Martina Hofmanová

Degenerate Parabolic Stochastic Partial Differential Equations: Quasilinear case

We study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an

A Regularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type

L^1

On Weak Solutions of Stochastic Differential Equations

-contraction property. In comparison to the first-order case (Debussche and Vovelle, 2010) and to the semilinear degenerate parabolic case (Hofmanova, 2013), the present result contains two new ingredients: a generalized Ito formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

A Bhatnagar–Gross–Krook approximation to stochastic scalar conservation laws

We consider a nondegenerate quasilinear parabolic stochastic partial differential equation (SPDE) with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We 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 Holder continuous in time and possesses spatial regularity. 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 problem 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.

Scalar conservation laws with rough flux and stochastic forcing

A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.

Weak solutions for a stochastic mean curvature flow of two-dimensional graphs

We study a BGK-like approximation to hyperbolic conservation laws forced by a multiplicative noise. First, we make use of the stochastic characteristics method and establish the existence of a solution for any fi xed parameter

Incompressible Limit for Compressible Fluids with Stochastic Forcing

\varepsilon

Quasilinear generalized parabolic Anderson model equation

. In the next step, we investigate the limit as

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

\varepsilon

Compressible Fluids Driven by Stochastic Forcing: The Relative Energy Inequality and Applications

tends to 0 and show the convergence to the kinetic solution of the limit problem.