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Transactions of the American Mathematical Society

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Formation of diffusion waves in a scalar conservation law with convection

Author: Kevin R. Zumbrun
Journal: Trans. Amer. Math. Soc. 347 (1995), 1023-1032
MSC: Primary 35L65
MathSciNet review: 1283568
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Abstract: We study the scalar conservation law, $ {u_t} + {[c(x)u]_x} + b{({u^2})_x} = {u_{xx}},\;c' \geqslant 0$, which is a model equation for the behavior of weak transverse waves near a viscous transitional shock. Solutions are shown to decay in $ {L^1}$ to a pair of diffusion waves, moving apart at speeds $ c( - \infty )$ and $ c( + \infty )$, behavior that has been observed numerically in solutions of the full equations. The interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori, in contrast to cases treated previously.

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Article copyright: © Copyright 1995 American Mathematical Society