In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique ...In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.展开更多
This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local ...This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.展开更多
In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. ...In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.展开更多
In this paper, we investigate the free boundary value problem （FBVP） for the cylindrically symmetric isentropic compressible Navier-Stokes equations （CNS） with density- dependent viscosity coefficients in the case...In this paper, we investigate the free boundary value problem （FBVP） for the cylindrically symmetric isentropic compressible Navier-Stokes equations （CNS） with density- dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.展开更多
This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P....This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Navier boundary value problem has at least a weak nontrivial solution for all λ>0?by using mountain pass theorem.展开更多
In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream fun...In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.展开更多
This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth so...This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.展开更多
In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that unde...In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that under given proper preconditioners,Uzawa algorithm is convergent for the stablization system.Bounds for the iteration error are provided.We show numerically that Uzawa algorithm is convergent as well for the stabilization systems when it is used in the steady-state Navier-Stokes problem(cf.[6]).展开更多
文摘In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.
基金partially supported by the NSFC(10871134)the AHRDIHL Project of Beijing Municipality (PHR201006107)
文摘This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.
基金supported by the National Natural Science Foundation of China(11331005,11471134)the Program for Changjiang Scholars and Innovative Research Team in University(IRT13066)the Scientific Research Funds of Huaqiao University(15BS201,15BS309)
文摘In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.
基金supported by NNSFC(11101145),supported by NNSFC(11326140 and11501323)China Postdoctoral Science Foundation(2012M520360)+1 种基金Doctoral Foundation of North China University of Water Sources and Electric Power(201032),Innovation Scientists and Technicians Troop Construction Projects of Henan Provincethe Doctoral Starting up Foundation of Quzhou University(BSYJ201314 and XNZQN201313)
文摘In this paper, we investigate the free boundary value problem （FBVP） for the cylindrically symmetric isentropic compressible Navier-Stokes equations （CNS） with density- dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.
文摘This paper is devoted to the following high order elliptic problems under the Navier boundary condition: ?Without assuming the standard subcritical polynomial growth condition ensuring the compactness of a bounded (P.S.) sequence, we show that the Navier boundary value problem has at least a weak nontrivial solution for all λ>0?by using mountain pass theorem.
基金supported by the National Natural Science Foundation of China (No.10871131)the Science and Technology Commission of Shanghai Municipality (No.075105118)+1 种基金the Shanghai Leading Academic Discipline Project (No.S30405)the Fund for E-institutes of Shanghai Universities(No.E03004)
文摘In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.
文摘This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.
文摘In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that under given proper preconditioners,Uzawa algorithm is convergent for the stablization system.Bounds for the iteration error are provided.We show numerically that Uzawa algorithm is convergent as well for the stabilization systems when it is used in the steady-state Navier-Stokes problem(cf.[6]).
