Kolmogorov Equations for Randomly Perturbed Generalized Newtonian Fluids

TL;DR

This paper studies the mathematical properties of stochastic models for shear-thinning fluids, proving existence and uniqueness of invariant measures for the associated Kolmogorov operators under certain conditions.

Contribution

It establishes the existence of invariant measures and proves $L^2$-uniqueness of the Kolmogorov operator for a class of stochastic generalized Newtonian fluids with shear-thinning behavior.

Findings

Existence of at least one invariant measure with exponential moment bounds.

$L^2$-uniqueness of the Kolmogorov operator for $p$ in a specific range.

Characterization of conditions under which the stochastic PDE admits unique invariant measures.

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 $p^\ast \approx 1.60407$.