Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum

The Navier-Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper. When the viscosity coefficient u is proportional to pθ with 0 〈 θ 〈 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang, T., Zhu, C. J.： Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192（2）, 475-501 （2003）]. Here p is the density. Moreover, a stabilization rate estimate for the density as t → ＋∞ for any θ 〉 0 is also given.     

粘性依赖于密度的可压缩Navier-Stokes方程

The global existence of solutions to the equations of one-dimensional compressible flow with density-dependent viscosity is proved. Specifically,the assumptions on initial data are that the modulo constant is stated at x=∞ ＋and x=-∞ ,which may be different ,the density and velocity are in L^z ,and the density is bounded above and below away from zero. The results also show that even under these conditions, neither vacuum states nor concentration states can be formed in finite time.     

Remark on compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial data

In this paper, we study the free boundary problem for 1D compressible Navier-Stokes equations with density-dependent viscosity. We focus on the case where the viscosity coefficient vanishes on vacuum. We prove the global existence and uniqueness for discontinuous solutions to the Navier-Stokes equations when the initial density is a bounded variation function, and give a decay result for the density as t→+∞.     

Local well-posedness to the cauchy problem of the 3-D compressible navier-stokes equations with density-dependent viscosity

In this article, we prove the local existence and uniqueness of the classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with large initial data and vacuum, if the shear viscosity μ is a positive constant and the bulk viscosity λ(ρ) = ρ^β with β ≥ 0. Note that the initial data can be arbitrarily large to contain vacuum states.     

Compressible navier-stokes equations with density-dependent viscosity, vacuum and gravitational force in the case of general pressure

This is a continuation of the article（Comm.Partial Differential Equations 26（2001）965）.In this article,the authors consider the one-dimensional compressible isentropic Navier-Stokes equations with gravitational force,fixed boundary condition,a general pressure and the density-dependent viscosity coefficient when the viscous gas connects to vacuum state with a jump in density.Precisely,the viscosity coefficient μ is proportional to ρ^θ and 0〈θ〈1/2,where ρ is the density,and the pressure P=P（ρ）is a general pressure.The global existence and the uniqueness of weak solution are proved.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

Regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with free boundary. The viscosity coefficient μ is proportional to ρθ with 0<θ<1, where ρ is the density. The existence and uniqueness of global weak solutions in H¹([0,1]) have been established in [S. Jiang, Z. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal. 12 (2005) 239-252]. We will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

Global existence of strong solutions of the Navier-Stokes equations for isentropic compressible fluids with density-dependent viscosity

This paper is concerned with global strong solutions of the isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in one-dimensional bounded intervals. Precisely, the viscosity coefficient μ=μ(ρ) and the pressure P is proportional to ργ with γ>1. The important point in this paper is that the initial density may vanish in an open subset. We also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition. Compared with former results obtained by Hyunseok Kim in [H. Kim, Global existence of strong solutions of the Navier-Stokes equations for one-dimensional isentropic compressible fluids, available at: http://com2mac.postech.ac.kr/papers/2001/01-38.pdf], we deal with density-dependent viscosity coefficient.     

A remark on the Cauchy problem of 1D compressible Navier-Stokes equations with density-dependent viscosity coefficients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1（R）,we will prove the existence of weak solutions to the Cauchy problems forθ〉0.This will improve results in Jiu and Xin’s paper（Kinet.Relat.Models,1（2）：313–330（2008））in whichθ〉12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

A remark on regularity of 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity

In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with the free boundary. The viscosity coefficient μ is proportional to ρθ with θ>0, where ρ is the density. The existence, uniqueness, regularity of global weak solutions in H¹([0,1]) have been established by Xin and Yao in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint]. Furthermore, under certain assumptions imposed on the initial data, we improve the regularity result obtained in [Z. Xin, Z. Yao, The existence, uniqueness and regularity for one-dimensional compressible Navier-Stokes equations, preprint] by driving some new a priori estimates.     

Global behavior of spherically symmetric Navier-Stokes equations with density-dependent viscosity

In this paper, we study a free boundary problem for compressible spherically symmetric Navier-Stokes equations without a solid core. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give some uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we obtain the stabilization rate estimates of exponential type in L^∞-norm and weighted H¹-norm of the solution by constructing some Lyapunov functionals. The results show that such system is stable under the small perturbations, and could be applied to the astrophysics.     

Discontinuous solutions of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum

In this paper, we study the evolutions of the interfaces between gas and the vacuum for one-dimensional viscous gas motions when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. Using some new a priori estimates, we establish the new local (in time) existence and uniqueness results under minimal hypotheses on the initial density, so that the interval for the power of the density in the viscosity coefficient is enlarged to (0,γ). In particular, we include the important case that the initial density could be piecewise smooth with arbitrarily large jump discontinuities, and could degenerate to zero.     

A note on barotropic compressible quantum Navier-Stokes equations

The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations has been proved very recently, by Jüngel (2009) [1], if the viscosity constant is smaller than the scaled Plank constant. This paper extends the results to the case that the viscosity constant equals the scaled Plank constant. By using a new estimate on the square root of the solution, apparently not available in [1], the semiclassical limit for the viscous quantum Euler equations (which are equivalent to the barotropic compressible quantum Navier-Stokes equations) can be performed; then the results of this paper are obtained easily.     

Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =b _ϱ ^β, whereb and β are positive constants. We prove that there exists an unique weak solution globally in time, provided that β<1/3.

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Sunto Si considera un problema di frontiera libera per l’equazione del moto unidimensionale isoentropico con viscosità dipendente dalla densità secondo la legge μ =b _ϱ ^β, doveb e β sono costanti positive. Si dimostra che esiste un’unica soluzione debole globale nel tempo, purché β<1/3.

     

A note on the non-formation of vacuum states for compressible Navier-Stokes equations

We give a refinement of Lemma 2.2 in [D. Hoff, J.A. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys. 216 (2001) 255-276] and complete the proof of non-formation of vacuum states for one-dimensional compressible Navier-Stokes equation given there.     

可压Navier-Stokes方程真空状态的动力学行为

本文主要考虑了一维可压Navier-Stokes方程真空状态的动力学行为.对于任意的熵弱解,如果初始状态不存在真空,我们证明了密度函数关于时间和空间变量是连续的且对于任意时间它是处处为正的.同时,我们还得到了含有间断连接的真空状态的整体熵弱解的存在性,结果显示其真空区域以代数速率被压缩,并在有限时间内消失.     

Global solutions of real compressible reactive gas with density-dependent viscosity and self-gravitation for higher-order kinetics

A mathematical model for viscous, real, compressible, reactive fluid flows is considered. The existence of global solutions for the free boundary problem with species diffusion in dynamic combustion is established when the viscosity λ depends on the density i.e., λ(ρ)=Aρα (), where A is a generic positive constant. Furthermore, the equations of state depend nonlinearly on density and temperature unlike the case of perfect gases or radiative flows. In addition, the shock wave, turbulence, vacuum, mass concentration or extremely hot spot will not be developed in any finite time if the initial data do not contain vacuum.