Related papers: A regularity criterion for the dissipative quasi-g…

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity with the same decay exponent as the linear…

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove global well-posedness result for small initial data lying in critical Besov spaces constructed…

In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of…

This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker…

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 this article, we study a Besov regularity estimate of weak solutions to a class of nonlinear elliptic equations in divergence form. The main purpose is to establish Calderon-Zygmund type estimate in Besov spaces with more general…

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 consider the modified quasi-geostrophic equation \begin{gather*} \del_t \theta + (u \cdot \grad) \theta + \kappa \Lambda^\alpha \theta = 0 u = \Lambda^{\alpha - 1} R^{\perp}\theta. \end{gather*} with $\kappa > 0$, $\alpha…

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…

The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,\theta )$ becomes regular provided that…

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 lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert…

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…

Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions…

We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions…

The dissipative solutions can be seen as a convenient generalization of the concept of weak solution to the isentropic Euler system. They can be seen as expectations of the Young measures associated to a suitable measure--valued solution of…

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and…

We prove new regularity criteria of the Prodi-Serrin type with weak Lebesgue integrability in both space and time for a viscous active chemical fluid in a bounded domain.

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 first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak…