Global well-posedness and temporal decay estimates for the viscous $\beta$-plane equations

TL;DR

This paper proves global existence, smoothing effects, and decay rates for solutions to the viscous β-plane equations, showing solutions behave like linear solutions asymptotically with faster decay than heat flow.

Contribution

It establishes the global well-posedness and decay estimates for the viscous β-plane equations, including asymptotic behavior and improved decay rates.

Findings

Global well-posedness for small initial data

Solutions decay faster than heat equation predictions

Asymptotic behavior matches linearized solutions

Abstract

We consider the Cauchy problem of the viscous $\beta$-plane equations. We first establish the global well-posedness of the system for the initial data sufficiently small compared to the Rossby parameter. The smoothing effect of the flow is then proved, and it is shown that the obtained global solution satisfies the equation in the classical sense. We also reveal that under some additional assumptions, the solution decays as fast as the corresponding linear solution and asymptotically behaves like a constant multiple of the integral kernel of the linearized equation. In particular, the decay rate of the solution is faster than expected from the flow of the heat equation.