Benjamin Melinand
email: melinand@ceremade.dauphine.fr

Kevin Zumbrun
email: kzumbrun@indiana.edu

Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions

We study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a finite interval with noncharacteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possesses (i) a spectral gap between neutral and growing/decaying modes, and (ii) an even number of nonstable eigenvalues λ (with a nonnegative real part). In the case that there are no nonstable eigenvalues, i.e., of spectral stability, we show this solution to be nonlinearly exponentially stable in H 2 × H 3 . Using "Goodman-type" weighted energy estimates, we establish spectral stability for small-amplitude data. For largeamplidude data, we obtain high-frequency stability, reducing stability investigations to a bounded frequency regime. On this remaining, bounded-frequency regime, we carry out a numerical Evans function study, with results again indicating universal stability of solutions.

Introduction

In this paper, we initiate in the simplest setting of 1D isentropic gas dynamics, a systematic study of existence and stability of steady solutions of systems of hyperbolic parabolic equations on a bounded domain, with noncharacteristic inflow or outflow boundary conditions, and data and solutions of amplitudes that are not necessarily small. We have in mind the scenario of a "shock tube", or finite-length channel with inflow-outflow boundary conditions, which in turn could be viewed as a generalization of the Poisseuille flow in the incompressible case.

Our conclusions in the present, isentropic case, obtained by rigorous nonlinear and spectral stability theory, augmented in the large-amplitude case by numerical Evans function analysis, are that for any choice of data there exists a unique solution, and this solution is linearly and nonlinearly time-exponentially stable in H 2 × H 3 . These results suggest a number of interesting directions for further investigation in 1 and multi-D.

Setting

We consider the 1D isentropic compressible Navier-Stokes equations ρ t + (ρu) x = 0, (ρu) t + ρu 2 + P (ρ) x = νu xx

(1) on the interval [0, 1], with the noncharacteristic boundary conditions

     ρ(t, 0) = ρ 0 > 0, u(t, 0) = u 0 > 0, u(t, 1) = u 1 > 0.
(2)

Notice that we have an inflow boundary condition at x = 0 and an outflow boundary condition at x = 1. We assume that the viscosity ν is positive and constant and that the pressure P is a smooth function satisfying

P > 0. ( 3
)
Stability of steady states for hyperbolic parabolic systems has been studied by many authors. For problems on the whole line, the reader can refer to [START_REF] Mascia | Pointwise green function bounds for shock profiles of systems with real viscosity[END_REF][START_REF] Barker | Stability of viscous shocks in isentropic gas dynamics[END_REF] and references within. In the case of noncharacteristic boundary conditions on the half line, see for instance [START_REF] Costanzino | Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers[END_REF][START_REF] Nguyen | Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems[END_REF]. For studies of scalar conservation laws on a bounded interval, one may see for instance [START_REF] Kreiss | Convergence to steady state of solutions of Burgers' equation[END_REF][START_REF] Jiu | Asymptotic behaviors of the solutions to scalar viscous conservation laws on bounded interval[END_REF]. Finally, we refer to [START_REF] Ervedoza | Local exact controllability for the one-dimensional compressible Navier-Stokes equation[END_REF] for the study of boundary controllability of the 1D Navier-Stokes equations.

In this paper, we study the existence and stability of steady states of (1) satisfying the boundary conditions (2). Section 2 is devoted to the existence and the uniqueness of such steady states. In Section 3, we study the corresponding linearized problem about the steady state. In section 4, we show that constant steady states and almost constant steady states, see Condition (5), are spectrally stable. We also show that general steady states are numerically spectrally stable. Section 5 is devoted to a local wellposedness result for problem (1)-(2). Then, in section 6, we show the nonlinear stability of steady states that are spectrally stable. Theorem 6.3 is the main result of this paper. Finally, in section 7, we improve the previous theorem under more restrictive assumptions.

Remark 1.1. It is worth noting that boundary conditions (2) are not the only ones we can deal with. For instance, the case

     ρ(t, 1) = ρ 0 > 0, u(t, 0) = u 0 < 0, u(t, 1) = u 1 < 0,
is equivalent by the change of variables x 1 -x and the change of unknowns (ρ, û) (ρ, -û). Moreover, these two possibilities are the only types of noncharacteristic boundary conditions yielding physically realizable steady states. For, the first equation of (1) yields that steady solutions have constant momentum ρu ≡ m, so that u(0) and u(1) necessarily agree in sign. By similar reasoning, characteristic boundary conditions u(0) = u(1) = 0 yield u ≡ 0 yield only trivial, constant steady states (ρ, u) ≡ (ρ 0 , 0).

Discussion and open problems

As mentioned earlier, our goal in this paper is to open a line of investigation of largeamplitude steady solutions for inflow-outflow problems on bounded domains. The main technical contribution is our argument for nonlinear exponential stability of spectrally stable solutions, which is both particularly simple and also applies to general hyperbolic parabolic systems of "Kawashima type", as considered on the whole-and half-line in [MZ03, BHRZ08, CHNZ09, NZ09]. Our goal, and the novelty of the argument as compared to those for the whole-and half-line, was to take advantage of the spectral gap to obtain a simple proof based on standard semigroup/energy methods. However, a close reading will reveal that this is deceptively difficult to accomplish, involving the introduction of a precisely chosen space (ρ, u) ∈ H 1 × L 2 with norm strong enough that we can carry out energy-based high-frequency resolvent estimates and different from the usual Kawashima type estimates, but weak enough that the range of nonlinear terms is densely contained.

The reduction of nonlinear to spectral stability gives a base for investigation of more general systems such as full (nonisentropic) gas dynamics or (isentropic or nonisentropic) MHD. Our results on uniqueness and universal stability on the other hand are likely accidents of low dimension. For example, the demonstration of unstable large-amplitude boundary layers in [START_REF] Serre | Boundary layer stability in real vanishing viscosity limit[END_REF][START_REF] Zumbrun | Stability of noncharacteristic boundary layers in the standing-shock limit[END_REF] is suggestive via the large-interval length limit from bounded interval toward the half-line, that unstable large-amplitude steady solutions might occur on bounded intervals for polytropic full gas dynamics in some parameter regimes. Definitely, the example of unstable shock waves on the whole line in [START_REF] Barker | Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability[END_REF] together with the asymptotic analysis in [START_REF] Sandstede | Absolute and convective instabilities of waves on unbounded and large bounded domains[END_REF][START_REF] Zumbrun | Stability of noncharacteristic boundary layers in the standing-shock limit[END_REF] of spectra in the whole-line limit shows that unstable steady solutions are possible on an interval for full gas dynamics with an artificial equation of state satisfying all of the usual requirements imposed in standard theory, including existence of a convex entropy, genuine nonlinearity of acoustic modes, etc. Moreover, due to the presence of spectral gap/absence of essential spectra in the bounded-interval problem, differently from the whole-and half-line problems, changes in stability of the type considered in [START_REF] Serre | Boundary layer stability in real vanishing viscosity limit[END_REF][START_REF] Zumbrun | Stability of noncharacteristic boundary layers in the standing-shock limit[END_REF], involving passage of a real eigenvalue through zero, are associated necessarily with bifurcation/nonuniqueness, by Lyapunov-Schmidt or center manifold reduction to the finite-dimensional case. (1) Thus, any such violations of stability should yield also examples of large-amplitude nonuniqueness at the same time. Small-amplitude uniqueness, on the other hand, follows readily by uniqueness of constant solutions, as follows by energy estimates like those here, plus continuity. The investigation of large-amplitude uniqueness and stability for larger systems thus 1 See in particular the center manifold theory for generators of C 0 semigroups in [START_REF] Haragus | Local bifurcations, center manifolds, and normal forms in infinitedimensional dynamical systems[END_REF] or the still more general Fredholm-based Lyapunov-Shmidt reduction of [Mon14, Appendix D] for closed densely defined operators with an isolated crossing eigenvalue, along with the general finite-dimensional bifurcation result of [START_REF] Barker | Convex entropy, hopf bifurcation, and viscous and inviscid shock stability[END_REF]Lemma 3.10]. This is to be contrasted with the case of the whole line discussed in [Zum01, §6.2], for which λ = 0 is embedded in essential spectrum and a crossing eigenvalue at λ = 0 may signal either steady state bifurcation as here or more complicated time-dependent bifurcations involving far-field behavior and solutions of an associated inviscid Riemann problem.

appears to be a very interesting direction for future exploration; likewise, the study of the corresponding multi-D problem, for which existence/uniqueness of small-amplitude solutions has been studied for example in [KK98, [START_REF] Kweon | Compressible Navier-Stokes equations in a bounded domain with inflow boundary condition[END_REF][START_REF] Mucha | Compressible perturbation of Poiseuille type flow[END_REF]. In both 1-and multi-D, a very interesting open problem would be to study the asymptotic structure of solutions in the small-viscosity limit, particularly in the multi-D case analogous to Poiseuille flow.

Notation

In this paper, C(•) denotes a nondecreasing and positive function and C a generic notation whose exact values are of no importance. | | 2 refers to the L 2 -norm on (0, 1) and

| | H n , for n ≥ 1, to the H n -norm. | | ∞ refers to the L ∞ -norm on [0, 1].

Acknowledgment

We would like to thank the anonymous referees for careful reading and for very valuable comments on the manuscript.

2 Existence and uniqueness of steady states

Analytical results

In this part, we prove the following result.

Proposition 2.1. Assume that P is a smooth function. For any (ρ 0 , u 0 , u 1 ), with ρ 0 > 0, u 0 > 0 and u 1 > 0, problem (1)-(2) has a unique steady solution (ρ, û) with ρ > 0.

Proof. A steady solution (ρ, û) of (1)-(2) satisfies

           (ρû) x = 0, ρû 2 + P (ρ) x = ν ûxx , ρ(0) = ρ 0 , û(0) = u 0 , û(1) = u 1 .
Thus, ρû = ρ 0 u 0 and

νρ 0 u 0 ρx = bρ 2 -(ρ 0 u 0 ) 2 ρ -ρ2 P (ρ), ρ(0) = ρ 0 ( 4
)
where b is a constant that has to be determined. We define the map

φ := b ρ(1) - ρ 0 u 0 u 1
where ρ is the unique solution of System (4). Notice that we only define φ when ρ is defined on [0, 1]. Then, we remark that φ(ρ 0 u 2 0 + P (ρ 0 )) = ρ 0 -ρ 0 u 0 u 1 and that φ is increasing. We also remark that if b 1 is in the domain of φ, any b < b 1 is in the domain of φ. Therefore, the domain of φ is an interval containing ρ 0 u 2 0 + P (ρ 0 ). Furthermore, φ is not bounded from above. Otherwise, one can show that ρ is bounded uniformly with respect to b and then that φ(b) > b for b large enough. One can also show that lim b -∞ φ(b) = -ρ 0 u 0 u 1 . Therefore, there exists a unique b such that ρ(1) = ρ 0 u 0 u 1 and ρ > 0.

Notice that a solution of (1) is constant if and only if u 0 = u 1 .

In the following, we study among other things the stability of almost constant steady solutions of (1), by which we mean solutions satisfying

∃ε > 0 , ε 1 and |u 0 -u 1 | ≤ ε. (5)
For this, the following lemma will be useful.

Lemma 2.2. Assume that we are under the assumptions of Proposition 2.1. Then, the unique steady solution (ρ, û) of problem (1)-( 2) is smooth and if u 1 = u 0 we have

ρ > 0 , û > 0, (u 1 -u 0 ) ρx < 0 , (u 1 -u 0 ) ûx > 0, lim u 1 u 0 (|ρ x | ∞ + |û x | ∞ ) = 0.
Proof. The first four inequalities are clear (notice that if

u 0 = u 1 , |û x | > 0)
. The last inequality follows from a comparison argument and the continuity of the map φ.

We denote solutions as compressive when ûx > 0 and expansive when ûx < 0.

Numerical simulations

A steady solution (ρ, û) of (1)-( 2) is characterized by system (4) where b is the unique zero of φ. The numerical computation of such a solution is carried out in two steps:

-We compute b with a Newton's method. We initiate the process with b = b 0 where b 0 = ρ 0 u 2 0 + P (ρ 0 ). Note that for a small viscosity (ν ≤ 1), the initial starting point b = b 0 ceases to be relevant. Thus, in this case, we use a dichotomy method to find a better starting point.

-The solution of system (4) is computed with a four-order Runge-Kutta method.

We display the results of numerical simulations for a monatomic pressure law P (ρ) = ρ 1.4 with ν = 1. Figure 1 represents the expansive solution for u 0 = 2, u 1 = 3 and ρ 0 = 3. Figure 2 represents the compressive solution when u 0 = 1.5, u 1 = 1 and ρ 0 = 2. In order to study the stability of steady states, we linearize system (1) about the steady state (ρ, û). Then, we study the corresponding eigenvalue problem for (r, v) λr + (ρv + ûr) x = 0,

λρv + ρûv + P (ρ)r x + ûx (ûr + ρv) = νv xx , (6)
with

r(0) = v(0) = v(1) = 0. ( 7
)
We define the linear unbounded operator

L (r, v) = -(ρv + ûr) x νv xx -(ρûv + P (ρ)r) x -ûx (ûr + ρv) , (8)
for (r, v) in the domain

D(L) = (r, v) ∈ H 1 × H 2 , r(0) = v(0) = v(1) = 0 and the matrix S = 1 0 0 ρ . ( 9
)
Remark 3.1. For constant steady states, the eigenvalue problem simplifies into

λr + ρv x + ûr x = 0, λρv + ρûv x + P (ρ)r x = νv xx . (10)
In the following, we denote by σ(S -1 L) the spectrum of S -1 L, D(L) in L 2 (0, 1). The following proposition shows that σ(S -1 L) only contains eigenvalues.

Proposition 3.2. The inverse of L exists and is compact and σ(S -1 L) only contains eigenvalues. Furthermore, the spectrum of

S -1 L, D(L) ∩ H 2 × H 3 in the space (r, v) ∈ H 1 , r(0) = v(0) = v(1) = 0 only contains eigenvalues.
Proof. First we show that 0 / ∈ σ(L). For f and g in L 2 (0, 1), we solve

L(ρ, v) = (f, g) with r(0) = v(0) = v(1) = 0.
This leads to the system

       ρv + ûr = - x 0 f (y), νv x = νv x (0) + ρûv - P (ρ) û ρv + x 0 f + x 0 g(y) - x 0 ûx y 0 f (y).
Then, we can solve the second equation with the initial condition v(0) = 0

v(x) = 1 ν x 0 exp 1 ν x y ρû - P (ρ) û ρdz νv x (0) - P (ρ) û y 0 f + y 0 g - y 0 ûx z 0 f dy.
Since v(1) = 0, we can compute v x (0) and it implies that L is invertible. Furthermore, if f and g are bounded in L 2 (0, 1), we get from the previous equality that v x (0) is bounded and that v is bounded in H 1 (0, 1). The first statement follows easily. The second statement follows from similar computations.

Remark 3.3. Notice that (ρ , û ) can not be an eigenfunction of S -1 L for the eigenvalue λ = 0 since it does not satisfy the boundary conditions (7). This differs from the whole line case.

In order to prove the spectral stability of steady solutions, we need high frequency estimates for problem (6)-(7). First, we establish a useful lemma.

Lemma 3.4. For any (r, v) satisfying the boundary conditions (7) and |λ| large enough, we have

|r| 2 2 ≤ C |λ| |r| 2 2 + |v| 2 2 + |(λS -L) (r, v) 1 | 2 2 , |r| 2 2 + |v| 2 2 ≤ C |λ| |r x | 2 2 + |v x | 2 2 + |(λS -L) (r, v)| 2 2 , |v x | 2 2 ≤ C |λ| |r x | 2 2 + |v xx | 2 2 + |(λS -L) (r, v)| 2 H 1 , where r(x) = x 0 r(y)dy. Proof. If we denote (λS -L) (r, v) = (f, g) and f (x) = x 0 f (y)dy, we have      λr + ρv + ûr = f , λr + (ρv + ûr) x = f, λρv + ρûv + P (ρ)r x + ûx (ûr + ρv) = νv xx + g. (11)
Thus, we easily see that

|r| 2 2 ≤ C |λ| |r| 2 2 + |v| 2 2 + |f | 2 2 .
Furthermore, by integrating by parts, we get

|r| 2 2 + ρv 2 2 = 1 |λ| 1 0 rλr + 1 |λ| 1 0 vλρv ≤ C |λ| |r| 2 2 + û |r| 2 1 0 + |v| 2 H 1 + |(f, g)| 2 2
and the second inequality follows from Lemma A.2. Finally, by differentiating the second equation of (11), we obtain

|(ρv) x | 2 2 = 1 |λ| 1 0 (ρv) x λ (ρv) x ≤ C |λ| |r| 2 H 1 + |v| 2 H 2 + [ρv x (νv xx -P (ρ)r x )] 1 0 + |(f, g)| 2 H 1 .
Then, we notice that

νv xx (0) -P (ρ 0 )r x (0) = ρ 0 u 0 v x (0) -g(0), νv xx (1) -P (ρ(1))r x (1) = ρ(1)u 1 v x (1) + P (ρ(1))ρ x (1)r(1) + ûx (1)u 1 r(1) -g(1)
and thanks to Lemma A.2, we get

ρv x νv xx -P (ρ)r x 1 0 ≤ C |v| 2 H 2 + C |r| 2 H 1 + C |g| 2 H 1 .
The result follows easily.

We can now establish a high frequency estimate in H 1 .

Proposition 3.5. Assume that P satisfies (3). There exists a constant α > 0 such that if (λ) > -α and |λ| is large enough,

|(r, v)| 2 H 1 ≤ C |(λS -L) (r, v)| 2 H 1 ×L 2 for any (r, v) ∈ (r, v) ∈ H 1 × H 1 , r(0) = v(0) = v(1) = 0 .
Proof. This proof is based on an appropriate Goodman-type energy estimate. In the following we denote (λS -L) (ρ, v) = (f, g). We define the energy

E (r, v) = 1 2 1 0 φ 1 |r x | 2 + φ 2 |(ρv) x | 2
where φ 1 and φ 2 satisfy

φ 1 > 0 , φ 2 > 0 , φ 1 = P (ρ)φ 2 , 1 2 (ûφ 1 ) x -2û x φ 1 < 0 (2) .
This energy is equivalent to the H 1 -norm by the Poincaré inequality (see Lemma A.1). Then, we compute

2 (λ)E (r, v) = 1 0 φ 1 r x λr x + 1 0 φ 2 (ρv) x λ (ρv) x .
Arguing by density, we assume that (r, v) ∈ H 2 × H 3 . We have

2 (λ)E (r, v) ≤ - 1 0 [φ 1 r x (ûr xx + ρv xx + ûx r x ]) + C |r x | 2 (|r| 2 + |v| H 1 + |f | H 1 ) + 1 0 φ 2 ρv x νv xxx -P (ρ)r xx -ρûv xx + 1 0 φ 2 ρv x g x + 1 0 φ 2 ρx v νv xxx -P (ρ)r xx + C |v| H 1 (|r| H 1 + |v| H 1 + |v xx | 2 ) .
Therefore, we have

2 (λ)E (r, v) ≤ 1 0 -ν ρφ 2 |v xx | 2 + 1 2 (ûφ 1 ) x -2û x φ 1 |r x | 2 + (r x v xx ) (-ρφ 1 + ρP (ρ)φ 2 ) + - 1 2 ûφ 1 |r x | 2 + φ 2 ρv x (νv xx -P (ρ)r x + g) 1 0 + C |r x | 2 (|r| 2 + |v| H 1 + |f | H 1 ) + C |v| H 1 (|r| H 1 + |v| H 1 + |v xx | 2 ) + C |g| 2 |v xx | 2 .
Then, we notice that

u 0 r x (0) = -ρ 0 v x (0) + f (0), νv xx (0) -P (ρ 0 )r x (0) + g(0) = ρ 0 u 0 v x (0), νv xx (1) -P (ρ(1))r x (1) + g(1) = ρ(1)u 1 v x (1) + P (ρ(1))ρ x (1) + ûx (1)u 1 r(1) (12)
and thanks to Lemma A.2, we obtain

- 1 2 ûφ 1 |r x | 2 + φ 2 ρv x (νv xx -P (ρ)r x + g) 1 0 ≤ C |r| 2 |r x | 2 + C |v x | 2 |v x | H 1 + C |f | 2 H 1 .
2 For instance, φ1(0) = 1 and ûφ 1 = 3û φ1 -δ û for δ > 0 small enough.

Then, using the second and the third inequality of Lemma 3.4 and Young's inequality, we can find a constant α > 0, such that for |λ| large enough,

2 (λ)E (r, v) ≤ -α |(r x , v xx )| 2 2 + C |(λS -L) (r, v)| 2 H 1 ×L 2 .
Since E is a norm equivalent to the H 1 -norm, the inequality follows from the Poincaré-Wirtinger inequality on v x .

The linear time evolution problem

In this part, we study the linearization of system (1) about the steady state (ρ, û)

         S r v t -L r v = 0, (r, v) |t=0 = (r 0 , v 0 ) , r(0) = v(0) = v(1) = 0.
We define for k ∈ N the spaces

H k = (r, v) ∈ H k , d l r dx l (0) = d l v dx l (0) = d l v dx l (1) = 0 , for any l < k H 1,0 = (r, v) ∈ H 1 × L 2 , r(0) = 0 .
The main goal of this subsection is to show a linear exponential stability in H 1,0 . This will help us to show the nonlinear exponential stability (see Remark 6.4). The following lemmas show that S -1 L generates a C 0 -semigroup on H k and H 1,0 .

Lemma 3.6. The operator S -1 L, D(L) is closed densely defined on L 2 and generates a C 0 -semigroup. Similarly S -1 L, H k ∩ H k+1 × H k+2 is closed densely defined on H k and generates a C 0 -semigroup.

Proof. The proof is similar to the proof of Proposition 2.2 in [START_REF] Mascia | Pointwise green function bounds for shock profiles of systems with real viscosity[END_REF]. In the following we denote (f, g) = (λ -S -1 L)(r, v). For λ > 0 and (r, v) ∈ D(L)

λ |r| 2 2 + λρv, 1 ρ v 2 = -(ρr x , r) 2 -(ûv x , r) 2 + (f, r) 2 + ν (v xx , v) 2 -P (ρ)r x , v 2 -(ρûv x , r) 2 + (g, v) 2 + C |(r, v)| 2 2 ≤ -ν |v x | 2 2 + (f, r) 2 + (g, v) 2 + C |(r, v)| 2 (|(r, v)| 2 + |v x | 2 )
where we have integrated by parts and we have noticed a good sign for |r(1)| 2 . Applying Young's inequality, there exists a constant C 0 > 0 such that

λ |r| 2 2 + λ |v| 2 2 ≤ C 0 |(r, v)| 2 2 + |(r, v)| 2 |(f, g)| 2 . ( 13
)
Dividing by |(r, v)| 2 , we get

λ |(r, v)| 2 ≤ C 0 |(r, v)| 2 + (λ -S -1 L)(r, v) 2 , hence for λ > C 0 |(r, v)| 2 ≤ 1 λ -C 0 (λ -S -1 L)(r, v) 2 . (14)
Similarly, we have for (r,

v) ∈ H 1 ∩ H 2 × H 3 λ |r x | 2 2 +λ |v x | 2 2 = (f x , r x ) 2 +(g x , v x ) 2 -(ûr xx , r x ) 2 +(v xxx , ν ρ v x ) 2 +C (|v xx | 2 + |(r x , v x )| 2 ) |(r, v)| H 1
and there exists a constant C 1 > 0 such that for any λ > 0

λ |(r x , v x )| 2 2 ≤ C 1 |(r, v)| 2 H 1 + |(r x , v x )| 2 |(f x , g x )| 2 . ( 15
)
Summing ( 13) and ( 15), and noting that

|(r, v)| 2 |(f, g)| 2 + |(r x , v x )| 2 |(f x , g x )| 2 ≤ |(r, v)| H 1 |(f, g)| H 1
by Cauchy-Schwarz' inequality, we get

λ |(r, v)| 2 H 1 ≤ (C 0 + C 1 ) |(r, v)| 2 H 1 + |(r, v)| H 1 (λ -S -1 L)(r, v) H 1 hence for λ > C 0 + C 1 , |(r, v)| H 1 ≤ 1 λ -C 0 -C 1 (λ -S -1 L)(r, v) H 1 . (16)
Since we know from Proposition 3.2 that the spectrum of S -1 L only contains eigenvalues, the inequalities ( 14) and ( 16) give resolvent bounds and it shows that S -1 L generates a C 0 -semigroup on L 2 and H 1 by the Hille-Yosida theorem (see also [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). The case k ≥ 2 is a small adaptation of the previous estimates.

In the following, we denote this C 0 -semigroup by e tS -1 L . Lemma 3.7. There exists a constant ω > 0 such that for any (r 0 , v 0 ) ∈ H 1,0

|e tS -1 L (r 0 , v 0 )| H 1 ×L 2 ≤ e ωt |(r 0 , v 0 )| H 1 ×L 2 . ( 17
)
Furthermore, e tS -1 L is a C 0 -semigroup on H 1,0 .

Proof. We argue by density and we take (r 0 , v 0 ) ∈ H 2 . In the following, we denote (r(t), v(t)) = e tS -1 L (r 0 , v 0 ). Notice that we have r(t, 0) = v(t, 0) = v(t, 1) = r x (t, 0) = 0 for any t ≥ 0. We define the energy

E(r, v) = A 2 |r| 2 2 + (ρv, v) 2 + v, ρ2 r x 2 + ν 2 |r x | 2 2 .
In the following we take A > 0 large enough. In particular, E is equivalent to the

H 1 × L 2 -norm. We get d dt E(r, v) ≤ νA (v xx , v) 2 + AC |(r, v)| 2 |(r, v)| H 1 + (νv xx , ρr x ) 2 -v, ρ3 v xx 2 -v, ρ2 ûr xx 2 -(ν ρv xx , r x ) 2 -ν (ûr xx , r x ) 2 + C (|v| 2 + |r x | 2 ) |(r, v)| H 1 ≤ -Aν |v x | 2 2 + v x , ρ3 v x 2 + ν 2 (û x r x , r x ) 2 + C (A |(r, v)| 2 + |r x | 2 ) |(r, v)| H 1
where we have used cancellation of the highest-order terms ± (νv xx , ρr x ) 2 , we have integrated by parts and we have noticed a good sign for r x (1) 2 . Then, applying Young's inequality, we obtain for some ω > 0 large enough,

d dt E(r, v) ≤ 2ω |r| 2 2 + |v| 2 2 + |r x | 2 2 ≤ 2ωE.
The inequality (17) follows easily. Finally for any U 0 ∈ H 1,0 and

V 0 ∈ H 1 e tS -1 L U 0 -U 0 H 1 ×L 2 ≤ e tS -1 L V 0 -V 0 H 1 + (1 + e ωt ) |U 0 -V 0 | H 1 ×L 2
and continuity at t = 0 follows since e tS -1 L is continuous at t = 0 on H 1 and H 1 is dense in H 1,0 .

The following proposition gives linear exponential stability under the assumption of a spectral gap. It is the main result of this subsection.

Proposition 3.8. Assume that P satisfies (3). Assume that there exists a constant α > 0, such that σ(S -1 L) < -α. Then, there exists θ and C, 0 < θ < α, such that for any (r 0 , v 0 ) ∈ H 1,0

e tS -1 L (r 0 , v 0 ) H 1 ×L 2 ≤ Ce -θt |(r 0 , v 0 )| H 1 ×L 2 .
Proof. If (r 0 , v 0 ) ∈ H 1 , Proposition 3.5 gives two constants C and θ with 0 < θ < α such that for any

λ ∈ C satisfying R(λ) = -θ λ -S -1 L -1 (r 0 , v 0 ) H 1 ×L 2 ≤ C |(r 0 , v 0 )| H 1 ×L 2 .
The result follows by density and Prüss' theorem (see for instance [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Yosida | Functional analysis[END_REF]).

4 Spectral stability

Constant and almost constant states

First, we study the spectral stability of constant states.

Proposition 4.1. Assume that (ρ, û) is a constant solution of (1) and that P satisfies (3). Then, there exists α > 0, σ(S -1 L) ≤ -α.

Proof. Computing ((10)

1 , P (ρ)r) L 2 + ((10) 2 , ρv) L 2 , we get (λ) P (ρ)r 2 2 + ρv 2 2 + ν |v x | 2 2 + 1 2 P (ρ(1))u 1 |r(1)| 2 = 0.
Thus, (λ) < 0. The result follows from Proposition 3.2 and Proposition 3.5.

We can now establish the main proposition of this part. We recall that (ρ, û) is a steady solution of (1)-(2). We introduce the Evans function associated to (ρ, v)

D [ρ 0 , u 0 , u 1 ] (λ) = v(1),
where (ρ, v) satisfies the ordinary differential equation

     r x = - λr + (ρv) x + ûx r û , νv xx = λρv + (ρûv) x + P (ρ)ρ x r -P (ρ) λr + (ρv) x + ûx r û + ûx (ûr + ρv) (18) with r(0) = v(0) = 0 , v (0) = 1.
One can easily show that D [ρ 0 , u 0 , u 1 ] (λ) = 0 if and only if λ is an eigenvalue of (6).

We now establish the spectral stability of almost constant steady solutions of (1).

Proposition 4.2. Assume that P satisfies (3). Let (ρ, û) be the unique steady solution of (1)-(2). Let ε > 0 be small enough. Then an eigenvalue λ of (6)-( 7) has a negative real part.

Proof. By Proposition 3.5, problem ( 6)-( 7) does not have any eigenvalue of nonnegative real part outside a compact set K. Furthermore, from Proposition 4.1, D [ρ 0 , u 0 , u 0 ] does not have any zero inside K ∩ { > 0}. Since the Evans function D depends continuously on the boundary conditions, D [ρ 0 , u 0 , u 1 ] never vanishes inside K ∩ { > 0} for ε small enough.

About general steady states

In the previous part, we only prove the spectral stability of almost constant states. In this part, we show some theoretical and numerical arguments that support the spectral stability of any steady states.

We know from previous works that the stability index criterion is a necessary condition for the spectral stability (see for instance [START_REF] Alexander | A topological invariant arising in the stability analysis of travelling waves[END_REF][START_REF] Pego | Asymptotic stability of solitary waves[END_REF][START_REF] Gardner | The gap lemma and geometric criteria for instability of viscous shock profiles[END_REF]). The stability index criterion states that sgn (D [ρ 0 , u 0 , u 1 ] (0)) sgn (D [ρ 0 , u 0 , u 1 ] (+∞)) = 1.

The following proposition shows that this criterion is satisfied.

Proposition 4.3. For all steady states of problem (1)-(2), the stability index criterion is satisfied.

Proof. First we compute sgn (D [ρ 0 , u 0 , u 1 ] (0)). Proceeding as in Proposition 3.2, we get the following system Then sgn v(1) = sgn v x (0) = 1. Secondly, we compute D [ρ 0 , u 0 , u 1 ] (+∞). We have

         ρv + ûr = 0, ρûv + ρ û P (ρ)v = νv x -νv x (0), r(0) = v(0) = 0 , v x (0) = 1,
     λr + ûr x = f, νv xx = λρv + P (ρ)r x + g, r(0) = v(0) = 0 , v x (0) = 1, ( 19
)
where |f | 2 + |g| 2 ≤ C (|r| 2 + |v| 2 + |v x | 2 )
. By solving the first equation of system (19) we get for λ large enough

|r| 2 ≤ C √ λ (|v| 2 +| v x | 2 ) .
We can rewrite the second equation of system (19) as

νv xx = λρv + P (ρ)r x + g with v(0) = 0 , v x (0) = 1, where | g| 2 ≤ C (|v| 2 + |v x | 2 )
. Then, we consider for s ∈ [0, 1] the equation

νw xx = λ ((1 -s)ρ + s) w + (1 -s)P (ρ)r x + (1 -s) g with w(0) = w(1) = 0.
Multiplying by w and integrating, we notice that when λ is large enough the only solution of this equation is w = 0. Therefore, for λ large enough, we define z solution of

νz xx = λz with z(0) = 0 , z x (0) = 1.
and v(1) and z(1) agree in sign. It follows that sgn v(1) = sgn z(1) = sgn z x (0) = 1.

This proposition also shows that problem (6)-( 7) has an even number of nonstable eigenvalues, i.e. eigenvalues with a nonnegative real part (see [START_REF] Alexander | A topological invariant arising in the stability analysis of travelling waves[END_REF][START_REF] Pego | Asymptotic stability of solitary waves[END_REF][START_REF] Gardner | The gap lemma and geometric criteria for instability of viscous shock profiles[END_REF]).

Thanks to Lemma 3.5, we can numerically check that σ(S -1 L) does not contain nonstable eigenvalues. Such verifications have for instance been done on the whole line (see [START_REF] Barker | Stability of viscous shocks in isentropic gas dynamics[END_REF]).

In the following, we display some numerical simulations for a monatomic pressure law P (ρ) = ρ 1.4 . For any λ, we can compute the associated Evans function thanks to system (18). We use a Runge Kutta 4 scheme. For each value of u 0 , u 1 , ρ 0 and ν, we compute the Evans function along semi-circular contours of radius M (see Figure 3). We choose M large enough such that our domain contains the half ball of Lemma 3.4. Figure 4 represents the image of the contour with M = 10, ν = 1, u 0 = 3 2 , u 1 = 1 and ρ 0 = 2. Figure 5 represents the image of the contour with M = 10, ν = 0.1, u 0 = 3 2 , u 1 = 1 and ρ 0 = 2. We can see on these examples that the winding number of these graphs are both zero. Several computations have been performed for other values of the parameters ν ∈ [0.1, 10], u 0 ∈ [1, 10], u 1 ∈ [1, 10] and ρ 0 ∈ [1, 10]. We could not find any nonstable eigenvalues.

u 0 = 3 2 , u 1 = 1, ρ 0 = 2.
u 0 = 3 2 , u 1 = 1, ρ 0 = 2.

Local existence

In this section, we state a local wellposedness result for problem (1)-(2) (see e.g. [START_REF] Matsumura | Initial-boundary value problems for the equations of motion of general fluids[END_REF][START_REF] Matsumura | Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas[END_REF]).

Proposition 5.1. Let ρ 0 > 0, u 0 > 0 and u 1 > 0. Assume that P satisfies (3). Let (ρ ini , u ini ) ∈ H 1 satisfying the boundary conditions (2) and ρ ini > 0. Then, there exists a time T > 0 such that problem (1)-(2) has a unique solution (ρ, u) in C [0, T ]; H 1 (0, 1) with

sup [0,T ] |(ρ, u) (t)| H 1 ≤ 2 |(ρ ini , u ini )| H 1 and ρ(t, x) ≥ ρ ini (x) 2 , 0 ≤ t ≤ T , 0 ≤ x ≤ 1.

Nonlinear stability

For a solution (ρ, u) of problem (1)-(2), we define (r, v) = (ρ -ρ, u -û). We notice that (r, v) satisfies the boundary conditions ( 7) and (L and S are defined in Section 3.1)

1 0 0 ρ r v t -L r v = -(rv) x -(ûv) x r -ρvv x -vrv x -(P (ρ) -P (ρ) -P (ρ)r) x . (20)
Then, we get

S r v t -L r v = N with r(t, 0) = u(t, 0) = u(t, 1) = 0 (21)
where

N 1 = -(rv) x and N 2 = - ρ ρ + r (ûv) x r + ρvv x + vrv x + P (r + ρ) -P (ρ) -P (ρ)r x + r ρ + r ρûv + P (ρ)r x + ûx (ûr + ρv) -νv xx .
Notice that N 1 (t, 0) = N 2 (t, 0) = 0 and that N 2 (t, 1) = -[ûv x r + (P (ρ + r) -P (ρ) -P (ρ)r) (ρ + r) x + P (ρ)rr x ] (t, 1).

The following proposition is a nonlinear damping estimate.

Proposition 6.1. Let T > 0 and consider a solution (r, v) ∈ C [0, T ]; H 1 of (21) on [0, T ]. Assume that P satisfies (3) and that there exists ε > 0 small enough such that

sup [0,T ] |(r, v)(t)| H 1 ≤ ε.
Then, there exists some constants C > 0 and θ 0 > 0 such that for all 0 ≤ t ≤ T and any

θ ≤ θ 0 , |(r, v)(t)| 2 H 1 ≤ Ce -θt |(r, v)(0)| 2 H 1 + C t 0 e -θ(t-s) |(r, v)(s)| 2 2 ds. Furthermore, if (r, v) ∈ C [0, T ]; H 2 × H 3 ∩ C 1 [0, T ]; H 1 and for ε small enough sup [0,T ] |(r, v)(t)| H 2 ×H 3 ≤ ε,
there exists some constants C > 0 and θ 1 > 0, for any 0 ≤ t ≤ T and θ < θ 1

|(r, v)(t)| 2 H 2 ×H 3 ≤ Ce -θt |(r, v)(0)| 2 H 2 ×H 3 + C t 0 e -θ(t-s) |(r, v)(s)| 2 2 ds.
Proof. This proof is based on an appropriate Goodman-type energy estimate and is similar to the proof of Proposition 3.5. We define the energy equivalent to the H 1 -norm (by the Poincaré inequality A.1)

E (r, v) = 1 2 1 0 φ 1 |r x | 2 + φ 2 |(ρv) x | 2
where φ 1 and φ 2 satisfy

φ 1 > 0 , φ 2 > 0 , φ 1 = P (ρ)φ 2 , 1 2 (ûφ 1 ) x -2û x φ 1 < 0.
Then, after some computations, we obtain

d dt E (r, v) ≤ -ν 1 0 ρφ 2 |v xx | 2 + 1 2 (ûφ 1 ) x -2û x φ 1 |r x | 2 + r x v xx (-ρφ 1 + ρP (ρ)φ 2 ) + - 1 2 ûφ 1 |r x | 2 + φ 2 ρv x (νv xx -P (ρ)r x ) 1 0 + 1 0 φ 1 r x (N 1 ) x + φ 2 (ρv) x (N 2 ) x + C |r x | 2 (|r| 2 + |v| H 1 ) + C |v| H 1 (|r| H 1 + |v| H 1 + |v xx | 2 ) .
Integrating by parts and using Lemma A.2 we get

1 0 φ 1 r x (N 1 ) x + φ 2 (ρv) x (N 2 ) x ≤ [φ 2 ρv x N 2 ] 1 0 + C |(r, v)| 2 H 1 (|(r, v)| H 1 + |v xx | 2 ).
Then, since N 1 (t, 0) = N 2 (t, 0) = 0, we have u 0 r x (t, 0) = -ρ 0 v x (t, 0), νv xx (t, 0) -P (ρ 0 )r x (t, 0) = ρ 0 u 0 v x (t, 0), νv xx (t, 1) -P (ρ(1))r x (t, 1) + N 2 (t, 1) = ρ(1)u 1 v x (t, 1) + P (ρ(1))ρ x (1)r(t, 1) + ûx (1)u 1 r(t, 1).

Finally, thanks to the previous boundary equalities, Lemma A.2, Lemma A.3, Young's inequality and the fact that |(r, v)| H 1 is small enough, we obtain

d dt E (r, v) ≤ -θ 0 |(r x , v xx )| 2 2 + C |(r, v)| 2 2 .
The first inequality easily follows from the Poincaré-Wirtinger inequality. Similarly, since (r t , v t ) satisfies the boundary conditions (7), we get for ε and θ 0 small enough

d dt E (r t , v t ) ≤ -θ 0 |(r tx , v txx )| 2 2 + C |(r t , v t )| 2 2 .
Then, using (20) we notice that

|(r t , v t )| 2 ≤ C |(r, v)| H 1 + C |v xx | 2 .
Therefore using the Poincaré inequalities, we get for δ and θ 1 small enough

d dt (E (r, v) + δE (r t , v t )) ≤ -θ 1 |(r x , v x )| 2 2 + |(r tx , v tx )| 2 2 + C |(r, v)| 2 2 (22)
and the result easily follows from the fact that

|(r, v)| H 2 ×H 3 ≤ C |(r, v)| H 1 + C |(r t , v t )| 2 .
Remark 6.2. By taking further time-derivatives, we could obtain an estimate similar to (22) in an arbitrarily high-regularity Sobolev space of mixed type H r × H s , with s ∼ 2r as r → ∞. This observation repairs a minor error in [START_REF] Costanzino | Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers[END_REF], citing an estimate with r = s.

We can now state the main result of this paper.

Theorem 6.3. Let ρ 0 > 0, u 0 > 0 and u 1 > 0. Let (ρ, û) be the unique steady solution of problem (1)-(2). Assume that P satisfies (3). Assume that there exists α > 0 such that (σ(S -1 L)) < -α. Then, there exists ε > 0 and θ > 0, for any (ρ ini , u ini ) ∈ H 2 × H 3 satisfying the boundary conditions (2), the compatibility conditions

(ρ ini u ini ) x (0) = 0, ρ ini u 2 ini + P (ρ ini ) -νu ini x x (0) = 0, ρ ini u 2 ini + P (ρ ini ) -νu ini x x (1) = 0 (23) and |(ρ ini , u ini ) -(ρ, û)| H 2 ×H 3 ≤ ε,
the unique solution (ρ, u) of problem (1)-(2) with the initial condition (ρ ini , u ini ) satisfies

|(ρ, u) (t) -(ρ, û)| H 2 ×H 3 ≤ C |(ρ ini , u ini ) -(ρ, û)| H 2 ×H 3 e -θt .
Remark 6.4. As we will see in the proof, since we do not know if N 2 (t, 1) = 0, the only way to use a linear damping estimate is to work in L 2 for the v component. That is why in Proposition 3.8 we used H 1 × L 2 and not H 1 . Notice also that we impose the compatibility conditions (23) in order to get enough regularity.

Proof. We denote by U (t, x) = (ρ, u) (t) -(ρ, û) (t, x). Let T be the existence time of Proposition 5.1. The Duhamel formulation of Equation ( 21) is, for 0 ≤ t ≤ T ,

U (t) = e tS -1 L U (0) + t 0 e (t-s)S -1 L S -1 N (s)ds. Noticing that N ∈ (r, v) ∈ H 1 × L 2 , r (
0) = 0 and that N contains at least quadratic terms, Proposition 3.8 gives the existence of θ > 0

|U (t)| 2 ≤ |U (t)| H 1 ×L 2 ≤ Ce -θt |U (0)| H 1 ×L 2 + t 0 e -θ(t-s) |U (s)| 2 H 2 C (|U (s)| H 2 ) ds.
Then, the equality (20) gives

|U (t)| 2 ≤ Ce -θt |U (0)| H 1 ×L 2 + t 0 C (|U (s)| H 2 ) e -θ(t-s) |U (s)| 2 H 1 + |U t (s)| 2 H 1 ds.
Furthermore, the compatibility conditions (23) imply that U ∈ C [0, T ]; H 2 × H 3 . Therefore, we can use the nonlinear damping estimate of Proposition 6.1 and by Proposition 5.1 the H 2 -norm of U is controlled by the initial condition. We get for ε and θ > 0 small enough

|U (t)| 2 ≤ C |U (0)| H 2 ×H 3 (1 + t)e -θt |U (0)| H 2 ×H 3 + t 0 (t -s)e -θ(t-s) |U (s)| 2 2 ds . Denoting ζ 0 (t) = sup [0,t] e θ 2 s |U (s)| 2 , we obtain that for 0 ≤ t ≤ T ζ 0 (t) ≤ C |U (0)| H 2 ×H 3 ε + ζ 0 (t) 2 . Furthermore,
With this additional assumption, we can establish a high frequency estimate in L 2 .

Proposition 7.1. Assume that P satisfies (3) and that Condition (24) is satisfied. There exists a constant α > 0 such that if (λ) > -α and |λ| is large enough,

|(ρ, v)| 2 2 ≤ C |(λS -L) (ρ, v)| 2 2 ,
for any (ρ, v) satisfying the boundary conditions (7).

Proof. This proof is based on an appropriate Goodman-type energy estimate. In the following we denote (λS -L) (ρ, v) = (f, g). We define the following energy

E (r, v) = 1 2 1 0 φ 1 |r| 2 + φ 2 ρ |v| 2
where φ 1 and φ 2 satisfy

φ 1 > 0 , φ 2 > 0 , ρφ 1 = P (ρ)φ 2 .
Then, we compute

2 (λ)E (r, v) = 1 0 φ 1 rλr + 1 0 φ 2 ρvλv .
After some computations we get

2 (λ)E (r, v) ≤ 1 0 -νφ 2 |v x | 2 + 1 2 û2 φ 1 û x |r| 2 + (rv x ) [P (ρ)φ 2 -ρφ 1 ] + C |(f, g)| 2 |(r, v)| 2 + 1 0 [ν(φ 2 ) xx -2φ 2 ûx ρ + (φ 2 ) x ρû] |v| 2 2 + (vr) [(φ 2 ) P (ρ) -φ 1 ρx -φ 2 ûx û].
Then, we separately consider the three situations ûx > 0 (compressive solution), ûx = 0 (constant solution) and ûx < 0 (expansive solution).

-If ûx > 0, we take φ 2 = 1, φ 1 = P (ρ) ρ and we get û2 φ 1 û x = û(φ 1 ) x -ûx φ 1 = P (ρ)ρ x û ρ < 0 , ν(φ 2 ) xx -2φ 2 ûx ρ + (φ 2 ) x ρû = -2û x ρ < 0.

-If ûx = 0, we take φ 1 = P (ρ) -βx, φ 2 = ρ -β ρ P (ρ) x with β > 0 small enough, and we get û(φ 1 ) x -ûx φ 1 = -β û < 0 , ν(φ 2 ) xx -2φ 2 ûx ρ + (φ 2 ) x ρû = -β ρ2 û P (ρ) < 0.

-If ûx < 0, we take φ 2 (x) = √ M -2x, M > 2 and φ 1 (x) = P (ρ) ρ φ 2 (x) and thanks to Condition (24) we get û(φ 1 ) x -ûx φ 1 = φ 2 ρ P (ρ)û

P (ρ) P (ρ) ρx - 1 M -2x < 0, ν(φ 2 ) xx -2φ 2 ûx ρ + (φ 2 ) x ρû ≤ φ 2 û 2ρ x - ρ M -2x < 0.
Moreover, in any case, we have (denoting r(x) = Thanks to this L 2 high frequency estimate, we can improve Proposition 3.8. Under the assumption that Re σ(S -1 L ) ≤ -α < 0, we get e tS -1 L (r, v) Finally, applying the Duhamel formulation in L 2 , we obtain the following theorem.

Theorem 7.2. Let ρ 0 > 0, u 0 > 0 and u 1 > 0. Let (ρ, û) be the unique steady solution of problem (1)-(2). Assume that P satisfies (3) and that Condition (24) is satisfied. Assume that there exists α > 0 such that (σ(S -1 L)) < -α. Then, there exists ε > 0 and θ > 0, for any (ρ ini , u ini ) ∈ H 1 × H 2 satisfying the boundary conditions (2) and

|(ρ ini , u ini ) -(ρ, û)| H 1 ×H 2 ≤ ε,

A L ∞ estimates and interpolation

In this appendix, we recall some basic results about Sobolev spaces in a bounded domain.

The first lemma is a Poincaré inequality.

Lemma A.1. For any f ∈ H 1 (0, 1) with f (0) = 0, we have

|f | 2 ≤ 2 |f x | 2 .
Proof. For any x, y ∈ [0, 1], we have

f (x) 2 = f (y) 2 + 2 x y f (z)f (z)dz. ( 25
)
Since f (0) = 0, we choose y = 0 and we obtain the result by integrating over x.

The following lemma allows us to control boundary terms and L ∞ -norms by appropriate Sobolev norms.

Lemma A.2. For any f ∈ H 1 (0, 1), we have

|f | ∞ ≤ 2 |f | 2 |f x | 2 , if f (0) = 0, |f | ∞ ≤ |f | 2 + 2 |f | 2 |f x | 2 .
Proof. It is a direct consequence of the equality (25).

We also have the following derivative-interpolation theorem.

Lemma A.3. For any v ∈ H 2 (0, 1),

|v x | 2 2 ≤ C |v| 2 2 + C |v| 2 |v xx | 2 .
Proof. Integrating by parts, we get

1 0 v 2 x dx = - 1 0 vv xx dx + [vv x ] 1 0 .
The result follows from the previous lemma.

Figure 1 :

 1 Figure 1: A steady expansive solution of (1). Left: ρ ; Right: u

  dy v x (0).

Figure 3 :

 3 Figure 3: Contour in the complex plane.

Figure 4 :

 4 Figure 4: Image of a contour mapped by the Evans function. ν = 1,u 0 = 3 2 , u 1 = 1, ρ 0 = 2.

Figure 5 :

 5 Figure 5: Image of a contour mapped by the Evans function. ν = 0.1,u 0 = 3 2 , u 1 = 1, ρ 0 = 2.

  (φ 2 ) x P (ρ) -φ 1 ρx -φ 2 ûx û ≤ C |r| 2 |v| H 1 .Thus, using the first inequality of Lemma 3.4, we can find a constant α > 0, for |λ| large enough,2 (λ)E (r, v) ≤ -α |(r, v)| 2 2 + C |(λ -L) (ρ, v)| 2 2, and the inequality follows.

2≤

  Ce -αt |(r, v)| 2 .Furthermore, thanks to the previous appropriate Goodman-type estimate, we can improve the nonlinear damping estimate in Proposition 6.1.If (r, v) in C [0, T ]; H 2 × H 1 is a solution of (21) [0, T ] and sup [0,T ] |(r, v)(t)| H 1 ×H 2 ≤ ε for ε small enough, we have |(r t , v t )(t)| 2 2 ≤ Ce -θt |(r, v)(0)| 2 H 1 ×H 2 + C t 0e -θ(t-s) |(r, v)(s)| 2 2 ds.

  the unique solution (ρ, u) of problem (1)-(2) with the initial condition (ρ ini , u ini ) satisfies|(ρ, u) (t) -(ρ, û)| H 1 ×H 2 ≤ C |(ρ ini , u ini ) -(ρ, û)| H 1 ×H 2 e -θt .

  denoting ζ 1 (t) = sup

				[0,t]
	P (y) P (y)	<	2 y	and ρx <	1 4	ρ if ûx < 0 (small expansive solutions).

e θ 2 s |U (s)| H 1 + e θ 2 |U t (s)| H 1 and using Proposition 6.1, ζ 1 is also controlled on [0, T ]. Finally, if ε is small enough, we can take T = +∞ and ζ 1 is bounded on R + .

7 An improvement in some situations

The main result of this paper, Theorem 6.3, states that spectrally stable steady states are stable in H 2 × H 3 . In this part, we prove that under more restrictive conditions, we can state a stability result in H 1 × H 2 . To achieve that, we add another assumption P > 0 if ûx > 0 (compressive solutions),


Annette Hermanns

Arno Gramatke

Klaus Henning

Implementing Blended Learning for 1000 Mechanical Engineering Students

Keywords: Blended Learning, Large Groups, VLE

The module Techniques of Programming of the RWTH Aachen University is one of five case studies within the EU funded project Blend-XL -Finding a Balance in Blended Learning with eXtra Large Students Groups. During the project the module changed from a face-to-face module into a blended learning one. Therefore a blended learning concept has been developed which includes a special virtual learning environment. With the help of the blended learning concept the main problem areas of this module, the different previous knowledge of the students and the motivation of them, have been faced. The first research cycle of the case study has been implemented. This paper describes the corresponding evaluation results and the outlook to the second cycle.

The Project Blend-XL

Teaching large student groups with more than 100 students is especially in the first year of study a common phenomenon. In the core modules of the study the large group is often taught in big lecture theatres. This leads to a lack of communication between the teacher and the students, which is often found de-motivating. As a result, the exam results have a lower success rate and the attrition rate is higher than in small-scale education [1]. The idea behind the project Blend-XL -Finding a Balance in Blended Learning with eXtra Large Student Groups is that the effective use of ICT can make the large-scale education more motivating and more personal. Designing pedagogical models for learning situations with blended learning -a mix of face-to-face and online activities -is the main focus of the project. The project started in October 2005 and is funded for three years by the Socrates Minerva Programme of the EU. The output of the project will be five blended courses, an educational design model for blended learning situations, scenarios and cases from educational practice, a toolkit for blended learning, a digital handbook for teachers and a virtual centre of expertise on blended learning. All the outputs are focused on large student groups. In the project five European partners are involved, the RWTH Aachen University in Germany, the Delft University of Technology in the Netherlands, the Academy of Humanities and Economics in Lodz in Poland, the University of Zilina in Slowakei and the University of Maribor in Slowenia. In the project the action research methodology is used [START_REF] O'brian | An Overview of the Methodological Approach of Action Research[END_REF]. The case studies will perform two cycles during the project, so that the experiences of the first cycle can directly improve the second cycle and the development of the pedagogical model is based on many experiences. In the following the first cycle of the case study Techniques of Programming of the RWTH Aachen University is described.

Case Study Aachen Techniques of Programming

Framework

The course Computer Science in Mechanical Engineering is a course for students in mechanical engineering in their first and second semester. It starts with the module Techniques of Programming (3 ECTS) in the first semester. In the second semester a lecture (3 ECTS), a practical class (1.5 ECTS) and a Project Work (3 ECTS) are following and are taking place in parallel. The module Techniques of Programming is the basis for the module Project Work. Both modules are mandatory in order to be allowed to take part in the written exam. About 1000 students attend the course [START_REF] Henning | Blended Learning for Students of Mechanical Engineering[END_REF]. Before the project started the module Techniques of Programming has been a face-to-face module in which the first semester students learn the basics of structured programming with the aid of the programming language C++. After an introduction into the working environment and the necessary tools, e. g. editing tools, the following questions are tackled: The students have got small theoretical inputs and gained practical experience by completing a range of tasks from simple exercises on a wide range of subjects to more complex problems later on. The students worked with a paper handout which includes the slides of the theoretical inputs and the exercises. The module is taking place in a computer pool, 160 computers are situated in four rooms for the students to work with. Linux is used as the operating system and all of the computers have internet access. To support the students with their work there are two student tutors in each room and a faculty tutor who oversees the entire computer pool.

Problem Areas

The main problem areas of the module Techniques of Programming are:

• Different previous knowledge: The participating students have different levels of previous knowledge. Some of them are already able to write a complex computer program, some of them have no experience at all. Therefore the module Techniques of Programming is boring for the first group and overstrains the second group. • Motivation: The module is mandatory. The students have to attend each module unit.

But they only have to attend and not to successfully end tasks. This and the above mentioned problem of different levels of previous knowledge can lead to unmotivated students. Therefore with the beginning of the project Blend-XL a blended learning concept has been developed in order to face the above mentioned problems.

Blended Learning Concept

The content and the structure of the module should not be changed, because the content is relevant for the module Project Work in the second semester and the structure is well tried in the last years. Because of the different level of previous knowledge of the students the decision was made that the theoretical inputs would be replaced with a virtual learning environment (VLE) and the exercises would be extended in number and in skill levels. Like this the students can work through the theory in their own way. Those who already know a lot can do it fast and the others have the possibility to read parts e. g. twice. The first group also has the possibility to do more exercises on a more difficult skill level than the second group. The VLE would also include multiple choice tests for each chapter. Like this the students can check, if they understand the main aspects of the chapter and can reread special parts if necessary. The used VLE has been developed from a student group with the focus on the special demands of the module. For this software different software development tools like PHP, mySQL, Macromedia Flash, AJAX and MediaWiki have been used [START_REF] Hermanns | ClaRa (C++ Learning at RWTH Aachen) Change from classical teaching to e-learning[END_REF]. As you can see in Figure 2-1 the VLE includes a menu with the chapters and the corresponding exercises and tests. One has the possibility to implement graphics, flash animations and videos.

Implementation

The existing content of the module has been worked over textual and graphical. As a result the complete theory of the course is now available in small chapters in the VLE. The exercises have been extended and divided into those, which all students should solve and those, which only the faster students could solve. The content, the tests and the exercises have been quality checked in two rounds in order to detect write errors, correctness and solvability. The whole student group involved in the case study was 480 students, which has been divided in three groups of 160 students, which are arranged one after the other in the semester as usual for this module. 120 of 160 students of the first group have had a normal face-to-face course. The rest of the group has been taught with the new blended learning concept. The students worked themselves through the theory with the help of the new VLE. The two coaching tutors were there to answer questions. Subsequent to the theory chapters the students solved the corresponding exercises themselves. Again the tutors were available to help and to answer questions. After the main part of the students has solved an exercise, the corresponding solution has been presented in detail. In this presentation the focus lies on the main questions and problems of the students during solving the exercise. The first 40 students mainly liked the blended learning approach. Therefore the second and the third group have been taught completely with the blended learning concept.

Evaluation

The evaluation has been undertaken with the help of different methods:

• "observing" the students behavior during the module,

• asking for direct feedback of the students via face-to-face or e-mail,

• discussions during regular meetings with the involved teachers,

• a teacher questionnaire and a cross-case questionnaire, which have been developed in the project Blend-XL.

The students set about the task in different ways: Working through one theory chapter before starting with the exercises or the test, starting with the test in order to see, which part of the chapter they have to read, starting with the exercises and reading the theory parts, if they were not able to solve it, or starting with the exercise and asking the tutors when they had problems in solving the task. This shows what different ways of learning the students have and that the use of a VLE gives the possibility to consider many of them.

In order to improve the VLE the students were asked to give feedback, what they liked or disliked at the VLE. Like this some writing errors in the textual parts have been filtered out.

Beside of this the only feedback was, that most of them like working with the VLE. From this we learnt that getting improvement for our VLE could only be reached by a special questionnaire, because in such a large group nobody feels really responsible for giving feedback.

In the cross-case questionnaire 216 of 360 students took part, of which 88% were male. Most of the participants were between 18 and 23 years old. In the questionnaire the following areas are treated: General questions, preparatory advice and course components, online materials and resources, teacher and peer support, group work (if provided), study time, the online learning environment as well as the programme as a whole. About 44% of the students say that it is important to have the possibility of time-and-place independent work and study. The working with the VLE assess 47.1% of the students as a positive experience, 41.4% are not sure and only 11.5% assess it as a negative experience. This might go hand in hand with the assess of the information and advice they received before the course started in terms of course description, explanation concerning what blended learning is, rationale for the way the blend was put together as well as the role and usage of blended learning tools. Here the students rate on a five point Likert scale from not at all to very mainly between very and fairly, which means that it is even nowadays very important to explain the use and handling of VLEs. The teachers were rated good up to excellent concerning their availability, explanations and dealing with questions. Some students mentioned that they would like to have more different exercises in order to be more motivated. Others would like to have more detailed explanations of the theory. This shows that especially in large groups the students really need the possibilities of a VLE, which means that one is able to offer various materials and content on different skill levels. The teachers have different impressions. Those who were responsible for the content of the VLE have a much higher workload than usually for this module. The others have had nearly the same workload as before. During the course the teachers were able to answer more individual questions than in the normal face-to-face module. They feel more than a facilitator 5(5)

of learning than a provider of content. This may correspond with the good feedback for the teachers in the cross-case questionnaire.

Outlook

The evaluation shows that using a blended learning concept is a feasible way to handle different previous knowledge and a lack of motivation of the students. With the help of VLEs especially in large groups different ways of learning can be considered and like this the motivation increased. Nowadays the students are used to get information via the computer. Therefore the acceptance of VLEs is high. But nevertheless a good instruction and personal contact to tutors are important.

Finishing the first cycle starts the planning phase for the second research cycle. There the focus will be on the different skill levels. Some students mentioned that they like to have more explanations. This will be considered by more detailed textual descriptions, more animations and small video sequences. Beside of this more exercises will be developed, so that the students have the possibility to select from more and more difficult ones. The next cycle will take place in spring 2008.

Figure 2 -

 2 Figure 2-1: Screenshot of the used VLE eClara

  What constitutes a computer program, what datatypes are there, what are loops and IF control structures, what are function calls?


Leyla Y Tokman

Disciplinary/ Interdiciplinary Virtual Design Studio

Keywords: Virtual Design Studio, Architecture, Education

The purpose of initiating Virtual Architectural Design Studio is the provision of a highly interactive studio environment independent of time and place by broadening the horizons of today's Turkish traditional architectural design studios through collaborations among universities thanks to European Credit Transfer System_ ECTS credit system. The method developed in virtual design studio (SMTS in Turkish/VDS in English) can be considered a sample for future virtual design studios and diciplinary/interdiciplinary interactions. It may also provide an opportunity for students and instructors to make use of the infrastructures learned and developed during the application as well as to widen their uses in future.

Introduction

Countries and continents are getting closer because of globalization and good opportunities on ICT. Global studios or courses and global internship for architectural education are very attractive tools for cooperative studies. In November 2006, There was an international forum (Architectural Education Forum III: ?Global Architectural Education Area?, Istanbul) that was pointed out the recent developments and oppurtunities of architectural education such as Mobility, Quality, Cooperative Studies, Global Studios, Global Courses, Global Internship. Regarding to this, exchange of students, academic personal, researchers and professionals all over the world can easily be realized in the third millennium. The Forum III emphasised that The European Higher Education Area and European Research Area was established after Bologna, and many important cooperative studies were realized in this process. ECTS in Europe is very systematic mobility tool for students. The mobility supporting programs like Erasmus, Socrates, Leonardo and agreements between schools are very helpful for students exchange not only in Europe but also in other continents. Regarding to the collaborative studies in architectural education, virtual courses/workshops, virtual studios, virtual design critcs and virtual conferences contribute to the globalization and interactive computer aided learning.

Virtual studio refer to "networked facilities that provide participants with access to the virtual organization's databases and computational resources, messaging and data exchange, and video conferencing, in a highly integrating fashion" [START_REF] Chen | A Virtual Studio Environment for Design Integration[END_REF]. Today, it will be necessary for architects to perform their vocational skills easily in virtual environments. Among those skills are an interactive/ a syncronous designing, drawing, modeling, an interactive/ a syncronous presentations, making a critique of their designs together with third parties (instructors, other students, users and interdisciplinary professionals such as engineers, artists etc.) and monitoring the construction process etc. as well as an asyncronous works.

Virtual Design Studio_VDS/SMTS in Anadolu University

VDS/SMTS2006 & SMTS2007

Virtual Architectural Design Studio as an virtual environment will be an important step towards more effective architecture education in Turkey based on the changes and improvements in the world of architecture both in national and international contexts. Accompanied with the experience of Istanbul Technical University, Suleyman Demirel University, Yildiz Technical University and as an interdiciplinary experience with the Fine Art Faculty of Anadolu University, this project will be an outstanding opportunity to integrate distance education to the field of architecture as well. The aim of this paper is to introduce the development process of the creative studio in terms of hardware, software and environment thanks to recent technological developments. The information to be presented here is based on the experience gained from the course SMTS2006 Virtual Architectural Design Studio carried out during 2005-2006 spring term with a total of 14 groups and SMTS2007 in 2006-2007 spring term with a total of 3-6 groups. In March 2006, the main virtual studio in the faculty is completed as an hardware and In June 2006, the softwares are completed to set up for the computers in the studio. This studio could be connected to seven points.

In order to realize this project, the course "Participatory Methods in Architecture" was carried out in virtual environment in the Department of Architecture of Anadolu University for the first time in 2005-2006 spring term. In the first phase, students were given theoretical information about information technologies and their uses as well as information about certain basic softwares such as Macromedia/Dreamweaver and Flash, Photoshop, Moviemaker, Sketch Up, Premier, CAD programs etc. Mixed project groups (Anadolu University -Istanbul Technical University -Fine ART Faculty) included one or two students from both universities. First of all each student prepared his/her own website, which gave him/her a chance to introduce his/her architectural identity to others and to gain experience in virtual environment. In the interdisciplinary team process, each member is encouraged to contribute, design and implement the group goals for the virtual project (Figure 1,2 and 3). The main titles in the course flow diagram are as follows [START_REF] Tokman | Global Studios: A Method From Online Architectural Design Education[END_REF] :

• Warm up and the formation of groups as well as the determination of the means of communication • Providing information about the uses of information technologies, possibilities, advantages and applications of virtual design studios and up-to-date samples • Instruction about softwares, in-class and out-of-class practice applications of softwares • Instruction about graphical interface design_GUI • Application: student websites • The determination of the design problem, its framework and the approach to be followed

• The design problem: suggestion and design critics • The design problem: presentations and publishing them on student websites • Feedback session on the content of the course and the method used as well as the differences between virtual and traditional exhibitions

The method developed in virtual design studio (SMTS in Turkish/VDS in English) can be considered a sample for future virtual design studios and diciplinary/interdiciplinary interactions. It may also provide an opportunity for students and instructors to make use of the infrastructures learned and developed during the application as well as to widen their uses in future.

Observations from the Virtual Studios_SMTS2006 and SMTS2007

The necessity that design studios should not be isolated and should be open to social interaction was also mentioned by [START_REF] Kvan | The Pedagogy Of Virtual Design Studio[END_REF], [START_REF] Sanoff | Community Participation Methods in Design and Planning[END_REF], [START_REF] Boyer | Building Community: A New Future For Architectural Education And Practise[END_REF], [START_REF] Tokman | Reality-Based Design Studio In Architectural Education[END_REF]. In this point of view, virtual design studios in design education is highly interactive learning environments. Another importance of VDS is its up-to-date technology (software and hardware). Therefore the architecture students should learn/use new skills based on ICT. According to Piaget, "Knowledge is experience that is acquired through interaction with the world, people and things" [START_REF] Ackermann | Piaget's Constructivism, Papert's Constructionism: What's The Difference[END_REF]. In SMTS2006 and SMTS2007, our students have these experiences:

• Synchronous online discussions are being increasingly used in developing the design process. The first result of these discussions facilitate learning the virtual design environment for students and group interaction between on-campus and off-campus students. For example, The groups were found to initiate design process through chat sessions and preliminary sketches, "imagination at work", whiteboards and modeling and regular transitions among these techniques. • Although Using the Technology is encouraging students to learn virtual environments, Technological competence of the students is crucial for a smooth flow of the lesson. For instance, within the scope of SMTS2006 and SMTS2007 courses, students were given instruction about certain software, such as "Macromedia-Dreamveawer" to prepare webpage's; "Moviemaker", "Adobe Photoshop" to create certain effects; and "Imagination At Work", "Whiteboard" to enable communication and an e-conference on shape grammer to create design ideas and e-workshop as an interdisciplinary collaboration on the early phases of creative design process. But today some software could have already known by some students. • The students are able to conduct joint studies with the students in different places.

• Learning to express their architectural identity by their own web sites encourages the students to present group projects and to develop the quality of design. • The students are more creative and productive in a syncronous design process because of interactive learning environment and interdisciplinary collaboration (figure 4). The software is also very easy to learn and easy to communicate with the partner, syncronously. After being developed a concept, the 3d modelling softwares, chat, Photoshop and "LogMeIn" are used. While Preparing the presentations by using MS Powerpoint and animations; learning about GUI_graphical user interface and designing the groups' web sites by using Macromedia/Dreamveawer, Adope/Photoshop The students developed their skills based on ICT.

There are also some problems faced in the virtual studio. For example, Since the academic calendars of both universities were not parallel in terms of set dates, it was necessary to make revisions, even reorganizations, on the common processes, such as, starting and finishing dates of the courses. The other problem is the firewalls of both universities. This problem is solved when "LogMeIn" software used.

These experiences could contribute to strengthen the foundations of research and education. Finally, virtual design studio provides unisolated interactive studio as a learning environment for students and lectures. Therefore, the students stated that they find it enjoyable to have contacts with people outside the university and to have professional discussions with them, which are the examples of socialization.

Conference ICL2007

September 26 -28, 2007 Villach, Austria 6(6)

Conclusion

Traditional design studios started to open to the world with virtual design studio. The opportunity of The Tripod: first_ECTS_European Credit Transfer System and second_international exchange programs such as ERASMUS, SOCRATES and third_collaboration such as VIRTUAL STUDIOS contributes the architecture programme. The national/international partnerships and also disciplinary /interdiciplinary partnerships of virtual studio provide a guiding light for the positive environment of the future. We believe that the interactive world of Virtual studio provides educators and students as active participants for keeping up the global quality of design education in an international academic and professional network.

Figure 1 .

 1 Figure 1. E-Design workshop between Faculty of Enginnering and Architecture and Fine Art Faculty on May 8, 2007: figure shows the virtual studio (photographed by the author).

Figure 2 .

 2 Figure 2. E-Design workshop: a syncronous design process (photographed by the author).

Figure 3 .

 3 Figure 3. E-Design workshop: e-sketch by using interactive LCD screen (dual-touch, DViT_Digital Vision Touch) with its electronic pen as an hardware and "imagination at work" as a software (photographed by the author).

5( 6 )Figure 4 .

 64 Figure 4. The groups includes two students accesed from two different places: one from department of Architecture in Virtual Studio, Iki Eylul Campus and the partner from department of Interior Design in Video Conference Hall, Yunus Emre Campus. The figure shows students'experiences in the early phases of syncronous design process. "Imagination at work" and chat are used as software and interactive LCD screen with electronic pen. This hardware is very helpful while developing the design ideas instead of a mouse and a monitor.The software is also very easy to learn and easy to communicate with the partner, syncronously. After being developed a concept, the 3d modelling softwares, chat, Photoshop and "LogMeIn" are used. While Preparing the presentations by using MS Powerpoint and animations; learning about GUI_graphical user interface and designing the groups' web sites by using Macromedia/Dreamveawer, Adope/Photoshop The students developed their skills based on ICT.

Acknowledgement:

"Information Technology In Architectural Education" is a Scientific Research Project (Project no. 060211) supported by Anadolu University [8]. The detailed information is avaliable on the web site: http://smts.anadolu.edu.tr . I would like to express my gratitude to Prof.Dr. Gulen Cagdas for sharing her virtual design studio experiences with us and for her invaluable support for the realization of this project. In addition, I would like to thank Prof.Dr. Rusen Yamacli for her support during studio studies. I would like to thank to all the students.

Author:

Leyla Y.


Maja Snajder

Mateja Verlic

Petra Povalej

Matjaz Debevc

Pedagogical evaluation of e-learning courses -Adapted pedagogical index

Keywords:

E-learning is becoming an important part of education and it can provide new possibilities for disabled people. Critical evaluation is becoming indispensable,

especially in e-learning systems supporting disabled people, as very little has been done in this area. This paper describes the development of adapted pedagogical index (AdaPI), which demonstrates the pedagogical effectiveness of different courses created and adapted for disabled.

Introduction

While only a few decades ago World Wide Web was considered a privilege, today it is a necessity. Everything went electronic; today we frequently encounter words like e-health, egovernment, e-shopping, e-business and e-learning among others. Furthermore, the initiatives and policies of European Union emphasise improvement of quality of life and e-inclusion of citizens, supported by advances in development of information and communication technologies. New and improved technologies opened the world of endless possibilities, especially for people with disabilities. In project DISNET one of the main goals was to bring new technologies and at least some of the new possibilities to people with different disabilities.

What does e-learning stand for? How does it fit into this picture of brighter future for people with special needs? E-learning can be defined as training or instructions delivered on a computer by using different media (including CD-ROM, Internet, and intranet) [START_REF] Clark | e-Learning and the Science of Instruction[END_REF]. E-learning courses are delivered via computer using multi-media materials -spoken or written text, sounds, pictures, animation or video [START_REF] Horton | E-Learning Tools and Technologies[END_REF][START_REF] Campbell | Effective Writing for E-Learning Environments[END_REF][START_REF] Horton | Using E-Learning[END_REF]. Different teaching and learning methods can be employed [START_REF] Clarke | e-Learning Skills[END_REF]: (i) e-learning is blended with traditional methods (usually traditional classroom learning), (ii) distance learning, (iii) any face-to-face contact with tutors and/or other e-course participants, and (iv) degree of freedom (participants can choose place, pace and time). Selection of method depends on characteristics of the target group (e.g. young people, adults, older population, persons with disability). A combination of appropriate e-learning methods and assistive technologies can help persons with disabilities to overcome the barriers by offering them the possibility to learn at their own pace from their homes and almost about everything they can imagine.

While the quality of multi-media materials and methods used is essential for successful elearning, the evaluation of this kind of learning is even more important. Various cultural, pragmatic and technological constraints affect the process of e-learning and therefore have influence on different aspects of the e-learning effectiveness and efficiency. Several models of evaluation from different perspectives (pedagogical and technical) exist, for example 2 [START_REF] Beer | Diagnosing the System for Organizations[END_REF] Kirkpatrick's Four Levels of Evaluation [START_REF] Galloway | Evaluating distance delivery and e-learning: Is Kirkpatrick's model relevant? Performance Improvement[END_REF], AHP approach by Colace et.al [START_REF] Colace | Evaluation Models for E-Learning Platform: an AHP approach[END_REF], Laurillard's conversational framework [START_REF] Laurillard | Rethinking University Teaching -a framework for the effective use of educational technology[END_REF][START_REF] Britain | A Framework for the Pedagogical Evaluation of eLearning Environments[END_REF], Viable System Model (VSM) [START_REF] Britain | A Framework for the Pedagogical Evaluation of eLearning Environments[END_REF][START_REF] Beer | Diagnosing the System for Organizations[END_REF], and also Pedagogy Effectiveness Index (PEI) [START_REF] Sonwalkar | A New Methodology for Evaluation: The Pedagogical Rating of Online Courses[END_REF], upon which our adapted pedagogical index (AdaPI) was based.

Project DISNET

Phare 2003 project DISNET (e-learning net for unemployed people with special needs) joined different Slovenian partners, whose effort was oriented towards developing e-learning system that would help unemployed people become equipped with computer knowledge and skills in their search for a job. In project we in some way continued the research of pioneers in this field: VISIOCOM [START_REF] Dugonik | Video production for distance education[END_REF], and BITEMA [START_REF]SOCRATES GRUNDTVIG project BITEMA[END_REF]. DISNET was specialized in training of disabled persons. Disabled people often have fewer opportunities to get desired jobs than others; however this project's goal was to reduce these differences. Over 300 people were included in courses based on blended learning. E-learning materials were prepared according to ECDL (European Computer Drivers License) curriculum [14] and adapted for people with disabilities, especially for heard-of-hearing and deaf persons. All engaged institutions have gained the status of Assistance e-learning centre; they all have tutors and adapted equipment to work with disabled.

Special goals of the project

Due to the specificity of the target group involved in the project DISNET five different goal aspects were considered: conceptual, didactical, practical, knowledge transfer and technological aspect.

Conceptual aspect:

• Development and description of e-learning models with qualified mentors and informing unemployed disabled people. • Development of e-learning platforms and didactical materials with help of several experts -teachers of computer science, ICT specialists and students, who tested the platform.

Didactical aspect:

• Presenting the platform to the mentors and tutors.

• Performing three test courses for each course and for each target group.

Practical aspect:

• Developing the network of institutions for helping unemployed disabled people getting ICT skills and enlarging their chances for employment. • Planning Centres of e-learning support in remote parts of country.

Knowledge transfer:

• Access to e-learning materials is always available for disabled people.

Technological aspect:

• Upgrading technical equipment of institutions involved in project and therefore improving their educational offers.

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Project target population

Primarily, unemployed persons with special needs were the target population for following reasons: there is an extremely high level of unemployment of disabled people, usually their special needs are ignored and they are excluded from the society. ICT offers possibilities to improve current situation and to mitigate problems mentioned above, for it can support their independence, alleviate communication and even offer possibilities to work at home. Secondarily, other unemployed persons without disabilities were given a chance to participate in the project.

ICT develops extremely fast and offers its users adapting technology to their personal needs.

Naturally, disabled people have special needs when it comes to ICT technology and also learning itself. Therefore special attention and adaptation is necessary. Today several assistive technologies and access services are available to overcome social, geographical and psychological barriers, people with special needs might encounter. However, even the most advanced technologies are not useful, if the potential users are not aware of them or don't know how to use them. Additional problem represents the lack of qualified teachers for people with special needs that would have sufficient computer skills and knowledge on how to effectively apply accessible technology. Solution for this problem was a tutoring system used during project. Participants were learning at home (or at other suitable location) and had nonstop support of tutors. Tutors were qualified for working with disabled people, had appropriate computer skills and were familiar with the functionalities of e-learning platform used.

Disabled people and their special needs

People with disabilities have very specific needs and concerns; therefore they are very special users of e-learning. Their needs and inclusiveness in the society depends on the type of the disability. There are many types of disabilities such as physical (difficulties with movement or self-care), sensory (blind and or partially sighted, deaf and hard-of-hearing) cognitive (learning problems, communication problems), psychiatric (behaviour and personality disorders), and health-related (chronically diseased).

Persons with special needs and characteristics as described above need adapted performance of regular educational programs, extra help by learning or even special educational programs.

In Slovenia, the problem of disabled people integration is still being solved. Although in the last years a great deal has been done in this area, it is still not enough. The problem of successful integration of deaf and hard-of-hearing in ICT environment still remainsstatistical research show the lowest integration results with this particular group.

In project DISNET we focused mainly on deaf and hard-of-hearing, blind and partially sighted and physically challenged people. All mentioned groups can nowadays, due to the advances in ICT, equally participate in education. However, we have to be aware that there are also differences between these groups and there are also differences even within individual group. Each group requires technology adapted to its specific needs in order to have equal conditions for learning.

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ECDL content and LMS system

Some people with special needs are familiar with the assistive technologies available on an average personal computer -however, they still need at least basic level of ICT skills to be able to use the computer and the technologies. Many e-learning courses on the topic of ICT are available online [START_REF]Learn that[END_REF][START_REF]Free computer tutorials[END_REF][START_REF]Key skills[END_REF][START_REF]Spletno učenje[END_REF], some of them are free, others require a fee to enrol, but very few of them are accessible. We decided to create our e-learning course materials according to ECDL curriculum and use an open-source learning management system to manage e-learning courses, users and user activities.

ECDL contents

The ECDL foundation is the global governing body in world-wide acknowledged computer skills certification programme European Computer Driving License (ECDL), also known as International Computer Driving License (ICDL) outside Europe [14]. This organization is dedicated to help raising the level of computer skills in society and they are setting and maintaining a reliable standard for ECDL computer skill certification programme. This was the reason to use ECDL syllabus 4.0 as a basis of e-learning materials developed for the participants in the DISNET project. Furthermore, materials have been adapted to meet special needs of the target population.

When creating and adapting materials we followed several guidelines for designing and implementing e-learning systems for persons with disabilities [START_REF] Achetemeier | Considerations for developing evaluations for online courses[END_REF][START_REF]Website accessibility guide -Understanding and implementing accessibility[END_REF][START_REF] Dringus | An adaptable Usability Heuristic Checklist for Online Courses[END_REF][START_REF] Gulliksen | Guidelines, standards, methods and processes for software accessibility[END_REF]. We tried to consider as much guidelines as possible; however it was impossible to include all of them.

LMS system

Moodle, the open source system, was used as a learning management system (LMS) for managing e-materials. Moodle's advantages over other LMSs were its user-friendliness, compatibility with different platforms and variety of integrated e-learning tools. Its main advantage was the fact that its developers endeavour to implement accessibility guidelines in the system. Moodle is also very suitable for managing multimedia contents and has a good and relatively simple mechanism for tracking participants' activities. E-mentors were advised to use all Moodle tools such as forums, chat rooms, tests, mailing lists, notifications etc.

Evaluation of project DISNET

Evaluation of e-learning system is necessary to assess the effectiveness, efficiency and after all a general success of e-learning implementation in the selected environment. Several aspects need to be included in the complete and thorough evaluation of e-learning coursese.g. quality of software (platform), contents, and user-system interaction, level of accessibility, usability etc. In this paper we focused on pedagogical evaluation of the elearning courses used in the project DISNET.

Conference ICL2007

September 26 -28, 2007 Villach, Austria 5(10)

Methods of evaluation

Choosing the right method of evaluation is the key for successful usage of educational system. However, not many methods are available when it comes to evaluating e-learning courses [START_REF] Zaharias | Designing On-Line Learning Courses: Implications for Usability[END_REF]. We have chosen some currently present methods from experts such as Squires and Preece [START_REF] Squires | Predicting quality in educational software: Evaluating for learning, usability and the synergy between them[END_REF][START_REF] Squires | Usability and learning: Evaluating the potential of educational software[END_REF], Ardito et al [START_REF] Ardito | An Approach to Usability Evaluation of e-Learning Applications[END_REF], Dringus and Cohen [START_REF] Dringus | An adaptable usability heuristic checklist for online courses[END_REF], Holzinger [START_REF] Holzinger | Incidental learning, motivation and the Tamagotchi Effect: VR-Friends, chances for new ways of learning with computers[END_REF], Sonwalkar [START_REF] Sonwalkar | A New Methodology for Evaluation: The Pedagogical Rating of Online Courses[END_REF] and Achtemeier [START_REF] Achtemeier | Considerations for developing evaluations of online courses[END_REF]. Different views of project were being evaluated and methods have been adapted for individual aspect of the evaluation. Areas that were evaluated were: evaluation of user friendliness of e-materials, estimation of general opinion of participants about course and its adaptation to disabled people, evaluating of pedagogical effectiveness and effectiveness of entire course. Table 1 presents methods of evaluation and the way they were used. This paper will mainly focus on presenting pedagogical effectiveness of project.

Pedagogical evaluation of the project

The basic pedagogical process can be defined as a systematic transfer of knowledge or skills from a mentor to a learner. However, the process is more complicated than it seems. It comprehends variety of different factors, such as motivation, individual's needs, learning materials, learners learning style, instructors teaching style, etc. For simple evaluation it would be sufficient to analyze pre-and post-knowledge exams -if exams show achieved knowledge, the course was effective. But as we already mentioned, pedagogical process is too complex and need for more thorough measurement of pedagogical effectiveness occurs.

Research and methodology

Our method for evaluating pedagogical effectiveness is based on Pedagogy Effectiveness Index (PEI) proposed by Sonwalkar [START_REF] Sonwalkar | A New Methodology for Evaluation: The Pedagogical Rating of Online Courses[END_REF]. However, some changes have been applied mainly because of the differences in populations that have been included in project. Our attempt was to develop similar index that would measure the pedagogical effectiveness of e-learning courses adapted for disabled people -adapted pedagogical index (AdaPI). AdaPI is similarly as PEI composed out of three dimensions and several factors. The dimensions are the same as were introduced by Sonwalkar. Main differences between PEI and AdaPI emerge from components that are basic parts of dimensions. Structure of AdaPI is presented in Table 2. Dimension learning styles retains first three components and changes last two (deduction and induction) with component VAK styles [START_REF] Dunn | Learning style inventory[END_REF]. Reason why we made the changes lies in disabilities that were common in our population. All materials were adapted for this specific population trying to follow the rule -if some sense organ is damaged -others will develop above average and relieve individual's communication and orientation. For instance, if one has vision difficulties, usually his hearing ability is above average. Questionnaire was therefore measuring, among other things, if courses took in consideration different learning styles by VAK theory -visual, auditory and kinaesthetic [START_REF] Marentič-Požarnik | Izziv raznolikosti. Stili spoznavanja, ucenja, mišljenja[END_REF]. This dimension was emphasizing meaning of active learning, which is supported by large amount of practical exercises and giving the possibility of incidental learning and independent discovery. Remaining components were measuring those factors.

6(10)
Dimension media contains five factors, which are referring to e-learning materials -text, graphics, audio materials, animation and organization of materials. Again, we are originating from the fact, that all people have different favourite channel trough which they perceive the world and environment. Some changes were in this dimension also. Components video and simulation, which are used in original index, are being joined with component animation. One new component was added, which it is very important, when it comes to adapting content for disabled people. It was mentioned earlier, that people with special needs often have also learning problems. That is why, material arrangement is important to ease their studying -it is important, if are materials arranged in logical sequence (from basic to advanced, etc.).

The third dimension is being called interaction, because it emphasizes the meaning of interaction between three factors of learning process -mentor, participant and material. Main difference between Sonwalkar's dimension and the one presented in this paper is inclusion of component mentors role, which is very important in every learning process.

Representation of each component was measured trough participants' opinions with help of a questionnaire. Afterwards AdaPI was calculated trough the procedure that will be described in detail later.

Developing measuring instrument and gaining data

Questionnaire was developed with the help of Sonwalkar's theory. Every component was measured trough several questions or statements. Questionnaire was composed out of closed type questions. Participants could choose from five answers (Likert scale [START_REF] Likert | Technique for the Measurement of Attitudes[END_REF]). Instrument was also verified with help of three random chosen individuals, who helped us develop more comprehensible and understandable questions.

Participants were then invited to fill the questionnaire in online classroom. After gathering of the data of 116 participants, the AdaPI was being calculated. Figure 3 is showing us age structure of population.

Developing AdaPI -adapted pedagogical index

AdaPI calculation process:

• Questionnaire was composed out of closed type questions. Participants could choose from five answers (Likert scale [START_REF] Likert | Technique for the Measurement of Attitudes[END_REF]). Answer one was always the negative dimension and five always positive. For the purpose of processing data all answers were transformed to interval from 0 to 4. • Each component was measured trough several questions. At this point, results were grouped by components. • Average values were calculated for each component.

• Final step included calculation of contribution (weight) for each component in particular dimension. The following formula was developed:

mn v v pi i 4 1 =
, where v pi stands for average value of dimension i, m is the number of components and n is the number of dimensions. • AdaPI is calculated as

∑ = n i v AdaPI 1 3.2.

Results and discussion

Value of AdaPI can lie in the interval from 0 to 1. The calculated AdaPI of our courses was 0.68. Since the value of AdaPI exceeded the value of 0.5, we can conclude that the project was successful from the pedagogical point of view. However, the improvement is always possible. Value of AdaPI and particular components is shown in Table 4. It can be seen that all components are balanced -they are achieving similar values. Interaction is the only dimension that is higher, so it can be concluded, that users were more satisfied with components included in this dimension -feedback, revision, e-mail, discussion and mentor's role.

Better results could be expected, if participants in courses would not have been novice users concerning computer knowledge, or perhaps if courses would have been performed completely as e-learning method and not as blended learning, as in our case. In next three tables (Tables 567) average values of each component are presented. In Table 5 it is shown, that component value of all four components lies above average, however, they are not very high. In dimension media, which can be seen in

Conclusions and future plans

We hope that with project DISNET an important step was made towards inclusiveness of people with special needs in education process in Slovenia. A new approach to learning for disabled has been introduced. According to the AdaPI results we obtained we can conclude that the project was successful from pedagogical perspective. It was interesting to see, that mentors/tutors played a crucial role in the learning process. If we consider all the special characteristics of primary target group -unemployed persons with different disabilities, this finding is not surprising. Disabled people need special attention because of their special needs, much higher levels of motivation than people without disabilities and especially a lot of help by getting familiar with all the technologies available.

In future it would be interesting to divide participants in different groups according to their knowledge, prior participation in DISNET project and then assess the pedagogical effectiveness with AdaPI. Furthermore, we would like continue our work by involving more disabled people in similar projects to improve the level of their computer skills and to increase their chances of employment.

Of course, it is necessary to conduct more studies regarding effectiveness and usefulness of elearning for people with special needs before we can claim that e-learning is the solution for most of the problems disabled people have to deal with.

Figure 3 :

 3 Figure 3: Structure of population by age

Table 1 :

 1 Evaluation methods

		SUMI	Pedagogical	Course	Automatically
		evaluation	effectiveness	effectiveness	evaluating with
					tools
	Purpose	Users testing	Test of	Test of	E-materials
			pedagogical	pedagogical	accessibility
			variables	variables
	Phase of the project	Final phase	Final phase	Final phase	Planning
	Object of testing	E-materials	Complete course Individual courses HTML code
	Target population	Users	Users	Users	Users
	Number of users	10 deaf people	116	116	-
		+ 65
		individuals
		without
		disabilities
	Number of evaluators	2	3	3	1

Table 2 :

 2 Structure of AdaPI -adapted pedagogical index

	LEARNING STYLES	MEDIA	INTERACTION
	Apprenticeship	Text	Feedback
	Incidental learning	Graphics	Revision
	Discovery	Audio	E-mail
	VAK styles	Animation	Discussion
		Material arrangement Mentor's role

Table 4 :

 4 Value of AdaPI

	Learning
	styles	Media	Interaction	AdaPI
	0.22	0.22	0.24	0.68

  Table 6 the most component values lay above average, except material arrangement component, that is showing almost maximum result. It can be concluded, that participants were very satisfied with arrangement of materials. Last table -Table7shows us the highest values of components included in dimension interaction. Once more it is proved, that participant needed mentor and that mentor's role is very important.

Table 5 :

 5 Average values of component learning styles

		Learning styles
	Apprenticeship	Incidental learning	Discovery	VAK styles
	3.49	3.68	3.93	3.65

Table 6 :

 6 Average values of component media

			Media
					Material
	Text	Graphics	Audio	Animation	arrangement
	3.78	3.98	3.41	3.35	4.24

Table 7 :

 7 Average values of component interaction

			Interaction
	Feedback	Revision	E-mail	Discussion	Mentor's role
	4.48	4.50	3.58	3.35	4.47


Ilya Leibo
email: leibo@metodist.ru

Tatiana Fialkova
email: fialkova@intmedia.ru

Irina Kaliteevskaya
email: ira.kaliteevskaya@gmail.com

Academic research for schools

Keywords: Distance education, e-enlightenment, podcasts

Academic subjects are usually extremely far from actual problems and questions of academic research. Material at school is often presented as a sum of fixed, immovable facts one has to learn, although in the world of research these facts may be regarded as questionable or even disproved. On the other hand, pupils rarely have an understanding of how facts are developed and understood. As it is impossible to bring a researcher to every school, we are working on the project that would help researchers to communicate with pupils and teachers via the Internet, as today's technological capabilities are able to facilitate new forms of learning.

A school and an academic research

School teachers in Russia, especially in the regions others than the capital cities, often suffer from a shortage of information. The main reason for this is lack of time and sometimes education, but also there are problems regarding library acquisitions. As a result there are teachers working with occasionally chosen text-books, usually depending on what a school library has. An average teacher likely had a connection with an academic research at best during his or her university studies. Some teachers collaborate with researchers and invite them to present a lecture or a series of lectures at school, but this is possible only in bigger university cities, or in special schools or classes, and only when there is an active and devoted teacher who has developed personal relationships with academics. There was an attempt to create an educational television program which could be used in the course of school lessons, thus making them more alive and diverse. But because of the obvious problems of schedules and the general shortage of the school courses (they are constantly decreasing) this didn't work. The problem became especially evident lately, specifically with history schoolteachers at the school conducted by Institute of Universal History of the Russian Academy of Sciences. They repeatedly asked about new discoveries, and questioned facts that are nowadays regarded false, but are still being repeated in textbooks. On the other hand the researchers who are usually writing textbooks for schools obtained the possibility to receive feedback concerning what teachers think about their production. All this demonstrated that some form of communication between teachers and researchers is necessary. However, a one-week school can, first of all, involve too few participants in comparison with the amount of schools and school teachers in the country, and, thus, every teacher may not participate in this kind of education. Second, this kind of course is too general to touch any question in great detail. A researcher would get an opportunity to discuss only the most general questions, giving a teacher an idea of his point of view on the situation as it is at the moment, but not in progress. New Internet technologies definitely are at the moment the only and very effective way to solve the problem. Toady the Institute of Open Education has commenced several projects with the Institute of Universal History and works on the other projects involving researchers from other fields. We would like to present a product based on the technology of podcast that we continue to develop together.

Lectures online

Our project "Lecture online" is made on the basis of podcasts. Researchers present lectures for school pupils on more or less general and disputable subjects that are then divided into smaller pieces. The podcasts are supplied with the multimedia hypertext structure that includes a text file with the transcript, encyclopaedic references, images and music illustrating the text, bibliography, and sometimes videofiles with a recording of the lecture or other materials. All this gives a teacher the possibility to select the information and to incorporate it into a lesson. This structure also gives the possibility to put a question or to express an opinion, and makes the communication between a teacher or a pupil and a researcher permanent. Moreover, the specificity of the product lets any user to be an active consumer of the information: one can decide on how deep he would like to get into the problem. The lectures are made in order to stimulate pupils to analyze, to speculate, and in order to demonstrate how researchers do their work, and where they gather their conclusions. For the time being the project is at its beginnings. And one of its most important advantages is that it can never be finished. Thus, it may help the school to overcome its obduracy and to vitalize school subjects.