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

Abstract

Abstract. We 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 two. 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.

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

“…by further employing the conservative form as considered in Lions-Perthame-Souganidis [78,79]; see Gess-Souganidis [59]. A generalization of this with a degenerate parabolic term ∇ · (A(u) · ∇u) has also been considered in [48,60].…”

“…One of the problems is the long-time behavior of solutions of nonlinear conservation laws driven by multiplicative noises. The noises of form ∇ · F (u) • dW have been considered in [59,60], in which the dynamics remains in the zero-spatial-average subspace of L 1 (T d ).…”

Pathwise Solutions for Fully Nonlinear First- and Second-Order Partial Differential Equations with Multiplicative Rough Time Dependence

“…The material about the first class of problems are part of the ongoing development of the theory in collaboration with P.-L. Lions [57,58,59,60,61,62,54,55]. The results about quasilinear divergence form equations are based on joint work with P.-L. Lions, B. Perthame and B. Gess [49,50,51,29,28,30,27]. Problems of the type discussed here arise in several applied contexts and models for a wide variety of phenomena and applications including mean field games, turbulence, phase transitions and front propagation with random velocity, nucleations in physics, macroscopic limits of particle systems, pathwise stochastic control theory, stochastic optimization with partial observations, stochastic selection, etc..…”

Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

“…They developed a pathwise well-posedness theory based on kinetic solutions. This theory was further extended by Gess and Souganidis [28,29].…”