Schedule, Notes/Handouts & Videos

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will focus on semilinear cases (and therefore we will not address the Prandtl equation in these lectures). We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid… We will also tackle some semilinear problems, adding for instance a convection term to the previous examples.

I will discuss spectral instability of a Stokes wave of small amplitude in the finite depth. Analysis of a periodic Evans function, which is new, near the origin of the spectral plane offers an alternative proof of the Benjamin-Feir instability. Analysis near the resonance of order 2 reveals spectral instability when 0.86430...1.3627..., so new unstable waves are found. This seems the first rigorous proof of such high-frequency instability. I will discuss extensions to capillary-gravity waves and others, if time permits. Joint work with Z. Yang.

In this talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. In particular, an example of such a system of bosons leads to a quintic nonlinear Schrodinger equation, which we rigorously derived in a joint work with Thomas Chen. An example of a system of classical particles that allows instantaneous ternary interactions leads to a new kinetic equation that can be understood as a step towards modeling a dense gas in non-equilibrium. We call this equation a ternary Boltzmann equation and we rigorously derive it in a recent work with Ioakeim Ampatzoglou. Time permitting, we will also discuss the recent work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this work introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.

The purpose of these lectures is to present general methods to construct boundary layers both in linear and nonlinear contexts. We will focus on semilinear cases (and therefore we will not address the Prandtl equation in these lectures). We will explain how the boundary layer sizes and profiles can be predicted in linear cases, together with some decay estimates. We will illustrate this method with several explicit examples: Ekman layers, reflection of internal waves in a stratified fluid… We will also tackle some semilinear problems, adding for instance a convection term to the previous examples.

In this talk we consider a minimizing movements type scheme for  the incompressible Navier-Stokes equations, combining the Lagrangian and Eulerian viewpoints. Our scheme is an improved version of the split scheme introduced in Ebin-Marsden. An essential ingredient is the H^1 projection problem, which is a viscous analogue of the L^2 projection introduced by Brenier for evolving the incompressible Euler equation.

 We will discuss the scheme and the regularity of solutions it produces.

The talk is based on a joint work with Wilfrid Gangbo and Matt Jacobs.

In this talk, I will discuss some results on radial symmetry property for stationary and uniformly-rotating solutions for the 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.