Title: Degenerate Dispersive Equations
Abstract: We discuss recent work on some quasilinear toy models for the phenomenon of degenerate dispersion, where the dispersion relation may degenerate at a point in physical space. In particular, we discuss the stationary states, as well as existence and uniqueness of solutions for degenerate KdV and NLS-type equations using a novel change of variables. This is joint work with Pierre Germain and Ben Harrop-Griffiths. Given time, we will discuss some work on Gibbs measures for a discrete version of this problem that is joint with with Jonathan Mattingly and Dana Mendelson.

Title: Global well-posedness and scattering in nonlinear wave equations
Abstract: In this talk we will consider the wave equation in odd space dimensions with energy-supercritical nonlinearity, in the radial setting. We will review the concentration compactness and rigidity arguments starting from the earlier work by Kenig-Merle and Duyckaerts-Kenig-Merle in the energy-critical and energy-supercritical cases, and outline the key ideas behind the proof of global well-posedness and scattering results in 3d and higher odd dimensions

Title: Nonlinear Schrödinger Equation with Repulsive Potentials and Rotations
Abstract: We will introduce the nonlinear Schrödinger equation (NLS) with repulsive harmonic potentials and rotations, and we will give a transform that connects this to the classical NLS in the mass-critical case. Then we will give some results on global well-posedness and blowup rates.

Title: A la Dhont method for solving Stokes equation with prescribed boundary conditions
Abstract: The motion of a sphere in stratified fluids is of importance to environmental applications involving the ocean and the atmosphere. Modeling this problem involves solving the Navier-Stokes equation with variable density field. In the viscous limit, the Oseen tensor provides the fundamental function solution to the Stokes equations for the flow external to a sphere located at the origin. Motivated by the difficulty that arises from the three dimensional convolution of the Oseen tensor and the forcing term, we attempt to derive the solution through a different method that would be more suitable for numerical implementation. We computed a new solution for the fluid flow applying La Dhont Method to the Stokes equation with prescribed boundary conditions at the sphere and at infinity. Our new tensor is a result of solving the PDE and matching the restrictions of the boundary conditions, and the rate of convergence of the solution at infinity. Comparisons between the Oseen tensor and the new solution using La Dhont Method approach will be discussed.