Applied Math Seminar: Ryan Aschoff (UC Riverside)

Title: Smooth non-decaying solutions to the 2D dissipative quasi-geostrophic equations

Abstract: In this talk we explore the two-dimensional dissipative surface quasi-geostrophic (SQG) equation with fractional diffusion of order 2α for α ∈ (1/2,1], focusing on the setting where the initial data does not decay at spatial infinity and periodicity is not assumed. In geophysical applications, the equations model shallow water currents with the scalar field θ is interpreted as the pressure, while the associated velocity field u governs the fluid motion. Traditionally, the transport velocity is recovered from the pressure via a constitutive law that fails when decay is absent. To overcome this, we replace it with a generalized, Serfati-type constitutive law—a method originally developed for the 2D Euler equations.

We will discuss how this approach enables us to prove the global existence and uniqueness of mild solutions, as well as classical solutions (with data bounded in C^k, for k≥2) without relying on spatial decay. The presentation will include an overview of the reformulated mild solution framework, which couples the pressure and velocity equations via the fractional heat operator and a modified convolution structure. In addition, we will outline extensions of this method to a Serfati-type SQG system and indicate how Littlewood-Paley techniques can be used to approach the inviscid case.