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January/February 2016 Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces
Igor Kukavica, Fei Wang, Mohammed Ziane
Adv. Differential Equations 21(1/2): 85-108 (January/February 2016).

Abstract

We address the global regularity of solutions to the Boussinesq equations with zero diffusivity in two spatial dimensions. Previously, the persistence in the space $H^{1+s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ for all $s\ge 0$ has been obtained. In this paper, we address the persistence in general Sobolev spaces, establishing it on a time interval which is almost independent of the size of the initial data. Namely, we prove that if $(u_0,\rho_0)\in W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $s\in(0,1)$ and $q\in[2,\infty)$, then the solution $(u(t),\rho(t))$ of the Boussinesq system stays in $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $t\in[0,T^*)$, where $T^*$ depends logarithmically on the size of initial data. If we furthermore assume that $sq>2$, then we get the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$. Moreover, we prove the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for the initial data with compact support, as well us for data in $W^{1+s,q}(\mathbb{T}^2)\times W^{s,q}(\mathbb{T}^2)$, without any restriction on $s\in(0,1)$ and $q\in[2,\infty)$.

Citation

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Igor Kukavica. Fei Wang. Mohammed Ziane. "Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces." Adv. Differential Equations 21 (1/2) 85 - 108, January/February 2016.

Information

Published: January/February 2016
First available in Project Euclid: 23 November 2015

zbMATH: 1334.35103
MathSciNet: MR3449331

Subjects:
Primary: 35K55 , 35M33 , 76B03 , 76D05

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 1/2 • January/February 2016
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