Global Sobolev regular solution for Boussinesq system

Abstract

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 ≔ R 2 × ( 0 , R ) \Omega := {{\mathbb{R}}}^{2}\times \left(0,R) . We construct a global finite energy Sobolev regularity solution ( v , ρ ) ∈ H s ( Ω ) × H s ( 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 H s + 2 ( Ω ) × H s + 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 s s 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.