Exponential blow-up of mild solutions to the fractional Boussinesq equations in the Gevrey class

TL;DR

This paper investigates the existence, uniqueness, and blow-up behavior of solutions to fractional Boussinesq equations within Sobolev-Gevrey spaces, revealing exponential growth and conditions leading to finite-time blow-up.

Contribution

It establishes new conditions for local mild solutions in Sobolev-Gevrey spaces and analyzes their blow-up behavior, including exponential growth rates.

Findings

Existence and uniqueness of solutions in Sobolev-Gevrey spaces.

Quantitative lower bounds on solution norm blow-up.

Solutions exhibit exponential growth near maximal time.

Abstract

This work establishes conditions for the existence and uniqueness of local mild solutions to the Boussinesq equations with fractional dissipations in Sobolev-Gevrey spaces. We prove that a unique mild solution exists in an appropriate Sobolev-Gevrey class and analyze its behavior up to the maximal time of existence. In particular, we derive quantitative lower bounds describing how the norm of the solution must blow up as it approaches a finite maximal time. As a corollary, we deduce that the solution exhibits exponential growth.