SBV-like regularity for general hyperbolic systems of conservation laws

TL;DR

This paper establishes SBV regularity for characteristic speeds in scalar and system hyperbolic conservation laws, showing that eigenvalues and related functions have controlled regularity properties under minimal assumptions.

Contribution

It proves SBV regularity of eigenvalues and characteristic speeds for general hyperbolic systems, extending previous results to more general settings with minimal smoothness assumptions.

Findings

Scalar flux derivative belongs to SBV.

Eigenvalue-related measures have no Cantor part.

Results apply to systems with distinct real eigenvalues.

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 $Df$.