Sharp decay estimates for an anisotropic linear semigroup and applications to the SQG and inviscid Boussinesq systems

TL;DR

This paper derives sharp decay estimates for an anisotropic linear semigroup relevant to the SQG and inviscid Boussinesq systems, demonstrating how dispersion influences long-term solution behavior and stability.

Contribution

It provides improved dispersive estimates for the anisotropic linear semigroup and applies these to establish long-time existence and stability results for related fluid systems.

Findings

Enhanced decay estimates for the anisotropic linear semigroup

Long-time existence results for the dispersive SQG equation

Stability of stationary solutions in the inviscid Boussinesq system

Abstract

At the core of this article is an improved, sharp dispersive estimate for the anisotropic linear semigroup $e^{R_1 t}$ arising in both the study of the dispersive SQG equation and the inviscid Boussinesq system. We combine the decay estimate with a blow-up criterion to show how dispersion leads to long-time existence of solutions to the dispersive SQG equation, improving the results obtained using hyperbolic methods. In the setting of the inviscid Boussinesq system it turns out that linearization around a specific stationary solution leads to the same linear semigroup, so that we can make use of analogous techniques to obtain stability of the stationary solution for an increased timespan.