Stochastic isentropic Euler equations

We study the stochastically forced system of isentropic Euler equations of gas dynamics with a $\gamma$-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section

Scalar conservation laws with stochastic forcing

We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation

Diffusion limit for the radiative transfer equation perturbed by a Wiener process

The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some diffculties caused by the random noise.Comment: 27 page

Invariant Measures for a Stochastic Fokker-Planck Equation

We study the kinetic Fokker-Planck equation perturbed by a stochastic Vlasov force term. When the noise intensity is not too large, we solve the Cauchy Problem in a class of well-localized (in velocity) functions. We also show that, when the noise intensity is sufficiently small, the system with prescribed mass admits a unique invariant measure which is exponentially mixing. The proof uses hypocoercive decay estimates and hypoelliptic gains of regularity. At last we also exhibit an explicit example showing that some restriction on the noise intensity is indeed required.Comment: Extended versio

A BGK approximation to scalar conservation laws with discontinuous flux

We study the BGK approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy Problem for the BGK approximation is well-posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem

Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions

28 pagesInternational audienceWe study the large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions. Under structural hypotheses on the flux of the equation, we describe the stationary solutions and show the convergence of the entropy solution to a stationary one. Numerical tests illustrate the theoretical results

Stochastic isentropic Euler equations

International audienceWe study the stochastically forced system of isentropic Euler equations of gas dynamics with a γ-law for the pressure. We show the existence of martingale weak entropy solutions; we also discuss the existence and characterization of invariant measures in the concluding section

Regularity of Stochastic Kinetic Equations

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow

On the strong convergence of the gradient in nonlinear parabolic equations

We consider the Cauchy-Dirichlet Problem for a nonlinear parabolic equation with L1 data. We show how the concept of kinetic formulation for conservation laws [Lions, Perthame, Tamor 94] can be be used to give a new proof of the existence of renormalized solutions. To illustrate this approach, we also extend the method to the case where the equation involves an additional gradient term