We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the γ-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff. We prove the existence and uniqueness of local classical solutions to the Cauchy problem, in the function space with Maxwellian type exponential decay with respect to the velocity variable. <b><i>To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).</i></b>

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In the paper, we study the Prandtl system with initial data admitting non-degenerate critical points. For any index \sigma\in [3/2, 2], we obtain the local in time well-posedness in the space of Gevrey class \sigma\in [3/2, 2], in the tangential variable and Sobolev class in the normal variable so that the monotonicity condition on the tangential velocity is not needed to overcome the loss of tangential derivative. This answers the open question raised in the paper of D. Grard-Varet and N. Masmoudi [{\it Ann. Sci. c. Norm. Supr}.(4) 48 (2015), no. 6, 1273-1325], in which the case \sigma\in [3/2, 2], is solved.

By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimal L p (1 p+) convergence rates to the nonlinear diffusion waves. The pointwise estimates and L p (1 p&lt; 2) convergence rates given in this paper are new.