Analytical smoothing effect of solution for the boussinesq equations

In this paper, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H 1-solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equation admet exactly same smoothing effect properties of incompressible Navier-Stokes equations

The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized \`a la Leray through a smoothing kernel of order $\alpha$ in the nonlinear term and a $\beta$-fractional Laplacian; we consider the critical case $\alpha+\beta=\frac{5}{4}$ and we assume $\frac 12 <\beta<\frac 54$. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order $\alpha$. We prove global well posedness when the initial velocity is in $H^r$ and the initial temperature is in $H^{r-\beta}$ for $r>\max(2\beta,\beta+1)$. This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of the solutions on the initial conditions.Comment: 28 pages; final version accepted for publication in Journal of Differential Equation

Global Regularity for an Inviscid Three-dimensional Slow Limiting Ocean Dynamics Model

We establish, for smooth enough initial data, the global well-posedness (existence, uniqueness and continuous dependence on initial data) of solutions, for an inviscid three-dimensional {\it slow limiting ocean dynamics} model. This model was derived as a strong rotation limit of the rotating and stratified Boussinesg equations with periodic boundary conditions. To establish our results we utilize the tools developed for investigating the two-dimensional incompressible Euler equations and linear transport equations. Using a weaker formulation of the model we also show the global existence and uniqueness of solutions, for less regular initial data