Kolmogorov equations for randomly perturbed generalized Newtonian fluids

Abstract

We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.e. a polynomial of degree p − 1, p ∈ (1, 2), i.e. the shear thinning case. We prove that the associated Kolmogorov operator K admits at least one infinitesimally invariant measure μ satisfying certain exponential moment estimates. Moreover, K is L 2 -unique w.r.t. μ provided p ∈ ( p

Stability and moment estimates for the stochastic singular $Φ$-Laplace equation

“…x, which yields a simplified model for stochastic singular power law fluids without convection term. It exhibits a similar nonlinearity as in an equation modeling non-Newtonian fluids with stochastic perturbation in the shear-thinning case, see [7,11,49].…”

Kolmogorov equations associated to the stochastic 2D Euler equations

“…We would like to mention that Kolmogorov equations in infinite dimensional spaces have been widely studied in the past, see e.g. [15,12,9,5,17,18,6]; however, the equation treated in this paper is not covered by those ones, and it requires some special techniques.…”