The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.

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Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

In this paper, we consider the compressible Navier–Stokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support. The viscosity coefficient is assumed to be a power function of the density so that the Cauchy problem is well-posed. New global existence results are established when the density function connects to the vacuum states continuously. For this, some new a priori estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum. We will also give a non-global existence theorem of regular solutions when the initial data are of compact support in Eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum.

Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum

In this paper, we study the one-dimensional motion of viscous gas connecting to vacuum state with a jump in density when the viscosity depends on the density. Precisely, when the viscosity coeffici...

Compressible navier-stokes equations with density-dependent viscosity and vacuum

We consider Navier–Stokes equations for compressible viscous fluids in one dimension. It is a well-known fact that if the initial datum are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists locally in time. In this paper, we show that under the same hypothesis, the density remains bounded by below by a positive constant uniformly in time, and that strong solutions therefore exist globally in time. Moreover, while most existence results are obtained for positive viscosity coefficients, the present result holds even if the viscosity coefficient vanishes with the density. Finally, we prove that the solution is unique in the class of weak solutions satisfying the usual entropy inequality. The key point of the paper is a new entropy-like inequality introduced by Bresch and Desjardins for the shallow water system of equations. This inequality gives additional regularity for the density (provided such regularity exists at initial time).

Existence and Uniqueness of Global Strong Solutions for One-Dimensional Compressible Navier–Stokes Equations

Due to the strong degeneracy at vacuum, both Euler and Navier-Stokes systems for compressible fluids (in which the viscosity is independent of density) behave singularly [7, 10, 16]. In particular, the classical one-dimensional isentropic Navier-Stokes system picks up unphysical solutions for two gases initially separated by vacuum states [7, 10]. To overcome this difficulty, Liu, Xin and Yang in [10] introduced the modified Navier-Stokes system (1.1) in which the viscosity coefficient depends on the density. It is shown in [10] that at least locally in time, the system (1.1) yields the physically relevant solution. As remarked by Liu, Xin and Yang in [10], the model is also motivated by the physical consideration that in the derivation of the compressible Navier-Stokes equations from the Boltzmann equations, the viscosity is not constant and depends on the temperature. For isentrpoic flow, this dependence is translated into the dependence of the viscosity on the density. For simplicity we consider in this paper

https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2005/0012/0003/MAA-2005-0012-0003-a002.pdf

Global Weak Solutions to 1D Compressible Isentropic Navier-Stokes Equations 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.

Free Boundary Problem for the Equation of One-Dimensional Motion of Compressible Gas with Density-Dependent Viscosity.

This paper deals with the existence and uniqueness of stationary solutions for the equations modelling the steady flow of compressible viscoelastic fluids of Oldroyd type in an exterior domain. The existence proof is based on an appropriate decomposition of the original nonlinear set of equations into auxiliar problems (Neumann problem for the Laplacian, Oseen problem, two transport equations) and on a fixed point argument in a suitable functional setting. Copyright © 1999 John Wiley & Sons, Ltd.

Existence of classical solutions for compressible viscoelastic fluids of Oldroyd type past an obstacle

We study the asymptotic behaviour of the certain class of non-Newtonian incompressible fluids-power law fluids in the whole space when the external force is zero Assuming that initial data belong toL
 1∩L
 2 we prove thatL
 2 decay in time ist
 −1/4

Remark on the L 2 decay for weak solution to equations of non-newtonian incompressible fluids in the whole space (I)

This article is concerned with the steady motion of a compressible viscoelastic non-Newtonian fluid of Maxwell type around a three-dimensional rigid body. Results on existence, uniqueness and asymptotic behaviour of the solution are obtained for small data.

Asymptotic Behaviour of Compressible Maxwell Fluids in Exterior Domains

The global existence and uniqueness of classical solution of steady motions of a third-grade fluid provided assumptions on positivness of p (coefficient of viscosity) and α 1 , γ (material coefficients) is proved.

Šárka Matušů-Nečasová

Papers

Free Boundary Problem for the Equation of One-Dimensional Motion of Compressible Gas with Density-Dependent Viscosity.

Existence of classical solutions for compressible viscoelastic fluids of Oldroyd type past an obstacle

Remark on the L 2 decay for weak solution to equations of non-newtonian incompressible fluids in the whole space (I)

Asymptotic Behaviour of Compressible Maxwell Fluids in Exterior Domains

Existence and uniqueness of steady motions of a third‐grade fluid