Boundary layers for compressible Navier-Stokes equations with outflow boundary condition

In this paper, we study the asymptotic relation between the solutions to the initial boundary value problem of the one-dimensional compressible full Navier-Stokes equations with outflow boundary condition and the associated Euler equations. We assume all the three characteristics to the corresponding Euler equations are all negative up to some small time, then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small.     

The Strong Solution of a Class of Generalized Navier-Stokes Equations

We study initial boundary value (lBV) problem for a class of generalized Navier-Stokes equations in L^q([0, T); L^p(Ω)). Our main tools are regularity of analytic semigroup by Stokes operator and space-time estimates. As an application we can obtain some classical results of the Navier-Stokes equations such as global classical solution of 2-dimensional Navier-Stokes equation etc.     

Navier-Stokes方程稳定性研究(Ⅲ)

本文给出Navier-Stokes方程某些初值问题存在C2解的必要条件,并给出其在{t=0}上的初值问题不适定的例证。Navier-Stokes方程的初值问题是研究这个方程的基础问题之一。国内外很多学者在这方面的研究曾取得了不同程度的结果。法国时J.Leray教授就曾在某种意义下证明过Navier-Stokes方程某种初边值问题解的存在性[3].本文根据J.Hadamard的偏微分方程的基础理论[1].给出某些关键问题的严格定义,叙述一个有关Navipr-Stokes方程不稳定的基本定理。最后给出若干例证,其证明可参见[4].     

Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves.It is shown that the unique solution to the Navier-Stokes equations exists for all time,and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks,as the viscosity vanishes.In contrast to previous related works,where either the composite wave is absent or the efects of initial layers are ignored,this gives the frst mathematical justifcation of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers.Our method of proof consists of a scaling argument,the construction of the approximate solution and delicate energy estimates.     

Asymptotic stability of viscous contact discontinuity to an inflow problem for compressible Navier-Stokes equations

This paper is concerned with an initial-boundary value problem for one-dimensional full compressible Navier-Stokes equations with inflow boundary conditions in the half space R₊=(0,+∞). The asymptotic stability of viscous contact discontinuity is established under the conditions that the initial perturbations and the strength of contact discontinuity are suitably small. Compared with the free-boundary and the initial value problems, the inflow problem is more complicated due to the additional boundary effects and the different structure of viscous contact discontinuity. The proofs are given by the elementary energy method.     

Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space

In the present paper, we investigate the large-time behavior of the solution to an initial-boundary value problem for the isentropic compressible Navier-Stokes equations in the Eulerian coordinate in the half space. This is one of the series of papers by the authors on the stability of nonlinear waves for the outflow problem of the compressible Navier-Stokes equations. Some suitable assumptions are made to guarantee that the time-asymptotic state is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Employing the L²-energy method and making use of the techniques from the paper [S. Kawashima, Y. Nikkuni, Stability of rarefaction waves for the discrete Boltzmann equations, Adv. Math. Sci. Appl. 12 (1) (2002) 327-353], we prove that this nonlinear wave is nonlinearly stable under a small perturbation. The complexity of nonlinear wave leads to many complicated terms in the course of establishing the a priori estimates, however those terms are of two basic types, and the terms of each type are “good” and can be evaluated suitably by using the decay (in both time and space variables) estimates of each component of nonlinear wave.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

Free Boundary Value Problem for the Cylindrically Symmetric Compressible Navier-Stokes Equations with a Constant Exterior Pressure

This paper is concerned with the free boundary value problem (FBVP) for the cylindrically symmetric barotropic 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, the unique cylindrically symmetric strong solution is shown to exist globally in time and tend to a non-vacuum equilibrium state exponentially as time tends to infinity.     

The Incompressible Limits of Compressible Navier-Stokes Equations in the Whole Space with General Initial Data

It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz＇s estimate of linear wave equation.     

一类修正的Navier-Stokes方程的长时间性态

该文主要讨论,Rⁿ上一类修正的 Navier-Stokes 方程弱解的长时间性态, 通过进一步改进Fourier分解方法, 得到了当初速度u_0∈ L² ∩L¹时其弱解在L² 范数下的最优衰减率为 (1+t)^n/4 同时该文也给出了修正的Navier-Stokes 方程与经典Navier-Stokes 方程的误差估计.     

Zero dissipation limit of the 1D linearized Navier-Stokes equations for a compressible fluid

In this paper we study the asymptotic limiting behavior of the solutions to the initial boundary value problem for linearized one-dimensional compressible Navier-Stokes equations. We consider the characteristic boundary conditions, that is we assume that an eigenvalue of the associated inviscid Euler system vanishes uniformly on the boundary. The aim of this paper is to understand the evolution of the boundary layer, to construct the asymptotic ansatz which is uniformly valid up to the boundary, and to obtain rigorously the uniform convergence to the solution of the Euler equations without the weakness assumption on the boundary layer.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations

For stochastic Navier-Stokes equations in a 3-dimensional bounded domain we first show that if the initial value is sufficiently regular, then martingale solutions are strong on a random time interval and we estimate its length. Then we prove the uniqueness of the strong solution in the class of all martingale solutions. Received November 15, 1995     

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.     

FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we investigate the free boundary value problem(FBVP) for the cylindrically symmetric isentropic compressible Navier-Stokes equations(CNS) with densitydependent 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.     

Stochastic Navier-stokes equations with multiplicative noise

An abstract stochastic Navier-Stokes equation with multiplicative white noise is considered. 2-dimensional Navier-Stokes equations with noise depending on first order derivatives of the solution are covered by the abstract model. Existence and uniqueness of a solution is proved for small initial data, and the associated local stochastic flow is constructed     

Convergence of compressible Navier-Stokes-Maxwell equations to incompressible Navier-Stokes equations

The combined quasi-neutral and non-relativistic limit of compressible Navier-Stokes-Maxwell equations for plasmas is studied. For well-prepared initial data, it is shown that the smooth solution of compressible Navier-Stokes-Maxwell equations converges to the smooth solution of incompressible Navier-Stokes equations by introducing new modulated energy functional.     

Equations stochastiques du type Navier-Stokes

We consider the problem of existence of solutions for stochastic Navier-Stokes equations. The random elements are the initial value of the solution and forcing terms which may be white noise in time.The method uses results on multi-valued functions. We also give energy type results and present examples.