Differential and Integral Equations

Smooth solutions of the vector Burgers equation in nonsmooth domains

John G. Heywood and Wenzheng Xie

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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


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