
Date Time 
Location  Speaker 
Title – click for abstract 

08/28 3:00pm 
BLOC 628 
Alexis Vasseur University of Texas 
Nonlinear PDE Seminar
Title: The 3D Quasigeostrophic equation: existence of solutions, lateral boundary conditions and regularity.
Abstract: The 3D Quasigeostropic 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 quasigeostrophic 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. 

09/07 1:50pm 
BLOC 628 
Irene Gamba University of Texas 
Nonlinear PDE's Seminar
Title: The Cauchy problem and BEC stability for the quantum BoltzmannCondensation System at very low temperature
Abstract: We discuss a quantum BoltzmannCondensation system that describes the evolution of the interaction between a well formed BoseEinstein Condensate (BEC) and the quasiparticles 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 BoseEinstein condensation critical temperature. In particular, the system couples the density of the condensate from a GrossPitaevskii 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 wellposedness 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 quasiparticles. This stability result leads to global in time existence of the initial value problem for the quantum BoltzmannCondensation system. 

09/18 3:00pm 
BLOC 628 
Dr. Edriss S. Titi Texas A&M University 
Nonlinear PDEs Seminar
Title: Determining the Global Dynamics of the Twodimensional NavierStokes Equations by a Scalar ODE
Abstract: One of the main characteristics of infinitedimensional dissipative evolution equations, such as the NavierStokes equations and reactiondiffusion systems, is that their longtime 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 finitedimensional feature of the longtime behavior of infinitedimensional dissipative systems to design finitedimensional 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 longtime dynamics of the NavierStokes equations can be imbedded in an infinitedimensional 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 NavierStokes equations is be determining by only one parameter that is governed by an ODE. The NavierStokes 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 reactiondiffusion systems and geophysical models. 

09/25 3:00pm 
BLOC 628 
Anna Mazzucato Penn State University 
Nonlinear PDE's Seminar
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 NavierStokes 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 noslip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseentype equation (linearization around a steady Euler flow) in general smooth domains.


10/30 3:00pm 
BLOC 628 
Alejandro Aceves Southern Methodist University 
Nonlinear PDE's Seminar
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 spatiotemporal dynamics and coherent modes described by nonlinear Schrӧdingerlike 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, multimode fibers and fiber arrays; most in need of a fresh theoretical formulation.


12/04 3:00pm 
BLOC 628 
Hakima Bessaih University of Wyoming 
Nonlinear PDE's Seminar
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 1currents.
This is achieved by first replacing the true Euler equation by a mollified one through the regularization of the BiotSavart 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 vectorvalued (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 1currents and their topologies. The
main results are based on the analysis of a nonlinear Lagrangiantype flow
equation. Most of the results are deterministic; as a byproduct, when the
initial conditions are given by families of independent random curves, we
prove a propagation of chaos result. 