Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces. (English) Zbl 1334.35103

Kukavica, Igor; Wang, Fei; Ziane, Mohammed

Adv. Differ. Equ. 21, No. 1-2, 85-108 (2016).

Summary: 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 \(s\geq 0\) 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 \(q\in[2,\infty)\).

MSC:

35K55	Nonlinear parabolic equations
35M33	Initial-boundary value problems for mixed-type systems of PDEs
76B03	Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76D05	Navier-Stokes equations for incompressible viscous fluids
35Q35	PDEs in connection with fluid mechanics
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

regularity; Boussinesq equations; Sobolev spaces