SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension

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 hyperbolic systems of conservation laws. More precisely, for the equation $$ u_t + f(u)_x = 0, \quad u : \mathbb{R}^+ \times \mathbb{R} \to \Omega \subset \mathbb{R}^N, $$ we only assume that the flux $f$ is a $C^2$ function in the scalar case ($N=1$) and Jacobian matrix $Df$ has distinct real eigenvalues in the system case $(N\geq 2)$. We show that for the scalar equation $f'(u)$ belongs to the SBV space, and for system of conservation laws the $i$-th component of $D_x\lambda_i(u)$ has no Cantor part, where $\lambda_i$ is the $i$-th eigenvalue of the matrix $Df$.