Constructing finite time singularities: Non-radial implosion for com- pressible Euler, Navier-Stokes and defocusing NLS in Td and Rd

Seminar in Mathematics - via zoom only

Abstract: The compressible Euler and Navier-Stokes equations describe the motion of compressible fluids. The defocusing nonlinear Schrödinger equation is a dispersive equation that has application in many physics areas. Through the Madelung transformation, the defocusing nonlinear Schrödinger equation is connected with the compressible Euler equation. In this colloquium I will start from the compressible Euler/Navier-Stokes equation and introduce the blow-up result called implosion. Then I will introduce the defocusing nonlinear Schrödinger equation and the longstanding open problem on the blow-up of its solutions in the energy supercritical regime. In the end I will talk about the Madelung transformation and its application to transfer the implosion from the compressible Euler to the defocusing nonlinear Schrödinger equation. During the talk I will mention our work with Gonzalo Cao Labora, Javier Gomez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.