This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.T...This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.The main ingredient are a novel and sharp L^(2) error estimate of discrete eigenfunctions,and a new error analysis of nonconforming finite element methods.展开更多
In this paper,we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator.We check and prove this condition for four nonc...In this paper,we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator.We check and prove this condition for four nonconforming methods and one conforming method.Hence they produce eigenvalues which are smaller than their exact counterparts.展开更多
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditi...We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.展开更多
The Stokes operator is a differential-integral operator induced by the Stokes equations. In this paper, we analyze the Stokes operator from the point of view of the Helmholtz minimum dissipation principle. We show tha...The Stokes operator is a differential-integral operator induced by the Stokes equations. In this paper, we analyze the Stokes operator from the point of view of the Helmholtz minimum dissipation principle. We show that, through the Hodge orthogonal decomposition, a pair of bounded linear operators, a restriction operator and an extension operator, are induced from the divergence-free constraint. As a consequence, we use it to calculate the eigenvalues of the Stokes operator.展开更多
In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthog...In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.展开更多
Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method.First,stream function form of the Navier-Stokes equations which homogeneous boundary conditio...Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method.First,stream function form of the Navier-Stokes equations which homogeneous boundary condition was given by introducing Couette flow.Second,the analytical expressions of the eigenfunction of the Stokes operator in the cylindrical gap region were given and its orthogonality was proved.The estimates of growth rate of the eigenvalue were presented.Finally,spectral Galerkin approximation of Couette-Taylor flow was discussed by introducing eigenfunctions of Stokes operator as basis of finite dimensional approximate subspaces.The existence,uniquence and convergence of spectral Galerkin approximation of nonsingular solution for the steady-state Navier-Stokes equations are proved.Moreover,the error estimates are given.Numerical result is presented.展开更多
In this paper,we prove that the generator of any bounded analytic semigroup in(θ,1)-type real interpolation of its domain and underlying Banach space has maximal L^(1)-regularity,using a duality argument combined wit...In this paper,we prove that the generator of any bounded analytic semigroup in(θ,1)-type real interpolation of its domain and underlying Banach space has maximal L^(1)-regularity,using a duality argument combined with the result of maximal continuous regularity.As an application,we consider maximal L^(1)-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous B_(q),^(s),1-type Besov spaces on domains of R^(n),n≥2.展开更多
Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Ar...Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability.It is shown that the Stokes operator can be inversed within an acceptable computational effort.This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix.It is shown,additionally,that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers,as well as for other problems where convergence of iterative methods slows down.Implementation of the Stokes operator inverse to time-steppingbased formulation of the Newton and Arnoldi iterations is discussed.展开更多
In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[J...In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[Jishan FAN,Fucai LI,G.NAKAMURA,Zhong TAN,Regularity criteria for the three-dimensional magnetohydrodynamic equations.J.Differential Equations,2014,256（8）：2858 2875]in some sense.The method is to establish a new bilinear estimate.展开更多
基金The author would like to thank Prof.Shangyou Zhang for helping the numerical experiments.The author was supported by the NSFC under Grants Nos.11571023 and 11401015.
文摘This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated Q_(1) and Crouzeix-Raviart elements of the Stokes eigenvalue problem.The main ingredient are a novel and sharp L^(2) error estimate of discrete eigenfunctions,and a new error analysis of nonconforming finite element methods.
基金supported by the NSFC Project 11271036the second author was supported in part by the NSFC Key Project 11031006Hunan Provincial NSF Project 10JJ7001.
文摘In this paper,we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes operator.We check and prove this condition for four nonconforming methods and one conforming method.Hence they produce eigenvalues which are smaller than their exact counterparts.
基金supported in part by the National Science Foundation under Grants DMS-0807551, DMS-0720925, and DMS-0505473the Natural Science Foundationof China (10728101)supported in part by EPSRC grant EP/F029578/1
文摘We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.
基金supported by the National Natural Science Foundation of China (No.10772103)the Shanghai Leading Academic Discipline Project (No.Y0103)
文摘The Stokes operator is a differential-integral operator induced by the Stokes equations. In this paper, we analyze the Stokes operator from the point of view of the Helmholtz minimum dissipation principle. We show that, through the Hodge orthogonal decomposition, a pair of bounded linear operators, a restriction operator and an extension operator, are induced from the divergence-free constraint. As a consequence, we use it to calculate the eigenvalues of the Stokes operator.
文摘In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.
文摘Axisymmetric Couette-Taylor flow between two concentric rotating cylinders was simulated numerically by the spectral method.First,stream function form of the Navier-Stokes equations which homogeneous boundary condition was given by introducing Couette flow.Second,the analytical expressions of the eigenfunction of the Stokes operator in the cylindrical gap region were given and its orthogonality was proved.The estimates of growth rate of the eigenvalue were presented.Finally,spectral Galerkin approximation of Couette-Taylor flow was discussed by introducing eigenfunctions of Stokes operator as basis of finite dimensional approximate subspaces.The existence,uniquence and convergence of spectral Galerkin approximation of nonsingular solution for the steady-state Navier-Stokes equations are proved.Moreover,the error estimates are given.Numerical result is presented.
文摘In this paper,we prove that the generator of any bounded analytic semigroup in(θ,1)-type real interpolation of its domain and underlying Banach space has maximal L^(1)-regularity,using a duality argument combined with the result of maximal continuous regularity.As an application,we consider maximal L^(1)-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous B_(q),^(s),1-type Besov spaces on domains of R^(n),n≥2.
文摘Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady three-dimensional flows and study of their stability.It is shown that the Stokes operator can be inversed within an acceptable computational effort.This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix.It is shown,additionally,that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers,as well as for other problems where convergence of iterative methods slows down.Implementation of the Stokes operator inverse to time-steppingbased formulation of the Newton and Arnoldi iterations is discussed.
基金Supported by the Natural Science Foundation of Jiangxi Province(Grant No.20151BAB201010)the National Natural Science Foundation of China(Grant Nos.1150112511361004)
文摘In this paper,we consider the density-dependent magnetohydrodynamic equations with vacuum,and provide a regularity criterion involving the velocity and magnetic fields in Besov space of negative order,which improves[Jishan FAN,Fucai LI,G.NAKAMURA,Zhong TAN,Regularity criteria for the three-dimensional magnetohydrodynamic equations.J.Differential Equations,2014,256（8）：2858 2875]in some sense.The method is to establish a new bilinear estimate.
