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 well-posedness for 2D Boussinesq system with general supercritical dissipation

In this paper, we consider the 2D incompressible generalized Boussinesq system with the general supercritical dissipation. Using the Fourier localization method, we obtain the local and global well-posedness for the system, and give some blow-up criterion with the velocity or the temperature.     

Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations

In this paper, we study the Cauchy problem for the 3D generalized Navier-Stokes-Boussinesq equations with fractional diffusion:

$$\left\{ {\begin{array}{*{20}{c}}{{u_t} + \left( {u \cdot \nabla } \right)u + v{\Lambda ^{2a}}u = -\nabla p + \theta {e_3},\;{e_3} = {{\left( {0,0,1} \right)}^T},} \\ {{\theta _t} + \left( {u \cdot \nabla } \right)t = 0,} \\ {Divu = 0.} \end{array}} \right.$$

With the help of the smoothing effect of the fractional diffusion operator and a logarithmic estimate, we prove the global well-posedness for this system with α ≥ 5/4. Moreover, the uniqueness and continuity of the solution with weaker initial data is based on Fourier localization technique. Our results extend ones on the 3D Navier-Stokes equations with fractional diffusion.

     

Global existence and decay estimate of solutions to magneto-micropolar fluid equations

We are concerned with magneto-micropolar fluid equations (1.3)–(1.4). The global existence of solutions to the Cauchy problem is investigated under certain conditions. Precisely, for the magneto-micropolar-Navier–Stokes (MMNS) system, we obtain global existence and large time behavior of solutions near a constant states in ${\mathbb{R}}^3$. Appealing to a refined pure energy method, we first obtain a global existence theorem by assuming that the $H^3$ norm of the initial data is small, but the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev norms ${\dot{H}}^{?s}$ $(0\le s<\frac{3}{2})$ or homogeneous Besov norms ${\dot{B}}_{2,\infty }^{?s}$ $(0<s\le \frac{3}{2})$, we obtain the optimal decay rates of the solutions and its higher order derivatives. As an immediate byproduct, we also obtain the usual $L^p?L^2$ $(1\le p\le 2)$ type of the decay rates without requiring that the $L^p$ norm of initial data is small. At last, we derive a weak solution to (1.3)–(1.4) in ${\mathbb{R}}^2$ with large initial data.     

Global well-posedness of the 3D generalized rotating magnetohydrodynamics equations

In this paper,we establish the global well-posedness of the generalized rotating magnetohydrodynamics equations if the initial data are in X~(1-2α) defined by X~(1-2α)={u∈D'(R~3):∫_(R~3)|ξ|~(1-2α)|(ξ)|dξ+∞}.In addition,we also give Gevrey class regularity of the solution.     

Covering mappings and well-posedness of nonlinear Volterra equations

The statements on solvability, solution estimates, and well-posedness of equations with conditionally covering mappings are proved. The results obtained are applied to the investigation of Volterra equations (including integral equations) unsolved for the unknown function.     

On the well-posedness of higher order viscous Burgers' equations

We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the L²$L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive multilinear estimates and use them in the contraction mapping principle   argument to prove local well-posedness for data with Sobolev regularity below L²$L^2$. We also prove ill-posedness for this type of models and show that the local well-posedness results are sharp in some particular cases viz., when the orders of dissipation p  , and nonlinearity k+1$k+1$, satisfy a relation p=2k+1$p=2k+1$.     

Global well-posedness of the Maxwell-Dirac system in two space dimensions

In recent work, Grünrock and Pecher proved that the Dirac-Klein-Gordon system in 2d is globally well-posed in the charge class (data in L² for the spinor and in a suitable Sobolev space for the scalar field). Here we obtain the analogous result for the full Maxwell-Dirac system in 2d. Making use of the null structure of the system, found in earlier joint work with Damiano Foschi, we first prove local well-posedness in the charge class. To extend the solutions globally we build on an idea due to Colliander, Holmer and Tzirakis. For this we rely on the fact that MD is charge subcritical in two space dimensions, and make use of the null structure of the Maxwell part.     

On well-posedness and blow-up for the full compressible Hall-MHD system

In this paper, we establish the local well-posedness and a blow-up criterion of strong solutions to the 3D compressible full Hall-MHD system with positive density.     

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier-Stokes equations with damping α|u|^β−1u(α>0) has global weak solutions for any β?1, global strong solution for any β?7/2 and that the strong solution is unique for any 7/2?β?5.     

Global and exponential attractors for 3‐D wave equations with displacement dependent damping

A weakly damped wave equation in the three‐dimensional (3‐D) space with a damping coefficient depending on the displacement is studied. This equation is shown to generate a dissipative semigroup in the energy phase space, which possesses finite‐dimensional global and exponential attractors in a slightly weaker topology. Copyright © 2006 John Wiley & Sons, Ltd.     

Global well-posedness of strong solutions to the 3D primitive equations with horizontal eddy diffusivity

In this paper, we consider the initial–boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H²$H^2$ initial data.     

Global well-posedness of the stochastic 2D Boussinesq equations with partial viscosity

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.     

Time-dependent propagation speed vs strong damping for degenerate linear hyperbolic equations

We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator.We discover a threshold effect. If the propagation speed is regular enough, then the damping prevails, and therefore the initial value problem is well-posed in Sobolev spaces. Solutions also exhibit a regularizing effect analogous to parabolic problems. As expected, the stronger is the damping, the lower is the required regularity.On the contrary, if the propagation speed is not regular enough, there are examples where the damping is ineffective, and the dissipative equation behaves as the non-dissipative one.     

Global smooth solution for the modified anisotropic 3D Boussinesq equations with damping

This paper is mainly concerned with the modified anisotropic three-dimensional Boussinesq equations with damping. We first prove the existence and uniqueness of global solution of velocity anisotropic equations. Then we establish the well-posedness of global solution of temperature anisotropic equations.