Sharp non-uniqueness for the Boussinesq equation with fractional dissipation
Zipeng Chen, Zhaoyang Yin
TL;DR
The paper proves sharp non-uniqueness for the d-dimensional Boussinesq system with fractional dissipation $(-\Delta)^{\alpha}$ on the torus by combining a gluing procedure that concentrates Reynolds stress in time with a space-time convex integration scheme. The method yields infinitely many weak solutions in subcritical spaces $L^p_TL^\infty_x$ (and $L^{\frac{2\alpha}{2\alpha-1}}_tL^q_x$ for $\alpha<\frac{d+1}{2}$) with the same initial data, while proving endpoint weak-strong uniqueness in $L^{\frac{2\alpha}{2\alpha-1}}_TL^\infty_x$. The construction also provides a weak solution whose singular set in time has Hausdorff dimension arbitrarily small. The results extend sharp non-uniqueness phenomena to the Boussinesq system and are adaptable to the Navier–Stokes setting, highlighting the delicate balance between nonlinearity and fractional dissipation at critical thresholds.
Abstract
This paper focuses on the $d$-dimensional ($d\geq2$) Boussinesq equation with fractional dissipation $(-Δ)^α$ on the torus. We show that the uniqueness property breaks down within the function space $L^p_tL^\infty_x$ for any $p<\frac{2α}{2α-1}$ when $1\leqα<\frac{d+1}{2}$ and the function space $L^\frac{2α}{2α-1}_tL^q_x$ for any $q<\infty$ when $1<α<\frac{d+1}{2}$. 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 $L^{\frac{2α}{2α-1}}_TL^\infty_x$.