Stochastic Entropy Solutions for Stochastic Nonlinear Transport Equations

This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of parabolic approximation motivated by the method of vanishing viscosity given by Feng and Nualart (J. Funct. Anal. 2008, 255, 313–373). Furthermore, we prove the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f.


Introduction
In this paper, we consider the existence and uniqueness of the solutions to the nonlinear transport equation with a stochastic forcing: dρ(t, x) + b(x) · ∇ x f (ρ(t, x))dt = A(ρ(t, x))dW t , t > 0, x ∈ R d , where W t is a one-dimensional Wiener process on a stochastic basis (Ω, F , P, {F t } t 0 ) and A : R → R is a real valued function. f : R → R and b : R d → R d are Borel functions, and the initial data ρ 0 is non-random.
When div x b = 0, then b(x) · ∇ x f (ρ(t, x)) = div x (b(x) f (ρ(t, x))), the equation in (1) models the phenomenon of complex fluid mixing in porous media flows and other problems in mathematics and physics [1][2][3][4][5]. A particular application of this model involves two-phase fluid flow, which has been used to study the flow of water through oil in a porous medium [6,7]. For the porous media flows, the spatial variations of porous formations occur on all length scales, but only the variations at the largest length scales are reliably reconstructed from data available. The heterogeneities occurring on the smaller length scales have to be incorporated stochastically. Consequently, the flows through such formations are stochastic [8].
Furthermore, stochastic entropy solution ρ is called a stochastic strong entropy solution if the below conditions hold: (iii) for each {F t } t 0 -adapted L 2 (R d )-valued stochastic processρ(t, x, ω), satisfying (3) and (4), we defineη through each entropy function η bỹ where r 0, v ∈ R, y ∈ R d and ψ ∈ C 2 0 (R 2d ), there is a deterministic function D(s, t), such that , y)A(ρ(r, y))dydr + D(s, t); (7) (iv) for each T > 0, there exist partitions 0 = t 0 < t 1 < · · · < t n = T such that We now state our main results. The first one is focused on the uniqueness.
Suppose that ρ 1 and ρ 2 are stochastic entropy solutions of (1), and one of them is a stochastic strong entropy solution. Then, for every t > 0, Remark 1. Compared with the uniqueness results given in [11,17], Theorem 1 is new since the 1/2-Hölder continuity of A is enough to ensure the uniqueness. Moreover, compared with the uniqueness result for stochastic differential equations in [32], the hypotheses of 1/2-Hölder continuity on A is optimal.
If b, f and A are more regular, we also have the following existence results.
Theorem 2. Let f ∈ C 2 (R) such that f is bounded and f (0) = 0. Assume that b ∈ W 1,∞ (R d ; R d ) and Then, (i) (1) has a stochastic strong entropy solution.
(ii) Moreover, in addition ρ 0 ∈ BV(R d ), for every T > 0, we have ρ ∈ L ∞ ([0, T]; L 1 (Ω; BV(R d ))) and there is a constant C depending only on b W 1,∞ (R d ) and f L ∞ (R) such that For a general function A, even for initial data ρ 0 ∈ L ∞ , the solution is not in L ∞ . To maintain the boundedness of solutions, additional assumptions on A should be added. Inspired by [9,10], we can suppose that A has compact support.
We now discuss the continuous dependence of the solutions on b, f and A. Some results for the continuity on A have established for the case of constant vector field b [17]. Here, we only give the continuous dependence of the solutions on b and f .
A meets the assumption (11). Let ρ be the unique stochastic strong entropy solution of (1) andρ be the unique stochastic strong entropy solution of For every T > 0, there exists a constant C > 0, which depends only on b W 1, and T, such that Remark 3. Without the noise, (1) has been discussed by Chen and Karlsen. Some results on the existence and uniqueness of solutions as well as continuous dependence on b and f have been obtained in [33]. Here, we get an analogue of [33] (Theorem 3.2) but simplify some assumptions on the velocity fields b andb.
The present paper is organized as follows. In Section 2, we give the proof of Theorem 1. Section 3 is devoted to the proof for Theorem 2. In Section 4, we prove the continuous dependence of solutions on b and f .
We end up this section by introducing some notations. N is natural numbers set. m ∈ N and C m 0 (R d ) stands for the vector space consisting of all functions φ, which, together with all their partial derivatives ∂ α φ of order |α| m, are continuous and have compact supports in R d . Given a measurable function ς, ς + = max{ς, 0} = ς ∨ 0 and ς − = − min{ς, 0} = −[ς ∧ 0]. The symbols ∇, div, ∆, if not differently specified, are referred to derivatives in x. For every R > 0, B R := {x ∈ R d : |x| < R}. It almost surely can be abbreviated to a.s.. The letter C will mean a positive constant, whose values may change in different places.

Proof of Theorem 2
(i) We prove the existence of stochastic strong entropy solutions for (1) by the method of vanishing viscosity, that is, we regard (1) as the ε ↓ 0 limit of the viscosity equation where ρ ε 0 is an approximation to ρ 0 . We now divide the proof into three steps.
Step 1. Existence and uniqueness of mild solutions to the Cauchy problem (22).
Here, ρ ε is said to be a mild solution of (22) With the help of Banach contraction mapping principle, there is a unique mild solution ρ ε to (22). Moreover, for every T > 0, Furthermore, for every 1 p < ∞, every T > 0, we have and We show that (4) holds for ρ ε . Let η θ be given by (15).
By using Itô's formula and the integration by parts, then where in (28) we have used the fact For θ, δ, M and ε be fixed, if one lets n approach to infinity, (28) turns to Observing that f is bounded, By virtue of (11), taking M > N, from (30) and (31), we have for all 0 t T (T > 0 is a given real number). Therefore, uniformly for ε 1.
Step 3. Existence of the stochastic strong entropy solution to the Cauchy problem (1).

Conclusions
In this paper, we have established three results on the existence and uniqueness of stochastic entropy solutions for a nonlinear transport equation by a stochastic perturbation, and the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f . Compared with the results on uniqueness given in [11,17], Theorem 1 is new since the 1/2-Hölder continuity of A is enough to ensure the uniqueness, and compared with the results on uniqueness for stochastic differential equations in [32], the hypotheses of 1/2-Hölder continuity on A is optimal. Moreover, we develop a new method of parabolic approximation to obtain the existence of solutions, which sheds some new light on the method of vanishing viscosity put forth by Feng and Nualart [11].
Author Contributions: All authors carried out the proofs and conceived the study. All authors read and approved the final manuscript.

Acknowledgments:
The authors are grateful to the anonymous referees for helpful comments and suggestions that greatly improved the presentation of this paper.

Conflicts of Interest:
The authors declare no conflict of interest.