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Transactions of the American Mathematical Society

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No $ L\sb 1$-contractive metrics for systems of conservation laws

Author: Blake Temple
Journal: Trans. Amer. Math. Soc. 288 (1985), 471-480
MSC: Primary 35L65; Secondary 76L05, 76N10
MathSciNet review: 776388
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Abstract: Let $ (\ast)$

$\displaystyle \quad {u_t} + F{(u)_x} = 0$

be any $ 2 \times 2$ system of conservation laws satisfying certain generic assumptions on $ F$ in a neighborhood $ \mathcal{N}$ of $ u$-space. We prove that for every nondegenerate metric $ D$ on $ u$-space there exists states $ {u_1}$ and $ {u_2}$ in $ \mathcal{N}$ such that $ \int_{ - \infty }^\infty {D(u(x,t),{u_1})\;dx} $ is a strictly increasing function of $ t$ in a neighborhood of $ t = 0$, where $ u$ is the admissible solution of $ ( \ast )$ with initial data

$\displaystyle u(x,0) = \left\{ {\begin{array}{*{20}{c}} {{u_1},} \hfill & {x \l... ...,} \hfill \\ {{u_1},} \hfill & {x \geqslant 1.} \hfill \\ \end{array} } \right.$

This contrasts with the case of a scalar equation in which $ \int_{ - \infty }^\infty {D(u(x,t),v(x,t))\;dx} $ is a decreasing function of $ t$ for all admissible solution pairs $ u$ and $ v$ when $ D$ is taken to be the absolute value norm.

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Article copyright: © Copyright 1985 American Mathematical Society