On the $\delta$-primitive and Boussinesq type equations

Abstract

In this article we consider the Primitive Equations without horizontal viscosity but with a mild vertical viscosity added in the hydrostatic equation, as in [13] and [16], which are the so-called $\delta-$Primitive Equations. We prove that the problem is well posed in the sense of Hadamard in certain types of spaces. This means that we prove the finite-in-time existence, uniqueness and continuous dependence on data for appropriate solutions. The results given in the 3D periodic space easily extend to dimension 2. We also consider a Boussinesq type of equation, meaning that the mild vertical viscosity present in the hydrostatic equation is replaced by the time derivative of the vertical velocity. We prove the same type of results as for the $\delta-$Primitive Equations; periodic boundary conditions are similarly considered.