# Meeting Details

Title: Concentration of entropy dissipation for scalar conservation laws Department of Mathematics Colloquium Stefano Bianchini, SISSA Let $u(t,x)$ be an entropy $L^\infty$-solution of the scalar conservation laws $u_t + f(u)_x = 0.$ By entropy solution we means that for every convex function $\eta$ it holds $\eta_t + q_x \leq 0,$ where the entropy flux $q$ is given by $q' = f' \eta'$. In particular it is a measure. Under no assumptions on the flux function $f$ the solution is in general only $L^\infty$, and thus questions regarding the regularity of the dissipation measure were open. We will review the basic theory of entropy solutions and show that the entropy dissipation is actually concentrated on a $1$-rectifiable set: there is a countable set of Lipschitz curves $\gamma_i(t)$ such that for all entropies $\eta$, entropy flux $q$ it holds $\eta_t + q_x = \sum_i c_{\eta,i}(t) \mathcal H1 \llcorner_{\gamma_i}.$ Corollaries of this results are regularity estimates for the original solution $u$: the existence of a Lagrangian representation, the structure of Young solutions, the BV regularity of $f'(u)$, the strong continuity in time.