Journal of the Mathematical Society of Japan

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

Reinhard FARWIG, Hideo KOZONO, and Hermann SOHR

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Abstract

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

Subjects
Primary: 76D05: Navier-Stokes equations [See also 35Q30]
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35K60: Nonlinear initial value problems for linear parabolic equations

Keywords
Stokes and Navier-Stokes equations very weak solutions nonhomogeneous data Serrin's class

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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