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 of R-3 and a time interval [0, T[, 0 < T <= infinity, with initial value u(0), external force f = divF, 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 u(0) and f or on the solution u itself such as Serrin's condition parallel to u parallel to(Ls(0, T; Lq(Omega))) < infinity with 2 < s < infinity, 2/s + 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 parallel to u parallel to(Ls') ((t-delta, t; Lq(Omega))) converges to 0 sufficiently fast as delta -> 0+, where 2/s' + 3/q > 1, then u is regular at t. The same conclusion holds when the kinetic energy 1/2 parallel to u(t)parallel to(2)(2) is locally Holder continuous with exponent alpha > 1/2.