Journal of the Mathematical Society of Japan

Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data

Abstract

We investigate the nonstationary Navier-Stokes equations for an exterior domain $\Omega\subset \bm{R}^3$ in a solution class $L^s (0,T;L^q(\Omega))$ of very low regularity in space and time, satisfying Serrin's condition $\frac{2}{s} + \frac{3}{q} = 1$ 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 $k = \div u \in L^s(0,T;L^r(\Omega))$, where $\frac{1}{3} + \frac{1}{q} = \frac{1}{r}$.

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 1 (2007), 127-150.

Dates
First available in Project Euclid: 25 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1180135504

Digital Object Identifier
doi:10.2969/jmsj/1180135504

Mathematical Reviews number (MathSciNet)
MR2302666

Zentralblatt MATH identifier
1107.76022

Citation

FARWIG, Reinhard; KOZONO, Hideo; SOHR, Hermann. Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Japan 59 (2007), no. 1, 127--150. doi:10.2969/jmsj/1180135504. https://projecteuclid.org/euclid.jmsj/1180135504

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