Formation of diffusion waves in a scalar conservation law with convection

Zumbrun, Kevin R.

doi:10.1090/S0002-9947-1995-1283568-8

Formation of diffusion waves in a scalar conservation law with convection
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by Kevin R. Zumbrun PDF

Trans. Amer. Math. Soc. 347 (1995), 1023-1032 Request permission

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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