Shock-layer bounds for a singularly perturbed equation
Author:
Jeffrey S. Scroggs
Journal:
Quart. Appl. Math. 53 (1995), 423-431
MSC:
Primary 35B25; Secondary 35B40, 35K57, 35L67
DOI:
https://doi.org/10.1090/qam/1343460
MathSciNet review:
MR1343460
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Abstract: The size of the shock-layer governed by a conservation law is studied. The conservation law is a parabolic reaction-convection-diffusion equation with a small parameter multiplying the diffusion term and convex flux. Rigorous upper and lower bounding functions for the solution of the conservation law are established based on maximum-principle arguments. The bounding functions demonstrate that the size of the shock-layer is proportional to the parameter multiplying the diffusion term.
J. R. Cannon, The One-Dimensional Heat Equation, vol. 23, Addison-Wesley Publishing Company, Reading, Massachusetts, 1984
F. A. Howes, Perturbed boundary value problems whose reduced solutions are nonsmooth, Indiana Univ. Math. J. 30, 267–280 (1981)
F. A. Howes, Multi-dimensional reaction-convection-diffusion equations, in Ordinary and Partial Differential Equations, Proceedings of the Eighth Conference, Dundee, Scotland, B. D. Sleeman and R. J. Jarvis, eds., vol. 1151, Springer-Verlag, New York, 1984, pp. 217–223
F. A. Howes, Multi-dimensional initial-boundary value problems with strong nonlinearities, Arch. Rat. Mech. Anal. 91, 153–168 (1986)
F. A. Howes, Nonlinear initial-boundary value problems in Rn, Lectures in Applied Mathematics 23, 259–273 (1986)
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973
W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970
J. R. Cannon, The One-Dimensional Heat Equation, vol. 23, Addison-Wesley Publishing Company, Reading, Massachusetts, 1984
F. A. Howes, Perturbed boundary value problems whose reduced solutions are nonsmooth, Indiana Univ. Math. J. 30, 267–280 (1981)
F. A. Howes, Multi-dimensional reaction-convection-diffusion equations, in Ordinary and Partial Differential Equations, Proceedings of the Eighth Conference, Dundee, Scotland, B. D. Sleeman and R. J. Jarvis, eds., vol. 1151, Springer-Verlag, New York, 1984, pp. 217–223
F. A. Howes, Multi-dimensional initial-boundary value problems with strong nonlinearities, Arch. Rat. Mech. Anal. 91, 153–168 (1986)
F. A. Howes, Nonlinear initial-boundary value problems in Rn, Lectures in Applied Mathematics 23, 259–273 (1986)
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973
W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970
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Article copyright:
© Copyright 1995
American Mathematical Society
