Cauchy-Gelfand problem for quasilinear conservation law

Guennadi Khenkine (1) , Alexandre Shananin (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 MIPT - Moscow Institute of Physics and Technology [Moscow]

We obtain the precise asymptotic $(t\to\infty)$ for solution $f(x,t)$ of Cauchy-Gelfand problem for quasilinear conservation law ${\pa f\over \pa t}+\v(f){\pa f\over \pa x}=0$ with initial data of bounded variation $f(x,0)=f^0(x)$. The main theorem develops results of T.-P.Liu (1981), Kruzhkov, Petrosjan (1987), Henkin, Shananin (2004), Henkin (2012). Proofs are based on vanishing viscosity method and localized Maxwell type conservation laws. The main application consists in the reconstruction of parameters of initial data responsible for location of inviscid shock waves in the solution $f(x,t)$

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Guennadi Khenkine, Alexandre Shananin. Cauchy-Gelfand problem for quasilinear conservation law. 2013. ⟨hal-00877591⟩