Stochastic Navier-Stokes equations for compressible fluids
Abstract:
We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function of momentum and density, which is affine linear in momentum and satisfies suitable growth assumptions with respect to density, and establish existence of the so-called finite energy weak martingale solution under the condition that the adiabatic constant satisfies $gamma>3/2$. The proof is based on a four layer approximation scheme together with a refined stochastic compactness method and a careful identification of the limit procedure.
17.09.14
09:00
Martina Hofmanova
A regularity result for quasilinear parabolic SPDE's
Abstract:
We consider a quasilinear parabolic stochastic partial differential equation driven by a multiplicative noise and study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is H"older continuous in time and possesses spatial regularity that is only limited by the regularity of the given data.
Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problém whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs. It is a joint work with Arnaud Debussche and Sylvain de Moor.
CZ
