SBV-like regularity for general hyperbolic systems of conservation laws

Abstract

We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general systems of conservation laws. More precisely, for the equation u_t + f(u)_x = 0, \quad u : \R^+ \times \R \to \Omega \subset \R^N, we only assume the flux $f$ is $C^2$ function in the scalar case (N=1) and Jacobian matrix $Df$ has distinct real eigenvalues in the system case $(N\geq 2)$. Using the modification of the main decay estimate in Lau and localization method applied in \cite{R}, we show that for the scalar equation $f'(u)$ belongs to SBV, and for system of conservation laws the scalar measure \[\big(D_u \lambda_i(u) \cdot r_i(u) \big) \big(l_i(u) \cdot u_x \big)] has no Cantor part, where $\lambda_i$, $r_i$, $l_i$ are the $i$-th eigenvalue, $i$-th right eigenvector and $i$-th left eigenvector of the matrix $Df$.