Related papers: Global regularity of 2D Rayleigh-B\'{e}nard equati…
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the…
This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-\Delta)^{\alpha}$ and magnetic diffusion given by reducing about logarithmic diffusion…
This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $\Lambda^{2\beta} b$. It is proved that when $\beta>1 $ there exists a unique global…
This paper aims at the global regularity problem concerning the 2D incompressible Boussinesq equations with general critical dissipation. The critical dissipation refers to $\alpha +\beta=1$ when $\Lambda^\alpha \equiv…
Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+…
In this paper we study the global regularity of the following 2D (two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*} \left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot \nabla b - \nu…
We show that in dimensions $n \geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is…
This paper is dedicated to the Oldroyd-B model with fractional dissipation $(-\Delta)^{\alpha}\tau$ for any $\alpha>0$. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small…
In this paper, we prove the global regularity of smooth solutions to 2D surface quasi-geostrophic (SQG) equations with super-critical dissipation for a class of large initial data, where the velocity and temperature can be arbitrarily large…
This paper examines the question for global regularity for the Boussinesq equation with critical fractional dissipation. The main result states that the system admits global regular solutions for all (reasonably) smooth and decaying data,…
In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium.…
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global…
Whether or not the solution to 2D resistive MHD equations is globally smooth remains open. This paper establishes the global regularity of solutions to the 2D almost resistive MHD equations, which require the dissipative operators…
In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the…
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in…
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 are concerned with global strong solutions and large time behavior for some inviscid Oldroyd-B models. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation inviscid Oldroyd-B model. Then, we…
As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global…
This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…