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 $p-1$, $p\in (1,2)$, i.\,e. the shear thinning case. We prove that the associated Kolmogorov operator $K$ admits at least one infinitesimally invariant measure $\mu$ satisfying certain exponential moment estimates. Moreover, $K$ is $L^2$-unique w.\,r.\,t. $\mu$ provided $p\in (p^{\ast },2)$, where $p^{\ast }$ is the second root of $p^3-8p^2+14p-6=0$, approximately $p^{\ast }\approx 1.60407$.

16 pages

References in corpus (1)

• SPDE in Hilbert Space with Locally Monotone Coefficients

Cited by in corpus (1)

• Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation