Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity

We prove the global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity.     

Global well-posedness for the 2-D nonhomogeneous incompressible MHD equations with large initial data

In this paper, we consider the 2-D nonhomogeneous incompressible magnetohydrodynamic equations with variable viscosity and variable conductivity. We obtain the global existence of solutions for this system with initial data in the scaling invariant Besov spaces and without size restriction for the initial velocity and magnetic field.     

Global well-posedness for the Euler-Boussinesq system with axisymmetric data

In this paper we prove the global well-posedness for the three-dimensional Euler-Boussinesq system with axisymmetric initial data without swirl. This system couples the Euler equation with a transport-diffusion equation governing the temperature.     

Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely,

     

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, -critical Hartree equation for the radial data in all dimensions (n5). We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.     

Global well-posedness for the viscous shallow water system with Korteweg type

In this paper, we study the Cauchy problem for a viscous shallow water system with Korteweg type in Sobolev spaces. We first establish the local well-posedness of the solution by using the Friedrich method and compactness arguments. Then, we prove the global existence of the solution to the system for the small initial data.     

Global well-posedness and time-decay for the full compressible non-resistive MHD equations in a 3D infinite slab

This paper is concerned with the global well-posedness and time-decay of the system of full compressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no compatibility conditions on the initial data.     

An Osgood type regularity criterion for the 2D Oldroyd‐B model

By means of the Littlewood‐Paley decomposition and the div‐curl Theorem by Coifman‐Lions‐Meyer‐Semmes, we prove an Osgood type regularity criterion for the 2D incompressible Oldroyd‐B model, that is, where denotes the Fourier localization operator whose spectrum is supported in the shell {|ξ|≈2^j}.     

Global regularity of the 2D micropolar fluid flows with zero angular viscosity

In this paper we prove the global existence and uniqueness of smooth solutions to the 2D micropolar fluid flows with zero angular viscosity.     

Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions

We are concerned with the global existence and uniqueness of strong solutions for the non-isentropic compressible Navier-Stokes-Poisson system in the framework of hybrid Besov spaces in three and higher dimensions. These results extend paper (Hao and Li, 2009 [4]) devoted to the barotropic case.     

Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity

In this paper, we consider an initial-boundary value problem for the 2D incompressible magnetomicropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium(0, e_1, 0).     

Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation

This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We are able to single out three special partial dissipation cases and establish the global regularity for each case. As special consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations, and the 2D micropolar equations with several types of partial dissipation always possess global classical solutions. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

Global regularity for the 2D magneto-micropolar system with partial and fractional dissipation

This paper focuses on the 2D incompressible magneto-micropolar sysytem with the kinematic dissipation given by the fractional operator (−Δ)^α, the magnetic diffusion by the fractional operator (−Δ)^β and the spin dissipation by the fractional operator (−Δ)^γ. α,β, and γ are nonnegative constants. We proved that this system with any α+β=2,1 ≤ α ≤ 2,γ=0, and α+γ ≥ 1,β=1 always possesses a unique global smooth solution $(\mathbf{u},\mathbf{b},\mathrm{w})\in H^s({\mathbb{R}}^2)(s>2)$ if the initial data is sufficiently smooth. In addition, we also obtained the global regularity results for several partial dissipation cases.     

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.