黏性系数依赖密度型可压缩Navier-Stokes 方程的零耗散极限问题

本文考虑黏性系数依赖密度的可压缩Navier-Stokes 方程解的零耗散极限问题. 假定Euler 方程的稀疏波解一端被真空状态连接, 我们证明Navier-Stokes 方程存在一列(依赖黏性的) 整体解, 且随着粘性的消失, 此整体解逐渐稳定于Euler 方程对应的稀疏波解和真空状态; 并且得到了一致衰减率估计. 此结果推广了常黏性系数的情形.     

Zero dissipation limit to rarefaction waves for the one-dimensional navier-stokes equations of compressible isentropic gases

We study the zero dissipation limit problem for the one-dimensional Navier-Stokes equations of compressible, isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions. We prove that the solutions of the Navier-Stokes equations with centered rarefaction wave data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. In the case that either the effects of initial layers are ignored or the rarefaction waves are smooth, we then obtain a rate of convergence which is valid uniformly for all time. Our method of proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure. © 1993 John Wiley & Sons, Inc.     

ZERO DISSIPATION LIMIT TO A RIEMANN SOLUTION FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.     

Zero Dissipation Limit to Rarefaction Waves for the 1-D Compressible Navier-Stokes Equations

The zero dissipation limit for the one-dimensional Navier-Stokes equations of compressible,isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions is investi...     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.     

微极热弹性无限板的轴对称自由振动

研究了在应力自由和刚性固定边界条件下,无能量耗散的均匀、各向同性微极热弹性无限板的轴对称自由振动波的传播,导出了相应的对称和斜对称模态波传播的闭合式特征方程和不同区域的特征方程.对短波的情况,应力自由热绝缘和等温板中对称和斜对称模态波传播的特征方程退化为Rayleigh表面波频率方程.根据导出的特征方程得到了热弹性、微极弹性和弹性板的结果.在对称和斜对称运动中计算了板的位移分量幅值、微转动幅值和温度分布,给出了对称和斜对称模式的频散曲线,并示出了位移分量和微转动幅值和温度分布的曲线.能够发现理论分析和数值结论是非常一致的.     

Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum states. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.     

The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one-dimensional isentropic drift-flux model of compressible two-phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.     

ZERO DISSIPATION LIMIT OF THE COMPRESSIBLE HEAT-CONDUCTING NAVIER-STOKES EQUATIONS IN HE PRESENCE OF THE SHOCK

The zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock is investigated. It is shown that when the heat ε→ 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier-Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid,assuming that it is in the thermodynamically sense perfect and polytropic.The fluid is between a static solid wall and a free boundary connected to a vacuum state.We take the homogeneous boundary conditions for velocity,microrotation and heat flux on the solid border and that the normal stress,heat flux and microrotation are equal to zero on the free boundary.The proof of the global existence of the solution is based on a limit procedure.We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.     

Stability of the planar rarefaction wave to two-dimensional Navier-Stokes-Korteweg equations of compressible fluids

This study is concerned with the large time behavior of the two-dimensional compressible Navier-Stokes-Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one-dimensional compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the two-dimensional compressible Navier-Stokes-Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi-dimensional viscous fluids with internal capillarity.     

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.     

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.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

微极各向同性弹性板中的Rayleigh-Lamb波

研究了无应力作用条件下，均匀、各向同性、圆柱形微极结构弹性板中波的传播．导出了对称和斜对称模式下波传播的特征方程．对短波这一极端情况，无应力圆板中对称和斜对称模态波的特征方程退化为Pmyle曲表面波频率方程．并得到薄板的计算结果．给出了位移和微转动分量，并绘制了相应图形．给出了若干特殊情况的研究结果及对称和斜对称模态特征方程的图示．     

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non‐isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial‐boundary value problem of the non‐isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero‐order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.     

Interaction of rarefaction waves in jet stream

In this paper we present a mathematical analysis of a supersonic jet stream out of an orifice into the atmosphere. The analysis involves the interaction of steady rarefaction waves, and the interaction of a rarefaction wave by the interface of the jet stream. The existence of the classical solution in the region of interactions of rarefaction waves is established. For small pressure difference the existence of the classical solution in the region of reflection is also obtained. Finally, for large pressure difference vacuum may be produced by strong expanding, and the corresponding wave structure with vacuum is also analyzed.