Stochastic Scalar Conservation Laws Research Articles

Invariant measures for stochastic conservation laws with Lipschitz flux in the space of almost periodic functions

We study the well-posedness of the initial value problem and the long-time behavior of almost periodic solutions to stochastic scalar conservation laws in any space dimension, under the assumption of Lipschitz continuity of the flux functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions.

Central limit theorem and moderate deviation principle for stochastic scalar conservation laws

We establish a central limit theorem and prove a moderate deviation principle for stochastic scalar conservation laws. Because of the lack of a viscous term, this is done in the framework of the kinetic solution. The weak convergence method and the doubling of variables method play a key role.

Uniqueness for stochastic scalar conservation laws on Riemannian manifolds revisited

We revise a uniqueness question for the scalar conservation law with stochastic forcing du + divgf(x,u)dt = ?(x,u)dWt, x ? M, t ? 0 on a smooth compact Riemannian manifold (M,g) whereWt is the Wiener process and x ? f(x,?) is a vector field on M for each ? ? R. We introduce admissibility conditions, derive the kinetic formulation and use it to prove uniqueness in a more straight-forward way than in the existing literature.

Well-posedness by noise for scalar conservation laws

We consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to well-posedness.

Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function.

Convergence of Approximations to Stochastic Scalar Conservation Laws

We develop a general framework for the analysis of approximations to stochastic scalar conservation laws. Our aim is to prove, under minimal consistency properties and bounds, that such approximations are converging to the solution to a stochastic scalar conservation law. The weak probabilistic convergence mode is convergence in law, the most natural in this context. We use also a kinetic formulation and martingale methods. Our result is applied to the convergence of the Finite Volume Method in the companion paper [15].

Well-posedness for stochastic scalar conservation laws with the initial-boundary condition

In this paper, we are interested in the initial-(non-homogeneous) Dirichlet boundary value problem for a multi-dimensional scalar non-linear conservation law with a multiplicative stochastic forcing. We introduce a notion of “renormalized” kinetic formulations in which the kinetic defect measures on the boundary of a domain are truncated. In such a kinetic formulation we establish a result of well-posedness of the initial-boundary value problem under only the assumptions (H1), (H2) and (H3) stated below, which are very similar ones in [6].

Stochastic non-isotropic degenerate parabolic–hyperbolic equations

We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic–hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of Brownian noise and periodic boundary conditions, we prove that the pathwise entropy solutions converge to their spatial average and provide an estimate on the rate of convergence. The third main result of the paper is a new regularization result in the spirit of averaging lemmata. This work extends both the framework of pathwise entropy solutions for stochastic scalar conservation laws introduced by Lions, Perthame and Souganidis and the analysis of the long time behavior of stochastic scalar conservation laws by the authors to a new class of equations.

Renormalized entropy solutions of stochastic scalar conservation laws with boundary condition

This paper is concerned with the renormalized stochastic entropy solutions of stochastic scalar conservation law forced by a multiplicative noise on a bounded domain with a non-homogeneous boundary condition. We first introduce a notion of renormalized stochastic entropy solutions and then establish the existence and uniqueness of the renormalized stochastic entropy solutions for a general L1-data. Our results allow us to give a positive answer to an open problem posed by Bauzet, Vallet and Wittbold in [4].

Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws

AbstractWe study the long‐time behavior and regularity of the pathwise entropy solutions to stochastic scalar conservation laws with random‐in‐time spatially homogeneous fluxes and periodic initial data. We prove that the solutions converge to their spatial average, which is the unique invariant measure of the associated random dynamical system, and provide a rate of convergence, the latter being new even in the deterministic case for dimensions higher than 2. The main tool is a new regularization result in the spirit of averaging lemmata for scalar conservation laws, which, in particular, implies a regularization by noise‐type result for pathwise quasi‐solutions.© 2016 Wiley Periodicals, Inc.

Semi-discretization for Stochastic Scalar Conservation Laws with Multiple Rough Fluxes

We develop a semi-discretization approximation for scalar conservation laws with multiple rough time dependence in inhomogeneous fluxes. The method is based on Brenier's transport-collapse algorithm and uses characteristics defined in the setting of rough paths. We prove strong L 1-convergence for inhomogeneous fluxes and provide a rate of convergence for homogeneous one's. The approximation scheme as well as the proofs are based on the recently developed theory of path-wise entropy solutions and uses the kinetic formulation which allows to define globally the (rough) characteristics.

The Equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Laws

The Equivalence Theorem of Kinetic Solutions and Entropy Solutions for Stochastic Scalar Conservation Laws

A Bhatnagar–Gross–Krook approximation to stochastic scalar conservation laws

Dans ce papier, nous etudions une approximation de type BGK pour des lois de conservations hyperboliques soumises a un bruit multiplicatif. Dans un premier temps, nous utilisons la methode des caracteristiques dans le cadre stochastique et etablissons l’existence d’une solution pour tout parametre $\varepsilon$ fixe. Nous nous interessons ensuite a la limite quand $\varepsilon$ tend vers $0$ et prouvons la convergence vers la solution cinetique du probleme limite.

Stochastic scalar conservation laws driven by rough paths

We prove the existence and uniqueness of solutions to a class of stochastic scalar conservation laws with joint space–time transport noise and affine-linear noise driven by a geometric p-rough path. In particular, stability of the solutions with respect to the driving rough path is obtained, leading to a robust approach to stochastic scalar conservation laws. As immediate corollaries we obtain support theorems, large deviation results and the generation of a random dynamical system.

Large deviations principles for stochastic scalar conservation laws

Large deviations principles for a family of scalar 1 + 1 dimensional conservative stochastic PDEs (viscous conservation laws) are investigated, in the limit of jointly vanishing noise and viscosity. A first large deviations principle is obtained in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. A second order large deviations principle is therefore investigated, however, this can be only partially proved. The second order rate functional provides a generalization for non-convex fluxes of the functional introduced by Jensen and Varadhan in a stochastic particles system setting.

Stochastic scalar conservation laws

We introduce a notion of stochastic entropic solution à la Kruzkov, but with Ito's calculus replacing deterministic calculus. This results in a rich family of stochastic inequalities defining what we mean by a solution. A uniqueness theory is then developed following a stochastic generalization of L 1 contraction estimate. An existence theory is also developed by adapting compensated compactness arguments to stochastic setting. We use approximating models of vanishing viscosity solution type for the construction. While the uniqueness result applies to any spatial dimensions, the existence result, in the absence of special structural assumptions, is restricted to one spatial dimension only.