On the structure of $L^{\infty }$-entropy solutions to scalar conservation laws in one-space dimension

Abstract

We prove that if $u$ is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. \\ In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a $C^0$-sense up to the degeneracy due to the segments where $f"=0$. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp.

Cited by in corpus (4)

• Regularity estimates for scalar conservation laws in one space dimension
• Optimal regularity for all time for entropy solutions of conservation laws in $B{V}^s$
• Structure and regularity of solutions to 1d scalar conservation laws
• Non existence of the BV regularizing effect for scalar conservation laws in several space dimension