Analyse, analyse numérique et contrôle des milieux continus

Comportement asymptotique des équations de convection-diffusion fractionnaires / Asymptotic behaviour for fractional diffusion-convection equations

21 mai 2018, 10:00

1h

Orateur

Liviu Ignat (IMAR, Bucarest, Roumanie)

Description

In this talk, we analyze the long time behaviour of the solutions of the equation $u_t(t,x)+(-\mathrm{\Delta }{)}^{\alpha /2}u(t,x)+(f(u){)}_x=0,t\in (0,\mathrm{\infty }),x\in \mathbf{R},$ where $\alpha \in (0,2)$ and $f(s)=|s{|}^{q-1}s/q$ with $q\in (1,\mathrm{\infty })$. We present some prvious results on the asymptotic expansion of the solutions when the time goes to infinity. We prove that in the one-dimensional case, for $q\in (1,\alpha )$ the asymptotic behaviour is given by the entropy solution of the conservation law $u_t(t,x)+(f(u){)}_x=0$, $u(0)=M{\delta }_0$ where $M$ is the mass of the initial data. The proof relies on tricky inequalities to guarantee an Oleinik type inequality $(u^{q-1}{)}_x\le 1/t$. This is a joint work with Diana Stan. This presentation is partially supported by CNCS-UEFISCDI No. PN-III-P4- ID-PCE-2016-0035.

Documents de présentation

Transparents

Liviu_Ignat.pdf