Long time decay to the solution to the 2D-DQG Equation

TL;DR

This paper investigates the long-term behavior of solutions to the 2D dissipative quasi-geostrophic equation, demonstrating that certain solution norms decay to zero over time, indicating stabilization.

Contribution

It extends previous well-posedness results by proving decay to zero of solution norms, revealing the long-time asymptotic behavior of the 2D-DQG equation.

Findings

Solution norms decay to zero as time approaches infinity

Establishes long-term stability of solutions

Provides decay rates in specific function spaces

Abstract

In \cite{JAMH1}, we prove the well posedness of the quasi-geostrophic equation $(QG)_{\alpha}\;,1/2<\alpha\leq 1$, in the space introduced by Z. Lei and F. Lin in \cite{ZY1}. In this chapter we discuss the long time behaviour. Mainly, we prove that $\|\theta\|_{{\mathcal X}^{1-2\alpha}}$ decays to zero as time goes to infinity.