Abstract: 

The stability of shear flows in the fluid mechanics is an old problem dating back to the famous Reynolds experiments in 1883. The question is to quantify the size of the basin of attraction of equilibria of the Navier-Stokes equations depending on the viscosity parameters, giving rise to the so-called stability threshold. In the case of a three-dimensional homogeneous fluid, it is known that the Couette flow has a stability threshold proportional to the viscosity, and this is sharp in view of a linear instability mechanism known as the lift-up effect. In this talk, I will explain how to exploit stratification (i.e. non-homogeneity in the fluid density) to improve this bound: the coupling between density and velocity gives rise to oscillations, which suppress the lift-effect. This can be captured at the linear level in an explicit manner, and at the nonlinear level by combining sharp energy estimates with suitable dispersive estimates.

Abstract: This blackboard talk is devoted to the semiclassical spectral analysis of the magnetic Laplacian in two dimensions. Assuming that the magnetic field is positive and has two symmetric radial wells, we establish an accurate tunneling formula, that is a one-term estimate of the spectral gap between the lowest two eigenvalues. This gap is exponentially small when the semiclassical parameter goes to zero, but positive.

Joint work with S. Fournais and L. Morin.

Abstract: 

Thin liquid films find various applications in industrial processes. These films, ranging from nanometers to micrometers in thickness, are often exposed to temperature gradients, which compete with stabilizing surface tension effects. A mathematical model describing the evolution of thin fluid film's height resting on a horizontal, heated plate can be derived via lubrication approximation, constituting a  quasilinear, degenerate, fourth-order partial differential equations. 

When a critical temperature disparity between the heated plate and the surrounding environment is reached, the flat steady state becomes spectrally unstable  and periodic stationary solutions bifurcate. Using a Hamiltonian formulation of the stationary problem and analytic bifurcation theory, we prove the existence of a global bifurcation curve converging to a weak stationary periodic solution exhibiting film rupture. 

This is a joint work with B. Hilder (TU Munich, Germany) and J. Jansen (Lund University, Sweden). 

Abstract: In this talk, I will present recent progress on the second order correction to the energy of the dilute Bose gas in the thermodynamic limit. By combining the original approach by Fournais and Solovej with recent advances by Haberberger, Hainzl, Nam, Seiringer and Triay on using Neumann bracketing in the localization step, one obtains a much shorter realization of the basic ideas of the original proof.

This talk is based on joint work with J.P. Solovej and T. Girardot, L. Junge, L. Morin, M. Olivieri and A. Triay.