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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].

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