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Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains

Published online by Cambridge University Press:  14 November 2011

Adrian T. Hill  and
Endre Süli
Adrian T. Hill
Affiliation:
School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K.
Endre Süli
Affiliation:
Oxford University Computing Laboratory, Numerical Analysis Group, 11 Keble Road, Oxford OX1 3QD, U.K.
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Abstract

The scalar nonlinear convection-diffusion equation

is considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 2 , 1995 , pp. 439 - 448
Copyright
Copyright © Royal Society of Edinburgh 1995

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