Strong traces for degenerate parabolic-hyperbolic  equations

Young-Sam Kwon

doi:10.3934/dcds.2009.25.1275

Article Contents

2009, Volume 25, Issue 4: 1275-1286. Doi: 10.3934/dcds.2009.25.1275

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Strong traces for degenerate parabolic-hyperbolic equations

Young-Sam Kwon^1,

1.
Department of Mathematics, Dong-A University, Busan 604-714

Received: March 31, 2008

Revised: June 30, 2009

Published: October 2009

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Abstract

In this paper we consider bounded weak solutions $u$ of degenerate parabolic-hyperbolic equations defined in a subset $]0,T[\times\Omega\subset \R^{+}\times \R^d$. We define a strong notion of trace at the boundary $]0,T[\times\partial\Omega$ reached by $L^1$ convergence for a large class of functionals of $u$ and at $0 \times \Omega$ reached by $L^1$ convergence for solution $u$. This result develops the strong trace results of Kwon, Vasseur [13] and Panov [19, 20] for more general equations, namely, degenerate parabolic-hyperbolic equations.

Keywords:

Degenerate parabolic-hyperbolic equation,
conservation law,
trace theorem,
$H$-measure,
kinetic formulation,
boundary value problem.

Mathematics Subject Classification: 35K20, 35K55, 35L65, 35K65.

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