24/04/2020   15h   Open Analysis & PDE Seminar (http://ivan.moyano.perso.math.cnrs.fr/OAPDES.html)  
 Cyril Letrouit (Ecole Normale Supérieure)   Subelliptic wave equations are never observable   In this talk, we explain a result we obtained recently, concerning the wave equation with a subRiemannian (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 BardosLebeauRauch 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 nonempty 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 $M\setminus \omega$.   Email: http://ivan.moyano.perso.math.cnrs.fr/OAPDES.html  Gestion: Iván Moyano 

24/04/2020   14h   Open Analysis & PDE Seminar (http://ivan.moyano.perso.math.cnrs.fr/OAPDES.html)  
 Clément Mouhot (University of Cambridge)   Unified approach to fluid approximation of linear kinetic equations with heavy tails   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 FokkerPlanck 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 infinitemass equilibrium distribution) but nonconstructive. 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 ParisDauphine).   Email: http://ivan.moyano.perso.math.cnrs.fr/OAPDES.html  Gestion: Iván Moyano 
