Large-Time Behaviour of Solutions to the Surface Quasi-Geostrophic Equations

TL;DR

This paper analyzes the long-term behavior of solutions to the Surface Quasi-Geostrophic equations by constructing a linear approximation and establishing convergence rates and bounds for the nonlinear term in $L^p$ spaces.

Contribution

It introduces a linear approximation method for solutions and provides precise convergence rates and bounds for the nonlinear component in the context of the Surface Quasi-Geostrophic equations.

Findings

Established a linear approximation with explicit convergence rates.

Proved sharp bounds for the nonlinear term in $L^p$ norms.

Demonstrated the asymptotic behavior of solutions over large times.

Abstract

We construct a linear approximation of the solution to the Surface Quasi-Geostrophic Equation in $\mathbb{R}^2$, and obtain a convergence rate in $L^p$ between the solution and this approximation with respect to time. We also demonstrate that the nonlinear term of the solution is bounded sharply in $L^p$ by the same function of time.