Singularity and Regularity in Active Scalar Equations

Abstract

One of the most fundamental questions in PDE is that of global existence. That is, givenan initial data, does a solution exist for all time? In this thesis, we explore this problem in the context of active scalar equations. We consider the motion of a scalar quantity, such as density, being transported by an incompressible fluid. The motion of the density, in turn, affects the motion of the fluid leading to a coupling between the fluid and the scalar quantity. The nonlinear interaction between the scalar and the fluid makes the question of global existence quite subtle. We explore the problem of global existence from both the positive and negative direction. In the first part of the thesis, we construct an instance of singularity formation in the incompressible porous medium equation in a setting that is not driven by the boundary. In the second part of the thesis, we study solutions to the surface quasi-geostrophic equation with one-homogeneous initial data. We derive a one- dimensional system and prove that positive solutions to the one-dimensional exist globally in time.