Global Sobolev regular solution for Boussinesq system

Abstract

This article is concerned with the study of the initial value problem for the three-dimensional viscous Boussinesq system in the thin domain Ω≔R2×(0,R)\Omega := {{\mathbb{R}}}^{2}\times \left(0,R). We construct a global finite energy Sobolev regularity solution (v,ρ)∈Hs(Ω)×Hs(Ω)\left({\bf{v}},\rho )\in {H}^{s}\left(\Omega )\times {{\mathbb{H}}}^{s}\left(\Omega ) with the small initial data in the Sobolev space Hs+2(Ω)×Hs+2(Ω){H}^{s+2}\left(\Omega )\times {{\mathbb{H}}}^{s+2}\left(\Omega ). Some features of this article are the following: (i) we do not require the initial data to be axisymmetric; (ii) the Sobolev exponent ss can be an arbitrary big positive integer; and (iii) the explicit asymptotic expansion formulas of Sobolev regular solution is given. The key point of the proof depends on the structure of the perturbation system by means of a suitable initial approximation function of the Nash-Moser iteration scheme.

Keywords

• boussinesq system
• global sobolev solution
• nash-moser iteration scheme
• 35q31
• 35a01