Non-Decaying Solutions to the 2D Dissipative Quasi-Geostrophic Equations

TL;DR

This paper proves the existence and uniqueness of global solutions to the 2D dissipative quasi-geostrophic equations without decay assumptions, including a generalized model, highlighting maximum principles and classical solutions for smooth data.

Contribution

It establishes the existence of non-decaying solutions for the 2D dissipative quasi-geostrophic equations with fractional diffusion, including a generalized model, without decay or periodicity assumptions.

Findings

Existence of solutions without decay at infinity.

Global classical solutions for smooth initial data.

Maximum principle holds for solutions with bounded initial data.

Abstract

We consider the surface quasi-geostrophic equation in two spatial dimensions, with subcritical diffusion (i.e. with fractional diffusion of order $2\alpha$ for $\alpha>\frac{1}{2}$.) We establish existence of solutions without assuming either decay at spatial infinity or spatial periodicity. One obstacle is that for $L^{\infty}$ data, the constitutive law may not be applicable, as Riesz transforms are unbounded. However, for $L^{\infty}$ initial data for which the constitutive law does converge, we demonstrate that there exists a unique solution locally in time, and that the constitutive law continues to hold at positive times. In the case that $\alpha\in(\frac{1}{2},1]$ and that the initial data has some smoothness (specifically, if the data is in $C^{2}$), we demonstrate a maximum principle and show that this unique solution is actually classical and global in time. Then, a density argument allows us to show that mild solutions with only $L^{\infty}$ data are also global in time, and also possess this maximum principle. Finally, we introduce a related problem in which we replace the usual constitutive law for the surface quasi-geostrophic equation with a generalization of Sertfati type, and prove the same results for this relaxed model.