Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity

TL;DR

This paper proves that solutions to the 2D fractional Boussinesq equations with positive viscosity and zero diffusivity maintain their Sobolev regularity over time, ensuring persistence of smoothness in these spaces.

Contribution

It establishes the global persistence of Sobolev regularity for solutions of the 2D fractional Boussinesq equations with zero diffusivity, a result previously unproven for this setting.

Findings

Solutions remain in Sobolev spaces for all positive times.

Regularity persists without loss over time.

Results hold for arbitrary fractional order $\alpha ext{ in }(1,2)$.

Abstract

We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$, where $s> 1$ and $q \in (2, \infty)$. We prove that the solution $(u(t), \rho(t))$ exists and belongs to $W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$ for all positive time $t$ for $q>2$, where $\alpha\in(1,2)$ is arbitrary.