The 2D Boussinesq equations with logarithmically supercritical velocities

Abstract

This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq-Navier-Stokes equations. The velocity $u$ in this system is related to the vorticity $\omega$ through the relations $u={\nabla }^{\perp }\psi$ and $\Delta \psi ={\Lambda }^{\sigma }(\log (I-\Delta ){)}^{\gamma }\omega$, which reduces to the standard velocity-vorticity relation when $\sigma =\gamma =0$. When either $\sigma >0$ or $\gamma >0$, the velocity $u$ is more singular. The "quasi-velocity" $v$ determined by $\nabla \times v=\omega$ satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when $\sigma =0$ and $\gamma \ge 0$. In addition, the vorticity $\omega$ is shown to be globally bounded in several functional settings such as $L^2$ for $\sigma >0$ in a suitable range.

27 pages

References in corpus (5)

• Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces
• Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation
• Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations
• Nonlinear maximum principles for dissipative linear nonlocal operators and applications
• Generalized surface quasi-geostrophic equations with singular velocities