Global persistence of geometrical structures for the boussinesq equation with no diffusion

Raphaël Danchin; Xin Zhang

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arXiv · 1603.07479

Global persistence of geometrical structures for the boussinesq equation with no diffusion

Raphaël Danchin·Xin Zhang

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Abstract

Here we investigate the so-called temperature patch problem for the incompressible Boussinesq system with partial viscosity, in the whole space $\mathbb{R}^N$ $(N \geq 2)$, where the initial temperature is the characteristic function of some simply connected domain with $C^{1, \varepsilon}$ H{\"o}lder regularity. Although recent results in [1, 15] ensure that an initially $C^1$ patch persists through the evolution, whether higher regularity is preserved has remained an open question. In the present paper, we give a positive answer to that issue globally in time, in the 2-D case for large initial data and in the higher dimension case for small initial data.

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Raphaël Danchin, Xin Zhang. 2016-03-24. Global persistence of geometrical structures for the boussinesq equation with no diffusion. https://arxiv.org/abs/1603.07479

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