GLOBAL PERSISTENCE OF GEOMETRICAL STRUCTURES FOR THE BOUSSINESQ EQUATION WITH NO DIFFUSION

Raphaël Danchin (1) , Xin Zhang (1)

1 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12

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 $C^{1, \varepsilon}$ Hölder 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.

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin, Xin Zhang. GLOBAL PERSISTENCE OF GEOMETRICAL STRUCTURES FOR THE BOUSSINESQ EQUATION WITH NO DIFFUSION. 2016. ⟨hal-01290221⟩