an:01705197
Zbl 0991.35070
Taniuchi, Yasushi
A note on the blow-up criterion for the inviscid 2-D Boussinesq equations
EN
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).
2002
a
35Q35 35B40 76B07
incompressible inviscid flows; passive scalar; 2-D Boussinesq equations; singularities
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].