Advances in Differential Equations

Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces

Igor Kukavica, Fei Wang, and Mohammed Ziane

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $q\in[2,\infty)$.

Article information

Source
Adv. Differential Equations Volume 21, Number 1/2 (2016), 85-108.

Dates
First available in Project Euclid: 23 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1448323165

Mathematical Reviews number (MathSciNet)
MR3449331

Zentralblatt MATH identifier
1334.35103

Subjects
Primary: 35K55: Nonlinear parabolic equations 35M33: Initial-boundary value problems for systems of mixed type 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76D05: Navier-Stokes equations [See also 35Q30]

Citation

Kukavica, Igor; Wang, Fei; Ziane, Mohammed. Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces. Adv. Differential Equations 21 (2016), no. 1/2, 85--108. https://projecteuclid.org/euclid.ade/1448323165.


Export citation