Stability properties of the steady state for the isentropic compressible Navier-Stokes equations with density dependent viscosity in bounded intervals

Stability properties of the steady state for the isentropic compressible Navier-Stokes equations with density dependent viscosity in bounded intervals

Abstract.

We prove existence and asymptotic stability of the stationary solution for the compressible Navier-Stokes equations for isentropic gas dynamics with a density dependent diffusion in a bounded interval. We present the necessary conditions to be imposed on the boundary data which ensure existence and uniqueness of the steady state, and we subsequent investigate its stability properties by means of the construction of a suitable Lyapunov functional for the system. The Saint-Venant system, modeling the dynamics of a shallow compressible fluid, fits into this general framework.

MARTA STRANI111Università Ca’ Foscari, Dipartimento di Scienze Molecolari e Nanosistemi, Venezia Mestre (Italy), E-mail address: marta.strani@unive.it, martastrani@gmail.com.


1. Introduction

In this paper we study existence and stability properties of the steady state for the one dimensional compressible Navier-Stokes equations with density-dependent viscosity, which describes the isentropic motion of compressible viscous fluids in a bounded interval. In terms of the variables mass density and velocity of the fluid , the problem reads as

(1.1)

to be complemented with boundary conditions

and initial data .

We restrict our analysis to the barotropic regime, where the pressure is a given function of the density satisfying the following assumptions

(1.2)

As concerning the viscosity term , we require for all .

A prototype for the term of pressure is given by the power law , , while the case with is known as the Lamé viscosity coefficient. In particular, the well known viscous Saint-Venant system, describing the motion of a shallow compressible fluid, corresponds to the choice and .

By considering the variables density and momentum , system (1.1) becomes

(1.3)

together with boundary conditions

(1.4)

In the following, we shall use both the formulations (1.1) and (1.3), depending on what is necessary; as an example, when studying the stationary problem, the variables appear to be more appropriate since in this case the second component of the steady state turns to be a constant.

Remark 1.1.

Given an initial datum , throughout the paper we will consider solutions to (1.1) . We refer the readers to Theorem A.1 in Appendix A for the proof of the existence of such a solution. We stress that, of course, also the solution to (1.3) belongs to the same functional space, since .

Depending on assumptions and approximations, the Navier-Stokes system may also contain other terms and gives raise to different types of partial differential equations. Indeed, natural modifications of the model emerge when additional physical effects are taken into account, like viscosity, friction or Coriolis forces; far from being exhaustive but only intended to give a small flavor of the huge number of references, see, for instance, [12, 15, 16, 24, 10] and the references therein for existence results of global weak and strong solutions, [20, 28, 34] for the problem with a density-dependent viscosity vanishing on vacuum, [6] for the full Navier-Stokes system for viscous compressible and heat conducting fluids. More recently, the interesting phenomenon of metastability (see, among others, [4, 9, 37] and the references therein) has been investigated both for the incompressible model [5, 27] and for the D compressible problem [31, 39]. As concerning system (1.1), there is a vast literature in both one and higher dimensions. Global existence results and asymptotic stability of the equilibrium states are obtained from Kawashima’s theory of parabolic-hyperbolic systems in [22], D. Bresch, B. Desjardins and G. Métivier in [8], P.L. Lions in [29] and W. Wang in [42] for the viscous model, and C.M. Dafermos (see [11]) for the inviscid model. As the compressible Navier-Stokes equations with density-dependent viscosity are suitable to model the dynamics of a compressible viscous flow in the appearance of vacuum [18], there are many literatures on the well-posedness theory of the solutions for the D model (see, for instance, [17, 21, 36, 43, 44] and the references therein). However, most of these results concern with free boundary conditions. Recently, the analysis of the dynamics in bounded domains has also been investigated (see, for instance, [7]); the initial-boundary value problem with , , has been studied by H.L. Li, J. Li and Z. Xin in [25]: here the authors are concerned with the phenomena of finite time vanishing of vacuum. We also quote the analysis performed in [26], where a particular attention is devoted to the dynamical behavior close to equilibrium configurations.

The existence of stationary solutions for system (1.1) and, in particular, for shallow water’s type systems, and the subsequent investigation of their stability properties has also been considered in the literature. To name some of these results, we recall here [1] and [13], where the authors are concerned with the inviscid case; in particular, in [13] the authors address the issue of stating sufficient boundary conditions for the exponential stability of linear hyperbolic systems of balance laws (for the investigation of the nonlinear problem, we refer to [2, 3]). The case with real viscosity has been addressed, for example, in [32]; we mention also the recent contributions [23, 33], where the authors investigate asymptotic stability of the steady state in the half line. We point out that when dealing with the open channel case (i.e. ), the study of the stationary problem presents less difficulties than the case of bounded domains, where one has to handle compatibility conditions on the boundary values coming from the study of the formal hyperbolic limit . In this direction, we quote the papers [35, 40], where the authors address the problem of the long time behavior of solutions for the Navier-Stokes system in one dimension and with Dirichlet boundary conditions (see also [41] for the extension of the results to the case of a density dependent diffusion). Finally, a recent contribution in the study of the stationary problem for the viscous model in a bounded interval and with nonhomogeneous Dirichlet boundary conditions is the paper [38], where the author considers (1.1) in the special case of and , corresponding to the viscous shallow water system. Being the literature on the subject so vast, we are aware that this list of references is far from being exhaustive.

Our aim in the present paper is at first to prove existence and uniqueness of a stationary solution to (1.3)-(1.4). Because of the discussion above, this results is likely to be achieved only if some appropriate assumptions on the boundary values are imposed; precisely, following the line of [38] where this problem has been addressed for the case of a linear diffusion , our first main contribution (for more details, see Section 3), is the following theorem.

Theorem 1.2.

Given and , let us consider the problem

(1.5)

and let us suppose that the following assumptions are satisfied: H1. The term of pressure and the viscosity term verify, for all

H2. Setting , there hold

where . Then there exists a unique stationary solution to (1.5), i.e. a unique solution independent on the time variable to the following boundary value problem

(1.6)
Remark 1.3.

We stress the the choice of the variables (instead of the most common choice density/velocity), is dictated by the fact that the second component of the steady state turns to be a constant, and this constant value is univocally determined once the boundary data are imposed.

Once the existence of a unique steady state for system (1.5) is proved, we devote the second part of this paper to investigate its stability properties. Precisely, we prove stability of the steady state in the sense of the following definition.

Definition 1.4.

A stationary solution to (1.5) is stable if for any there exists such that, if , then, for all , it holds

where is the solution to (1.5).

Our second main result is stated in the next theorem, showing that the stability of the stationary solution constructed in Theorem 1.2 can be proved only if some additional assumptions on the boundary data are imposed.

Theorem 1.5.

Let the assumptions of Theorem 1.2 be satisfied, and let us also assume the following additional hypotheses. H3. There hold, for all

being the unique steady state of (1.5) given in Theorem 1.2. H4.The boundary values are chosen such that

for some positive small enough.

Then the steady state is stable in the sense of Definition (1.4).

Remark 1.6.

It is worth notice that Theorem 1.5 prove stability of the steady state for all time (since the constant in Definition 1.4 in independent on ). We also point out that the strategy used here do not provide stability of in the case the boundary values do not satisfy any smallness condition, while its existence is assured also in this setting (cfr Theorem 1.2); however, our guess is that “large”  solutions are not stable (see also the analysis of [33] and [45], where a similar smallness condition has been imposed in order to have stability of the steady state to a Navier-Stokes system in the half line), and this will be the object of further investigations.

We close this Introduction with a short plan of the paper. In Section 2 we study the inviscid problem, obtained formally by setting in (1.3); we show that, at the hyperbolic level, some compatibility conditions on the boundary data are needed in order to ensure the existence of a weak solution. In particular, such conditions follow from the definition of a couple entropy/entropy flux which, in the present setting, are given by

being .

Section 3 is devoted to the study of the stationary problem for (1.5), and in particular to the proof of Theorem (1.2); to this aim we will state and prove several Lemmas showing that, once the boundary conditions are imposed and assumption H2 is satisfied, there exists a unique positive connection for (1.5), i.e. a unique stationary solution connecting the boundary data. Such analysis deeply relies on the strategy firstly performed in [38], where the author addresses the same problem in the easiest case of a linear diffusion, namely .

In Section 4, we turn our attention to the stability properties of the steady state, proving that it is stable in the sense of Definition 1.4; the key point to achieve such result is the construction of a Lypaunov functional, which, in the present setting, is defined as

It is easy to check that is positive defined and null only when computed on the steady state; the tricky part will be the computation of the sign of its time derivative along the solutions, needed in order to apply a Lyapunov type stability theorem.

Finally, in Appendix A we prove the existence of a solution to (1.1) belonging to ; part of the computations are inspired by [19].

As stressed in the introduction, results relative to the existence and stability properties of the steady state for the Navier-Stokes equations in bounded intervals appear to be rare; the study of the stationary problem for (1.5) (with generic pressure and viscosity ) and, mostly, the subsequent investigation of the stability properties of the steady state are, to the best of our knowledge, new.

It is also worth noticing that this analysis is meaningful in view of the possible investigation of the phenomenon of metastability for the one dimensional compressible Navier-Stokes system; indeed, all the informations on the stability properties of the steady state can be useful for the study of the slow motion of the corresponding time dependent solution (see [39, Section 3.1]).

2. The inviscid problem

We start our analysis by studying the limiting regime , obtained formally by putting in (1.1); we obtain the following hyperbolic system for unviscous isentropic fluids

(2.1)

System (2.1) is complemented with the same boundary and initial conditions of (1.1). We recall that the usual setting where such a system is studied is given by the entropy formulation, hence non classical discontinuous solutions can appear; thus, we primarily concentrate on the problem of determining the entropy jump conditions for the hyperbolic system (2.1). As previously done in [38] (see also [30]), such conditions are dictated by the choice of a couple entropy/entropy flux and such that

  • the mapping is convex;

  • in any region where is a solution to (2.1).

In particular, if and only if

(2.2)

To start with, let us consider the easiest case of a power law type of pressure, i.e. with and . In this case (see, for instance, [14]) the entropy corresponds to the physical energy of the system and it is defined as

(2.3)

By solving (2.2), it turns out that is defined as

(2.4)

In the case of a general term of pressure satisfying assumptions (1.2), the couple entropy/entropy flux is given by

(2.5)

being

Of course we observe that, when , we recover (2.3)-(2.4). Following the line of [38], given , and , let and be an entropic discontinuity of (2.1) with speed , that is we assume the function

(2.6)

to be a weak solution to (2.1) satisfying, in the sense of distributions, the entropy inequality

(2.7)

On one side, with the change of variable , system (2.1) reads

and the request of weak solution translates into the Rankine-Hugoniot conditions, that read

(2.8)

On the other side, the entropy condition (2.7) reads , where

Setting , we have

so that, recalling

and by using (2.8) , the entropy condition translates into

(2.9)

By squaring the first condition in (2.8), we obtain a system for the quantities , whose solutions are given by

(2.10)

When looking for the stationary solutions to (2.1), i.e. , (2.10) translates into the following conditions for the boundary values

(2.11)

that, together with (2.9), univocally determine the possible choices of the boundary data for the jump solution (2.6) with to be an admissible steady state for the system.

In particular, for all , we can state that the one-parameter family

is a family of stationary solutions to (2.1) if and only if both (2.9) and (2.11) are satisfied.

Finally, we point out that, in terms of the variables density/momentum, conditions (2.9)-(2.11) read as

Example 2.1.

In the case of the scalar Saint-Venant system, i.e. , stationary solutions to

to be considered with boundary data and , solve

where

Moreover, only entropy solutions are admitted, so that, from (2.9)

Since , then , and this condition describes the realistic phenomenon of the hydraulic jump consisting in an abrupt rise of the fluid surface and a corresponding decrease of the velocity.

3. Stationary solutions for the viscous problem

This section is devoted to the study of the existence and uniqueness of a stationary solution for the Navier-Stokes system (1.3). As stressed in the introduction, we here prefer to use the variables density/momentum rather than the most common choice density/velocity since in this case the second component of steady state turns to be a constant, which is univocally determined by the boundary values. We are thus left with a single equation for the variable which can be integrated with respect to , by paying the price of the appearance of an integration constant. For , the stationary equations read

(3.1)

which is a couple of ordinary differential equations; by integrating in we can lower the order of the system obtaining the following stationary problem for the couple :

(3.2)

being an integration constant that depends on the values of the solution and its derivative on the boundary, while is univocally determined by the boundary datum ; indeed, since the component of the steady state turns to be constant, the values and are forced to be equal to a common value, named here .

Let us define

since there hold

where indicates the derivative of with respect to . Thus, the second equation in (3.2) can be rewritten as

Setting , with the change of variable , and since is invertible, we have

(3.3)

We thus end up with an autonomous first order differential equation of the form ; in this case it is not possible to obtain an explicit expression for the solution, and in order to provide qualitative properties of the solution we have to study the function .

The problem of studying properties of the right hand side of (3.3) has been previously addressed in [38] in the case of a linear diffusion (that is, ). Precisely, the author states and proves a set of results describing the behavior of the function both with respect to and with respect to .

We recall here some of these results for completeness, since they will be useful to describe the qualitative properties of the function ; for more details we refer to [38, Lemma 3.1, Lemma 3.2]. From now on, we will always suppose the pressure term to satisfy assumptions (1.2). We also recall that, by definition

(3.4)
Lemma 3.1.

For every , there exists at least a value such that there exist two positive solutions to the equation .

Remark 3.2.

As enlightened in the proof of [38, Lemma 3.1], a sufficient condition on the constant for the existence of two positive solutions to is given by

(3.5)

where solves , while . Indeed, since and , if is such that

then the thesis follows. By exploiting the conditions and , we end up with (3.5).

Lemma 3.3.

Let be such that there exist two positive solutions to the equation . Hence, given , the set defined as

is such that , for some .

Lemmas 3.1-3.3 assure that, once the boundary conditions are imposed, there always exists a value for the integration constant such that there exist two positive solutions to the equation satisfying

This is of course a necessary condition for the existence of an increasing positive connection between and , as enlightened in Figure 1 in the specific example of and .

Figure 1. Plot of the solutions to . The choice for and is such that . In the plane we can see that the solution starting from can not reach , since is an equilibrium solution for the equation. The same holds if .

3.1. The stationary problem

By taking advantage of the already known properties of the function , we now study the function . We first notice that the function is increasing for and decreasing for , where , implicitly defined as

is such that . Moreover, as already stressed in Remark 3.2, if is such that , that is

then there exist two positive solutions to the equation . Given , since and , we have

proving that is a positive increasing function as well.

Let us now consider ; we prove the following lemma.

Lemma 3.4.

Let , with defined in (3.4). For every there exist such that . Moreover, the function in increasing in the interval , and decreasing in the interval , being .

Proof.

Lemma 3.1 assures the existence of two positive values and such that and, as a consequence, and has to be defined as

(3.6)

Since and , there exist and they are unique and such that (3.6) holds. Hence, has exactly two positive zeros for all the choices of . Furthermore

so that the sign of is univocally determined by the sign of . Therefore, if is such that , then

We finally notice that condition (3.5) for the existence of two positive solutions to the equation , also assures that has to positive zeros. Indeed

so that if and only if , where is such that . Furthermore,

which is exactly (3.5).

Example 3.5 (The Saint-Venant system with density dependent viscosity).

When , the stationary equation (3.2) for reads

Let us consider the simplest case , , and let us plot the function . We have

so that

Figure 2. Plots of different with , and . The dashed line plots , the dashed-point line plots , while the black line plots .

Figure 2 shows the plot of for different choice of , compared with the plot of (where ); the dashed line and the dashed point line plot with and respectively. As proved in Lemma 3.4, we can see that the monotonicity properties of the function are preserved, as well as the existence of two positive zeros.

3.2. Existence and uniqueness of a positive connection

Let us go back to the problem of the existence and uniqueness of the solution to the stationary problem (3.1). As already shown, once the boundary conditions for the function are imposed, problem (3.1) reads

where and , being .

Hence, the equation for the variable is an equation on the form

where is an integration constant depending on the boundary data. Once the boundary conditions are imposed, a positive connection between and (i.e. a positive solution to connecting and ) exists only if

being and such that .

The following Lemma (to be compared with Lemma 3.3) aims at showing some properties of the function as a function of ; precisely, we describe how the distance between the two zeroes of the function changes with respect to this parameter.

Lemma 3.6.

Let with defined in (3.4), and let be such that (3.5) holds, so that there exist two positive solutions to the equation . Given , the set defined as

is such that , for some .

Proof.

Since and , we want to show that is an increasing function with respect to . Indeed, this would imply that, if there exists a value such that

then, for all

being and the two positive zeros of .

Since and is an increasing function that does not depend on , is an increasing function in the variable if so it is for . We have

so that, since , when .

Thus, we only need to prove that there exist a value such that . We know that and for all . Moreover

so that . Furthermore, if we ask for

(3.7)

we have . Condition (3.7) can be rewritten as

that is, since

Since and are increasing function in the interval and respectively, we obtain the following condition for the constant

If this condition holds, then

showing that as . On the other hand we know that where is such that . Hence

Since as , and since is an increasing and continuous function, we know that as , implying as . We have thus proved that, if we choose large enough, then for every choice of . More precisely, is chosen in such a way that

where and are such that either or .

Definition 3.7.

We define the region of admissible values as the set of all the values such that there exists two positive solutions to the equation and Lemma 3.6 holds. In the plane , is determined by the equations

We recall that is such that and .

Proposition 3.8.

The region is the epigraph of an increasing function , i.e.

Proof.

Setting , we have

meaning that is an increasing function in the plane . Moreover, the condition is equivalent to

and we get

whose equality defines two parabolas. Finally, the function is such that

since is a decreasing function. Hence , since is obtained by matching increasing functions.

We now prove the existence of a -connection, i.e. we prove the existence of a solution to

satisfying

We first notice that