Abstract
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 u 0, 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 u 0 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 δ → 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 Hölder continuous with exponent \({\alpha > \frac{1}{2}}\).
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Farwig, R., Komo, C. Regularity of weak solutions to the Navier–Stokes equations in exterior domains. Nonlinear Differ. Equ. Appl. 17, 303–321 (2010). https://doi.org/10.1007/s00030-010-0055-4
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DOI: https://doi.org/10.1007/s00030-010-0055-4