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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$L^1$-framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations


Authors: Gui-Qiang Chen and Kenneth H. Karlsen
Journal: Trans. Amer. Math. Soc. 358 (2006), 937-963
MSC (2000): Primary 35K65, 35B35, 35G25, 35D99
Published electronically: December 28, 2004
MathSciNet review: 2187640
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Abstract: We develop a general $L^1$-framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our $L^1$-framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an $L^1$ error estimate for an upwind-central finite difference scheme.


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Additional Information

Gui-Qiang Chen
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730
Email: gqchen@math.northwestern.edu

Kenneth H. Karlsen
Affiliation: Centre of Mathematics for Applications, Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email: kennethk@math.uio.no

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03689-X
PII: S 0002-9947(04)03689-X
Keywords: $L^1$--framework, degenerate parabolic equations, quasilinear, anisotropic, entropy solutions, kinetic formulation, continuous dependence, error estimates, vanishing viscosity, difference schemes
Received by editor(s): January 11, 2004
Published electronically: December 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.