We construct an incompressible velocity field on the two dimensional unit sphere by alternating two zonal flows with random amplitudes. We show that the time evolution of any mean-free initial data passively advected by the velocity field is exponentially mixed in time.
This is a joint work with Marc Nualart.
We consider the 2D incompressible Euler equation on a bounded simply connected domain. We give sufficient conditions on the domain so that for all bounded initial vorticity, the weak solutions are unique. Our sufficient conditions allow us to prove uniqueness for a large subclass of $C^{1,\alpha}$ domains and convex domains. Previously uniqueness for general bounded initial vorticity was only...
In this talk we consider the classical water wave problem for an incompressible inviscid fluid occupying a time-dependent domain in the plane, whose boundary consists of a fixed horizontal bed together with an unknown free boundary separating the fluid from the air outside the confining region.
We provide the first construction of overhanging gravity water waves having the approximate form...
We will present a new criterion to study the non purely-imaginary spectrum of linear Hamiltonian operators. We will apply it to prove linear stability or instability of steady solutions of the Euler equations.
In this talk, I will talk about some of our recent results regarding singularity formation in incompressible fluid dynamics, including models such as 3 dimensional Euler and Incompressible Porous Media.
In this talk, we present a non-uniqueness result for the forced SQG equation in supercritical Sobolev spaces. A key step is the construction of smooth, compactly supported vortices that exhibit nonlinear instability. This is joint work with Á. Castro, D. Faraco, and M. Solera.
Intermittency is a remarkable feature of three-dimensional turbulence for which we still lack explanation from first principles. It will be shown how a dissipation with a non-trivial lower-dimensional part induces a quantitative intermittent regularity of the weak solution. The result is in fact more general than that.
In this talk I will present a recent result in collaboration with Thomas Alazard (CNRS-École Polytechnique) showing generic growth of sobolev norms of the vorticity in the 2d Euler equations.
In this talk, I will discuss asymptotic stability in the incompressible porous media equation in a periodic channel. It is well known that a stratified density, which strictly decreases in the vertical direction, is asymptotically stable under sufficiently small, smooth perturbations. We achieve optimality in the regularity assumptions on the perturbation and in the convergence rate. We apply...
