2020/11/07 by Andres A. Contreras Hip, Hip, Andres A. Contreras, Xavier Lamy +1
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2011.03710
openalex publication_date 2020/11/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We prove L2 stability estimates for entropic shocks among weak, possibly\n\non-entropic, solutions of scalar conservation laws \∂tν+\∂x f(u)=0 with strictly convex flux function f. This generalizes\nprevious results by Leger and Vasseur, who proved L2 stability among entropy\nsolutions. Our main result, the estimate \
int
mathbb R\n|u(t,
cdot)-u0shock(
cdot -x(t))|2
,dx
leq
int
mathbb\nR|u0-u0shock|2 +C
mu+([0,t]
times
R), for some Lipschitz\nshift x(t), includes an error term accounting for the positive part of the\nentropy production measure \μ=\∂t(u2/2)+\∂x q(u), where\nq'(u)=uf'(u). Stability estimates in this general non-entropic setting are of\ninterest in connection with large deviation principles for the hydrodynamic\nlimit of asymmetric interacting particle systems.\n Our proof adapts the scheme devised by Leger and Vasseur, where one\nconstructs a shift x(t) which allows to bound from above the time-derivative\nof the left-hand side. The main difference lies in the fact that our solution\nu(t,\⋅) may present a non-entropic shock at x=x(t) and new bounds are\nneeded in that situation. We also generalize this stability estimate to initial\ndata with bounded variation.\n