Density-dependent Viscosity Research Articles

Time-discretized steady compressible Navier–Stokes equations with inflow and outflow boundaries

The time-discretized steady compressible Navier–Stokes equations in n-dimensional bounded domains with the velocity specified only at the inflow boundary are considered. The existence and uniqueness of L p solutions are proved for p > n. For time-discretized steady flows, results of Kweon and Kellogg and of Kweon and Song are extended in a manner that allows for more general domains and for density-dependent viscosity coefficients. Moreover, we only require p > n which is a critical barrier in the previous works.

Global classical solutions for 3D compressible Navier–Stokes equations with vacuum and a density-dependent viscosity coefficient

In this paper, we prove the global existence of classical solutions to the three-dimensional (3D) compressible Navier–Stokes equations with a density-dependent viscosity coefficient (λ=λ(ρ)) provided the initial data is of small energy. This in particular implies that the solutions may have large oscillations and contain vacuum states. As a result of the uniform estimates, the large-time behavior of the solution is also studied. The result obtained generalizes those results in Zhang (2011) [39] and Huang et al. (2012) [17] where the non-vacuum initial data and the constant viscosity coefficients are considered, respectively.

Vacuum Behaviors around Rarefaction Waves to 1D Compressible Navier--Stokes Equations with Density-Dependent Viscosity

In this paper, we study the large time asymptotic behavior toward rarefaction waves for solutions to the one-dimensional compressible Navier--Stokes equations with density-dependent viscosities for general initial data whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. First, a global-in-time weak solution around the rarefaction wave is constructed by approximating the system and regularizing the initial data with general perturbations, and some a priori uniform-in-time estimates for the energy and entropy are obtained. Then it is shown that the density of any weak solution satisfying the natural energy and entropy estimates will converge to the rarefaction wave connected to vacuum with arbitrary strength in sup-norm time-asymptotically. Our results imply, in particular, that the initial vacuum at far field will remain for all the time, which is in sharp contrast to the case of nonvacuum rarefaction waves studied in [Q. S. Jiu, Y. Wang, and Z. P. Xin, Comm. Partial Differential Equations, 36 (2011), pp. 602--634], where all the possible vacuum states will vanish in finite time. Finally, it is proved that the weak solution becomes regular away from the vacuum region of the rarefaction wave.

Self-Similar Solutions of the Compressible Flow in One-Space Dimension

For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Discontinuous Initial Data

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient and discontinuous initial data in this paper. For piecewise regular initial density, we show that there exists a unique global piecewise regular 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 at an algebraic time-rate, and the jump discontinuity of density also decays at an algebraic time-rate as the time tends to infinity.

Global Solutions to the One-Dimensional Compressible Navier--Stokes--Poisson Equations with Large Data

In this paper, we study the global solutions with large data away from vacuum to the Cauchy problem of the one-dimensional compressible Navier--Stokes--Poisson system with density-dependent viscosity coefficient and density- and temperature-dependent heat-conductivity coefficient. The proof is based on some detailed analysis on the bounds on the density and temperature functions.

Existence of global weak solutions to finitely extensible nonlinear bead–spring chain models for dilute polymers with variable density and viscosity

We prove the existence of global-in-time weak solutions to a general class of coupled bead–spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains, with finitely extensible nonlinear elastic (FENE) spring potentials. The class of models under consideration involves the unsteady incompressible Navier–Stokes equations with variable density and density-dependent dynamic viscosity in a bounded domain in Rd, d=2 or 3, for the density, the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term and a nonlinear density-dependent drag coefficient. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With initial density ρ0∈[ρmin,ρmax] for the continuity equation, where ρmin>0; a square-integrable and divergence-free initial velocity datum u˜0 for the Navier–Stokes equation; and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t↦(ρ(t),u˜(t),ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ρ(0),u˜(0),ψ(0))=(ρ0,u˜0,ψ0), such that t↦ρ(t)∈[ρmin,ρmax], t↦u˜(t) belongs to the classical Leray space and t↦ψ(t) has bounded relative entropy with respect to M and t↦ψ(t)/M has integrable Fisher information (w.r.t. the Gibbs measure dν:=M(q˜)dq˜dx˜) over any time interval [0,T], T>0.

The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients

We consider the Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, we are mainly concerned with the decay estimates of the density and velocity as $t \rightarrow \infty$. Firstly, we obtain the decay estimates of $\rho-\bar{\rho}$ and u in $L^2(R)$ norm, then we obtain the decay estimate of $\rho-\bar{\rho}$ in $L^{\infty}(R)$ norm as $\bar{\rho}>0$. Secondly, we construct a functional and use the energy method to obtain the decay estimate of $\rho$ in $L^{\infty}(R)$ norm as $\bar{\rho}=0$.

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 multi-dimensional 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.

Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries

In this paper, we study a class of analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients, which describe compressible fluids moving into outer vacuum. For suitable viscous polytropic fluids, we construct a class of radial symmetric and self-similar analytical solutions in RN(N⩾2) with both continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum. Such solutions exhibit interesting new information such as the formation of vacuum at the center of the symmetry as time tends to infinity and explicit regularities and large time decay estimates of the velocity field.

Zero dissipation limit of the rarefaction wave with vacuum for the compressible Navier-Stokes equations with density-dependent viscosity

In this paper, we study the zero dissipation limit of solutions to the compressible isentropic Navier- Stokes equations with density-dependent viscosity ∈(ρ) = ∈ρα. Given a rarefaction wave with one-side vacuum state to the compressible Euler equations, we prove that the Navier-Stokes equations admit a sequence of global solutions, which converge to the above rarefaction and vacuum as the viscosity tends toward zero. Moreover, some decay rates depending on α with respect to viscosity ∈ are obtained.

Some exact blowup or global solutions for the non-isentropic Navier–Stokes equations with density-dependent viscosity

In this paper, we construct some exact solutions for the non-isentropic Navier–Stokes equations with density-dependent viscosity in RN. A class of exact solutions is obtained for γ=θ+1N⩾1:(1)ρ(t,x→)=fx1+d1a(t),x2+d2a(t),….,xN+dNa(t)a(t)Nui(t,x→)=ȧ(t)a(t)(xi+di)fori=1,2,….,NS(t,x→)=-1Nlnfx1+d1a(t),x2+d2a(t),…,xN+dNa(t)a(t)=tκN+a0with an arbitrary scalar C1 function f>0; constants a0N>0 and di. In particular, for a0<0 and the even dimensions, the solutions blow up in the finite time T=−κNa0. For a0>0, the constructed solutions are global. Here the main contribution is that we free the density function to be an arbitrary positive C1 functions for the non-isentropic fluids. We note that our new method can work only for the non-isentropic fluids. In addition, the constructed exact solutions are useful for testing numerical methods for the system.

Compressible flows: New existence results and justification of the Reynolds asymptotic in thin films

In this paper, we are first interested in the compressible Navier-Stokes equations with density dependent viscosities in bounded domains with non-homogeneous Dirichlet conditions. We study the well posedness of such models with density dependent viscosity coefficients in the non-stationary and in the stationary cases. We apply the last result in thin domains context, justifying the compressible Reynolds equations.

Cauchy problem for the one-dimensional compressible Navier-Stokes equations

We consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient. For regular initial data, we show that the unique strong solution exits globally in time and converges to the equilibrium state time asymptotically. When initial density is piecewise regular with jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump discontinuity of the density decays exponentially and the piecewise regular solution tends to the equilibrium state as t → + ∞.

Global behavior of 1D compressible isentropic Navier-Stokes equations with a non-autonomous external force

In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity and a non-autonomous external force. The viscosity coefficient μ is proportional to r θ with 0 <θ < 1, where r is the density. Under certain assumptions imposed on the initial data and external force f ,w e obtain the global existence and regularity. Some ideas and more delicate estimates are introduced to prove these results.

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 coefficients. By assuming ρ0 ∈ L1(R), we will prove the existence of weak solutions to the Cauchy problems for θ > 0. This will improve results in Jiu and Xin’s paper (Kinet. Relat. Models, 1(2): 313–330 (2008)) in which θ > \(\tfrac{1} {2} \) is required. In addition, We will study the large time asymptotic behavior of such weak solutions.

Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations

The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.

Global existence of strong solutions to the micro-polar, compressible flow with density-dependent viscosities

This article is concerned with global strong solutions of the micro-polar, compressible flow with density-dependent viscosity coefficients in one-dimensional bounded intervals. The important point in this article is that the initial density may vanish in an open subset.

Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier–Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution ( ρ , u ) where ρ ∈ C 1 ( [ 0 , T ] ; H 1 ( [ 0 , 1 ] ) ) and u ∈ H 1 ( [ 0 , T ] ; H 2 ( [ 0 , 1 ] ) ) for any T > 0 . As it is pointed out in Xin (1998) [31] that the smooth solution ( ρ , u ) ∈ C 1 ( [ 0 , T ] ; H 3 ( R 1 ) ) ( T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ 2 u t t , such as Lemmas 3.2–3.6. This leads to further regularities of ( ρ , u ) where ρ ∈ C 1 ( [ 0 , T ] ; H 3 ( [ 0 , 1 ] ) ) , u ∈ H 1 ( [ 0 , T ] ; H 3 ( [ 0 , 1 ] ) ) . It is still open whether the regularity of u could be improved to C 1 ( [ 0 , T ] ; H 3 ( [ 0 , 1 ] ) ) with the appearance of vacuum, since it is not obvious that the solutions in C 1 ( [ 0 , T ] ; H 3 ( [ 0 , 1 ] ) ) to the initial boundary value problem must blow up in finite time.

Boundary layers for compressible Navier–Stokes equations with density-dependent viscosity and cylindrical symmetry

In this paper, we consider the zero shear viscosity limit for the Navier–Stokes equations of compressible flows with density-dependent viscosity coefficient and cylindrical symmetry. The boundary layer effect as the shear viscosity μ=ερθ goes to zero (in fact, ε→0 in this paper, which implies μ→0) is studied. We prove that the boundary layer thickness is of the order O(εα), where 0<α<12 for the constant initial data and 0<α<14 for the general initial data, which extend the result in Frid and Shelukhin (1999) [4] to the case of density-dependent viscosity coefficient.