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We investigate the well-posedness of \alpha-surface quasi-geostrophic (\alpha-SQG) equations in the half-plane, where \alpha =0 and \alpha =1 correspond to the 2D Euler and SQG equations respectively. For 0<\alpha \le 1/2, we prove local well-posedness in certain weighted anisotropic Hölder spaces. We also show that such a well-posedness result is sharp: for any 0<\alpha \le 1, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on \alpha) for initial data smooth up to the boundary. [ABSTRACT FROM AUTHOR]

We consider the 3D incompressible Navier–Stokes equations under the following 2+\frac {1}{2}-dimensional situation: vertical vortex blob (quasi-streamwise vortices) being stretched by two-dimensional shear flow. We prove enhanced dissipation induced by such quasi-streamwise vortices. [ABSTRACT FROM AUTHOR]

In a companion paper [arXiv:1903.00833], we gave a detailed account of the well-posedness theory for singular vortex patches. Here, we discuss the long-time dynamics of some of the classes of vortex patches we showed to be globally well-posed in the above-mentioned paper. In particular, we give examples of time-periodic behavior, cusp formation in infinite time at an exponential rate, and spiral formation in infinite time. [ABSTRACT FROM AUTHOR]