Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. (English) Zbl 1107.76022

Farwig, Reinhard; Kozono, Hideo; Sohr, Hermann

J. Math. Soc. Japan 59, No. 1, 127-150 (2007).

Summary: We investigate nonstationary Navier-Stokes equations for an exterior domain \(\Omega\subset \mathbb R^3\) in a solution class \(L^s(0,T;L^q(\Omega))\) of very low regularity in space and time, satisfying Serrin’s condition \(\frac 2s+\frac 3q=1\) but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by \(u|_{\partial\Omega}= g\in L^s(0,T; W^{-1/q,q} (\partial\Omega))\), and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition \(k= \operatorname{div} u\in L^s(0,T;L^r(\Omega))\), where \(\frac 13+ \frac 1q= \frac 1r\).

MSC:

76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

low regularity; Serrin’s condition; divergence condition