THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THERMOMECHANICAL CUCKER-SMALE-STOKES EQUATIONS IN THE WHOLE DOMAIN

We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.

REMARK ON UNIQUE CONTINUATION OF SOLUTIONS TO THE STOKES AND THE NAVIER-STOKES EQUATIONS

New simple proofs of unique continuation of solutions for the Stokes equation and Navier-Stokes equations is presented under weaker conditions.

Global Classical Solutions for One Dimensional Navier Stokes Equations

In this article the author considers Cauchy problem for one dimensional Navier Stokes equations and the global smooth resolvablity for classical solutions is obtained.

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.

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.

Some Similarity Reduction Solutions to Two-Dimensional Incompressible Navier-Stokes Equation

We investigate the symmetry reduction for the two-dimensional incompressible Navier-Stokes equationin conventional stream function form through Lie symmetry method and construct some similarity reduction solutions.Two special cases in [D.K.Ludlow,P.A.Clarkson,and A.P.Bassom,Stud.Appl.Math.103 (1999) 183] and a theoremin [S.Y.Lou,M.Jia,X.Y.Tang,and F.Huang,Phys.Rev.E 75 (2007) 056318] are retrieved.

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.

Spatial Decay Estimates for the Solutions to Stokes Equations in Four Kinds of Semi-Infinite Cylinders

This paper investigates the spatial behavior of the solutions of the Stokes equations in a semi-infinite cylinder.We consider four kinds of semi-infinite cylinders with boundary conditions of Dirichlet type.For each type of cylinder we obtain the spatial decay estimates for the solutions.To make the attenuation meaningful,we derive the explicit bound for the total energy in terms of the initial boundary data.

Quadratic nonconforming finite element method for 3D Stokes equations on cuboid meshes

In this paper, the quadratic nonconforming brick element （MSLK element） intro- duced in [10] is used for the 3D Stokes equations. The instability for the mixed element pair MSLK-P1 is analyzed, where the vector-valued MSLK element approximates the velocity and the piecewise P1 element approximates the pressure. As a cure, we adopt the piecewise P1 macroelement to discretize the pressure instead of the standard piecewise P1 element on cuboid meshes. This new pair is stable and the optimal error estimate is achieved. Numerical examples verify our theoretical analysis.

Parallel finite element algorithm based on full domain partition for stationary Stokes equations

Based on the full domain partition, a parallel finite element algorithm for the stationary Stokes equations is proposed and analyzed. In this algorithm, each subproblem is defined in the entire domain. Majority of the degrees of freedom are associated with the relevant subdomain. Therefore, it can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. This allows the algorithm to be implemented easily with low communication costs. Numerical results are given showing the high efficiency of the parallel algorithm.

An Uzawa-type algorithm for the coupled Stokes equations

An Uzawa-type algorithm is designed for the coupled Stokes equations discretized by the mixed finite element method.The velocity solved by the presented algorithm is weakly divergence-free,which is different from the one solved by the common Uzawa method.Besides,an optimal relaxation parameter of the presented algorithm is provided.

A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).

Application of Splitting Extrapolation to Stokes Equation

This paper deals with the splitting extrapolation for mixed finite element used in the approximation of the steady Stokes equation. Applying the multi variate asymptotic expansion of the error on independent grid parameters, we can get a parallel algorithm and a global fine grid approximations with high accuracy.

THE FUNDAMENTAL SOLUTION METHOD FOR INCOMPRESSIBLE NAVIER STOKES EQUATIONS

A complete boundary integral formulation for incompressible Navier Stokes equations with time discretization by operator splitting is developed by using the fundamental solutions of the Helmhotz operator equation with different orders. The numerical results for the lift and the drag hysteresis associated with a NACA0012 aerofoil oscillating in pitch are good in comparison with available experimental data.

BOUNDARY LAYER ESTIMATION OF THE QUASILINEAR STOKES EQUATIONS

We first prove the existence and uniqueness of solution of quasilinear Stokes equations. Then it is shown when the viscosity vanishes, the solution of the quasilinear Stokes equations tends to the solution of the degenerate equations, in which the viscous term is omitted from the quasilinear Stokes equations and the boundary condition is weakened. In the end, we obtain the boundary layer estimation, Our result shows that the thickness of the boundary layer is proportional to epsilon(1/4).

A Note on Similarity Solutions of Navier-Stokes Equations

This note looks at the two similarity solutions of the Navier-Stokes equations in polar coordinates. In the second solution an initial value problem is reduced into generalized stationary KDV and hence integrable.

Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation

The proper orthogonal decomposition （POD） is a model reduction technique for the simulation Of physical processes governed by partial differential equations （e.g., fluid flows）. It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference （FD） scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

We propose a p-multilevel preconditioner for hybrid high-order(HHO)discretizations of the Stokes equation,numerically assess its performance on two variants of the method,and compare with a classical discontinuous Galerkin scheme.An efficient implementa-tion is proposed where coarse level operators are inherited using L2-orthogonal projec-tions defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free.Both h-and k-dependency are investigated tackling two-and three-dimensional problems on standard meshes and graded meshes.For the two HHO for-mulations,featuring discontinuous or hybrid pressure,we study how the combination of p-coarsening and static condensation influences the V-cycle iteration.In particular,two dif-ferent static condensation procedures are considered for the discontinuous pressure HHO variant,resulting in global linear systems with a different number of unknowns and matrix non-zero entries.Interestingly,we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

A Two-Grid Technique for the Penalty Method of the Steady Navier-Stokes Equations

A two grid technique for solving the steady incompressible Navier Stokes equations in a penalty method was presented and the convergence of numerical solutions was analyzed. If a coarse size H and a fine size h satisfy H=O(h 13-s )(s=0(n=2);s=12(n=3), where n is a space dimension), this method has the same convergence accuracy as the usual finite element method. But the two grid method can save a lot of computation time for its brief calculation. Moreover, a numerical test was couducted in order to verify the correctness of above theoretical analysis.

Approximate Analytical Solutions to the Heat and Stokes Equations on the Half-Line Obtained by Fokas’ Transform

In this paper, we evaluate the integrals that are solutions of the heat and Stokes’ equations obtained by Fokas’ transform method by deriving exact formulas. Our method is more accurate and efficient than the contour deformation and parametrization used by Fokas to compute these integrals. In fact, for the heat equation, our solution is exact up to the imaginary error function and for the Stokes equation, our solution is exact up to the incomplete Airy function. In addition, our solutions extend to the lateral boundary without convergence issues, allow for asymptotic expansions, and are much faster than those obtained by other methods.