# zbMATH — the first resource for mathematics

A note on the blow-up criterion for the inviscid 2-D Boussinesq equations. (English) Zbl 0991.35070
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 223, 131-140 (2002).
Summary: We show that a smooth solution of the 2-D Boussinesq equations $\partial_tu+ u\cdot\nabla u+\nabla p=\theta f,\quad \partial_t \theta+ u\cdot \nabla\theta =0,\quad \text{div }u=0,$ in the whole plane $$\mathbb{R}^2$$ breaks down if and only if a certain norm of $$\nabla\theta$$ blows up at the same time. Here the norm is weaker than the $$L^\infty$$-norm and generates a Banach space including singularities of $$\log\log 1/ |x|$$. Roughly speaking, when a smooth solution breaks down, $$\nabla\theta$$ has stronger singularities than $$\log\log 1/ |x|$$ or has an infinite number of singularities.
For the entire collection see [Zbl 0972.00046].

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 76B07 Free-surface potential flows for incompressible inviscid fluids