Asymptotic behavior of 2D stably stratified fluids with a damping term in the velocity equation

TL;DR

This paper investigates the long-term decay behavior of solutions to 2D inviscid Boussinesq equations with damping, using Fourier analysis and bilinear estimates to establish decay rates.

Contribution

It provides the first detailed analysis of decay rates for this system, combining Fourier kernel analysis with nonlinear estimates.

Findings

Established explicit decay rates for smooth solutions.

Analyzed the Green kernel of the linearized problem in Fourier space.

Demonstrated the effectiveness of bilinear estimates in handling nonlinearity.

Abstract

This article is concerned with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.