Existence of turbulent weak solutions to the generalized Navier-Stokes equations in exterior domains and large time behaviour

Jörg Wolf

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 1-35

MSC 2000

35Q30 Stokes and Navier-Stokes equations

35Q40 Equations from quantum mechanics

Abstract
Let $\Omega$ be an exterior domain in $\R^n\,(n=2,3,4)$ , with boundary being not necessarily smooth. For any initial velocity ${\bf u}_0\in L^2(\Omega)^n$ such that $\nabla\cdot {\bf u}_0=0$ (in sense of distribution) and external forces \[ \bfF\in L^1(0,\infty; L^2(\Omega )^n)+ L^2(0,\infty; W^{-1,2} (\Omega )^n) \] we are able to construct a turbulent weak solution ${\bf u}\in C_w([0,\infty); L^2(\Omega)^n)\cap L^2(0,\infty; W^{1, 2}_0(\Omega )^n)$ to the equations of motion of a non-Newtonian fluid. Simultaneously, we prove that this solution fulfils the non-uniform decay condition \[ \|\bfu(t)\|_{L^2(\Omega)} \rightarrow 0 \quad \mbox{as}\quad t \rightarrow \infty. \]