Global Well‐Posedness of Three‐Dimensional Boussinesq‐Wave System in a Thin Domain

ABSTRACT This paper aims to study the hydrostatic approximation of three‐dimensional Boussinesq‐wave system in a thin domain. We derive the asymptotic limit of the three‐dimensional Boussinesq‐wave system in a thin domain, when the depth of the domain and viscosity converge to zero in a related way. Unlike the hyperbolic hydrostatic Navier‐Stokes system, here the temperature leads to the new loss of derivative. In the setting of Gevrey space with index 2, we address the global well‐posedness for the hydrostatic Boussinesq‐wave system and justify the hydrostatic limit.