EXISTENCE OF WEAK SOLUTIONS OF 3-D EULER EQUATIONS WITH AXIAL SYMMETRV

1IntroductionInthispaper,weconsidertilefollowillgeqllatiolls:wilerstheunknownfullctionsarevelocityfields'd(x.t)=(.UI(x.t)..uZ(x.t)..it3(x,t))itlldscalarpressurefunctiollp(x,t).Asweallknow,(1.l)istheNavier-Stokesequatiollswithviscosityealld(1.2)istheEulerequationsobtainedbyvanishingtheviscosityill(1.1).Tilershavebeedalargellumberofresultsabout2-DEulerequatiolls.Butillthreedilllellsioll,illersisstillfewresults.TillsispartiallybecauseofthefactthattwodillleusiollalNavier-Stokescquatiollshasbe…     

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.     

GLOBAL STABILITY OF SOLUTIONS WITH DISCONTINUOUS INITIAL DATA CONTAINING VACUUM STATES FOR THE RELATIVISTIC EULER EQUATIONS

The global stability of Lipschitz continuous solutions with discontinuous initial data for the relativistic Euler equations is established in a broad class of entropy solutions in L∞containing vacuum states. As a corollary, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in     

EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS

We are concerned with the global existence of entropy solutions of the twodimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ(T) is assumed to have a positive lower bound. We first consider the Cauchy problem(the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is suffciently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave(weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

THE REGULAR SOLUTIONS OF THE ISENTROPIC EULER EQUATIONS WITH DEGENERATE LINEAR DAMPING

The regular solutions of the isentropic Euler equations with degenerate linear damping for a perfect gas are studied in this paper. And a critical degenerate linear damping coefficient is found, such that if the degenerate linear damping coefficient is larger than it and the gas lies in a compact domain initially, then the regular solution will blow up in finite time; if the degenerate linear damping coefficient is less than it, then under some hypotheses on the initial data, the regular solution exists globally.     

带线性退化阻尼项的可压缩欧拉方程组的整体正规解

本文研究了理想气体的带线性退化阻尼项的可压缩欧拉方程组的真空初值问题.利用能量估计的方法,在适当的初始条件下,获得了初值问题的正无偏见解整体存在的结果.推广了可压缩等熵欧拉方程组的结果.     

ON A NEW 3D MODEL FOR INCOMPRESSIBLE EULER AND NAVIER-STOKES EQUATIONS

In this paper, we investigate some new properties of the incompressible Euler and Navier-Stokes equations by studying a 3D model for axisymmetric 3D incompressible Euler and Navier-Stokes equations with swirl. The 3D model is derived by reformulating the axisymmetric 3D incompressible Euler and Navier-Stokes equations and then neglecting the convection term of the resulting equations. Some properties of this 3D model are reviewed. Finally, some potential features of the incompressible Euler and Navier-Stokes equations such as the stabilizing effect of the convection are presented.     

REMARKS ON THE EXISTENCE OF WEAK SOLUTIONS TO 2-D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

1.IntroductionConsiderthefollowinghomogeneous2-DincompressibleNavier-Stokesequations:whereV=(VI(t,x),VZ(t,x))isthevelocityofthefluid,Pisthescalarpressure,V'V=V101 V202,andtheconstantu>0isthekineticviscosityofthefluid(whenu=0,thissystemiscalledEulerequations,andwedenoteitby(1.1)).Byactingtheoperationofcurlon(1.1),wecanobtaintheevolutionequationforvorticityw=(a1VZ--02VI).Itisthevorticity-streamformof(1.1)(whenu=0,wedenoteitby(1.2)).For(1.1),in[4],R.DipernaandA.Majdaprovedtheglobalexiste…     

THE GLOBAL L~2 STABILITY OF SOLUTIONS TO THREE DIMENSIONAL MHD EQUATIONS

In this paper,we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains.Under suitable conditions of the large solutions,it is shown that the large solutions are stable.And we obtain the equivalent condition of this stability condition.Moreover,the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.     

CONVERGENCE OF VORTEX METHODS FOR 3-D EULER EQUATIONS

1. IntroductionThe convergence problem of vortex methods for the Euler equations has been studied by many authors. Hald and Delprete proved the convergence for two--dimensionalinitial value problems [3]. Three-dimensional initial value problems were studied byBeale and Majda [2] and Beale [1]. Ying [4] and Ying and Zhang [sl, [61 provedthe convergence of vortex methods for two--dimensional initial-boundary value problems of the Euler equations. Ying [7] proved the convergence of vortex met…     

非线性随机分数阶微分方程Euler方法的弱收敛性

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方...     

POSITIVE SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＯＦＮＥＵＴＲＡＬＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥（黄振勋，高国柱）ＨｕａｎｇＺｈｅｎｘｕｎ（Ｄｅｐｔ．ｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２０...     

ON THE SELF-SIMILAR SOLUTIONS OF THE MAGNETO-HYDRO-DYNAMIC EQUATIONS

In this paper, we show that, for the three dimensional incompressible magnetohydro-dynamic equations, there exists only trivial backward self-similar solution in L^p（R^3） for p ≥ 3, under some smallness assumption on either the kinetic energy of the self-similar solution related to the velocity field, or the magnetic field. Second, we construct a class of global unique forward self-similar solutions to the three-dimensional MHD equations with small initial data in some sense, being homogeneous of degree -1 and belonging to some Besov space, or the Lorentz space or pseudo-measure space, as motivated by the work in [5].     

LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode. In [3, 4, 5] it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in [3]. Their fundamental nature thus makes them of interest in their own right.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME NONLINEAR DIFFERENCE EQUATIONS

In this paper, we investigate the asymptotic behavior of the extremal solutions of a difference equation and their character and prove the existence of the non-extremal solutions.     

高阶Duffing方程的周期解

在本文中,我们将二阶Duffing方程周期解存在的结果推广到高阶Duffing方程 x~((2n))+g(x)=e(t), (n≥1).     

ON SOLUTIONS OF TWO-SCALE DIFFERENCE EQUATIONS

ＯＮ ＳＯＬＵＴＩＯＮＳ ＯＦ ＴＷＯ－ＳＣＡＬＥ ＤＩＦＦＥＲＥＮＣＥ ＥＱＵＡＴＩＯＮＳ ￥ＷＡＮＧＪＩＡＮＺＨＯＮＧＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓｃｏｎｃｅｒｎｅｄｗｉｔｈｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｔｗｏ－ｓｃａｌ...     

L2 DECAY OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DAMPING

In this article, we show large time behavior of weak solutions to the Cauchy problem of the Navier-Stokes equations with damping α|u|β-1u (α0).