Global Existence and Blow-up of Solutions to Multi-dimensional (n ≤ 5) Viscous Cahn-Hilliard Equation

In this paper we consider the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5, and established global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial data.

Viscous Cahn-Hilliard Equation with Concentration Dependent Mobility

This paper is devoted to viscous Cahn-Hiliiard equation with concentration dependent mobility. Some results on the existence, uniqueness and large time behavior are established.

A NOTE ON LARGE TIME BEHAVIOUR OF SOLUTIONS FOR VISCOUS CAHN-HILLIARD EQUATION

This article is devoted to the discussion of large time behaviour of solutions for viscous Cahn-Hilliard equation with spatial dimension n 〈 5. Some results on global existence of weak solutions for small initial value and blow-up of solutions for any nontrivial initial value are established.

A Provable Positivity-Preserving Local Discontinuous Galerkin Method for the Viscous and Resistive MHD Equations

In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.

Fast algorithm for viscous Cahn-Hilliard equation

The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh(TT-M)finite element(FE)method to ease the problem caused by strong nonlinearities.The TT-M FE algorithm includes the following main computing steps.First,a nonlinear FE method is applied on a coarse time-meshτ_(c).Here,the FE method is used for spatial discretization and the implicit second-orderθscheme(containing both implicit Crank-Nicolson and second-order backward difference)is used for temporal discretization.Second,based on the chosen initial iterative value,a linearized FE system on time fine mesh is solved,where some useful coarse numerical solutions are found by Lagrange’s interpolation formula.The analysis for both stability and a priori error estimates is made in detail.Numerical examples are given to demonstrate the validity of the proposed algorithm.Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time.

Global Existence and L_p Decay Estimate of Solutions for Viscous Cahn-Hilliard Equation with Inertial Term

In this paper,Cauchy problem of viscous Cahn-Hilliard equation with inertial term in multi-space dimension is considered.Based on the detailed analysis of Green function,using fixed point theorem,we get the global in-time existence of classical solution.Furthermore,we get Lp decay rate of the solution.

New algorithm for solving 3D incompressible viscous equations based on projection method

A new algorithm based on the projection method with the implicit finite difference technique was established to calculate the velocity fields and pressure.The calculation region can be divided into different regions according to Reynolds number.In the far-wall region,the thermal melt flow was calculated as Newtonian flow.In the near-wall region,the thermal melt flow was calculated as non-Newtonian flow.It was proved that the new algorithm based on the projection method with the implicit technique was correct through nonparametric statistics method and experiment.The simulation results show that the new algorithm based on the projection method with the implicit technique calculates more quickly than the solution algorithm-volume of fluid method using the explicit difference method.

ALMOST PERIODIC SOLUTION OF ONE DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATION

This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

This article is concerned with the existence of maximal attractors in Hi （i = 1, 2, 4） for the compressible Navier-Stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains Ωn in Rn（n = 2,3）. One of the important features is that the metric spaces H（1）, H（2）, and H（4） we work with are three incomplete metric spaces, as can be seen from the constraints θ 〉 0 and u 〉 0, with θand u being absolute temperature and specific volume respectively. For any constants δ1, δ2……,δ8 verifying some conditions, a sequence of closed subspaces Hδ（4） H（i） （i = 1, 2, 4） is found, and the existence of maximal （universal） attractors in Hδ（i） （i = 1.2.4） is established.

Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier Expansion Basis

The differential quadrature method based on Fourier expansion basis is applied in this work to solve coupled viscous Burgers’ equation with appropriate initial and boundary conditions. In the first step for the given problem we have discretized the interval and replaced the differential equation by the Differential quadrature method based on Fourier expansion basis to obtain a system of ordinary differential equation (ODE) then we implement the numerical scheme by computer programing and perform numerical solution. Finally the validation of the present scheme is demonstrated by numerical example and compared with some existing numerical methods in literature. The method is analyzed for stability and convergence. It is found that the proposed numerical scheme produces a good result as compared to other researcher’s result and even generates a value at the nodes or mesh points that the results have not seen yet.

NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

POINTWISE ESTIMATES OF SOLUTIONS FOR THE NONLINEAR VISCOUS WAVE EQUATION IN EVEN DIMENSIONS

In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.

GLOBAL WEAK SOLUTIONS TO ONE-DIMENSIONAL COMPRESSIBLE VISCOUS HYDRODYNAMIC EQUATIONS

In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial con- ditions. The diffusion term εuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1] （α=1/2） to 0 〈 α ≤ 1. In addition, we perform the limit ε→0 with respect to 0 〈 α ≤1/2.

Least Square Finite Element Method for Viscous Splitting of Unsteady Incompressible Navier–Stokes Equations

In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton’s method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.

AN IMPLICIT ALGORITHM OF THIN LAYER EQUATIONS IN VISCOUS，TRANSONIC，TWO－PHASE NOZZLE FLOW

Omitting viscosity along flow direction, we have simplified the dimensionless N-Sequations in arbitrary curved coordinate system as the thin layer equations. Using theimplicit approximate-factorization algorithm to solve the gas-phase governing equ-ations and the characteristic method to follow the tracks of particles, we then obtainedthe full coupled numerical method of two-phase.transonic, turbulent flow. Here, par- ticle size may be grouped, the subsonic boundary condition at entry of nozzle is ireatedby quasi-characteristic method in reference plane and the algebraic model is used forturbulent flow. These methods are applied in viscous two-phase flow. calculation of ro-cket nozzle and in the prediciton of thrust and specific impulse for solid propellant ro-cket motor. The calculation results are in good agreement with the measurerment va-lues. Moreover, the influences of different particle radius, different particle mass frac-tion and particle size grouped on flow field have been discussed, and the influences of particle two-dimensional radial velosity component and viscosity on specific impulse ofrocket motor have been analysed.The method of this paper possesses the advantage of saving computer time. More important, the effect is more obvious for the calculation of particle size being grouped.

ZERO KINEMATIC VISCOSITY-MAGNETIC DIFFUSION LIMIT OF THE INCOMPRESSIBLE VISCOUS MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

We investigate the uniform regularity and zero kinematic viscosity-magnetic diffusion limit for the incompressible viscous magnetohydrodynamic equations with the Navier boundary conditions on the velocity and perfectly conducting conditions on the magnetic field in a smooth bounded domain Ω⊂R^(3).It is shown that there exists a unique strong solution to the incompressible viscous magnetohydrodynamic equations in a finite time interval which is independent of the viscosity coefficient and the magnetic diffusivity coefficient.The solution is uniformly bounded in a conormal Sobolev space and W^(1,∞)(Ω)which allows us to take the zero kinematic viscosity-magnetic diffusion limit.Moreover,we also get the rates of convergence in L^(∞)(0,T;L^(2)),L^(∞)(0,T;W^(1,p))(2≤p<∞),and L^(∞)((0,T)×Ω)for some T>0.

Existence of Solutions to a Viscous Thin Film Equation

A fourth-order degenerate parabolic equation with a viscous term: ?is studied with the initial-boundary conditions ux=wx=0?on {-1,1}×(0,T), u(x,0)=u0(x)?in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u)?and the viscosity term. The existence of nonnegative weak solution is obtained.

THE STABILITY OF THE DELTA WAVE TO PRESSURELESS EULER EQUATIONS WITH VISCOUS AND FLUX PERTURBATIONS

This paper is concerned with the pressureless Euler equations with viscous and flux perturbations.The existence of Riemann solutions to the pressureless Euler equations with viscous and flux perturbations is obtained.We show the stability of the delta wave of the pressureless Euler equations to the perturbations;that is,the limit solution of the pressureless Euler equations with viscous and flux perturbations is the delta wave solution of the pressureless Euler equations as the viscous and flux perturbation simultaneously vanish in the case u_(-)> u_(+).

An ADI Finite Volume Element Method for a Viscous Wave Equation with Variable Coefficients

Based on rectangular partition and bilinear interpolation,we construct an alternating-direction implicit(ADI)finite volume element method,which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients.This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes.Optimal error estimate in L2 norm is obtained for the schemes.Compared with the finite volume element method of the same convergence order,our method is more effective in terms of running time with the increasing of the computing scale.Numerical experiments are presented to show the efficiency of our method and numerical results are provided to support our theoretical analysis.