## Advances in Differential Equations

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

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

Igor Kukavica, Fei Wang, and Mohammed Ziane

#### 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