Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Behaviour at time t = 0 of the solutions of semi-linear evolution equations

We investigate the regularity at time t = 0 of the solutions of linear and semi-linear evolutions equations (including the Stokes and Navier-Stokes equations), Necessary and sufficient conditions on the data for an arbitrary order of regularity are given (the classical “compatibility conditions”). In the case of the Stokes and Navier-Stokes equations the compatibility conditions which we find are global conditions on the data. The presentation given here seems to improve and generalize the known results even in the simplest case of linear evolution equations.     

Contrôlabilité exacte des approximations de Galerkin des équations de Navier-Stokes

We consider Navier-Stokes equations in a smooth domain of ℝⁿ, n = 2, 3, with Dirichlet boundary conditions. We introduce the classical Galerkin approximation and study its exact controllability when the control acts on an open non-empty subset of the domain. Under suitable assumptions on the elements of the Galerkin basis, we prove that this finite-dimensional system is exactly controllable. The proof combines HUM together with a classical fixed point argument, and relies on the cancellation properties of the non-linearity appearing in the Navier-Stokes equations.     

On the blow up criterion for the 2-D compressible Navier-Stokes equations

Motivated by [10], we prove that the upper bound of the density function j9 controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.     

Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

     

Global well-posedness of the stochastic 2D Boussinesq equations with partial viscosity

This paper deals with the stochastic 2D Boussinesq equations with partial viscosity. This is a coupled system of Navier-Stokes/Euler equations and the transport equation for temperature under additive noise. Global well-posedness result of this system under partial viscosity is proved by using classical energy estimates method.     

On dispersive effect of the Coriolis force for the stationary Navier-Stokes equations

The dispersive effect of the Coriolis force for the stationary and non-stationary Navier-Stokes equations is investigated. Existence of a unique solution is shown for arbitrary large external force provided the Coriolis force is large enough. In addition to the stationary case, counterparts of several classical results for the non-stationary Navier-Stokes problem have been proven. The analysis is carried out in a new framework of the Fourier-Besov spaces.     

Navier-Stokes方程的精确解——Dirac-Pauli表象的复变函数理论及其在流体力学中的应用(Ⅱ)

本文是文[1]的继续。在文[1]中我们应用Dirac-Pauli表象的复变函数理论并引入Kaluza“鬼”坐标,将不可压缩粘流动力学的Navier-Stokes方程化成只有一对复未知函数的非线性方程。在本文中,我们将除时间之外的复自变量进行重新组合,从而成对地减少了复自变量的数目。最后,我们将Navier-Stokes方程化成经典的Burgers方程。联结Burgers方程与扩散方程的Cole-Hopf变换实际上是B?cklund变换,而扩散方程众所周知是具有通解的。于是,我们利用B?cklund变换求得了Navier-Stokes方程的精确解。     

A new defect correction method for the Navier-Stokes equations at high Reynolds numbers

In this paper, a new defect correction method for the Navier-Stokes equations is presented. With solving an artificial viscosity stabilized nonlinear problem in the defect step, and correcting the residual by linearized equations in the correction step for a few steps, this combination is particularly efficient for the Navier-Stokes equations at high Reynolds numbers. In both the defect and correction steps, we use the Oseen iterative scheme to solve the discrete nonlinear equations. Furthermore, the stability and convergence of this new method are deduced, which are better than that of the classical ones. Finally, some numerical experiments are performed to verify the theoretical predictions and show the efficiency of the new combination.     

Existence and uniqueness of global classical solutions to 3D isentropic compressible Navier-Stokes equations with general initial data

In this paper we study the global existence and uniqueness of classical solutions to the Cauchy problem for 3D isentropic compressible Navier-Stokes equations with general initial data which could contain vacuum.We give the relation between the viscosity coefficients and the initial energy,which implies that the Cauchy problem under consideration has a global classical solution.     

流面上的Navier-Stokes方程及其维数分裂方法

该文首先提出了流面和流层的概念,然后推导出了半测地坐标系下流层内的三维NS (Navier-Stokes)方程,以及流面上的二维NS方程.通过引入流面上的流函数,得到了流函数方程的非线性初边值问题,并讨论了方程解的存在性和唯一性.基于以上讨论,提出了求解三维NS方程的维数分裂方法, 并给出了算例.     

Weak Dissipative Structure for Compressible Navier-Stokes Equations

This paper concerns the Cauchy problem for compressible Navier-Stokes equations.The weak dissipative structure is explored and a new proof for the classical solutions are shown to exist globally in time if the initial data is sufficiently small.     

The penalty method for equations of viscoelastic media

In the present paper, we study the global classical solvability of the first initial-boundary value problem for some three-dimensional equations and the convergence of solutions of the equations to the classical solutions of the first initial-boundary value problem for the Navier-Stokes equations as ε→0. Bibliography:35 titles. Dedicated to the memory of V. N. Popov Published inZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 267–278. Translated by A. P. Oskolkov.     

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (?1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are local-in-space regularity estimates near the initial time, which are of independent interest.     

Modified Navier-Stokes equations for finite-difference computation of viscous flows

The classical Navier-Stokes model is modified so as to increase its accuracy. A system of third-order differential equations is proposed describing viscous flow of liquids and gases. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 189–198.     

Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation

This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We are able to single out three special partial dissipation cases and establish the global regularity for each case. As special consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations, and the 2D micropolar equations with several types of partial dissipation always possess global classical solutions. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

Small-time existence for the Navier-Stokes equations with a free surface

We prove the small-time existence of a solution of the Navier-Stokes equations, for any initial data, for a free boundary fluid with surface tension taken into account. A fixed point method is used. The linearized problem is hyperbolic and dissipative. The classical methods to solve it seem to fail and the method used here could perhaps be applied for equations of the same kind.     

Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of R3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.     

Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density

We present a sufficient condition on the blowup of smooth solutions to the compressible Navier-Stokes equations in arbitrary space dimensions with initial density of compact support. As an immediate application, it is shown that any smooth solutions to the compressible Navier-Stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support, and an upper bound, which depends only on the initial data, on the blowup time follows from our elementary analysis immediately. Another implication is that there is no global small (decay in time) or even bounded (in the case that all the viscosity coefficients are positive) smooth solutions to the compressible Navier-Stokes equations for polytropic fluids, no matter how small the initial data are, as long as the initial density is of compact support. This is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum. The blowup of smooth solutions to the compressible Euler system with initial density and velocity of compact support is a simple consequence of our argument. © 1998 John Wiley & Sons, Inc.     

The Compressible boundary layer and the Boltzmann equation

We consider the flow of ideal gas in half space described by the system of compressible Navier-Stokes equations. We apply the Prandtl scaling and we obtain the system of compressible Prandtl equations. In this article, a modification of the classical Chapman-Enskog method is proposed, which allows us to derive the system of compressible Prandtl equations directly from the Boltzmann equation without the use of the Knudsen-layer correction. Different types of boundary conditions are discussed.