SOME EXACT SOLUTIONS OF 3-DIMENSIONAL ZERO-PRESSURE GAS DYNAMICS SYSTEM

The 3-dimensional zero-pressure gas dynamics system appears in the modeling for the large scale structure formation in the universe. The aim of this paper is to construct spherically symmetric solutions to the system. The radial component of the velocity and density satisfy a simpler one dimensional problem. First we construct explicit solutions of this one dimensional case with initial and boundary conditions. Then we get special radial solutions with different behaviours at the origin.     

DIRECTIONAL DO-NOTHING CONDITION FOR THE NAVIER-STOKES EQUATIONS

The numerical solution of flow problems usually requires bounded domains although the physical problem may take place in an unbounded or substantially larger domain. In this case, artificial boundaries are necessary. A well established artificial boundary condition for the Navier-Stokes equations diseretized by finite elements is the “do-nothing” condition. The reason for this is the fact that this condition appears automatically in the variational formulation after partial integration of the viscous term and the pressure gradient. This condition is one of the most established outflow conditions for Navier-Stokes but there are very few analytical insight into this boundary condition. We address the question of existence and stability of weak solutions for the Navier-Stokes equations with a “directional do-nothing” condition. In contrast to the usual “do-nothing” condition this boundary condition has enhanced stability properties. In the case of pure outflow, the condition is equivalent to the original one, whereas in the case of inflow a dissipative effect appears. We show existence of weak solutions and illustrate the effect of this boundary condition by computation of steady and non-steady flows.     

HYPERBOLIC-PARABOLIC CHEMOTAXIS SYSTEM WITH NONLINEAR PRODUCT TERMS

We prove the local existence and uniqueness of week solution of the hyperbolic-parabolic Chemotaxis system with some nonlinear product terms. For one dimensional case, we prove also the global existence and uniqueness of the solution for the problem.     

Ground states of a nonlinear Schrdinger system with nonconstant potentials

In the case where either the potentials Vj,μj andβare periodic or Vj are well-shaped andμj andβare anti-well-shaped,existence of a positive ground state of the Schrdinger system -Δu1+V1(x)u1=μ1(x)u31+β(x)u1u22in RN,-Δu2+V2(x)u2=β(x)u21u2+μ2(x)u32in RN,uj∈H1(RN),j=1,2,where N=1,2,3,is proved provided thatβis either small or large in terms of Vj andμj.The system with constant coefficients has been studied extensively in the last ten years,and the nonconstant coefficients case has seldom been studied.It turns out that new technical machineries in the setting of variational methods are needed in dealing with the nonconstant coefficients case.     

COMPARISONS OF ABSOLUTELY CONVERGENT TRIGONOMETRIC SERIES

In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then     

Laws of Large Numbers for Arrays of Rowwise Negatively Associated Random Variables under Nonlinear Probabilities北大核心CSCD

In this paper, a strong law of large numbers for arrays of rowwise negatively associated random variables is obtained under nonlinear probabilities, from which Kolmogorov type and Marcinkiewicz–Zygmund type strong laws of large numbers are derived. And the notion of negative association is weaker than some existing notions of dependence in nonlinear probabilities. Furthermore, an extension of strong law of large numbers for arrays of rowwise independent random variables under nonlinear probabilities is obtained. As a special case, a Kolmogorov type strong law indicates that not only the cluster points of empirical averages lie in the interval between the lower expectation and upper expectation quasi-surely, but such an interval is also the smallest one that covers the empirical averages quasi-surely. Furthermore, the strong law also states that the upper and lower limits of the empirical averages will converge to the upper and lower expectations with upper probabilities one, respectively. © 2022 Chinese Academy of Sciences. All rights reserved.     

FURTHER APPLICATIONS OF POLAR COORDINATES IN C~n

Simplify the proof on the domain of convergence of multiple power series and consider the case where some of z1, ··· , zn are contained only in a finite number of terms of the series. Obtain some results on holomorphic functions in Cn.     

FURTHER APPLICATIONS OF POLAR COORDINATES IN Cn

Simplify the proof on the domain of convergence of multiple power series and consider the case where some of z1, · · · , zn are contained only in a finite number of terms of the series. Obtain some results on holomorphic functions in Cn.     

An oscillating multiplier on Herz type spaces

In this paper, the boundedness of an oscillating multiplier m γ,β for different β on the Herz type spaces is obtained. This operator was initially studied by Wainger and Fefferman-Stein. Our results extend one of the main results in a paper by Xiaochun Li and Shanzhen Lu for the non-weighted case, if β is close to 1 or α is suitably large. For β ≥ 1, the results with no weights on the Herz type spaces are also new.     

Bingham Flows in Periodic Domains of Infinite Length

The Bingham fluid model has been successfully used in modeling a large class of non-Newtonian fluids. In this paper, the authors extend to the case of Bingham fluids the results previously obtained by Chipot and Mardare, who studied the asymptotics of the Stokes flow in a cylindrical domain that becomes unbounded in one direction, and prove the convergence of the solution to the Bingham problem in a finite periodic domain, to the solution of the Bingham problem in the infinite periodic domain, as the length of the finite domain goes to infinity. As a consequence of this convergence, the existence of a solution to a Bingham problem in the infinite periodic domain is obtained, and the uniqueness of the velocity field for this problem is also shown. Finally, they show that the error in approximating the velocity field in the infinite domain with the velocity in a periodic domain of length 2? has a polynomial decay in ?, unlike in the Stokes case(see [Chipot,M. and Mardare, S., Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction, Journal de Math′ematiques Pures et Appliqu′ees, 90(2), 2008,133–159]) where it has an exponential decay. This is in itself an important result for the numerical simulations of non-Newtonian flows in long tubes.     

Navier-Stokes方程的最佳有限元非线性Galerkin算法

1．引言对于Navier－Stokes方程有限元数值求解方面的研究已有很多的文章和专著,多数是采用有限元Galerkin算法,例见文献[1-4]．然而,由于Navier-Stokes方程在大雷诺数时有其强的非线性性和对时间土的长期依赖性,用计算机求解Navier－Stokes方程在速度和容量方面是难以承受的．为了克服这些困难,最近人们提出了有限元非线性Galerkin算法,见文献卜8］,然而这种算法只是在某一有限时刻之后具有好的收敛速度,在初始时刻的某一区间不能达到好的收敛速度．本文应用Taylor展开技术导出了数值求解二维非定常Navier－Stokes方程的最佳…     

NONLINEAR GALERKIN METHOD FOR NAVIER-STOKES EQUATIONS WITH STREAM-VORTICITY FORM

I Let D be a launder domain in Rz with Lipehitz continuous bodare I'. ac consider the stream-vortidty form of hmedependent Navier-stokes Muahons describing the 'flow Of a ~ incompreSSible nuid confined in Dwhere to and & are vorticity and stream function. BecaUSe the action (1) doeS not include the differentialten z, ~ tie ~ con&tion of' dab at In tthe Paper, we give a ho element n~ Galerkin ~, acs ~ is ~ an tab finite element spaceS X. and X* for the aPPmxhaation of the ~ ac v~ fUncti…     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

1．引言本文的工作主要是讨论非定常的热传导一对流问题的向后一步的Euler全离散化的非线性Galerkin混合元解的存在性及其误差估计．该工作是对山中的同一问题研究的第二部分．在第一部分［1］，我们已经讨论了此问题的半离散化的情形．由于所研究的目标都是非定常的热传导一对流问题，其背景是相同的，在此将不重复了，请参考［1］．本文的安排如下，52先回顾非定常的热传导一对流问题的混合元解的经典性质．53回顾半离散化的非线性Galerkin混合元解的性质，并导出后续讨论需要的一些关于时间导数的估计．54讨论向后一步的Euler全离散化…     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

In this paper, a fully discrete format of nonlinear Galerkin mixed element method with backward one-step Euler discretization of time for the non stationary conduction-convection problems is presented. The scheme is based on two finite element spaces XH and Xh for the approximation of the velocity, defined respectively on a coarse grid with grids size H and another fine grid with grid size h<< H, a finite element space Mh for the approximation of the pressure and two finite element spaces AH and Wh, for the approximation of the temperature,also defined respectivply on the coarse grid with grid size H and another fine grid with grid size h. The existence and the convergence of the fully discrete mixed element solution are shown. The scheme consists in using standard backward one step Euler-Galerkin fully discrete format at first L0 steps (L0 2) on fine grid with grid size h, but using nonlinear Galerkin mixed element method of backward one step Euler-Galerkin fully discrete format through L0 + 1 step to end step. We have proved that the fully discrete nonlinear Galerkin mixed element procedure with respect to the coarse grid spaces with grid size H holds superconvergence.     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

非线性Sobolev方程Galerkin解法的后处理与超收敛

研究非线性Sobolev方程Galerkin解法的后处理与超收敛．对半离散及全离散格式,证明了当有限元空间次数,r≥2时,有限元解经过后处理,H1-模和L2-模误差估计可分别提高一阶．     

对流-扩散问题的Galerkin部分迎风有限元方法

时(其中h表示典型的网格尺寸),将会出现数值解的伪振荡.为了克服这种数值不稳定性,人们提出了多种解决途径,例如采用迎风型的差分格式.Zienkiewicz等人首先提出用Petrov-Galerkin有限元法求解对流-扩散问题.他们通过分别选择解空间和检验函数空间,克服了数值不稳定性.但这类方法由于解空间和检验函数空间的基函数比较     

A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION

We propose and analyze a $C^0$-weak Galerkin (WG) finite element method for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, the system of Navier-Stokes equations is reduced to a single fourth-order nonlinear partial differential equation and the incompressibility constraint is automatically satisfied. The proposed method uses continuous piecewise-polynomial approximations of degree $k+2$ for the stream-function $\psi$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $\nabla\psi$ on the interelement boundaries. The existence of a discrete solution is proved by means of a topological degree argument, while the uniqueness is obtained under a data smallness condition. An optimal error estimate is obtained in $L^2$-norm, $H^1$-norm and broken $H^2$-norm. Numerical tests are presented to demonstrate the theoretical results.     

Navier-Stokes方程带Backtracking技巧的两重网格算法

1 引 言考虑二维不可压 Navier-Stokes方程: