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The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0, Comment: Theorem 1.1-1.3 are restated. The paper is accepted by Commun. Math. Sci

We consider the initial value problem for the Nernst-Planck equations coupled to the incompressible Euler equations in $\mathbb T^2$. We prove global existence of weak solutions for vorticity in $L^p$. We also obtain global existence and uniqueness of smooth solutions. We show that smooth solutions of the Nernst-Planck-Navier-Stokes equations converge to solutions of the Nernst-Planck-Euler equations as viscosity tends to zero. All the results hold for large data.

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes., 18 pages, 1 figure

We use contour dynamics to derive equations of motion for infinite planar surface quasi-geostrophic (SQG) fronts, and show that it leads to the same result as a regularization procedure introduced previously by Hunter and Shu (2018)., Comment: 12 pages, 2 figures

We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (GSQG) equations. We derive a cubic approximation of the contour dynamics equation, and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.