## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 38, Number 1 (2009), 111-136.

### On the stokes operator in general unbounded domains

Reinhard FARWIG, Hideo KOZONO, and Hermann SOHR

#### Abstract

It is known that the Stokes operator is not well-defined in $L^q$-spaces for certain unbounded smooth domains unless $q=2$. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general unbounded smooth domains from the three-dimensional case, see \cite{FKS1}, to the $n$-dimensional one, $n\geq 2$, replacing the space $L^q, 1\ltq\lt\infty$, by $\s{L}^q$ where $\s{L}^q = L^q\cap L^2$ for $q\geq 2$ and $\s{L}^q = L^q+L^2$ for $1\ltq\lt2$. In particular, we show that the Stokes operator is well-defined in $\s{L}^q$ for every unbounded domain of uniform $C^{1,1}$-type in $\R^n$, $n\geq 2$, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.

#### Article information

**Source**

Hokkaido Math. J. Volume 38, Number 1 (2009), 111-136.

**Dates**

First available in Project Euclid: 28 July 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1248787007

**Digital Object Identifier**

doi:10.14492/hokmj/1248787007

**Mathematical Reviews number (MathSciNet)**

MR2501897

**Zentralblatt MATH identifier**

1170.76011

**Subjects**

Primary: 76D05: Navier-Stokes equations [See also 35Q30]

Secondary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

**Keywords**

General unbounded domains domains of uniform C^{1,1}-type Stokes operator Stokes resolvent Stokes semigroup maximal regularity

#### Citation

FARWIG, Reinhard; KOZONO, Hideo; SOHR, Hermann. On the stokes operator in general unbounded domains. Hokkaido Math. J. 38 (2009), no. 1, 111--136. doi:10.14492/hokmj/1248787007. https://projecteuclid.org/euclid.hokmj/1248787007.