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Finite time blow up in the hyperbolic Boussinesq system. (English) Zbl 1382.35054
Summary: In recent work of G. Luo and T. Y. Hou [Multiscale Model. Simul. 12, No. 4, 1722–1776 (2014; Zbl 1316.35235)], a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call “hyperbolic Boussinesq system”. This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot-Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of initial data.

MSC:
35B44 Blow-up in context of PDEs
35Q31 Euler equations
35Q35 PDEs in connection with fluid mechanics
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