Title: The 3D Quasi-geostrophic equation: existence of solutions, lateral boundary conditions and regularity.

Abstract:
The 3D Quasi-geostropic equation is a model used in climatology to model the evolution of the atmosphere for small Rossby numbers. It can be derived from the primitive equation. The surface quasi-geostrophic equation (SQG) is a special case where the atmosphere above the earth is at rest. The evolution then depends only on the boundary condition, and can be reduced to a 2D model.
In this talk, we will show how we can derive the physical lateral boundary conditions for the inviscid 3D QG, and construct global in time weak solutions. Finally, we will discuss the global regularity of solutions to the QG equation with Ekman pumping.

Title: The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation System at very low temperature

Abstract: We discuss a quantum Boltzmann-Condensation system that describes the evolution of the interaction between a well formed Bose-Einstein Condensate (BEC) and the quasi-particles cloud. The kinetic model, derived as weak turbulence kinetic model from a quantum Hamiltonian, is valid for a dilute regime at which the temperature of a bosonic gas is very low compared to the Bose-Einstein condensation critical temperature. In particular, the system couples the density of the condensate from a Gross-Pitaevskii type equation to the kinetic equation through the dispersion relation in the kinetic model and the corresponding transition probability rate from pre to post collision momentum states.

We show the well-posedness of the Cauchy problem for the system, find qualitative properties of the solution such as instantaneous creation of exponential tails, and prove the uniform condensate stability related to the initial mass ratio between condensed particles and quasi-particles. This stability result leads to global in time existence of the initial value problem for the quantum Boltzmann-Condensation system.

Title: Determining the Global Dynamics of the Two-dimensional Navier-Stokes Equations by a Scalar ODE

Abstract:
One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters - finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.

Title: On the vanishing viscosity limit in incompressible flows

Abstract:
I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.

Title: Mathematical modeling of light filamentation

Abstract:
Since the first observation of a nonlinear process in light matter interaction in 1961, better lasers and designs of photonic structures have opened new ways to explore nonlinear phenomena with many important technological applications. In this talk we will focus our attention to spatio-temporal dynamics and coherent modes described by nonlinear Schrӧdinger-like equations. While the presentation will center on models developed to explain experiments of light filament propagation in air, we will also discuss recent optical experiments in quadratic media, multi-mode fibers and fiber arrays; most in need of a fresh theoretical formulation.

Date: Tuesday, December 4, 2018 (please notice this is different date than what we have announced earlier)

Title: Mean field limit of interacting filaments for 3d Euler equation

Abstract: The 3D Euler equation, precisely local smooth solutions of class $H^s$ with $s>5/2$ are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments, described by means of the concept of 1-currents. This is achieved by first replacing the true Euler equation by a mollified one through the regularization of the Biot-Savart law through a small coefficient $\epsilon$. Families of N interacting curves are considered, with long range mean field type interaction, that depends on the coefficient $\epsilon$. When $N$ goes to infinity, the limit PDE is vector-valued (mollified Euler equation) and each curve interacts with a mean field solution of the PDE.

This target is reached by a careful formulation of curves and weak solutions of the PDE which makes use of 1-currents and their topologies. The main results are based on the analysis of a nonlinear Lagrangian-type flow equation. Most of the results are deterministic; as a by-product, when the initial conditions are given by families of independent random curves, we prove a propagation of chaos result.