Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects

TL;DR

This paper investigates the long-term solvability of the 2D inviscid Boussinesq equations with borderline regularity, employing dispersive effects and advanced estimates to extend solutions over longer times.

Contribution

It establishes long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity, using Strichartz estimates and a blow-up criterion for large stratification parameter.

Findings

Long-time solvability for large stratification parameter $$.

Uniform local solvability in Besov spaces.

Applicability to initial data with borderline regularity.

Abstract

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter $\kappa$ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large $\kappa$ regardless of the size of initial data.