Nonlinear maximum principles for dissipative linear nonlocal operators and applications

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations

A global regularity result for the 2D Boussinesq equations with critical dissipation

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by $\Lambda^\alpha u$ in the velocity equation and by $\Lambda^\beta \theta$ in the temperature equation, where $\Lambda=\sqrt{-\Delta}$ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when $(\alpha,\beta)$ is in the critical range: $\alpha>\frac{\sqrt{1777}-23}{24} =0.798103..$, $\beta>0$ and $\alpha+ \beta =1$. This result improves the previous work of Jiu, Miao, Wu and Zhang \cite{JMWZ} which obtained the global regularity for $\alpha> \frac{23-\sqrt{145}}{12} \approx 0.9132$, $\beta>0$ and $\alpha+ \beta =1$.Comment: This version fix a minor error in the previous versio

2D Boussinesq equations with logarithmically super-critical conditions

This thesis focuses on the regularity problem of two generalized two dimensional Boussinesq equations. The first model contains the critical level of diffusion and a double logarithmically super-critical velocity. The second model contains logarithmically super-critical dissipation. The proof takes the advantage of the two equivalent definitions of the dissipative operator. We also extend the Besov spaces to better suit the new operator. In Chapter 5, we give a small data regularity result for super-critical Surface Quasi-Geostrophic equations. This is achieved by generalize the definition of Only Small Shock first introduced in [21]. The proof also use the modulus of continuity approach in [53]. The last chapter deal with an axisymmetric Navier-Stokes model by Hou and Li in n-dimensional setting. The local and global regularity result is achieved by requiring a strong enough fractional Laplacian dissipation

Regularity and blow up for active scalars

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasi-geostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime.Comment: 33 page

Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation

Abstract The incompressible Boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important. The well-posedness and related problem on the Boussinesq equations have recently attracted considerable interest. This paper examines the global regularity issue on the 2D Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the case when the thermal diffusion dominates. We establish the global wellposedness for the 2D Boussinesq equations with a new range of fractional powers of the Laplacian