GLOBAL SOLUTIONS TO THE NAVIER-STOKES EQUATIONS FOR COMPRESSIBLE HEAT-CONDUCTING FLOW WITH SYMMETRY AND FREE BOUNDARY

ABSTRACT

Global solutions of the multidimensional Navier-Stokes equations for compressible heat-conducting flow are constructed, with spherically symmetric initial data of large oscillation between a static solid core and a free boundary connected to a surrounding vacuum state. The free boundary connects the compressible heat-conducting fluids to the vacuum state with free normal stress and zero normal heat flux. The fluids are initially assumed to fill with a finite volume and zero density at the free boundary, and with bounded positive density and temperature between the solid core and the initial position of the free boundary. One of the main features of this problem is the singularity of solutions near the free boundary. Our approach is to combine an effective difference scheme to construct approximate solutions with the energy methods and the pointwise estimate techniques to deal with the singularity of solutions near the free boundary and to obtain the bounded estimates of the solutions and the free boundary as time evolves. The convergence of the difference scheme is established. It is also proved that no vacuum develops between the solid core and the free boundary, and the free boundary expands with finite speed.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated.The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wav...     

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.     

LOCAL EXISTENCE OF SOLUTION TO FREE BOUNDARY VALUE PROBLEM FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS

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.     

GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY

In this paper,we consider the 3 D compressible isentropic Navier-Stokes equations when the shear viscosity μ is a positive constant and the bulk viscosity is λ（ρ）=ρ^β with β> 2,which is a situation that was first introduced by Vaigant and Kazhikhov in [1].The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain ■ is obtained.Here the initial density keeps a non-vacuum state ■> 0 when z→±∞.Our results also show that the solution will not ...     

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.     

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with densitydependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel′and the continuation argument.     

GLOBAL EXISTENCE OF STRONG SOLUTIONS OF NAVIER-STOKES EQUATIONS WITH NON-NEWTONIAN POTENTIAL FOR ONE-DIMENSIONAL ISENTROPIC COMPRESSIBLE FLUIDS

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.     

INTERFACE BEHAVIOR OF COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DISCONTINUOUS BOUNDARY CONDITIONS AND VACUM

In this paper,we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum,across which the density changes discontinuosly.Smoothness of the solutions and the uniqueness of the weak solutions are also discussed.The present paper extends results in Luo-Xin-Yang[12] to the jump boundary conditions case.     

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.     

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form of η(ρ)=ρ~α.The existence of unique global H^2m-solutions(m ∈ N) to the free boundary problem is proven for when 0 <α<1/4.Furthermore,we obtain the global C~∞-solutions if the initial data is smooth.     

直接边界积分方法计算水波的稳定性

1.引 言 数值模拟流体自由界面运动一直是研究水波的主要方法.海浪攀爬海岸的研究是水动力学中的一个很经典且很具有挑战性的课题,因为在水面附近的方程是高度非线性的.对于二维情形下有一致倾斜度的海岸上的水波,Carrier＆Greeenspan[7]建立了基于浅水模型的非线性理论,Tuck＆Hwang引入变量代换,把最初的非线性方程转变成更容易分析的线性方程.这种直接对单一流体用浅水方程计算自由界面的办法仍然被广泛应用.Zhang,Wu,Hou[23]给出了这个问题的Euler-Langrange混合格式,Li ＆ Zhang[13]借助这个格式,并引入人工边界条件对海浪攀爬海岸问题进行了整体的数值模拟.     

GLOBAL NONEXISTENCE FOR A VISCOELASTIC WAVE EQUATION WITH ACOUSTIC BOUNDARY CONDITIONS

This paper deals with a class of nonlinear viscoelastic wave equation with damping and source terms ■ with acoustic boundary conditions. Under some appropriate assumption on relaxation function g and the initial data, we prove that the solution blows up in finite time if the positive initial energy satisfies a suitable condition.     

NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY

In this paper,we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension.If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only,it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves,while the initial per...     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated.The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions.To do this,we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave.By using them together with an inequality concerning the heat kernel in the half space,we obtain the desired a priori estimates.The proof is based on the elementary energy method by the anti-derivative argument.     

GLOBAL SOLVABILITY OF NONLINEAR WAVE EQUATION WITH A VISCOELASTIC BOUNDARY CONDITION

The paper deals with the global solvability of nonlinear wave equation with a viscoelastic boundary condition. The problem is a mathematical mode for nonlinare one-dimensional motion of an elastic bar connected with a viscoelastic element at one end of the bar. Under some physically reasonable assumptions, the boundary condition is dissipative and the existence of global smooth solution of the problem is proved for small data.     

声波方程吸收边界条件的稳定性分析

引言对于无界区域中波动现象的数值模拟，必需引进人工边界将计算限制在一个有界区域上．为了确定解，需要在人工边界上加适当的边界条件．对于声波和弹性波方程，这样的一组人工边界条件，也叫吸收边界条件，在［1，2］中被系统地构造出来．对于声波方程，这些吸收边界条件恰好是单程波方程的近似．如山中所指出，减少边界反射，便于在计算中应用和稳定性是构造吸收边界条件的三点关键．Ellgqllist和Maid。用模态分析方法15］证明，带有［IJ中构造的吸收边界条件的波动方程初边值问题是适定的，并且估计了人工边界所产生的误差．对于更广…     

CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.     

ZERO DISSIPATION LIMIT TO CONTACT DISCONTINUITY FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

The zero dissipation limit to the contact discontinuities for one-dimensional compressible Navier-Stokes equations was recently proved for ideal polytropic gas(see Huang et al. [15, 22] and Ma [31]), but there is few result for general gases including ideal polytropic gas. We prove that if the solution to the corresponding Euler system of general gas satisfying(1.4) is piecewise constant with a contact discontinuity, then there exist smooth solutions to Navier-Stokes equations which converge to the inviscid solutions at a rate of κ14 as the heat-conductivity coefficient κ tends to zero. The key is to construct a viscous contact wave of general gas suitable to our proof(see Section 2). Notice that we have no need to restrict the strength of the contact discontinuity to be small.     

GLOBAL SOLUTIONS TO THE 2D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SOME LARGE INITIAL DATA

We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in R². By exploiting the global-in-time estimate to the two-dimensional(2D for short) classical incompressible Navier-Stokes equations and using techniques developed in(SIAM J Math Anal, 2020, 52(2): 1806–1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity...