Global Sobolev regular solution for Boussinesq system

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