Regularity of weak solutions to the Navier Stokes equations in exterior domains
Let u be a weak solution of the Navier Stokes equations in an exterior domain \Omega \subset \mathbb{R}^3} and a time interval [0, T[ 0 T with initial value u0, external force f div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u0 and f or on the solution u itself such as Serrin's condition u |_{L^s(0,T; L^q(\Omega) \infty} with {2 s \infty, \frac{2}{s} \frac{3}{q} =1} In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111 2131, 2007; J Math Fluid Mech 11:1 14, 2008; Banach Center Publ 81:175 184, 2008) to exterior domains. If e.g. u fulfills Serrin's condition in a left side neighborhood of t or if the norm u |_{L^{s'(t \delta,t; L^q(\Omega) converges to 0 sufficiently fast as d 0 where \frac{2}{s' \frac{3}{q} 1} then u is regular at t. The same conclusion holds when the kinetic energy \frac{1}{2} u(t) |_2^2} is locally Holder continuous with exponent \alpha \frac{1}{2}