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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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References

1Asplund, E. and Bungart, L.. A First Course in Integration (New York: Holt, 1966).Google Scholar
2Burgers, J.. Application of a model system to illustrate some points of the statistical theory of free turbulence. Netherl. Akad. Wefensh. Proc. 43 (1940), 2–12.Google Scholar
3Friedman, A.. Partial Differential Equations of Parabolic Type (Englewood Cliffs, N.J.: Prentice Hall, 1964).Google Scholar
4Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn. (Berlin: Springer, 1983).Google Scholar
5Henry, D.. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981).CrossRefGoogle Scholar
6Hill, A. T.. Attractors for nonlinear convection-diffusion equations and their numerical approximation (D. Phil. Thesis, Oxford, 1992).Google Scholar
7Hirsch, M. W.. Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal. 16 (1985), 432–39.CrossRefGoogle Scholar
8Hopf, E.. The equation u, + uux = µuxx. Comm. Pure Appl. Math. 3 (1950), 201–30.CrossRefGoogle Scholar
9Howes, F.. The asymptotic stability of steady solutions of reaction-convection-diffusion equations. J. Reine Angew. Math. 388 (1988), 212–20.Google Scholar
10Kazdan, J. and Kramer, R. J.. Invariant criteria for existence of solutions to second order quasilinear elliptic equations. Comm. Pure Appl. Math. 31 (1978), 619–45.CrossRefGoogle Scholar
11Kružkov, S. N.. First order quasilinear equations in several independent variables. Math. USSR Sb. 10(1970), 217–43.CrossRefGoogle Scholar
12Ladyzhenskaya, O. A. and Ural'ceva, N.. Linear and Quasilinear Elliptic Equations (New York: Academic Press, 1968).Google Scholar
13Ladyzhenskaya, O., Solonnikov, V. and Ural'ceva, N.. Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs 23 (New York: American Mathematical Society, 1968).CrossRefGoogle Scholar
14Levine, H. A., Payne, L. E., Sacks, P. E. and Straughan, B.. Analysis of a convective reaction-diffusion equation II. SIAM J. Math. Anal. 20 (1989), 133–47.CrossRefGoogle Scholar
15Matano, H.. Existence of nontrivial unstable sets for equilibriums of strongly order preserving systems. J. Fac. Sci. Univ. Tokyo 30 (1983), 645–73.Google Scholar
16Oleinik, O.. Uniqueness and stability of the generalised solution of the Cauchy problem for a quasilinear equation. Amer. Math. Soc. Transl. Ser. 33 (1964), 285–90.Google Scholar
17Porter, M. H. and Weinberger, H. F.. On the spectrum of second order operators. Bull. Amer. Math. Soc. 72(1966), 251–5.CrossRefGoogle Scholar
18Stampacchia, G.. Contribuiti alia regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secundo ordine ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 12 (1958), 223–44.Google Scholar
19Whitham, G. B.. Linear and Nonlinear Waves (New York: Wiley-Interscience, 1974).Google Scholar