• Harmonic Analysis and Partial Differential Equations
• Mini-workshop: Euler equation and turbulence

• Schedule

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will give an overview of recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder 1/3-norm is Lebesgue integrable in time.

We consider weak stationary solutions to the incompressible Euler equations and show that the analogue of the h-principle obtained by De Lellis and Szekelyhidi for time-dependent weak solutions continues to hold. The key difference arises in dimension d = 2, where it turns out that the relaxation is strictly smaller than what one obtains in the time-dependent case. This is joint work with Laszlo Szekelyhidi Jr.

In this talk the regularity of solutions to the classical water wave problem for two-dimensional Euler flows with vorticity is addressed. It is shown that the profile together with all streamlines beneath a periodic water wave travelling over a flat bed are real-analytic curves, provided that the vorticity function is merely integrable and that there are stagnation points in the flow. The presented method allows to treat besides classical waves of finite depth also solitary waves, waves with infinite depth, capillary waves and waves over stratified flows.

The symmetry of periodic gravity water waves is an intriguing problem whose study goes back to a paper of P. Garabedian from 1965. In the last decade there has been some progress in the study of this problem. Namely, it is known now that the symmetry property is implied by the existence of a unique crest per period. In this talk we present a new characterization of the symmetric waves in terms of the underlying flow: a wave is symmetric if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed. This intrinsic characterization is new even for Stokes waves.

The Theory of Weak Turbulence is a Physical Theory which describes the distribution of energy among the different wavelengths of several Wave Equations. One of the most popular examples is the Theory of Weak Turbulence for the Nonlinear Schrödinger Equation. The main mathematical object of the theories of Weak Turbulence is a kinetic equation which allows to interpret the transfers of energy between different wavelengths as collisions between groups of particles, in a manner analogous to Boltzmann equation. One of the most peculiar features of the theories of Weak Turbulence is the fact that they have power law solutions for the density of energy in the space of different wavelengths. In this talk I will describe several mathematical results that can be derived for the Weak Turbulence kinetic equation associated to the cubic Schrödinger Equation. In particular phenomena like blow-up in finite time, long time asymptotics and self-similar solutions describing the transfer of energy towards large wavelengths numbers will be considered.

Linear inviscid damping for the Couette shear flow (y,0) with algebraic rates is a classical result, however passing to nonlinear damping or even the setting of more general shear flows has been mostly open. Recently, following the works of Villani and Mohout on nonlinear Landau damping, Masmoudi and Bedrossian proved nonlinear inviscid damping for Couette flow for small Gevrey regular perturbations in the setting (x,y) ∈ T×R. In this talk we discuss both works and further present new results on linear inviscid damping, scattering and stability for monotone shear flows under Sobolev regular perturbations both in the usual setting of y ∈ R and the physically more natural setting y ∈ [0,1] with impermeable boundary.