## Differential and Integral Equations

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

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

Yoshihiro Shibata and Senjo Shimizu

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060651

**Mathematical Reviews number (MathSciNet)**

MR1972873

**Zentralblatt MATH identifier**

1054.35056

**Subjects**

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

#### Citation

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