The 2D inviscid Boussinesq equations with fractional diffusion in bounded domain

Abstract

The incompressible Boussinesq equations serve as an important model in geophysics especially in the study of Rayleigh–Bénard convection. This paper focuses on the 2D incompressible inviscid Boussinesq equations with fractional diffusion (−Δ)βθ in bounded domain, equipping with the slip boundary condition for velocity vector field and Dirichlet boundary condition for temperature. We obtain the global existence and uniqueness of classical solutions in the range of β∈[3/4,1) and also show the local corresponding result for β∈[0,1). To the best of our knowledge, this is the first paper considering the Boussinesq equation with fractional diffusion in bounded domain. This result extends the work by K. Zhao (2010) to a fractional dissipation for temperature in bounded domain.