The purpose of this talk is to describe the quantum limits of eigenstates of magnetic Laplacians on hyperbolic surfaces. We will see that three distinct regimes appear, depending on the energy level: low, critical, and high. The main result is a quantitative version of Quantum Unique Ergodicity at the critical energy: at this energy level, eigenstates concentrate at a quantitative rate towards the Liouville measure. This is joint work with Laurent Charles.

The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effects of the viscosity. Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities (Annals of Math. 2005). However, achieving this limit with physical viscosities remained an open question up to our recent result together with Geng Chen and Moon-Jin Kang. In this presentation, we will provide a basic overview of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. We will describe the basic ideas and difficulties involved in the study of physical inviscid limits in the contex