## Differential and Integral Equations

- Differential Integral Equations
- Volume 10, Number 5 (1997), 961-974.

### Smooth solutions of the vector Burgers equation in nonsmooth domains

John G. Heywood and Wenzheng Xie

#### Abstract

We prove the existence and uniqueness of smooth solutions of the vector Burgers equation in arbitrary two- and three-dimensional domains. The only assumption about the spatial domain is that it should be an open set. The underlying estimates for these results are proved using new "elliptic-Sobolev" inequalities of Xie ([13], [15]) for the Laplacian. Our purpose in giving these results is to develop methods that we think can be eventually transferred to the Navier-Stokes equations. Indeed, the only missing point is the proof of analogous "elliptic-Sobolev" inequalities for the Stokes operator, which we conjecture to be valid.

#### Article information

**Source**

Differential Integral Equations, Volume 10, Number 5 (1997), 961-974.

**Dates**

First available in Project Euclid: 1 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367438628

**Mathematical Reviews number (MathSciNet)**

MR1741761

**Zentralblatt MATH identifier**

0891.35136

**Subjects**

Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Secondary: 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Heywood, John G.; Xie, Wenzheng. Smooth solutions of the vector Burgers equation in nonsmooth domains. Differential Integral Equations 10 (1997), no. 5, 961--974. https://projecteuclid.org/euclid.die/1367438628