GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u + 3 x u 5 x u + 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s (T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.     

GLOBAL WELL-POSEDNESS AND SCATTERING FOR THE MASS-CRITICAL HARTREE EQUATION IN HIGH DIMENSIONS

We obtain global well-posedness and scattering, and global Lt,x2(d+2)/d spacetime bounds for solutions to the defocusing mass-critical Hartree equation in Rt × Rxd, d ≥ 5.     

GLOBAL WELL-POSEDNESS IN ENERGY SPACE OF SMALL AMPLITUDE SOLUTIONS FOR KLEIN-GORDON-ZAKHAROV EQUATION IN THREE SPACE DIMENSION

The Cauchy problem of the Klein-Gordon-Zakharov equation in three dimensional space {utt-?u + u =-nu,(x, t) ∈ R~3× R_+,ntt-?n= ?|u|~2,(x, t) ∈ R~3× R_+,u(x, 0) = u_0(x), ?_tu(x, 0) = u_1(x),n(x, 0) = n_0(x), ?_tn(x,0) =n_1(x),(0.1) is considered. It is shown that it is globally well-posed in energy space H~1× L~2× L~2× H~(-1) if small initial data(u_0(x), u_1(x), n_0(x), n_1(x)) ∈(H~1× L~2× L~2× H~(-1)). It answers an open problem: Is it globally well-posed in energy space H~1× L~2× L~2× H~(-1) for 3D Klein-GordonZakharov equation with small initial data [1, 2]? The method in this article combines the linear property of the equation( dispersive property) with nonlinear property of the equation(energy inequalities). We mainly extend the spaces F~s and N~s in one dimension [3] to higher dimension.     

关于Hirota-Satsuma系统初值问题的局部和整体适定性

本文利用ＫＤＶ方程所对应的线性方程解所具有的光滑效应及压缩映像原理,得到了Ｈｉｒｏｔａ－Ｓａｔｓｕｍａ系统初值问题的局部和整体适定性结果．     

A LEGENDRE PSEUDOSPECTRAL METHOD FOR SOLVINGNONLINEAR KLEIN-GORDON EQUATION

1.IntroductionAsweknow,theKlein-Gordonequationisanimportantmathematicalmodelinquantummechanics.Itisoftheformwherefi=(--1,1)",x=(xl,x25'jx.),bisarealnumber.AssumethatU000=UI(x)=0onOnandAsin[1],itcanbeshownthatifUOEHI(n)nH(n),UIEL'(fl)andfEL'(o,T;L'(n)),then(1.1)hasuniquesolutionUELoo(o,T;Hi(n)nH(fl)).oftheotherhand,somefinitedifferenceschemeswereproposedin12,31withstrictproo]ofgeneralizedstabilityandconvergence.TheirnumericalsolutionskeepthediscretEconservations.Oneofspecialcases(…     

三维地磁流方程的整体吸引子

在本文中,我们通过引入一类Ｆｒｅｃｈｅｔ空间,得到了三维有界光滑区域上的地磁流方程的弱解的整体吸引子的存在性。     

A FOURIER SPECTRAL SCHEME FOR SOLVING NONLINEAR KLEIN-GORDON EQUATION

A Fourier spectral scheme is proposed for solving the periodic problem of nonlinear Klein-Gordon equation. Its stability and convergence are investigated. Numerical results are also presented.     

WELL-POSEDNESS FOR THE CAUCHY PROBLEM TO THE HIROTA EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES

The local well-posedness of the Cauchy problem for the Hirota equation is established for low regularity data in Sobolev spaces Hs(s≥-1/4). Moreover, the global well-posedness for L2 data follows from the local well-posedness and the conserved quantity. For data in Hs(s > 0), the global well-posedness is also proved. The main idea is to use the generalized trilinear estimates, associated with the Fourier restriction norm method.     

GLOBAL L^p ESTIMATES FOR THE PARABOLIC EQUATION OF THE BI-HARMONIC TYPE

In this paper similarly to the second-order case, we give an elementary and straightforward proof of global Lp estimates for the initial-value parabolic problem of the bi-harmonic type. Moreover, we obtain the existence and uniqueness of the solutions in the suitable space using the potential theory, Marcinkiewicz interpolation theorem and approximation argument. Meanwhile, by the same approach we can deal with the general polyharmonic cases.     

WELL-POSEDNESS AND GLOBAL ATTRACTORS FOR LIQUID CRYSTALS ON RIEMANNIAN MANIFOLDS

ABSTRACT

We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version ofthe Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, deriving the system from a variational principle, and provide a very simple proof of local well-posedness for this coupled system using a contraction mapping argument. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two. A small data result in n dimensions follows easily. Finally, we introduce the Lagrangian averaged liquid crystal equations, which arise from averaging the Navier-Stokes fluid motion over small spatial scales in the variational principle. We show that this averaged system is globally well-posed and has compact global attractors even when M is three-dimensional.     

THE GLOBAL SOLUTIONS FOR A FOURTH ORDER NONLINEAR SCHRaINGER EQUATION

ＴＨＥＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＡＦＯＵＲＴＨＯＲＤＥＲＮＯＮＬＩＮＥＡＲＳＣＨＲǒＩＮＧＥＲＥＱＵＡＴＩＯＮＦｅｎｇＸｕｅｓｈａｎｇ（冯学尚）（ＮｕｍｅｒｉｃａｌＬａｂｏｒａｔｏｒｙｆｏｒＨｅｌｉｏｓｐｈｅｒｉｃｐｈｙｓｉｃｓ，Ｐ．Ｏ．Ｂ...     

Sine—Gordon方程的全局吸引子的维数估计

本文得到了阻尼Ｓｉｎｅ－Ｇｏｒｄｏｎ方程的狄氏问题的全局吸引子的Ｈａｕｓｄｏｒｆｆ维数以偶数上界的参数条件，特别地，当阻尼与Ｌａｐｌａｅ算子的第一个特征值适当大时，全局吸引子是零维的，零维吸引子恰是系统的唯一平衡解并且指数吸引相空间的有界集。     

GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES

In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^s(·)p(·),h(·),q(R³)with s(·)=4-2α-3/p(·).We prove global well-posedness result with small initial data belonging to FN^4-2α-3/p(·)p(·),h(·)q(R³).The result of this paper extends some recent work.     

GLOBAL EXISTENCE OF THE SOLUTION FOR THE LANDAU-LIFSHITZ EQUATION OF THE FERROMAGNETIC SPIN CHAIN

1IntroductionInthepresentpaper,wecollsidertileLandau-LifSllitzequationoftileferromagneticspinchainwhichtakestheform:where"x'ldenotesthevectorcrossproductinR',fi=(--D,D),D>0,u=(ul,u3,u3):fixR -R3istilespinvector,or>0isaGilbertdampingconstant,Pisarealparameter,J=diag(ri,acfac)isaconstantmatrix.TheaboveequationarisesinthedoublyanisotropicHeisenbergspinsystem.Thissystemisimpliedbytileconservationofenergyandmagnitudeofu.SeeDanielandLab.hmanan[1]please.Insomeferromagneticchainsystems,thematri…     

求解抛物型方程的混合元法及其拟最优最大模估计

讨论如下抛物型方程模型问题: 这里Ω为具充分光滑边界(?)Ω的平面有界区域.假设对于讨论的f和A,(1.1)的解存在唯一,且具有所需要的正则性.引进q=-grad u,V=H(div,Ω),W=L~2(Ω),则(1.1)的变分形式如下:求{q,u}:t∈[0,T]→V×W,使     

ON THE GLOBAL ATTRACTIVITY FOR A NEUTRAL DIFFERENCE EQUATION WITH TWO DELAYS

ＯＮ　ＴＨＥ　ＧＬＯＢＡＬ　ＡＴＴＲＡＣＴＩＶＩＴＹ　ＦＯＲ　Ａ　ＮＥＵＴＲＡＬ　ＤＩＦＦＥＲＥＮＣＥ　ＥＱＵＡＴＩＯＮ　ＷＩＴＨ　ＴＷＯ　ＤＥＬＡＹＳＯＮＴＨＥＧＬＯＢＡＬＡＴＴＲＡＣＴＩＶＩＴＹＦＯＲＡＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＣＥＥＱＵＡＴ...     

带第三类边界条件的热传导方程的守恒型格式和最大模原理

§1.引言 在二维区域Ω(边界为?Ω)上,根据热量守恒原理,描述变温过程的积分关系式为初条件为     

高阶非线性Schrödinger方程的整体适定性

本文研究了一类非线性高阶Schr(o)dinger方程Cauchy问题的整体适定性.利用不动点定理,获得了整体解的存在唯一性及解关于初值的连续依赖性和解具有较强的衰减估计.推广了文献[4]中的结果.