ))|Δyψ(s,y)dγdαdyds].

Proof.

Let ωϕ be a modulus of continuity of ϕ. Then, thanks to Lemma 4.13, we obtain

|1|C𝔼[ΠTd0101|ωϕ(|ϑ|)|2|u(s,y,α)-u~(s,x,γ)|ψ(s,y)|Δyϱδ(x-y)|𝑑γ𝑑α𝑑x𝑑y𝑑s]C(ψ)|ωϕ(|ϑ|)|2δ2

and

|2|C𝔼[ΠTd0101|ωϕ(|ϑ|)|2|u(s,y,α)-u~(s,x,γ)||yψ(s,y)||yϱδ(x-y)|𝑑γ𝑑α𝑑x𝑑y𝑑s]+𝔼[ΠTdCϑ|yψ(s,y)||yϱδ(x-y)|𝑑x𝑑y𝑑s]C(ψ)|ωϕ(|ϑ|)|2δ+Cϑδ.

Hence, we have

|1|+|2|C(ψ)|ωϕ(|ϑ|)|2δ2+C(ψ)|ωϕ(|ϑ|)|2δ+Cϑδ.

Put δ=ϑ2/3. Then, by our assumption (i), we see that

lim(ϑ,δ)(0,0)(1+2)=0.

To conclude the proof, it is now required to show

lim(ϑ,δ)(0,0)3=𝔼[ΠT0101|ϕ(u(s,y,α))-ϕ(u(s,y,γ))|Δyψ(s,y)𝑑γ𝑑α𝑑y𝑑s],

which follows easily from the fact that (a,b)|ϕ(a)-ϕ(b)| is Lipschitz continuous and that

|ϕβϑ(a,b)-|ϕ(a)-ϕ(b)||Cϑfor any a,b.

The following proposition combines all of the above results.

Proposition 4.15.

Let u~(t,x,γ) and u(t,x,α) be the predictable process with initial data v(0,x) and u(0,x), respectively, which have been extracted out of Young measure-valued sub-sequential limits of the sequences {uθ(t,x)}θ>0 and {uε(t,x)}ε>0, respectively. Then, for any non-negative H1([0,)×Rd) function ψ(t,x) with compact support, the following inequality holds:

0𝔼[d|v0(x)-u0(x)|ψ(0,x)𝑑x]+𝔼[ΠT0101|u~(t,x,γ)-u(t,x,α)|tψ(t,x)𝑑γ𝑑α𝑑x𝑑t]-𝔼[ΠT0101F(u(t,x,α),u~(t,x,γ))xψ(t,x)𝑑γ𝑑α𝑑x𝑑t]-𝔼[ΠT(0101|ϕ(u(t,x,α))-ϕ(u~(t,x,γ))|𝑑γ𝑑α)xψ(t,x)𝑑x𝑑t].(4.10)

Proof.

First we add (4.2) and (4.3) and then pass to the limit limε0liml0limκ0limδ00. Invoking Lemmas 4.2, 4.3, 4.4, 4.5, 4.6, 4.8 and 4.12, we put δ=ϑ2/3 in the resulting expression and then let ϑ0 with the second parts of Lemmas 4.3 and 4.4. Keeping in mind Lemmas 4.7, 4.12 and 4.14, we conclude that (4.10) holds for any non-negative test function ψCc2([0,)×d). It now follows by routine approximation arguments that (4.10) holds for any ψ with compact support such that ψH1([0,)×d). This completes the proof. ∎

Remark 4.16.

Note that the same proposition holds without assuming the existence of a modulus of continuity for ϕ if η is not a function of x. Indeed, it is possible to pass to the limit first in the parameter ϑ, then δ, in Lemmas 4.3, 4.4, 4.7 and 4.12. Thus, if one assumes that η is not a function of the space variable x, it is also possible to pass to the limit first in the parameter ϑ, then δ, in Lemma 4.12 since one would have

liml0limκ0limδ00((I3+J3)+(I4+J4))C1ϑT

in its proof. Then the result holds following the first situation on [3, p. 523].

Our aim is to show the uniqueness of u(t,x,α) and u~(t,x,γ). To do this, we follow the ideas of [1, 3], and define for each n,

ϕn(x)={1if |x|n,na|x|aif |x|>n,

where a=d2+ε~ in which ε~>0 could be chosen in such a way that ϕnL2(d). Also, for each h>0 and fixed t0, we define

ψht(s)={1if st,1-s-thif tst+h,0if s>t+h.

A straightforward calculation revels that

ϕn(x)=-aϕn(x)|x|x|x|𝟏|x|>nL2(d)d,Δϕn(x)=a(2+2ε~-a)ϕn(x)|x|2L2({|x|>n}).

Clearly, (4.10) holds with ψ(s,x)=ϕn(x)ψht(s). Thus, for a.e t0, we obtain

1htt+h𝔼[d0101|u(s,x,α)-u~(s,x,γ)|ϕn(x)𝑑γ𝑑α𝑑x]𝑑s𝔼[0T{|x|>n}0101|ϕ(u(s,x,α))-ϕ(u~(s,x,γ))|Δϕn(x)ψht(s)𝑑γ𝑑α𝑑x𝑑s]   -𝔼[0T{|x|>n}0101F(u(s,x,α),u~(s,x,γ))ϕn(x)ψht(s)𝑑γ𝑑α𝑑x𝑑s]   -𝔼[0Tψht(s)({|x|>n}0101|ϕ(u(s,x,α))-ϕ(u~(s,x,γ))|ϕn(x)n~𝑑γ𝑑α𝑑x)𝑑s]   +𝔼[d|v0(x)-u0(x)|ϕn(x)𝑑x].(4.11)

Since ϕn(x)n~=an>0 on the set {|x|>n}, we have from (4.11),

1htt+h𝔼[d0101|u(s,x,α)-u~(s,x,γ)|ϕn(x)𝑑γ𝑑α𝑑x]𝑑s𝔼[0T{|x|>n}0101a(2+2ε~-a)ϕn(x)|x|2|ϕ(u(s,x,α))-ϕ(u~(s,x,γ))|ψht(s)𝑑γ𝑑α𝑑x𝑑s]   +𝔼[0T{|x|>n}0101aϕn(x)|x|F(u(s,x,α),u~(s,x,γ))x|x|ψht(s)𝑑γ𝑑α𝑑x𝑑s]   +𝔼[d|v0(x)-u0(x)|ϕn(x)𝑑x].(4.12)

Note that |F(a,b)|cf|a-b| for any a,b. Since ϕ is a Lipschitz continuous function and n1, inequality (4.12) gives

1htt+h𝔼[d0101|u(s,x,α)-u~(s,x,γ)|ϕn(x)𝑑γ𝑑α𝑑x]𝑑sC𝔼[ΠT[0,1]2|u(s,x,α)-u~(s,x,γ)|ϕn(x)ψht(s)𝑑γ𝑑α𝑑x𝑑s]+𝔼[d|v0(x)-u0(x)|ϕn(x)𝑑x].

Now passing to the limit as h0, and then using a weaker version of Gronwall’s inequality, we obtain for a.e. t>0,

𝔼[d0101|u(t,x,α)-u~(t,x,γ)|ϕn(x)𝑑γ𝑑α𝑑x]eCT𝔼[d|v0(x)-u0(x)|ϕn(x)𝑑x].

Thus, if we assume that v0(x)=u0(x), then we arrive at the conclusion

𝔼[d0101|u(t,x,α)-u~(t,x,γ)|ϕn(x)𝑑γ𝑑α𝑑x]=0,

which says that for almost all ωΩ, a.e. (t,x)(0,T]×d and a.e. (α,γ)[0,1]2, we have the equation u(t,x,α)=u~(t,x,γ). On the other hand, we conclude that the whole sequence of viscous approximation converges weakly in L2(Ω×ΠT). Since the limit process is independent of the additional (dummy) variable, the viscous approximation converges strongly in Lp(Ω×(0,T);Lp(Θ)) for any p<2 and any bounded open set Θd.

4.2 Existence of an entropy solution

In this subsection, using the strong convergence of the sequence of viscous solutions and the a priori bounds (3.17), we establish the existence of an entropy solution to problem (1.1).

Fix a non-negative test function ψCc([0,)×d), BT and a convex entropy flux triple (β,ζ,ν). Now apply the Itô–Lévy formula (3.1) and conclude

𝔼[𝟏BΠTβ′′(uε(t,x))|G(uε(t,x))|2ψ(t,x)𝑑x𝑑t]𝔼[𝟏Bdβ(uε(0,x))ψ(0,x)𝑑x]-ε𝔼[𝟏BΠTβ(uε(t,x))uε(t,x)ψ(t,x)𝑑x𝑑t]   +𝔼[𝟏BΠT(β(uε(t,x))tψ(t,x)+ν(uε(t,x))Δψ(t,x)-ψ(t,x)ζ(uε(t,x)))𝑑x𝑑t]   +𝔼[𝟏B0TEd01η(x,uε(t,x);z)β(uε(t,x)+θη(x,uε(t,x);z))ψ(t,x)𝑑θ𝑑xN~(dz,dt)]   +𝔼[𝟏B0TEd01(1-θ)η2(x,uε(t,x);z)β′′(uε(t,x)+θη(x,uε(t,x);z))      ×ψ(t,x)dθdxm(dz)dt].(4.13)

Let the predictable process u(t,x) be the pointwise limit of uε(t,x) for a.e. (t,x)(0,T)×d almost surely. One can now pass to the limit in (4.13) (same argument as in [5]) except the first term. The pointwise limit of uε(t,x) is not enough to pass to the limit in the first term of the inequality because uε is in a gradient term. For this, we proceed as follows: Fix vL2(Ω×ΠT). Define

fε=β′′(uε(t,x))ψ(t,x)𝟏Bandgε=G(uε(t,x)).

Note that fε is uniformly bounded and gεg=G(u(t,x)) in L2(Ω×ΠT). Also, fε converges to f pointwise (up to a subsequence), where

f=β′′(u(t,x))ψ(t,x)𝟏B.

Since

|fεv|β′′ψ(t,x)|v(t,x)|

and the right-hand side is L2 integrable, one can apply the dominated convergence theorem to conclude fεvfv in L2(Ω×ΠT). Moreover, we have fεgεfg in L2(Ω×ΠT), and therefore, by Fatou’s lemma for weak convergences,

𝔼[𝟏BΠTβ′′(u(t,x))|G(u(t,x))|2ψ(t,x)𝑑x𝑑t]lim infε0𝔼[𝟏BΠTβ′′(uε(t,x))|G(uε(t,x))|2ψ(t,x)𝑑x𝑑t].

Thus, we can pass to the limit in (4.13) as ε0 and arrive at following inequality:

𝔼[𝟏BΠTβ′′(u(t,x))|G(u(t,x))|2ψ(t,x)𝑑x𝑑t]-𝔼[𝟏Bdβ(u0(x))ψ(0,x)𝑑x]𝔼[𝟏BΠT(β(u(t,x))tψ(t,x)+ν(u(t,x))Δψ(t,x)-ψ(t,x)ζ(uε(t,x)))𝑑x𝑑t]   +𝔼[𝟏B0TEd01η(x,u(t,x);z)β(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θ𝑑xN~(dz,dt)]   +𝔼[𝟏B0TEd01(1-θ)η2(x,u(t,x);z)β′′(u(t,x)+θη(x,u(t,x);z))      ×ψ(t,x)dθdxm(dz)dt].(4.14)

We are now in a position to prove the result of existence of an entropy solution for the original problem (1.1).

Proof of the theorem 2.5.

The uniform moment estimate (3.17) together with a general version of Fatou’s lemma give

sup0tT𝔼[u(t,)22]<andG(u)L2(Ω×ΠT)2<.

For any 0ψCc([0,)×d) and any given convex entropy flux triple (β,ζ,ν), inequality (4.14) holds for every BT. Hence, the inequality

dβ(u0(x))ψ(0,x)𝑑x+ΠTβ(u(t,x))tψ(t,x)𝑑x𝑑t+ΠTν(u(t,x))Δψ(t,x)𝑑x𝑑t-ΠTψ(t,x)ζ(u(t,x))𝑑x𝑑t   +0TEd01η(x,u(t,x);z)β(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θ𝑑xN~(dz,dt)   +EΠT01(1-θ)η2(x,u(t,x);z)β′′(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θ𝑑x𝑑tm(dz)ΠTβ′′(u(t,x))|G(u(t,x))|2ψ(t,x)𝑑x𝑑t

holds P-almost surely. This shows that u(t,x) is an entropy solution of (1.1) in the sense of Definition 2.2. This completes the proof. ∎

We now close this section with a sketch of the justification of our claim in Remark 2.7. To see this, let hδ denote a smooth even convex approximation of ||p defined for positive x by the following: hδ vanishes at 0, and uniquely recovered from its second-order derivative defined as hδ′′(x)=xp-2 if x[0,1δ] and 1/δp-2 if x>1δ. It holds that 0hδ(x)h(x)=Kp|x|p, and there exists Cp such that 0hδ′′(x)Cph(x). Furthermore, it is easily seen that

hδ′′(x+y)C~p(hδ′′(x)+hδ′′(y)).

Note that the weak Itô–Lévy formula in Theorem A.1 makes sense for β=hδ as hδ′′ is bounded. This enables us to write, for almost every t>0,

Edhδ(uϵ)𝑑x-Edhδ(u0)𝑑x+E0td(ϕ(uϵ)+ϵ)hδ′′(uϵ)|uϵ|2𝑑x𝑑t=E0tEd(hδ(uϵ+η(x,uϵ;z))-hδ(uϵ)-η(x,uϵ;z)hδ(uϵ))𝑑xm(dz)𝑑t.=E0tEd01(1-θ)(η(x,uϵ;z))2hδ′′(uϵ+θη(x,uϵ;z))𝑑θ𝑑xm(dz)𝑑s.

We can now use the properties of hδ and the assumptions on η to arrive at

Edhδ(uϵ)𝑑xEdhδ(u0)𝑑x+CηE0td(1+uϵ2)hδ′′(uϵ)𝑑sEd|u0|p𝑑x+KηE0tdhδ(uϵ)𝑑s,

and, by a weak Gronwall inequality,

Edhδ(uϵ)𝑑xeKηtEd|u0|p𝑑x

for almost all t. This implies

Ed|uϵ|p𝑑xeCηtEd|u0|p𝑑x

by the monotone convergence theorem. The solution u will inherit the same property by Fatou’s lemma.

If u0 is bounded and η(x,u;z)=0 for |u|M, with given M, then consider the non-negative regular convex function xh(x)=[(x+K)-]2+[(x-K)+]2, where K=max(M+M1,u0). Since h(u0)=0 and h vanishes where η is active, Itô’s formula yields

Ed|h(uϵ)|𝑑x=0

and uϵ is uniformly bounded by K. Again, the solution u will inherit the same property by passing to the limit.

5 Uniqueness of the entropy solution

To prove the uniqueness of the entropy solution, we compare any entropy solution to the viscous solution via Kruzhkov’s doubling variables method and then pass to the limit as the viscosity parameter goes to zero. We have already shown that the limit of the sequence of viscous solutions serves to prove the existence of an entropy solution to the underlying problem. Now, let v(t,x) be any entropy solution and let uε(t,x) be a viscous solution for problem (3.1). Then one can use exactly the same arguments as in Section 4, and end up with the following equality:

𝔼[d01|u(t,x,α)-v(t,x)|ϕn(x)𝑑α𝑑x]=0.

This implies that, for almost every t[0,), one has v(t,x)=u(t,x,α) for almost every xd and (ω,α)Ω×(0,1). In other words, this proves the uniqueness of the entropy solution.

A Weak Itô–Lévy formula

Let u be an H1(d)-valued t-predictable process and assume that it is a weak solution to the SPDE

du(t,x)-Δϕ(u(t,x))dt=divxf(u(t,x))dt+Eη(x,u(t,x);z)N~(dz,dt)+εΔu(t,x)dt,t>0,xd.(A.1)

In addition, in view of (3.7), we further assume that uL2((0,T)×Ω;H1(d)). Moreover, u satisfies the initial condition u0L2(d) in the following sense: P -almost surely

limh01h0hdu(t,x)ϕ(x)𝑑x=du0(x)ϕ(x)𝑑x(A.2)

for every ϕCc(d). We have the following weak version of the Itô–Lévy formula for u(t,).

Theorem A.1.

Let assumptions (i)(v) hold and let u(t,) be an H1(Rd)-valued weak solution of equation (A.1), as described in Section 3.1.3, which satisfies equation (A.2). Then for every entropy triplet (β,ζ,ν) and ψCc1,2([0,)×Rd), it holds P-almost surely that

dβ(u(T,x))ψ(T,x)𝑑x-dβ(u(0,x))ψ(0,x)𝑑x=ΠTβ(u(t,x))tψ(t,x)𝑑x𝑑t-ΠTψ(t,x)ζ(u(t,x))𝑑x𝑑t   +ΠTE01η(x,u(t,x);z)β(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θN~(dz,dt)𝑑x   +ΠTE01(1-θ)η2(x,u(t,x);z)β′′(u(t,x)+θη(x,u(t,x);z))ψ(t,x)𝑑θm(dz)𝑑x𝑑t   -ΠT(εψ(t,x).xβ(u(t,x))+εβ′′(u(t,x))|xu(t,x)|2ψ(t,x))dxdt   -ΠTϕ(u(t,x))β′′(u(t,x))|u(t,x)|2ψ(t,x)𝑑x𝑑t+ΠTν(u(t,x))Δψ(t,x)𝑑x𝑑t

for almost every T>0.

Proof.

Let {τk} be a standard sequence of mollifiers on d. Then for every ρ()Cc1((0,T)) we have that

-0Tu(s,)*τkρ(s)𝑑s=0Tρ(s)Δ(ϕ(u(s,))*τk)𝑑s+0Tρ(s)divx(f(u)τκ)𝑑s+0TEρ(s)(η(x,u,z)*τk)N~(dz,ds)+ϵ0TΔ(u*τk(s,x))ρ(s)𝑑s(A.3)

holds P-almost surely. For every n, define

ρnt(s)={nsif 0s1n,1if 1ns<t,1-n(s-t)if t+1n>st,0elsewhere.

It follows by standard approximation arguments that (A.3) is still valid if we replace ρ() by ρnt(). Afterwards, we invoke the right continuity of the stochastic integral and standard facts related to Lebesgue points of Banach space valued functions to pass to the limit n and conclude that, for almost all t>0,

u*τk(t,)-u0*ρk=0tΔ(ϕ(u(s,))*τk)𝑑s+0tdivx(f(u)τκ)𝑑s+0tE(η(x,u,z)*τk)N~(dz,ds)+ϵ0tΔ(u*τk(s,x))𝑑s

holds P-almost surely. Above, we have used that the weak solution satisfies the initial condition in the sense of (A.2). Let β be the entropy function mentioned in the statement and let ψ be the test function specified. Now we apply the Itô–Lévy chain rule to β(u*τk(t,)) to have, for almost every t>0,

β(u*τk(t,))=β(u0*ρk)+0tβ(u*τk(t,))Δ(ϕ(u(s,))*τk)𝑑s+0tβ(u*τk(t,))divx(f(u)τκ)𝑑s+ϵ0tβ(u*τk(t,))Δ(u*τk(s,x))𝑑s+0tE(β(u*τk+η(x,u,z)*τk)-β(u*τ))N~(dz,ds)+0tE(β(u*τk+η(x,u,z)*τk)-β(u*τ)-η(x,u,z)*τkβ(u*τk))m(dz)𝑑t

P-almost surely. We now apply the Itô–Lévy product rule to β(u*τk)ψ(t,x) and integrate with respect to x to obtain for almost every T>0,

dβ(u*τk(T,x))ψ(T,x)𝑑x=dβ(u0*ρk)ψ(0,x)𝑑x+0Tdβ(u*τk)sψ(s,x)𝑑x𝑑s+0Tdβ(u*τk(s,))Δ(ϕ(u(s,))*τk)ψ(s,x)𝑑x𝑑s+0Tdβ(u*τk(s,))divx(f(u)τκ)ψ(s,x)𝑑x𝑑s

+ϵ0TRdψ(s,x)β(u*τk(s,))Δ(u*τk(s,x))𝑑x𝑑s+0TdEψ(s,x)(β(u*τk+η(x,u,z)*τk)-β(u*τ))𝑑xN~(dz,ds)+0TdEψ(s,x)(β(u*τk+η(x,u,z)*τk)-β(u*τ)-η(x,u,z)*τkβ(u*τk))m(dz)𝑑x𝑑s(A.4)

almost surely. Note that u*τk(T,)u(T,) and u0*τku0 in L2(Ω×d) as k0. Therefore, by the Lipschitz continuity of β, we have

dβ(u*τk(T,x))ψ(T,x)𝑑xdβ(u(T,x))ψ(T,x)𝑑x

and

dβ(u0*ρk)ψ(0,x)𝑑xdβ(u0)ψ(0,x)𝑑x

in L2(Ω). By a similar reasoning,

0Tdβ(u*τk)sψ(s,x)𝑑x𝑑s0Tdβ(u)sψ(s,x)𝑑x𝑑sas k0.

Furthermore, note that

0Tdβ(u*τk(t,))Δ(ϕ(u(s,))*τk)ψ(s,x)𝑑s𝑑x=-0Tdx(ψ(t,x)β(u*τk(t,x))).(ϕ(u())*τk)(s,x))dxds

and u,ϕ(u)L2(0,T;L2(Ω×d)). Therefore,

x(ψ(t,x)β(u*τk(t,x)))x(ψ(t,x)β(u(t,x)))

and ϕ(u)*τkϕ(u) in L2(0,T;L2(Ω×d)) as k0. Therefore,

0Tdβ(u*τk(t,))Δ(ϕ(u(s,))*τk)ψ(s,x)𝑑s𝑑x-0Tdx(ψ(t,x)β(u(t,x))).(ϕ(u(s,x)))dxds

in L1(Ω) as k0. By the same reasoning,

ΠTψ(s,x)β(u*τk(s,x))Δ(u*τk(s,x))𝑑x𝑑s-ΠTx(ψ(s,x)β(u(s,x))).(u(s,x))dxds

in L1(Ω) as k0.

Also, it may be recalled that divxf(u)L2(0,T;L2(Ω×d)) and β(u*τk)β(u) in L2(0,T;L2(Ω×d)) as κ0. Therefore,

0Tdβ(u*τk(t,))divx(f(u)τκ)ψ(s,x)𝑑x𝑑s0Tdβ(u)divx(f(u))ψ(s,x)𝑑x𝑑s

as k0 in L1(Ω). To this end, we denote

Ik(s,z)=dψ(s,x)(β(u*τk+η(x,u,z)*τk)-β(u*τ))𝑑x,I(s,z)=dψ(s,x)(β(u+η(x,u,z))-β(u*τ))𝑑x.

It follows from straightforward computations that

0TE|Ik(s,z)-I(s,z)|2m(dz)𝑑s0as k0.

Therefore, we can invoke the Itô–Lévy isometry and pass to the limit as k0, in the martingale term in (A.4). This completes the validation of the passage to the limit as k0 in every term of (A.4). The assertion is now concluded by simply letting k0 in (A.4) and rearranging the terms. ∎

References

  • [1]

    B. Andreianov and M. Maliki, A note on uniqueness of entropy solutions to degenerate parabolic equations in N, NoDEA Nonlinear Differential Equations Appl. 17 (2010), no. 1, 109–118.  Google Scholar

  • [2]

    C. Bauzet, G. Vallet and P. Wittbold, The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperbolic Differ. Equ. 9 (2012), no. 4, 661–709.  Web of ScienceCrossrefGoogle Scholar

  • [3]

    C. Bauzet, G. Vallet and P. Wittbold, A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force, J. Hyperbolic Differ. Equ. 12 (2015), no. 3, 501–533.  CrossrefWeb of ScienceGoogle Scholar

  • [4]

    M. Bendahmane and K. H. Karlsen, Renormalized entropy solutions for quasi-linear anisotropic degenerate parabolic equations, SIAM J. Math. Anal. 36 (2004), no. 2, 405–422.  CrossrefGoogle Scholar

  • [5]

    I. H. Biswas, K. H. Karlsen and A. K. Majee, Conservation laws driven by Lévy white noise, J. Hyperbolic Differ. Equ. 12 (2015), no. 3, 581–654.  Web of ScienceCrossrefGoogle Scholar

  • [6]

    I. H. Biswas, U. Koley and A. K. Majee, Continuous dependence estimate for conservation laws with Lévy noise, J. Differential Equations 259 (2015), no. 9, 4683–4706.  Web of ScienceCrossrefGoogle Scholar

  • [7]

    I. H. Biswas and A. K. Majee, Stochastic conservation laws: Weak-in-time formulation and strong entropy condition, J. Funct. Anal. 267 (2014), no. 7, 2199–2252.  Web of ScienceCrossrefGoogle Scholar

  • [8]

    H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Publishing, Amsterdam, 1973.  Google Scholar

  • [9]

    J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999), no. 4, 269–361.  CrossrefGoogle Scholar

  • [10]

    G.-Q. Chen, Q. Ding and K. H. Karlsen, On nonlinear stochastic balance laws, Arch. Ration. Mech. Anal. 204 (2012), no. 3, 707–743.  Web of ScienceCrossrefGoogle Scholar

  • [11]

    G.-Q. Chen and K. H. Karlsen, Quasilinear anisotropic degenerate parabolic equations with time-space dependent diffusion coefficients, Commun. Pure Appl. Anal. 4 (2005), no. 2, 241–266.  CrossrefGoogle Scholar

  • [12]

    G.-Q. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 4, 645–668.  CrossrefGoogle Scholar

  • [13]

    A. Debussche, J. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, prepint (2013), https://arxiv.org/abs/1309.5817v1.  

  • [14]

    A. Debussche and J. Vovelle, Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259 (2010), no. 4, 1014–1042.  CrossrefWeb of ScienceGoogle Scholar

  • [15]

    W. E, K. Khanin, A. Mazel and Y. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151 (2000), no. 3, 877–960.  CrossrefGoogle Scholar

  • [16]

    J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal. 255 (2008), no. 2, 313–373.  Web of ScienceCrossrefGoogle Scholar

  • [17]

    H. Holden and N. H. Risebro, Conservation laws with a random source, Appl. Math. Optim. 36 (1997), no. 2, 229–241.  CrossrefGoogle Scholar

  • [18]

    J. U. Kim, On a stochastic scalar conservation law, Indiana Univ. Math. J. 52 (2003), no. 1, 227–256.  CrossrefGoogle Scholar

  • [19]

    S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise, Encyclopedia Math. Appl. 113, Cambridge University Press, Cambridge, 2007.  Google Scholar

  • [20]

    G. Vallet, Dirichlet problem for a degenerated hyperbolic-parabolic equation, Adv. Math. Sci. Appl. 15 (2005), no. 2, 423–450.  Google Scholar

  • [21]

    G. Vallet and P. Wittbold, On a stochastic first-order hyperbolic equation in a bounded domain, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 4, 613–651.  Web of ScienceCrossrefGoogle Scholar

About the article

Received: 2017-05-17

Revised: 2017-07-17

Accepted: 2017-07-30

Published Online: 2017-09-12


The authors are profoundly thankful for the generous support from IFCAM, which allowed them to travel between India and France and made this collaboration possible. In addition, the first author acknowledges the support of INSA. The third author acknowledges the support of ISIFoR.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 809–844, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0113.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Ananta K. Majee
Applied Mathematics and Computation, 2018, Volume 338, Page 676

Comments (0)

Please log in or register to comment.
Log in