| Abstract: |
| In this talk, we will talk about the large time behavior of kinetic equations without spatial confinement and with fat tailed local thermodynamic equilibria. It has been proved in most of the cases that such operators can have an anomalous diffusion limit, meaning that in the appropriate scaling, the macroscopic equation is the fractional heat equation.
At the level of the kinetic equation, we develop an $L^2$ hypocoercivity approach to obtain decay rates towards $0$. It requires to find the good expression for the entropy and new functional inequalities of Poincar\`{e} type. The method is applied to kinetic equations with various linear collision operators: the Fokker-Planck operator, the Linear Boltzmann operator and the fractional Fokker-Planck operator. The result let appear a competition between the micro-scale and the macro-scale behavior. |
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