24 avril 2020

C. Mouhot (University of Cambridge)
Unified approach to fluid approximation of linear kinetic equations with heavy tails

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Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : The rigorous fluid approximation of linear kinetic equations was first obtained in the late 70s when the equilibrium distribution decays faster than polynomials. In this case the limit is a diffusion equation. In the case of heavy tail equilibrium distribution (with infinite variance), the first rigorous derivation was obtained in 2011 in a joint paper with Mellet and Mischler, in the case of scattering operators. The limit shows then anomalous diffusion ;
it is a governed by a fractional diffusion equation. Lebeau and Puel proved last year the first similar result for Fokker-Planck operator, in dimension 1 and assuming that the equilibrium distribution has finite mass. Fournier and Tardif gave an alternative probabilistic proof, more general (covering any dimension and infinite-mass equilibrium distribution) but non-constructive. We present a unified elementary approach, fully quantitative, that covers all previous cases as well as new ones. This is a joint work with Emeric Bouin (Université Paris-Dauphine).

Unified approach to fluid approximation of linear kinetic equations with heavy tails  Version PDF

Cyril Letrouit (Ecole Normale Supérieure (Paris))
Subelliptic wave equations are never observable

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Lieu : https://bbb2.imo.universite-paris-saclay.fr/b/nic-m3v-7dt

Résumé : In this talk, we explain a result we obtained recently, concerning the wave equation with a sub-Riemannian (i.e. subelliptic) Laplacian. Given a manifold $M$, a measurable subset $\omega\subset M$, a time $T_0$ and a subelliptic Laplacian $\Delta$ on $M$, we say that the wave equation with Laplacian $\Delta$ is observable on $\omega$ in time $T_0$ if any solution $u$ of $\partial_tt^2u-\Delta u=0$ with fixed initial energy satisfies $\int_0^T_0\int_\omega |u|^2dxdt\geq C$ for some constant $C>0$ independent on $u$.
It is known since the work of Bardos-Lebeau-Rauch that the observability of the elliptic wave equation, i.e. with a Riemannian Laplacian, in time $T_0$ is almost equivalent to the geometric control condition (GCC), which stipulates that any geodesic ray meets $\omega$ within time $T_0$. We show that in the subelliptic case, as soon as $M\backslash \omega$ has non-empty interior and $\Delta$ is ``subelliptic but not elliptic", GCC is never verified, which implies that subelliptic wave equations are never observable. The proof is based on the construction of sequences of solutions of the wave equation concentrating on geodesics (for the distance on $M$ associated to $\Delta$) spending a long time in $M\setminus \omega$.

Subelliptic wave equations are never observable  Version PDF