No $L_ 1$-contractive metrics for systems of conservation laws

Abstract:

Let $(\ast )$ \[ \quad {u_t} + F{(u)_x} = 0\] 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 \[ u(x,0) = \left \{ {\begin {array}{*{20}{c}} {{u_1},} \hfill & {x \leqslant 0,} \hfill \\ {{u_2},} \hfill & {0 < x < 1,} \hfill \\ {{u_1},} \hfill & {x \geqslant 1.} \hfill \\ \end {array} } \right .\] 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.