Sharp non-uniqueness for the Boussinesq equation with fractional dissipation

TL;DR

This paper demonstrates the failure of uniqueness for solutions to the fractional dissipation Boussinesq equation in certain function spaces, revealing sharp thresholds and constructing smooth solutions outside small singular sets.

Contribution

It establishes sharp non-uniqueness results for the fractional Boussinesq equation in specific function spaces, extending understanding of solution behavior under fractional dissipation.

Findings

Uniqueness breaks down in $L^p_tL^ abla_x$ for certain $p$ and $ abla$ ranges.

Constructed solutions are smooth outside small Hausdorff dimension sets.

Weak-strong uniqueness holds in the space $L^{rac{2eta}{2eta-1}}_T L^ abla_x$.

Abstract

This paper focuses on the $d$-dimensional ($d\geq2$) Boussinesq equation with fractional dissipation $(-\Delta)^{\alpha}$ on the torus. We show that the uniqueness property breaks down within the function space $L^p_tL^\infty_x$ for any $p<\frac{2\alpha}{2\alpha-1}$ when $1\leq\alpha<\frac{d+1}{2}$ and the function space $L^\frac{2\alpha}{2\alpha-1}_tL^q_x$ for any $q<\infty$ when $1<\alpha<\frac{d+1}{2}$. Moreover, the weak solutions we construct are smooth outside a set of singular times with Hausdorff dimension arbitrarily small. This result is sharp, as weak-strong uniqueness holds in the space $L^{\frac{2\alpha}{2\alpha-1}}_TL^\infty_x$.