Regularity of weak solutions to the Navier-Stokes equations in exterior domains. (English) Zbl 1189.76115

Farwig, Reinhard; Komo, Christian

NoDEA, Nonlinear Differ. Equ. Appl. 17, No. 3, 303-321 (2010).

Summary: Let \(u\) be a weak solution of the Navier-Stokes equations in an exterior domain \({\Omega \subset \mathbb{R}^3}\) and a time interval \([0, T[\), \(0 < T \leq \infty \), with initial value \(u _{0}\), external force \(f = \text{div} F\), 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 R. Farwig et al., Indiana Univ. Math. J. 56, No. 5, 2111–2131 (2007; Zbl 1175.35100); J. Math. Fluid Mech. 11, No. 3, 428–442 (2009; Zbl 1185.35162); in: Parabolic and Navier-Stokes equations. Part 1. Proceedings of the confererence, Bȩdlewo, Poland, September 10–17, 2006. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 81, Pt. 1, 175–184 (2008; Zbl 1154.35416)] 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 \(\delta \rightarrow 0+\), 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 \({\alpha > \frac{1}{2}}\).

MSC:

76D05	Navier-Stokes equations for incompressible viscous fluids
35Q30	Navier-Stokes equations
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

instationary Navier-Stokes equations; very weak solutions; weak solutions; Serrin’s class; local in time regularity; exterior domain

Citations:

Zbl 1175.35100; Zbl 1185.35162; Zbl 1154.35416

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