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

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.

14 pages; revised proof of Proposition 2.1, corrected typos

References in corpus (2)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation

Cited by in corpus (14)

• Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation
• Convergence to Stratified Flow for an Inviscid 3D Boussinesq System
• Linear Inviscid Damping for Couette Flow in Stratified Fluid
• On the Boussinesq equations with non-monotone temperature profiles
• On asymptotic stability of the 3D Boussinesq equations without thermal conduction
• Regularity results for viscous 3D Boussinesq temperature fronts
• Asymptotic Stability for some Stratified Stationary Solutions of the Inviscid Incompressible Porous Medium Equation
• Asymptotic stability of the 2D Boussinesq equations without thermal conduction
• On variable viscosity and enhanced dissipation
• Global regularity of non-diffusive temperature fronts for the 2D viscous Boussinesq system
• Asymptotics for small data solutions of the Ablowitz-Ladik equation
• Global well-posedness for the 2D Boussinesq equations with a velocity damping term
• Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon }\cap L^2$ force
• Long Time Stability for Solutions of a Beta-Plane Equation