Speaker: Minh-Binh Tran Texas A&M University
Title: Evolution of finite temperature Bose-Einstein Condensates: Some rigorous studies on condensate growth
Abstract: In trapped Bose-Einstein condensates (BECs), condensate growth refers to the process in which an increasing number of quasi-particles are immediately transferred from the non-condensate state (the thermal cloud) into the condensate state following the initial formation of the BEC. Despite its physical significance, this phenomenon has not yet been studied rigorously from a mathematical standpoint. In this work, we investigate a kinetic equation whose collision operator includes three types of wave interactions: one corresponding to a 3-wave process, and two classified as 4-wave processes. This wave kinetic equation models the evolution of the density function of the thermal cloud. We establish the immediate formation of condensation in solutions to this equation, thus providing a rigorous demonstration of the condensate growth phenomenon.
Speaker: David Ambrose Drexel University
Title: Some non-decaying, non-periodic existence theory for fluid equations
Abstract: We consider the irrotational Euler equations and the surface quasi-geostrophic equation in the case that the unknowns do not decay and are not spatially periodic. In such settings, constitutive laws of convolution type (such as the Biot-Savart law) do not apply directly, as the convolution integral does not converge. These can be replaced with identities of Serfati type, which separate the integrals into near-field and far-field pieces, with the far field contribution being able to be manipulated for better convergence properties. We use these identities to find existence of solutions for the 2D Euler equations with bounded velocity and vorticity (generalizing a result of Serfati), for the 3D Euler equations in uniformly local Sobolev spaces, and for SQG in Holder spaces and in uniformly local Sobolev spaces. This includes joint work with Elaine Cozzi, Daniel Erickson, James Kelliher, Milton Lopes Filho, and Helena Nussenzveig Lopes.
Speaker: Wilhelm Schlag Yale University
Title: On the long-term dynamics of nonlinear wave equations on the line with a critical potential
Abstract: We will present recent results with J. Krieger and K. Widmayer on a cubic NLS on the line with a repulsive inverse square potential. Some of the context in the wider space-time resonance and wave packet methods will be provided.