FORMULA OF GLOBAL SMOOTH SOLUTION FOR NON-HOMOGENEOUS M-D CONSERVATION LAW WITH UNBOUNDED INITIAL VALUE

In this article, we prove the existence and obtain the expression of its solution formula of global smooth solution for non-homogeneous multi-dimensional（m-D） conservation law with unbounded initial value; our methods are new and essentially different with the situation of bounded initial value.

Global Smooth Solution for the Quasi-linear Wave Equation

In this paper, we study the Cauchy problem for the following quasi-linear wave equation utt-2kuxxt=β（ux^n）x, where k〉0 and β are real numbers, and n ≥ 2 is an integer. We prove that for any T〉0, the Cauchy problem admits a unique global smooth solution u∈C^∞（（0, T]; H^∞（R）） ∩ C（[0, T]; H^2（R）） ∩ C^1（[0, T]; L^2（R）） under suitable assumptions on the initial data.

ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS FOR BIPOLAR COMPRESSIBLE NAVIER-STOKES-MAXWELL SYSTEM FROM PLASMAS

This paper is concerned with the bipolar compressible Navier-Stokes-Maxwell system for plasmas. We investigated, by means of the techniques of symmetrizer and elaborate energy method, the Cauchy problem in R^3. Under the assumption that the initial values are close to a equilibrium solutions, we prove that the smooth solutions of this problem converge to a steady state as the time goes to the infinity. It is shown that the difference of densities of two carriers converge to the equilibrium states with the norm ｜｜·｜｜H^s-1, while the velocities and the electromagnetic fields converge to the equilibrium states with weaker norms than ｜｜·｜｜H^s-1. This phenomenon on the charge transport shows the essential difference between the unipolar Navier-Stokes-Maxwell and the bipolar Navier-Stokes-Maxwell system.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

Global Smooth Solution to the Incompressible Navier-Stokes-Landau-Lifshitz Equations

In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in T2or R2and R3.

THE ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS TO THE MACROSCOPIC MODELS FOR SEMICONDUCTORS

The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assump- tion on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.

Global Smooth Solutions to the 2-D Inhomogeneous Navier-Stokes Equations with Variable Viscosity

Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0-1∈Hs+1(R2);u0∈Hs(R2)∩H_∈(R2) for s>2 and 0<ε<1;the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid.Furthermore,the L2 decay rate of the velocity field is obtained.

Global smooth solution to a coupled Schrodinger system in atomic Bose-Einstein condensates with two-dimensional spaces

We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.

EXISTENCE OF GLOBAL SMOOTH SOLUTIONS FOR TWO IMPORTANT NONSTRICTLY QUASILINEAR HYPERBOLIC SYSTEMS

In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.

Existence and Uniqueness of the Global Smooth Solution to the Periodic Boundary Value Problem of Fractional Nonlinear Schr^dinger System

In this paper, we study a class of coupled fractional nonlinear Schr^dinger system with periodic boundary condition. Using Galerkin method, the existence of global smooth solution is obtained. We also prove the uniqueness of the solution.

Global Existence of Smooth Solutions of Compressible Bipolar Euler-Maxwell Equations

The bipolar compressible Euler-Maxwell equations as a fluid dynamic model arising from plasma physics to describe the dynamics of the compressible electrons and ions is investigated. This work is concerned with three-dimensional Euler-Maxwell equations with smooth periodic solutions. With the help of the symmetry operator techniques and energy method, the global smooth solution with small amplitude is constructed around a constant equilibrium solution with asymptotic stability property.

GLOBAL EXISTENCE OF SOLUTIONS FOR A STRONGLY COUPLED REACTION-DIFFUSION SYSTEM

This paper is concerned with an Initial Boundary Value Problem (IBVP) for a strongly coupled semilinear reaction-diffusion system. By using the upper and lower solutions method and Leray-Schauder fixed point theorem and so on, the authors prove the global existence and uniqueness of a. smooth. solution for this IBVP under some appropriate conditions.

THE GLOBAL COMBINED QUASI-NEUTRAL AND ZERO-ELECTRON-MASS LIMIT OF NON-ISENTROPIC EULER-POISSON SYSTEMS

We consider a non-isentropic Euler-Poisson system with two small parameters arising in the modeling of unmagnetized plasmas and semiconductors.On the basis of the energy estimates and the compactness theorem,the uniform global existence of the solutions and the combined quasi-neutral and zero-electron-mass limit of the system are proved when the initial data are close to the constant equilibrium state.In particular,the limit is rigorously justified as the two parameters tend to zero independently.

Asymptotic Behavior of the Solutions to the One-Dimensional Nonisentropic Hydrodynamic Model for Semiconductors

In this paper, the asymptotic behavior of the global smooth solutions to the Cauchy problem for the one-dimensional nonisentropic Euler-Poisson （or full hydrodynamic） model for semiconductors, where the energy equation with non-zero thermal conductivity coefficient are contained, is discussed. The global existence of smooth solutions for the Cauchy problem with small perturbed initial data is proved. In particular, that the solutions converge to the corresponding stationary solutions exponentially fast as t → ∞ is showed.

具有非线性边界条件的不可压MHD-Boussinesq方程组初边值问题的整体适定性

The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq equation with the given pressure-velocity’s relation boundary condition for the fluid field,one generalized perfectly conducting boundary condition for the magnetic field and one density/temperature-velocity’s relation boundary condition for the density/temapture at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also proven.

Smooth Solution of Multi-dimensional Nonhomogeneous Conservation Law: Its Formula, and Necessary and Sufficient Blowup Criterion

In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term,where the initial data lies in W1,∞(Rn) ∩ C1(Rn). We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

SOME RESULTS ON HYPERBOLIC SYSTEMS WITH RELAXATION

In this article, first, the authors prove that there exists a unique global smooth solution for the Cauthy problem to the hyperbolic conservation laws systems with relaxation; second, in the large time station, they prove that the global smooth solutions of the hyperbolic conservation laws systems with relaxation converge to rarefaction waves solution at a determined L^P（p ≥ 2） decay rate.

Cauchy Problem of Quasilinear Equation with Forced Terms

In this paper, we consider Cauchy problem for a class of quasilinear hyperbolic equations with forced terms, extend and improve the existence in paper[2].

Global Existence and Time-decay Rates of Solutions to 2D Magneto-micropolar Fluid Equations with Partial Viscosity

In this paper,we investigate the initial value problem for the two-dimensiona magneto-micropolar fluid equations with partial viscosity.We prove that global existence of smooth large solutions by the energy method.Furthermore,with aid of the Fourier splitting methods,optimal time-decay rates of global smooth large solutions are also established.

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.