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

Abstract

In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à 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 $\frac{1}{2}<\beta <\frac{5}{4}$. 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.

28 pages; final version accepted for publication in Journal of Differential Equations

References in corpus (2)

• Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
• Global regularity for a slightly supercritical hyperdissipative Navier-Stokes system

Cited by in corpus (2)

• Remarks on the non-uniqueness in law of the Navier-Stokes equations up to the J.-L. Lions' exponent
• Enhanced dissipation for stochastic Navier-Stokes equations with transport noise