On the sharp time decay rates for the 2D generalized quasi-geostrophic equation and the Boussinesq system

TL;DR

This paper determines the precise decay rates over time for solutions to the 2D quasi-geostrophic and Boussinesq equations with fractional dissipation, identifying their asymptotic profiles related to stable processes.

Contribution

It provides explicit decay rates and asymptotic profiles for these equations, enhancing understanding of their long-term behavior with fractional dissipation.

Findings

Sharp decay rates for solutions are established.

Asymptotic profiles are explicitly identified.

Connections to stable process kernels are demonstrated.

Abstract

We compute the sharp time decay rates of the solutions of the IVP for quasi-geostrophic equation and the Boussinesq model, subject to fractional dissipation. Moreover, we explicitly identify the asymptotic profiles, the kernel of the $\alpha$ stable processes, which are analogues of the Oseen vortices.