Differential and Integral Equations

On a resolvent estimate for the Stokes system with Neumann boundary condition

Yoshihiro Shibata and Senjo Shimizu

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We obtain the $L_p$ estimate of solutions to the resolvent problem for Stokes system with Neumann type boundary condition in a bounded or exterior domain in $\mathbb R^n$. The result has been obtained by Grubb and Solonnikov [7, 8, 9, 10] by the systematic use of theory of pseudo-differential operators. In this paper, we give an essentially different proof from [7, 8, 9, 10]. The point of our proof is to use the space $W^{-1}_p$ in order to handle with the equation $\nabla\cdot u = g$ in the half-space model problem as well as in the whole space one. Our result is an extension of the paper by Farwig and Sohr [5] about the Dirichlet zero condition case to the Neumann boundary condition case.

Article information

Differential Integral Equations, Volume 16, Number 4 (2003), 385-426.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B45: A priori estimates 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 76D07: Stokes and related (Oseen, etc.) flows


Shibata, Yoshihiro; Shimizu, Senjo. On a resolvent estimate for the Stokes system with Neumann boundary condition. Differential Integral Equations 16 (2003), no. 4, 385--426. https://projecteuclid.org/euclid.die/1356060651

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