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

@article{Gess2014LongTimeBI,
title={Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws},
author={Benjamin Gess and Panagiotis E. Souganidis},
journal={Communications on Pure and Applied Mathematics},
year={2014},
volume={70}
}
• Published 14 November 2014
• Mathematics
• Communications on Pure and Applied Mathematics
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 2. The main tool is a new regularization…
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## References

SHOWING 1-10 OF 62 REFERENCES
Invariant measure of scalar first-order conservation laws with stochastic forcing
• Mathematics
• 2013
Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic
Scalar conservation laws with rough (stochastic) fluxes
• Mathematics
• 2013
We develop a pathwise theory for scalar conservation laws with quasilinear multiplicative rough path dependence, a special case being stochastic conservation laws with quasilinear stochastic
Stochastic averaging lemmas for kinetic equations
• Mathematics
• 2011
We develop a class of averaging lemmas for stochastic kinetic equations. The velocity is multiplied by a white noise which produces a remarkable change in time scale. Compared to the deterministic
Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case
• Mathematics
• 2014
We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $${\mathbb {R}}^N$$RN with quasilinear multiplicative “rough path” dependence by
On a stochastic scalar conservation law
In this paper, we discuss the Cauchy problem for a scalar conservation law with a random noise. When the flux function is quadratic (e.g., Burgers' equation), the well-known existence result of
Scalar conservation laws with multiple rough fluxes
• Mathematics
• 2014
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and
Large time behavior and homogenization of solutions of two-dimensional conservation laws
• Mathematics
• 1993
We study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data. Under a very weak nonlinearity condition, we prove that the
Long-time behavior in scalar conservation laws
• Mathematics
• 2008
We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine