Stochastic shear thickening fluids: Strong convergence of the Galerkin approximation and the energy equality

Abstract

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial of degree p-1 of the rate of strain tensor, while the colored noise is considered as a random force. We focus on the shear thickening case, more precisely, on the case $p\in [1+{\frac{d}{2}},{\frac{2d}{d-2}})$, where d is the dimension of the space. We prove that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.