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

Abstract

Abstract. Let Ω be an exterior domain in R (n = 2, 3, 4) , with boundary being not necessarily smooth. For any initial velocity u0 ∈ L(Ω) such that ∇ · u0 = 0 (in sense of distribution) and external forces F ∈ L(0,∞; L(Ω)) + L(0,∞; W−1,2(Ω)n) we are able to construct a turbulent weak solution u ∈ Cw([0,∞); L(Ω)) ∩ L(0,∞; W 1,2 0 (Ω)) to the equations of motion of a non-Newtonian fluid. Simultaneously, we prove that this solution fulfils the non-uniform decay condition ‖u(t)‖L2(Ω) → 0 as t →∞.