Georgia Institute of Technology College of Sciences

This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.

In this talk we report on recent works (with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato) on the optimal control of (possibly path dependent) McKean-Vlasov equations valued in Hilbert spaces. On the other side we present the first ideas of a work with S. Federico, D. Ghilli and M. Rosestolato, on Mean Field Games in infinite dimension.

The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.

For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation.

A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.

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The linear stability of a family of Kelvin-Stuart Cat's eyes flows of 2D Euler equation was studied both analytically and numerically. We proved linear stability under co-periodic perturbations and linear instability under multi-periodic perturbations. These results were first obtained numerically using spectral methods and then proved analytically.

The Bluejeans link is: https://bluejeans.com/353383769/0224

We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.

We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation.