Well-posedness of the non-local conservation law by stochastic perturbation

TL;DR

This paper establishes the existence and uniqueness of strong solutions for stochastic non-local conservation laws with discontinuous flux functions, extending previous linear results to nonlinear cases using probabilistic methods.

Contribution

It provides the first nonlinear extension of linear stochastic conservation laws, proving well-posedness in an $L^{1}igcap L^{2}$ setting with constructive proofs.

Findings

Proved existence and uniqueness of weak solutions.

Solutions are strong in the probabilistic sense.

Extended linear results to nonlinear stochastic conservation laws.

Abstract

Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an $L^{1}\cap L^{2}$ setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove existence and uniqueness of weak solutions. The solution is strong solution in the probabilistic sense. The proofs are constructive and based on the method of characteristics (in the presence of noise), It\^o-Wentzell-Kunita formula and commutators. Our results are new , to the best of our knowledge, and are the first nonlinear extension of the seminar paper [20] where the linear case was addressed.

Equations

$$\{\begin{array}{rl}& {\partial }_{t}u(t,x)+Div(F(t,x,u))=0,\\ & u{|}_{t=0}=u_0.\end{array}$$

$$du(t,x)+Div(F(u))dt=g(u)dW(t,x),$$

$$du(t,x)+Div(F(t,x,u))\circ d{z}_t=0,$$

$$\left\{\begin{aligned} &\partial_{t}u(t,x)+Div\big{(}(F(t,x,(K\ast u)(x))+\frac{dB_{t}}{dt})\cdot u(t,x)\big{)}=0,\,,\\[5.0pt] &u|_{t=0}=u_{0}\,.\end{aligned}\right.$$

$$F\in L^{\infty }([0,T],L^1(\mathbb{R},L^{\infty }(\mathbb{R}))),$$

$$F\in L^{\infty }([0,T]\times \mathbb{R}\times \mathbb{R}),$$

$$F_1\in L^{\infty }([0,T],L^1(\mathbb{R},L^{\infty }(\mathbb{R}))),$$

$$F_3\in L^{\infty }([0,T]\times \mathbb{R}\times \mathbb{R}),$$

$$F_{3,3}\in L^2([0,T],L^1(\mathbb{R},L^{\infty }(\mathbb{R}))),$$

$$u_0\in L^2(\mathbb{R})\cap L^1(\mathbb{R}),$$

$$K\in C_0^{\infty }(\mathbb{R}).$$

$${\int }_0^t{X}_{s}d{B}_s=\underset{n\to \infty }{\lim }\sum _{t_i\in {\pi }_n,t_i\le t}{X}_{t_i}(B_{t_{i+1}\wedge t}-B_{t_i}),$$

$${\int }_0^t{X}_s\circ d{B}_s=\underset{n\to \infty }{\lim }\sum _{t_i\in {\pi }_n,t_i\le t}\frac{(X_{t_{i+1}\wedge t}+X_{t_i})}{2}(B_{t_{i+1}\wedge t}-B_{t_i}),$$

$$[X,Y{]}_t=\underset{n\to \infty }{\lim }\sum _{t_i\in {\pi }_n,t_i\le t}(X_{t_{i+1}\wedge t}-X_{t_i})(Y_{t_{i+1}\wedge t}-Y_{t_i}),$$

$$X_{s,t}(x)=x+{\int }_s^{t}b(r,X_{s,r}(x))dr+B_t-B_s.$$

$$b\in L^1((0,T);C_b^{m,\alpha }({\mathbb{R}}^d))$$

$$Y_{s,t}(x):=X_{s,t}^{-1}(x)$$

$$Y_{s,t}(x)=x-{\int }_s^{t}b(r,Y_{r,t}(x))dr-(B_t-B_s).$$

$${\int }_{\mathbb{R}}u(t,x)\phi (x)dx=$$

$$+{\int }_0^t{\int }_{\mathbb{R}}u(s,x)F(s,x,(K\ast u)){\partial }_x\phi (x)dxds$$

$$+{\int }_0^t{\int }_{\mathbb{R}}u(s,x){\partial }_x\phi (x)dx\circ d{B}_s.$$

$${\int }_{\mathbb{R}}u(t,x)\phi (x)dx={\int }_{\mathbb{R}}{u}_0(x)\phi (x)dx+{\int }_0^t{\int }_{\mathbb{R}}u(s,x)F(s,x,(K\ast u)){\partial }_x\phi (x)dxds$$

$$+{\int }_0^t{\int }_{\mathbb{R}}u(s,x){\partial }_x\phi (x)dxd{B}_s+\frac{1}{2}{\int }_0^t{\int }_{\mathbb{R}}u(s,x){\partial }_x^2\phi (x)dxds.$$

$$F^n(t,.,z)=(F{\ast }_x{\rho }_n)(t,.,z)$$

$$u_0^n=u_0\ast {\rho }_n.$$

$$\left\{\begin{aligned} &du^{1}(t,x)+Div\bigg{(}u^{1}(t,x)\cdot\big{(}F^{1}(t,x,(u_{0}^{1}\ast K))dt+\circ dB_{t}\big{)}\bigg{)}=0\,,\\[5.0pt] &u^{\varepsilon}\big{|}_{t=0}=u_{0}^{1}.\end{aligned}\right.$$

$$\left\{\begin{aligned} &du^{n+1}(t,x)+Div\bigg{(}u^{n+1}(t,x)\cdot\big{(}F^{n+1}(t,x,(u^{n}\ast K))dt+\circ dB_{t}\big{)}\bigg{)}=0\,,\\[5.0pt] &u^{n+1}\big{|}_{t=0}=u_{0}^{n+1}.\end{aligned}\right.$$

$$u^{n+1}(t,x)=u_0^{n+1}(Y_t^{n+1}(x))J{Y}_t^{n+1}(x)$$

$$d{X}_t^{n+11}=F^{n+1}(t,X_t,(u^n\ast K)(t,X_t))dt+d{B}_t,X_0=x.$$

$${\int }_{\mathbb{R}}|u^{n+1}(t,x)|dx={\int }_{\mathbb{R}}|u_0^{n+1}(Y_t^{n+1}(x))|J{Y}_t^{n+1}(x)dx={\int }_{\mathbb{R}}|u_0^{n+1}(y)|dx\le C.$$

$$\mathbb{E}\bigg{[}\bigg{|}\partial_{x}X_{s,t}^{n}(x)\bigg{|}^{-1}\bigg{]}\leq C,$$

$${\int }_{\mathbb{R}}\mathbb{E}[|u^{n+1}(t,x){|}^2]dx$$

Full text

Key words and phrases. Stochastic partial differential equation, Conservation Law, Regularization by noise, Itô-Wentzell-Kunita formula, Irregular flux.

MSC2010 subject classification: 60H15, 35R60, 35F10, 60H30.

1 Introduction

We consider the conservation law

[TABLE]

Here u is called the conserved quantity while F is the flux. These type of the equations express the balance equations of continuum physics, when small dissipation effects are neglected. A basic example is provided by the equations of gas and fluid dynamics, traffic flow and sedimentation of solid particles in a liquid. The well-posedness theorems within the class of entropy solutions were established by Kruzkov, see [26]. The selection of the physically relevant solution is based on the so called entropy condition that assert that a shock is formed only when the characteristics carry information toward the shock.

In 1995 was introduced by Lions, Perthame and Tadmor [29] the notions of called kinetic solution and relies on a new equation, the so-called kinetic formulation, that is derived from the conservation law at hand and that (unlike the original problem) possesses a very important feature - linearity. The two notions of solution, i.e. entropy and kinetic, are equivalent whenever both of them exist, nevertheless, kinetic solutions are more general as they are well defined even in situations when neither the original conservation law or the corresponding entropy inequalities can be understood in the sense of distributions. Among other significant references in this direction, let us emphasize the works of Chen and Perthame [9], Perthame [37] and Lions, Benoit and Souganidis [32].

Recently there has been an interest in studying the effect of stochastic forcing on nonlinear conservation laws [10, 14, 18, 25]. These papers consider the following stochastic scalar conservation laws

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with particular emphasis on existence and uniqueness questions (well-posedness).

For other hand, in [30] and [31] Lions, Benoit and Souganidis, introduced the theory of pathwise solutions to study the following stochastic conservation law

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where $z_{t}$ is continuous noise. They defined a new concept of the solution and proved existence and uniqueness via kinetic formulation. When the noise $z_{t}$ is the standard Brownian motion the pathwise solutions corresponding formally on the Stratonovich interpretation of the equation. See also Gess and Souganidis in [24].

The purpose of the present paper is a contribution to the following general question: can one hope for uniqueness of weak solutions of some class of conservation laws with irregular flux under stochastic perturbation. We present the first positive result. More precisely, we study the following stochastic conservation law

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Here, $(t,x)\in[0,T]\times\mathbb{R}$, $\omega\in\Omega$ is an element of the probability space $(\Omega,\mathbb{P},\mathcal{F})$, $F:[0,T]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ , $B_{t}$ is a standard Brownian motion and $K$ is a regular kernel. The stochastic integration is to be understood in the Stratonovich sense. The Stratonovich form is the natural one for several reasons , including physical intuition related to the Wong-Zakai principle.

The novelty of our results is to show existence and uniqueness of weak solutions for one-dimensional stochastic nonlocal conservation law (1.2) when $F$ has low regularity in the spatial variable. Conservation laws with non-local fluxes have appeared recently in the literature, arising naturally in many fields of application, such as in crowd dynamics (see [6] and the references therein), or in models inspired from biology( see [5, 22]).

In recent years there has been an increasing interest in random influences on PDE. The questions of regularizing effects and well-posedness by noise for partial differential equations have attracted much interest in recent years. The literature on regularization (i.e. improvement on uniqueness) by noise is vast and giving a complete survey at this point would exceed the purpose of this paper. Concerning the case of transport/continuity equations with irregular drift, we mention the works [3, 4, 16, 17, 20, 33, 34, 35, 36]. In particular, we would like to emphasize the papers [34] and [36] since the proofs have served as an inspiration for some steps of this work.

The proofs are based in the method of characteristics, some estimation on the flow associated to characteristics, in commutators and stochastic calculus techniques.

1.1 Hypothesis.

We denoted $F_{1}=\partial_{1}F$, $F_{3}=\partial_{3}F$ and $F_{3,3}=\partial_{3}^{2}F$ In this paper we assume the following hypothesis:

Hypothesis 1.1**.**

The flux $F$ satisfies

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Moreover, the initial condition verifies

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and the kernel satisfies

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1.2 Notations

First, through of this paper, we fix a stochastic basis with a one-dimensional Brownian motion \big{(}\Omega,\mathcal{F},\{\mathcal{F}_{t}:t\in[0,T]\},\mathbb{P},(B_{t})\big{)}. Then, we recall to help the intuition, the following definitions

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where $\pi_{n}$ is a sequence of finite partitions of $[0,T]$ with size $|\pi_{n}|\rightarrow 0$ and elements $0=t_{0}<t_{1}<\ldots$. The limits are in probability, uniformly in time on compact intervals. Details about these facts can be found in Kunita [27]. Also we address from that book, Itô’s formula, the chain rule for the stochastic integral, for any continuous d-dimensional semimartingale $X=(X_{1},X_{2},\ldots,X_{d})$, and twice continuously differentiable and real valued function f on $\mathbb{R}^{d}$.

1.3 Stochastic flow.

We consider $0\leq s\leq t$ and $x\in\mathbb{R}^{d}$, consider the following stochastic differential equation in $\mathbb{R}^{d}$

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and denote by $X_{t}(x):=X_{0,t}(x),t\in[0,T],x\in\mathbb{R}^{d}$.

For $m\in\mathbb{N}$ and $0<\alpha<1$, let us assume the following hypothesis on $b$:

[TABLE]

where $C^{m,\alpha}(\mathbb{R}^{d})$ denotes the class of functions of class $C^{m}$ on $\mathbb{R}^{d}$ such that the last derivative is Hölder continuous of order $\alpha$.

It is well known that under condition (1.11), $X_{s,t}(x)$ is a stochastic flow of $C^{m}$-diffeomorphism (see for example [11] and [27]). Moreover, the inverse flow

[TABLE]

satisfies the following backward stochastic differential equation

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for every $0\leq s\leq t\leq T$. We will denote $Y_{0,t}(x):=Y_{t}(x)$ for every $t\in[0,T],x\in\mathbb{R}^{d}$.

2 Definition and Existence of Solutions.

2.1 Definition of solutions

Definition 2.1**.**

A stochastic process $u\in L^{\infty}([0,T],L^{2}(\Omega\times\mathbb{R}))\cap L^{1}(\Omega\times[0,T]\times\mathbb{R})\cap L^{\infty}(\Omega\times[0,T],L^{1}(\mathbb{R}))$ is called a $L^{2}$- weak solution of the Cauchy problem (1.2) when: For any $\varphi\in C_{0}^{\infty}(\mathbb{R})$, the real valued process $\int u(t,x)\varphi(x)dx$ has a continuous modification which is an $\mathcal{F}_{t}$-semimartingale, and for all $t\in[0,T]$, we have $\mathbb{P}$-almost surely

[TABLE]

Remark 2.2**.**

Using the same idea as in Lemma 13 [20], one can write the problem (1.2) in Itô form as follows, a stochastic process $u\in L^{\infty}([0,T],L^{2}(\Omega\times\mathbb{R}))\cap L^{1}(\Omega\times[0,T]\times\mathbb{R})\cap L^{\infty}(\Omega\times[0,T],L^{1}(\mathbb{R}))$ is a $L^{2}$- weak solution of the SPDE (1.2) iff for every test function $\varphi\in C_{0}^{\infty}(\mathbb{R})$, the process $\int u(t,x)\varphi(x)dx$ has a continuous modification which is a $\mathcal{F}_{t}$-semimartingale and satisfies the following Itô’s formulation

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[TABLE]

2.2 Existence.

The goal of this section is to prove general existence result for stochastic conservation law with low regularity of the flux function.

Lemma 2.3**.**

Assume that hypothesis 1.1 holds. Then there exists $L^{2}$-weak solutions of the Cauchy problem (1.2).

Proof.

Step 1: Regularization.

Let $\{\rho_{n}\}_{n}$ be a family of standard symmetric mollifiers, We define the family of regularized coefficients given by

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and

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Clearly we observe that, for every $n\in\mathbb{N}$, any element $F^{n}$, $u_{0}^{n}$ are smooth with bounded derivatives of all orders. We define $u^{1}$ as the solution of the following SPDE

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Moreover, inductively we define

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Following the classical theory of H. Kunita [28] we obtain that

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is the unique solution to the regularized equation (2.15), where $Y_{t}^{n+1}$ is the inverse to the following stochastic differential equation (SDE):

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Step 2: Boundedness. We observe that

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We postpone the following estimation for the appendix

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where the constant does not depend on $n$.

Then making the change of variables $y=Y_{t}^{\varepsilon}(x)$ we have that

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Now, by inequality (2.17) we conclude

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Therefore, the sequence $\{u^{n}\}$ is bounded in $u\in L^{2}(\Omega\times[0,T]\times\mathbb{R})\cap L^{\infty}([0,T],L^{2}(\Omega\times\mathbb{R}))$ . Then there exists a convergent subsequence, which we denote also by $u^{n}$, such that converge weakly in $L^{2}(\Omega\times[0,T]\times\mathbb{R})$ and weak-star in $L^{\infty}([0,T],L^{2}(\Omega\times\mathbb{R}))$ to some process $u\in L^{2}(\Omega\times[0,T]\times\mathbb{R})\cap L^{\infty}([0,T],L^{2}(\Omega\times\mathbb{R}))$. Since this subsequence is bounded in $L^{1}(\Omega\times[0,T]\times\mathbb{R})$ we follows that $u^{n}$ converge to one measure $\mu$ and $\mu=u$. From estimation (2.16) we deduce that $u\in L^{\infty}(\Omega\times[0,T],L^{1}(\mathbb{R}))$.

Step 3: Passing to the Limit. Now, if $u^{n+1}$ is a solution of (2.15), it is also a weak solution, that is, for any test function $\varphi\in C_{0}^{\infty}(\mathbb{R})$, $u^{n+1}$ verifies (written in the Itô form):

[TABLE]

Now, we observe that $G^{n+1}=u^{n+1}(s,x)\,F^{n+1}(s,x,(u^{n}\ast K))$ is uniformly bounded in $L^{2}(\Omega\times[0,T]\times\mathbb{R})$. Then there exists a convergent subsequence, which we denote also by $G^{n}$, such that converge weakly in $L^{2}(\Omega\times[0,T]\times\mathbb{R})$ to some process $G\in L^{2}(\Omega\times[0,T]\times\mathbb{R})$.

Then passing to the limit in equation (2.2) along the convergent subsequences found, we have

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From the last equality we have that $\int_{\mathbb{R}}u(t,x)\varphi(x)dx$ is continuous semimartingale for any test function $\varphi\in C_{0}^{\infty}(\mathbb{R})$. Thus $u\ast K$ is continuous semimartingale. We observe that $u^{n+1}\ast K$ converge to $u\ast K$ and that $F^{n+1}(s,x,(u^{n}\ast K))$ strong converge to $F(s,x,(u\ast K))$ in $L^{2}([0,T]\times\Omega,L_{loc}^{2}(\mathbb{R}))$. Then passing to the limit in equation (2.2) along the convergent subsequences found, we conclude that

$\begin{aligned} \int_{\mathbb{R}}u(t,x)\varphi(x)dx=&\int_{\mathbb{R}}u_{0}(x)\varphi(x)\ dx+\int_{0}^{t}\!\!\int_{\mathbb{R}}u(s,x)F(s,x,(u\ast K))\partial_{x}\varphi(x)\ dxds\\[5.0pt] &+\int_{0}^{t}\!\!\int_{\mathbb{R}}u(s,x)\ \partial_{x}\varphi(x)\ dx\,dB_{s}\,+\frac{1}{2}\int_{0}^{t}\!\!\int_{\mathbb{R}}u(s,x)\ \partial_{x}^{2}\varphi(x)\ dx\,ds.\end{aligned}$ ∎

3 Uniqueness.

3.1 Estimation on the flow.

Assume that $u$ is a $L^{2}$-solution of the Cauchy problem (1.2), . Let $\{\rho_{\varepsilon}\}_{\varepsilon}$ be a family of standard symmetric mollifiers. We denoted $F_{\varepsilon}(t,.,(u\ast K))=(F\ast_{x}\rho_{\varepsilon})(t,.,(u\ast K))$. We observe that the regularization is done in the second variable. Following the same steps as in the Lemma 4.1 we obtain:

Lemma 3.1**.**

Assume the hypothesis 1.1. Then for $T>0$ there exist a constant $C$ such that

[TABLE]

where C depend on $\|F\|_{L^{\infty}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$, $\|F\|_{L^{\infty}([0,T]\times\mathbb{R}\times\mathbb{R})}$, $\|F_{1}\|_{L^{\infty}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$, $\|F_{3}\|_{L^{\infty}([0,T]\times\mathbb{R}\times\mathbb{R})}$ and $\|F_{3,3}\|_{L^{2}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$.

Remark 3.2**.**

The same results is valid for the backward flow $Y_{s,t}$ since it is solution of the same SDE driven by the drifts $-b$.

3.2 Main result.

In this section, we shall present a uniqueness theorem for the SPDE (1.2). We pointed that similar arguments was used in previous works [34] , [36] for stochastic linear continuity equation.

Theorem 3.3**.**

Under the conditions of hypothesis 1.1, uniqueness holds for $L^{2}$- weak solutions of the Cauchy problem (1.2) in the following sense: if $u,v$ are $L^{2}$- weak solutions with the same initial data $u_{0}\in L^{2}(\mathbb{R})\cap L^{1}(\mathbb{R})$, then $u=v$ almost everywhere in $\Omega\times[0,T]\times\mathbb{R}$.

Proof.

Step 1: Two solutions. Let $v$ and $w$ are two $L^{2}$- weak solutions with initial conditions equal to $u_{0}$. We denoted $u=v-w$. Then $u$ verifies

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Step 2: Primitive of the solution. We set

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We observe that $\partial_{x}V(t,x)=u(t,x)$ belong to $L^{2}(\Omega\times[0,T]\times\mathbb{R})$. Now, we consider a nonnegative smooth cut-off function $\eta$ supported on the ball of radius 2 and such that $\eta=1$ on the ball of radius 1. For any $R>0$, we introduce the rescaled functions $\eta_{R}(\cdot)=\eta(\frac{.}{R})$.

For all test functions $\varphi\in C_{0}^{\infty}(\mathbb{R})$ we obtain

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where $\theta(x)=\int_{-\infty}^{x}\varphi(y)\ dy$. By definition of $L^{2}$-solutions , taking as test function $\theta(x)\eta_{R}(x)$ we get

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We observe that

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as $R\rightarrow\infty$. Passing to the limit in equation (3.2) we have that

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Step 3: Smoothing. Let $\{\rho_{\varepsilon}(x)\}_{\varepsilon}$ be a family of standard symmetric mollifiers. For any $\varepsilon>0$ and $x\in\mathbb{R}^{d}$ we use $\rho_{\varepsilon}(x-\cdot)$ as test function, then we deduce

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We denote $V_{\varepsilon}(t,.)=(V\ast_{x}\rho_{\varepsilon})(t,.)$, $F_{\varepsilon}(t,.,(K\ast v)(.))=(F\ast_{x}\rho_{\varepsilon})(t,.)$ and $(FV)_{\varepsilon}(t,.)=(F.V\ast_{x}\rho_{\varepsilon})(t,.)$. Thus we have

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[TABLE]

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where we denote $\mathcal{R}_{\epsilon}(V,F)=F_{\varepsilon}\ \partial_{x}V_{\varepsilon}-(F\partial_{x}V)_{\varepsilon}$.

Step 4: Method of Characteristics. Now, we consider the flow

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Applying the Itô-Wentzell-Kunita formula to $V_{\varepsilon}(t,X_{t}^{\epsilon})$ , see Theorem 8.3 of [27], we have

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Then, considering that $X_{t}^{\epsilon}=X_{0,t}^{\epsilon}$ and $Y_{t}^{\epsilon}=Y_{0,t}^{\epsilon}=(X_{0,t}^{\epsilon})^{-1}$ we deduce that

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[TABLE]

Step 5: Localization. Now, we consider a nonnegative smooth cut-off function $\eta$ supported on the ball of radius 2 and such that $\eta=1$ on the ball of radius 1. For any $R>0$, we introduce the rescaled functions $\eta_{R}(\cdot)=\eta(\frac{.}{R})$. From the last step we have

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Step 6: Convergence of the commutator I. Now, we observe that $\mathcal{R}_{\epsilon}(V,F)$ converge to zero in $L^{2}([0,T]\times\Omega\times\mathbb{R})$. In fact, we get that

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Moreover, we have

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and

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Then by the dominated convergence theorem we obtain

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Step 7: Convergence of the conmutator II. We observe that

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By Hölder’s inequality we obtain

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From step 6 we follow

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From lemma 3.1 we deduce

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Step 8: Conclusion . From step 5 we have

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Now we observe that

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where $\tilde{K}$ is one compact set. If we take $R$ sufficiently large we have

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Taking the limit as $\epsilon$ converge to zero we deduce

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[TABLE]

By Gronwall lemma we deduce that $V=0$. Then we have $u=0$.

∎

4 Appendix

Lemma 4.1**.**

Assume the hypothesis 1.1. Then for $T>0$ there exist a constant $C$ such that

[TABLE]

where C depend on $\|F\|_{L^{\infty}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$, $\|F\|_{L^{\infty}([0,T]\times\mathbb{R}\times\mathbb{R})}$, $\|F_{1}\|_{L^{\infty}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$, $\|F_{3}\|_{L^{\infty}([0,T]\times\mathbb{R}\times\mathbb{R})}$ and $\|F_{3,3}\|_{L^{2}([0,T],L^{1}(\mathbb{R},L^{\infty}(\mathbb{R})))}$.

Proof.

For simplicity we assume $s=0$.

Step 1: Regularization.

We consider the SDE :

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We observe that $\partial_{x}X_{t}^{n+1}$ verifies

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Step 2: Semimartingale representation. By definition of solution we get that $(K\ast u^{n})(t,x)$ satisfies

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Then by Itô formula we have

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In order to write the last equality in Itô formulation we have to calculate the covariation

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Now, we obtain

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and

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We set $u^{n,k}(t,x):=(K\ast u^{n})(t,x)$. From the Itô formula for the product of two semimartingales we obtain

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Applying covariation in the last equality we deduce

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From formulas (4.24) and (4.25) we obtain

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Integrating the last equality we get

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Step 3: Itô-Wentzell-Kunita formula. Applying the Itô-Wentzell-Kunita formula to $\int_{-\infty}^{X_{t}^{n+1}}F(t,x,(K\ast u^{n})(x))dx$ , see Theorem 8.3 of [27], we deduce

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Step 4: Boundedness.

We have that

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Step 5: Conclusion.

From step 3 we have

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where

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and

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amd

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We set

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and

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From (4.26) and inequalities in step 4 we obtain

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Finally by Hölder inequality we deduce

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Finally we observe that the processes

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and

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are martingales with expectation equal to one. From this we conclude our lemma. ∎

5 Final remarks.

Remark 5.1**.**

(Linear case) Now suppose that $F(t,x,z)=b(x)$. If $b\in L^{1}\cap L^{\infty}$ then it satisfies the hypothesis 1. Thus, we have the uniqueness for the stochastic continuity equation with irregular drift. Then our result is the nonlinear extension of the theory of regularization by noise for transport/continuity equation initiated by Flandoli, Gubinelli and Priola in [20].

We pointed of according to the theory of Diperna-Lions (see [15]) the uniqueness of deterministic transport/continuity equation holds when $b$ has $W^{1,1}$ spatial regularity together with a condition of boundedness on the divergence. The theory has been generalized by L. Ambrosio [1] to the case of only $BV$ regularity for b instead of $W^{1,1}$. We refer the readers to two excellent summaries in [2] and [13].

Remark 5.2**.**

The main our tool in order to have estimations on the derivative of the flow was the Itô-Wentzell-Kunita formula. However, it is possible only to apply this formula for compositions of semimartingales. In order to generalized our result for more general $F$ we have in mind to work in the context of the theory of stochastic calculus via regularization. This calculus was introduced by by F. Russo and P. Vallois ( see [39] as general reference ) and it have been studied and developed by many authors. In the papes of F.Flandoli and F. Russo and R. Coviello and F.Russo they obtain a Itô-Wentzell-Kunita formula for more general process, see [21] and [7]. We also mention the recent extension of the Itô-Wentzell-Kunita formula by R. Duboscq and A. Reveillac in [8].

Remark 5.3**.**

We pointed that multiplicative noise as the one used in the stochastic conservation law (1.2) is not enough to improve the regularity of solutions of the following stochastic Burgers equation

[TABLE]

Indeed, for this equation one can observe the appearance of shocks in finite time, just as for the deterministic Burgers equation. We address the reader to [19] for a more detailed discussion of this topic.

Remark 5.4**.**

Finally we point after the writing of this paper appeared in arxiv the paper of Gess and Maurelli [23] where the authors consider stochastic scalar conservation laws with spatially inhomogeneous flux. Assuming low regularity of the flux function with respect to its spatial variable they proved uniqueness of stochastic kinetic entropy solutions when are not necessarily uniqueness in the corresponding deterministic scalar conservation law. Our result is in one-dimension but we prove uniqueness in the class of weak solutions and we assume very low regularity on the flux functions (in spatial variable).

Acknowledgements

Christian Olivera is partially supported by FAPESP by the grants 2017/17670-0 and 2015/07278-0 and by CNPq by the grant 426747/2018-6.