Related papers: Eventual regularization for the slightly supercrit…

Recently, Silvestre proved that certain weak solutions of the slightly supercritical surface quasi-geostrophic equation eventually become smooth. To prove this, he employed a De Giorgi type argument originated in the work of Caffarelli and…

Motivated by the De Giorgi type argument used in a recent paper by Caffarelli and Vasseur, we prove H\"older-regularity for weak solutions of the supercritical quasi-geostrophic equation with minimal assumptions on the initial datum.

We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak…

We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Holder continuous for large time.

We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical…

We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions…

We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a (strictly positive) finite time.

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after…

We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a corollary, we obtain a global…

In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $\kappa (-\Delta)^\alpha$…

In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial…

We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime,…

We consider a family of singular surface quasi-geostrophic equations $$ \partial_{t}\theta+u\cdot\nabla\theta=-\nu (-\Delta)^{\gamma/2}\theta+(-\Delta)^{\alpha/2}\xi,\qquad u=\nabla^{\perp}(-\Delta)^{-1/2}\theta, $$ on…

Consider an axis-symmetric suitable weak solution of 3D incompressible Navier-Stokes equation with nontrivial swirl. If the solution satisfies a slightly supercritical assumption, we will prove that v is regular. This extends the results of…

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations in mixed time-space Besov spaces.

We prove a conjecture by De Giorgi on the elliptic regularization of semilinear wave equations in the finite-time case.

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in "critical" Sobolev spaces.…

In this paper we provide regularity results for active scalars that are weak solutions of almost critical drift-diffusion equations in general surfaces. This includes models of anisotropic non-homogeneous media and the physically motivated…

We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol…

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the…