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.     

Regularity and asymptotic behavior of 1D compressible Navier-Stokes-Poisson equations with free boundary

In this paper, firstly, we consider the regularity of solutions in to the 1D Navier-Stokes-Poisson equations with density-dependent viscosity and the initial density that is connected to vacuum with discontinuities, and the viscosity coefficient is proportional to ρθ with 0<θ<1. Furthermore, we get the asymptotic behavior of the solutions when the viscosity coefficient is a constant. This is a continuation of [S.J. Ding, H.Y. Wen, L. Yao, C.J. Zhu, Global solutions to one-dimensional compressible Navier-Stokes-Poisson equations with density-dependent viscosity, J. Math. Phys. 50 (2009) 023101], where the existence and uniqueness of global weak solutions in H¹([0,1]) for both cases: μ(ρ)=ρθ, 0<θ<1 and μ=constant have been established.     

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.     

Global existence of solutions for compressible Navier-Stokes equations with vacuum

In this paper, we will investigate the global existence of solutions for the one-dimensional compressible Navier-Stokes equations when the density is in contact with vacuum continuously. More precisely, the viscosity coefficient is assumed to be a power function of density, i.e., μ(ρ)=Aρθ, where A and θ are positive constants. New global existence result is established for 0<θ<1 when the initial density appears vacuum in the interior of the gas, which is the novelty of the presentation.     

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.     

Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).     

Global solutions to isentropic planar magnetohydrodynamic equations with density‐dependent viscosity

In this paper, we study the isentropic compressible planar magnetohydrodynamic equations with viscosity depending on density and with free boundaries. Precisely, when the viscosity coefficient λ(ρ) is proportional to ρ^θ with θ > 0, where ρ is the density, we establish the existence of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.     

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.     

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→+∞.     

A remark on free boundary problem of 1‐D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we prove the existence and uniqueness of the weak solution of the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity µ(ρ)=ρ^θ with θ∈(0, γ?2], γ>1. The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results apply for the one‐dimensional Siant–Venant model of shallow water and generalize ones in (Arch. Rational Mech. Anal. 2006; 182: 223–253). Copyright © 2009 John Wiley & Sons, Ltd.     

Compressible Navier–Stokes equations with vacuum state in the case of general pressure law

In this paper, we consider the one‐dimensional compressible isentropic Navier–Stokes equations with a general ‘pressure law’ and the density‐dependent viscosity coefficient when the density connects to vacuum continuously. Precisely, the viscosity coefficient µ is proportional to ρ^θ and 0<θ<1, where ρ is the density. And the pressure P = P(ρ) is a general ‘pressure law’. The global existence and the uniqueness of weak solutions is proved, and a decay result for the pressure as t→ + ∞ is given. It is also proved that no vacuum states and no concentration states develop, and the free boundary do not expand to infinite. Copyright © 2006 John Wiley & Sons, Ltd.     

Global strong solutions of the Cauchy problem for 1D compressible Navier-Stokes equations with density-dependent viscosity

We consider the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = Aρ ^α , where α > 0 and A > 0. The global existence of strong solutions is obtained, which improves the previous results by enlarging the interval of α. Moreover, our result shows that no vacuum is developed in a finite time provided the initial data does not contain vacuum.     

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.     

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...     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance

This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes equations effected by external force of general form in R³.     

Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aim of this paper is to discuss the global existence and uniqueness of strong solution for a class of the isentropic compressible Navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of the 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.     

Macroscopic regularity for the Boltzmann equation

The regularity of solutions to the Boltzmann equation is a fundamental problem in the kinetic theory. In this paper, the case with angular cut-off is investigated. It is shown that the macroscopic parts of solutions to the Boltzmann equation, i.e., the density, momentum and total energy are continuous functions of(x, t) in the region R3×(0, +∞). More precisely, these macroscopic quantities immediately become continuous in any positive time even though they are initially discontinuous and the discontinuities of solutions propagate only in the microscopic level. It should be noted that such kind of phenomenon can not happen for the compressible Navier-Stokes equations in which the initial discontinuities of the density never vanish in any finite time, see [22]. This hints that the Boltzmann equation has better regularity effect in the macroscopic level than compressible Navier-Stokes equations.