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The 2D Boussinesq equations with logarithmically supercritical velocities. (English) Zbl 1248.35156
Summary: 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 \(\gamma\geq 0\). 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B35 Stability in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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