On the two-dimensional compressible isentropic Navier-Stokes equations

Mod´elisation Math´ematique et Analyse Num´erique M2AN, Vol. 36, N^o 6, 2002, pp. 1091–1109 DOI: 10.1051/m2an:2003007

ON THE TWO-DIMENSIONAL COMPRESSIBLE ISENTROPIC NAVIER–STOKES EQUATIONS

Catherine Giacomoni

and Pierre Orenga

Abstract. We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two- dimensional case with γ =cp/cv = 2. These equations also modelize the shallow water problem in height-flow rate formulation used to solve the flow in lakes and perfectly well-mixed sea. We establish a convergence result for the time-discretized problem when the momentum equation and the continuity equation are solved with the Galerkin method, without adding a penalization term in the continuity equation as it is made in Lions (1998). The second part is devoted to the numerical analysis and mainly deals with problems of geophysical fluids. We compare the simulations obtained with this compressible isentropic Navier–Stokes model and those obtained with a shallow water model (Di Martino et al., 1999). At first, the computations are executed on a simplified domain in order to validate the method by comparison with existing numerical results and then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analytical example presented by Weigant (1995) which shows that even if the data are rather smooth, we cannot have bounds onρinL^pforplarge ifγ <2 whenN= 2.

Mathematics Subject Classification. 35Q30.

Received: September 7, 2001. Revised June 26, 2002.

Introduction

This paper is devoted to the analysis of the two-dimensional compressible isentropic Navier–Stokes equations for which existence results have been established by Lions [6]. We introduce another construction of approximate solutions of the problem by using the Galerkin method which induces a simpler numerical approximation. Let Ω be a bounded simply connected (to simplify) smooth open domain inR² with boundary Γ and letQbe the cylinderQ= Ω×]0, T[ with its boundary Σ = Γ×]0, T[.

We consider the following Cauchy problem

∂ρ

∂t + div (ρu) = 0 inQ, (1)

∂ρu_i

∂t + div (ρuu_i)−µ∆u_i+∂_i (aρ^γ) =ρf_i, 1≤i≤2, inQ, (2)

Keywords and phrases. Navier–Stokes, compressible, shallow water, time-discretisation, Galerkin.

1 Syst`emes Physiques de l’Environnement, UMR CNRS 6134, Universit´e de Corse, Quartier Grossetti, BP 52, 20250 Corte, France. e-mail: [email protected], [email protected]

c EDP Sciences, SMAI 2003

where ρ ≥ 0 corresponds to the fluid density and ua vector-valued (in R²) function that corresponds to its velocity. The function f =f(x, t) is a given function corresponding to the force terms on Q, µ is a positive viscous coefficient and we consider the caseγ= 2 in order to compare this model with the shallow water model in the numerical part. These equations modelize the shallow water problem in height-flow rate formulation.

We specify initial conditions

ρ(t= 0) =ρ₀(x)∈Ω, ρu(t= 0) =m₀(x)∈Ω (3) and we assume thatρ₀ andm₀satisfy

ρ₀∈L^γ(Ω), ρ₀≥0 a.e.in Ω, ρ₀≡0, (4)

m₀∈L^γ+12γ (Ω), m₀= 0 a.e.on{ρ₀= 0}, (5)

|m₀|²

ρ₀ ∈L¹(Ω),

|m₀|²

ρ₀ = 0 on{ρ₀= 0}

· (6)

We must add boundary conditions to the system (1, 2). We use classical boundary conditions to geophysical fluids and particularly, for the shallow water models

u·n = 0 and curlu= 0 on Σ. (7)

The first condition is a natural condition of impermeability type on the normal velocity where n denotes the unit outward normal to Γ. The second condition can be interpreted as a viscous term dissipation at the boundary (more exactly a non-dissipation since this term is equal to zero) [8]. These conditions are more suitable to geophysical fluid equations than a classical Dirichlet condition which generates very expensive calculations of boundary layer. Moreover, these conditions allows to solve the problem with the Galerkin method using a special basis which permits to write the Hodge–Helmoltz decomposition of vector fields as sum of gradient and Curl of scalar fields [4]. It is this decomposition which allows to obtain the necessary compactness to pass to the limit in the equations.

Notice that the resolution of these equations can be extended to the case of periodic boundary conditions for instance, and for values of γ different from 2 using the notion of renormalized solutions of the continuity equation [2]. Indeed, in this case, the Galerkin method is not directly valid because the energy estimates are not immediately obtained. In order to avoid this difficulty, we can solve the system (1, 2) by making the change of variableβ(ρ) =ρ^γ/2. The continuity equation is then solved under the following form

∂ρ^γ/2

∂t + div

ρ^γ/2u

+ γ

2 −1

(divu)ρ^γ/2= 0 and we verify easily that we obtain the estimates.

We use a time-discretization method to solve this problem and the main differences with the existence proof developed in [6] are that we use the Galerkin method to solve the time-discretized problem and we do not need to penalize the continuity equation. In particular, we show that we have enough compactness on the Galerkin solutions to pass to the limit in the equations. Let us note that this compactness is obtained thanks to the boundary conditions used and in the case of Dirichlet boundary conditions, we do not manage to obtain it.

In the numerical part, we solve this problem that we compare with a shallow water model in which the diffusion term depends onρ. At last, we consider an analytic case introduced by Weigant [11] that shows the formation of singularities in finite time for smooth solutions of (1, 2).

The numerical results show that the proposed numerical method is well adapted to this kind of problems.

For the following analysis, we define the functional spaceV by V =

ϕ∈L²(Ω)²/divϕ∈L²(Ω),curlϕ∈L²(Ω), ϕ·n = 0 onγ

=

ϕ∈H¹(Ω)², ϕ·n = 0 onγ equipped with the graph-norm

||ϕ||²_V =||ϕ||²_L2(Ω)²+||divϕ||²_L2(Ω)+||curlϕ||²_L2(Ω). (8) Notice thatV corresponds to the spaceH₀(div ; Ω)∩H(curl ; Ω) [4] and v∈ V can be split as follows:

v=∇p+ Curlq

where ^∂p_∂n = 0 andq= 0 onγ. We denote byA(u, ϕ) the bilinear formA(u, ϕ) = (divu,divϕ) + (curlu,curlϕ) where (·,·) represents the scalar product in L²(Ω) orL²(Ω)². Notice that when Ω is simply connected then A(u, u) is an equivalent norm onV. However, if Ω is not simply connected, one can prove [8] that we can obtain a bound onuinV.

We present the special basis ofV used afterwards [9]

(B)





−∆u=λu in Ω, u·n = 0 on Γ, curlu= 0 on Γ,

for which the solutions can be obtained by solving the following scalar problems (when Ω is simply connected)

(N)





−∆p_i =λ_ip_i in Ω,

∂p_i

∂n = 0 on Γ,

(D)

−∆qi =µ_iq_i in Ω, q_i = 0 on Γ.

The set of {∇p_i, i∈N^∗} ∪ {Curlq_i, i∈N^∗} is an orthogonal basis of L²(Ω)² and V, and is dense in L^p(Ω)², p <∞. Notice that if Γ is of class C^m−1,1, then∇p_i and Curlq_i∈H^m(Ω)².

Let us recall the following theorem which is the main existence result established in [6]. Assuming the conditions (3) and f ∈L¹

0, T;L^γ−12γ (Ω)²

, the solutions (ρ, u) satisfying (1, 2) in the sense of distributions are such that

ρ≥0, ρ∈L^∞(0, T;L^γ(Ω))∩C([0, T];L^q(Ω)),1≤q < γ, u∈L²(0, T;V),

ρ|u|²∈L^∞(0, T;L¹(Ω)), ρu∈C

[0, T];L^γ+12γ (Ω)²−ω

.

Theorem 0.1. Under the above conditions, there exists a solution (ρ, u)of (1, 2) satisfying the initial condi- tions (3) and such that ρ ∈ L^p(Ω×(0, T))where p =γ+_N²γ−1. In addition, (ρ, u) satisfies the following energy inequality for almost all t∈[0, T]

Ω

1

2ρ|u|²+ a

γ−1ρ^γ dx+µA(u, u) ≤

Ω

1 2

|m₀|² ρ₀ + a

γ−1ρ^γ₀ dx+ _t

0 ds

Ωdx ρf·u. (9) One can see that, with these estimates, the passage to the limit in the pressure termρ^γ is the main difficulty.

1. Time-discretization

In [6], a first step to solve the equations (1, 2) is based on a classical Euler time-discretization. In this section, we prove the existence of solutions of this time-discretized problem by using the Galerkin method. So, in this work we analyze the following stationary problem

αρ+ div (ρu) =hin Ω (10)

αρu+ div (ρu⊗u)−µ∆u+a∇ ρ²

=ρf+g in Ω (11)

where α = _∆t¹ > 0 and f, g, h are given functions defined by h = _∆t¹ ρ(t−∆t), h ≥ 0 in Ω, h ≡ 0 and g= _∆t¹ ρ(t−∆t)u(t−∆t).

Always in [6], in order to establish the existence of solutions for this problem, the author approximates (10, 11) by the following penalized problem

αρ+ div (ρu)− ∆ρ=h (12)

hu 2 +1

2ρu· ∇u+αρu 2 +1

2div (ρu⊗u)−µ∆u+a∇

ρ²

=ρf+g (13)

where > 0. The existence of solutions to this problem is shown by a Leray–Schauder fixed point method.

Notice that one adds to (12, 13) the Neumann boundary condition (for instance) onρ, ^∂ρ_∂n = 0 on Γ.

In this paper, in order to approximate the solution of discretized problem (10, 11), we use the Galerkin method for the momentum and continuity equations which allows to circumvent the use of the penalization in the continuity equation thanks to the regularity of the special basis. We apply the Leray–Schauder theorem in order to construct the approximate solutions and moreover, we show that we have enough compactness onρto pass to the limit.

The momentum equation can be written in the following way u(αρ+ div (ρu)) +ρu· ∇u−µ∆u+a∇

ρ²

=ρf+g in Ω (14)

and since, we have

αρ+ div (ρu) =hin Ω, (15)

it is equivalent to solve the following problem

αρ+ div (ρu) =hin Ω, (16)

hu+ρu· ∇u−µ∆u+a∇

ρ²

=ρf+g in Ω. (17)

This problem is solved under the following weak formulation (W)

α(ρ, ψ) + (div (ρu), ψ) = (h, ψ), ∀ψ∈H¹(Ω), (18)

(hu, ϕ) + (ρu· ∇u, ϕ) +µA(u, ϕ) +a

∇ ρ²

, ϕ

= (ρf, ϕ) + (g, ϕ), ∀ϕ∈ V ∩H⁴(Ω)². (19) We considerVn the space generated by the nfirst elements of the basis (B) of V and H¹(Ω)_m represents the subspace ofH¹(Ω) generated by the functions{p₁, . . . , p_m}, solutions of problem (N). So,u_n,mand ρ_n,mare under the form

u_n,m= n i=1

a_i(t)ϕ_i(x), ρ_n,m= m

i=1

b_i(t)p_i(x).

We approach (W) by the following problem (W_n,m),nfixed

α(ρ_n,m, p_i) + (div (ρ_n,mu_n,m), p_i) = (h_n,m, p_i), ∀p_i∈H¹(Ω)_m (20) (h_n,mu_n,m, ϕ_i) + (ρ_n,mu_n,m· ∇u_n,m, ϕ_i) +µA(u_n,m, ϕ_i) +a

∇(ρ_n,m)², ϕ_i

= (ρ_n,mf, ϕ_i)

+ (g_n,m, ϕ_i), ∀ϕ_i∈ Vn (21) and we note (W_n) the problem after the passage to the limit onm:

α(ρ_n, p_i) + (div (ρ_nu_n), p_i) = (h_n, p_i), ∀pi∈H¹(Ω) (22) (h_nu_n, ϕ_i) + (ρ_nu_n· ∇un, ϕ_i) +µA(u_n, ϕ_i) +a

∇(ρn)², ϕ_i

= (ρ_nf, ϕ_i) + (g_n, ϕ_i), ∀ϕi∈ Vn. (23) Notice thatW_n verifies the continuity equation exactly, that is necessary for the passage to the limit inn.

We assume that the data verify

h≥0, h≡0, h∈L²(Ω), g∈L^p(Ω)², p <2 andf ∈L⁴(Ω)². (24) Replacingϕbyuin equation (19) and using the relationu· ∇

ρ²

= 2

αρ²−hρ+ div ρ²u

, we obtain

Ωh|u|²

2 +αρ|u|² 2 + 2a

αρ²−hρ

+µA(u, u) dx=

Ωρu·f +u·g dx. (25) We can now state the following theorem:

Theorem 1.1. Assuming (24), the sequence of solutions (ρ_n,m, u_n,m) of (W_n,m) converges uniformly in nto (ρ_n, u_n)solution of problem(W_n) with(ρ_n, u_n)satisfying

ρ_n is bounded in L⁴(Ω), (26)

∇

divu_n− a µρ²_n

is bounded in L^r(Ω)², r < 4

3, (27)

and(ρ_n, u_n)converges to (ρ, u)which verifies the energy equation (25).

The proof of this theorem is built up on these following steps

• Construction of approximate solutions of (18, 19) (Sect. 1.1).

• Bound onρ_n inL⁴(Ω) (Sect. 1.2).

• Bound on divu_n−bρ²_n in W^1,r(Ω), r < ⁴₃, whereb= ^a_µ (Sect. 1.3).

• Compactness result onρ_n (Sect. 1.4).

1.1. Construction of approximate solutions

Lemma 1.2. (W_n)has a solution (ρ_n, u_n)satisfying

(ρ_n, u_n)∈H¹(Ω)×H⁴(Ω)². (28)

Proof. We consider the following problem where vis fixed, v∈H³(Ω)²:

α(ρ_n,m, p_i) + (div (ρ_n,mu_n,m), p_i) = (h_n,m, p_i), ∀p_i∈H¹(Ω)_m (29) (h_n,mu_n,m, ϕ_i) + (ρ_n,mv· ∇v, ϕ_i) +µA(u_n,m, ϕ_i) +a

∇(ρ_n,m)², ϕ_i

= (ρ_n,mf, ϕ_i) + (g_n,m, ϕ_i),∀ϕ_i∈ Vn. (30)

Let us write the estimates onρ_n,m andu_n,m. On the one hand, replacingp_i byρ_n,min (29), we obtain:

α||ρn,m||²_L2(Ω)+1 2

Ωdivu_n,mρ²_n,mdx=

Ωh_n,mρ_n,mdx. (31)

On the other hand, replacingϕ_i byu_n,m in (30), we obtain

Ωh_n,mu²_n,mdx+

Ωρ_n,mu_n,mv· ∇vdx+µA(u_n,m, u_n,m)−a

Ωρ²_n,mdivu_n,mdx=

Ωρ_n,mu_n,m·f dx +

Ωg_n,m·u_n,mdx. (32) Multiplying (31) by 2aand summing with (32), we get

2aα||ρ_n,m||²_L2(Ω)+

Ωh_n,mu²_n,mdx+

Ωρ_n,mu_n,mv·∇vdx+µ||u_n,m||²_Vn = 2a

Ωh_n,mρ_n,mdx+

Ωρ_n,mu_n,m·f dx +

Ωg_n,m·u_n,mdx. (33) Thus, we obtain bounds on u_n,m in V_n and onρ_n,m in L²(Ω). We can extract two sequences still notedu_n,m andρ_n,mwhich converge weakly respectively tou^∗_n, ρ^∗_n, whenm goes to +∞.

The following step consists in the passage to the limit when m goes to +∞. The main difficulty is due to the term∇ρ²_n,m. The bound onρ_n,min L²(Ω) is not sufficient to pass to the limit in this term. We must show thatρ_n,mis bounded inH¹(Ω) in order to obtain a strong convergence onρ_n,m. Then we can pass to the limit in each term of (29, 30).

Taking the gradient of the continuity equation, multiplying by∇ρ_n,m and integrating over Ω, we obtain

||∇ρ_n,m||²_L2(Ω)²

α− −C||Du_n,m||_L^∞_(Ω)

≤C||ρ_n,m||_L²_(Ω)||∇divu_n,m||²_L2(Ω)². (34) Choosing ∆t such thatα− −C||Dun,m||_L^∞_(Ω)>0 whereC is a positive constant, we get the bound onρ_n,m inH¹(Ω) thanks to the regularity of the functions basis. Indeed,u_n,mis bounded inH⁴(Ω)²then∇divu_n,mis bounded inL^∞(Ω).

Finally, we prove the existence of solutions to the problem (W_n) with the Leray–Schauder fixed point theo- rem [12]. Fort∈[0,1] and for all (φ, v)∈L²(Ω)×H³(Ω)², we defineA_t(φ, v) = (ρ_n, u_n) as the solution of the following problem (W_n)^t

α(ρ_n, p_i) + (div (ρ_nv), p_i) = (th_n, p_i), ρ_n∈H¹(Ω) (h_nu_n, ϕ_i) + (ρ_nv· ∇v, ϕi) +µA(u_n, ϕ_i) +a

∇ ρ²_n

, ϕ_i

= (ρ_n(tf), ϕ_i) + (tg_n, ϕ_i), u_n∈H⁴(Ω)². It is easy to check thatA_tis a compact mapping onL²(Ω)×H³(Ω)² that depends continuously on parameter t∈[0,1] and thatA₀(φ, v) = (0,0). Moreover, each fixed point (ρ_n, u_n) ofA_tsolves

α(ρ_n, p_i) + (div (ρ_nu_n), p_i) = (th_n, p_i), ∀p_i∈H¹(Ω) (h_nu_n, ϕ_i) + (ρ_nu_n· ∇u_n, ϕ_i) +µA(u_n, ϕ_i) +a

∇ ρ²_n

, ϕ_i

= (ρ_n(tf), ϕ_i) + (tg_n, ϕ_i),∀ϕ_i∈ Vn∩H⁴(Ω)² where (ρ_n, u_n) is bounded inH¹(Ω)×H⁴(Ω)² uniformly int∈[0,1]. Applying the Leray–Schauder principle, we deduce the existence of a solution of the previous problem for t= 1,i.e. for (22, 23).

1.2. Bounds on ρ

in L

(Ω)

We prove in this section a regularity result which shows that ρ_n solution of (22) is bounded in L⁴(Ω).

Replacing ϕby∇p_i in (23), using the special basis defined in the previous section and multiplying by ⁻¹_λ

i, we have

a

Ωρ²_np_i dx=

Ω(−h_nu_n−ρ_nu_n· ∇u_n+ρ_nf+g_n)∇pi

λ_i dx+µ

Ωdivu_n p_i dx. (35) We first prove that ρ_n is bounded in L³(Ω). Multiplying (35) by

P_n ρ²_n1/2

p_i where P_n is the projection operator on{p₁, . . . , p_n} and summing oni, we obtain

P_n ρ²_n3/2

= 1

a P_n

ρ²_n1/2

i

Ω(−h_nu_n−ρ_nu_n· ∇u_n+ρ_nf+g_n)∇p_i

λ_i dx+µ

Ωdivu_n p_i dx

p_i. (36) In this equation, we just need to bound the term ρ_nu_n· ∇u_n. We adapt the strategy of Lions in [6] which consists in decomposingρ_nu_n=∇θ_n+ Curlq_n. Sinceρ_nu_n∈L^r(Ω)², ¹_r = ¹₂+¹_q, q <∞, there exists∇θ_n and Curlq_n ∈L^r1(Ω)²,1< r₁≤r <2 such that

ρ_n(u_n· ∇)u_n= (∇θ_n· ∇)u_n+ (Curlq_n· ∇)u_n.

Since div∇θ_n = div (ρ_nu_n) = h_n−αρ_n is bounded in L²(Ω) and in addition θ_n ∈ W^1,4/3(Ω), then ∇θ_n is bounded inH¹(Ω)²→L^β(Ω)², β <∞. So (∇θ_n· ∇)u_n is bounded inL¹(Ω)² (at least) and there exists∇θ_n,1 and Curlq_n,1 such that

(∇θn· ∇)un =∇θn,1+ Curlq_n,1 withθ_n,1 bounded inL²(Ω) andq_n,1∈H₀¹(Ω).

Next, since div Curlq_n = curl (∇u^j_n) = 0, 1 ≤j ≤N, we can apply the results of Coifman et al. [1] and deduce that the product of Curlq_n by∇u^j_n belongs to the Hardy spaceH^q(Ω)² where 1< ¹_q =¹_r+¹₂ <1 + _N¹,

23 < q <1. Since Sobolev’s embeddings are also valid inH^q(Ω)²(equals toL^q(Ω)² ifq >1) with _N+1^N < q <1, thenH^q(Ω)²→W^−1,q∗(Ω)² where _q¹_∗ =¹_q −_N¹ and we can write

(Curlq_n· ∇)u_n=∇θ_n,2+ Curlq_n,2 whereθ_n,2 is bounded in L^q∗(Ω) withq^∗= 2− andq_n,2∈H₀¹(Ω).

Finally, we can write that

ρ_n(u_n· ∇)u_n=∇(θ_n,1+θ_n,2) + Curl (q_n,1+q_n,2) =∇θ_n,3+ Curlq_n,3 withθ_n,3 bounded inL²⁻(Ω) andq_n,3∈H₀¹(Ω).

These properties verified byθ_n,3 andq_n,3allow us to show thatρ_n is bounded inL³(Ω). Indeed, in (36) we write

P_n

ρ²_n1/2

i

Ω−ρ_nu_n· ∇u_n∇p_i λ_i dx

p_i=− P_n

ρ²_n1/2

i

Ω∇θ_n,3∇p_i λ_i dx

p_i

= P_n

ρ²_n1/2

i

Ωθ_n,3p_i dx

p_i

= P_n

ρ²_n1/2

P_n(θ_n,3)

and integrating over Ω, we deduce

Ω

P_n ρ²_n1/2

P_n(θ_n,3) dx≤ P_n

ρ²_n1/2

L³(Ω)||P_n(θ_n,3)||_L3/2(Ω). (37) So, in view of the L²⁻(Ω) bound onθ_n,3, (36) yields

P_n ρ²_n^3/2

L^3/2(Ω)≤CP_n ρ²_n^1/2

L^3/2(Ω)+ 1

(38) andP_n

ρ²_n

is bounded inL^3/2(Ω). Thus, P_n ρ²_n

converges weakly toin L^3/2(Ω) and we have

ΩP_n ρ²_n

p_i dx=

Ωρ²_np_i dx−→

Ωp_i dx, ∀p_i ∈L³(Ω), (39) that allows to deduce thatρ²_n converges weakly toin L^3/2(Ω) and so thatρ_n is bounded inL³(Ω).

Using the same previous arguments, we show that ρ_n is bounded in L⁴(Ω). Indeed, multiplying (35) by P_n

ρ²_n

p_iand summing oni, we obtain P_n

ρ²_n2

= 1 a

P_n ρ²_n

i

Ω(−hnu_n−ρ_nu_n· ∇un+ρ_nf+g_n)∇pi

λ_i dx+µ

Ωdivu_n p_i dx

p_i, i= 1, . . . , n.

(40)

Since ρ_n is bounded in L³(Ω), ρ_n(u_n· ∇)u_n = ∇θ_n,3+ Curlq_n,3 is bounded in L¹(Ω)² and we can findθ_n,3 bounded inL²(Ω). Using the same computations that in (37), we obtain

ΩP_n ρ²_n

i

Ω−ρ_nu_n· ∇u_n∇pi

λ_i dx

p_i dx≤P_n ρ²_n

L²(Ω)||P_n(θ_n,3)||_L²_(Ω) and finally, (40) yields

P_n ρ²_n

L²(Ω)≤C. (41)

We conclude thatP_n ρ²_n

is bounded inL²(Ω) andρ_n is bounded in L⁴(Ω).

1.3. Bounds on div u

− bρ

in W

(Ω)

Let us consider once more the equation (23) with ϕ = ^∇p√_λⁱ

i. Let P_n^∇ be the L²-projection operator on ∇p√ 1

λ1, . . . ,^∇p√_λⁿ

n

, we write P_n^∇

µ∇divu_n−a∇

ρ²_n

=P_n^∇(h_nu_n+ρ_nu_n· ∇u_n−ρ_nf−g_n), (42) so

P_n^∇

∇

µdivu_n−aρ²_n

=P_n^∇(h_nu_n+ρ_nu_n· ∇u_n−ρ_nf−g_n). (43) We show that the right member terms are bounded inL^r(Ω)², r < ⁴₃. Indeed,ρ_nu_n·∇u_n is bounded inL^r(Ω)², with ¹₄+¹_p+¹₂ =¹_r andp <∞. In addition,h_nu_n andρ_nf are bounded inL^4/3(Ω)² andg_n inL^p(Ω)², p <2.

SinceP_n^∇

∇

divu_n−bρ²_n

=P_n^∇(∇(π_n)) is bounded inL^r(Ω)², with b=_µ^a then there existsπsuch that P_n^∇(∇(π_n))−→ ∇πweakly inL^r(Ω)²,i.e.

Ω

P_n^∇(∇π_n)αdx−→

Ω∇π αdx, ∀α∈L^r∗(Ω)², 1

r^∗ = 1−1

r, (44)

Ω

n i=1

∇π_n,∇p√ _i λ_i

∇p√ _i

λ_i αdx−→

Ω∇π αdx, ∀α∈L^r∗(Ω)², (45) then choosingα= ∇p_i

√λ_i ∈L^r∗(Ω)² and using the orthonormality of the basis, we have

Ω∇πn∇p_i

√λ_i dx−→

Ω∇π∇p_i

√λ_i dx. (46)

Then, ∇π_n is bounded inL^r(Ω)² and π_n converges strongly toπ= divu−bρ² in L^r(Ω) whereρ²is the weak limit ofρ²_n.

1.4. Compactness Result on ρ

The crucial bounds on ρ_n and divu_n−bρ²_n respectively shown in Sections 1.2 and 1.3 permit to obtain an important compactness result allowing to pass to the limit in the equations. We recall the theorem shown in [6]

(Chap. 6, p. 81)

Theorem 1.3. ρ_n converges toρstrongly inL^p(Ω)for all 1≤p <4, when ngoes to+∞.

The proof of this result uses in particular the regularization Lemma 2.3 described in [3] and [5] which allows to build up renormalized solutions and uses some convexity properties of these solutions. We just precise the importance of the bound on divu_n−bρ²_ninW^1,r(Ω) which is necessary to prove this result. Indeed, divu_n−bρ²_n is relatively compact inL^r(Ω) and converges strongly inL^r(Ω) to divu−bρ². This property allows to pass to the limit in the term ( +ρ_n)^θ

divu_n−bρ²_n

where >0 and 0< θ ≤1. Notice that a key for several proofs described in [6] is the convergence of products of this type (see Appendix B of [6]). Indeed, the author shows that the product of β(ρ_n) by [divu_n−bρ^γ_n] converges weakly to the product of the weak limits of each term i.e. to ¯β[divu−bρ^γ], for any continuous functionsβ on [0,∞) such thatβ(t)t^(q−γ)andβ(t)t^(q/2)goes to 0 ast goes to +∞.

Finally, this strong convergence result on ρ_n, given above, allows to pass to the limit in the term∇ρ^γ, the real difficulty.

2. Numerical applications

We present in this section the numerical results obtained with the compressible and isentropic Navier–Stokes model (18, 19) denoted NSCI. In view to obtain a numerical validation, we compare these results with a comparable shallow water model denoted SW, in which the diffusion term depends onρand must be evaluated at each time step. Next, we present the numerical results obtained with an analytical case verifying (1, 2) introduced by Weigant [11].

At first, the tests are executed on a simplified studied domain, a square of one unit in length. This particular geometry allows to obtain an analytical expression for the eigenfunctions, solutions of problems (N) and (D) defined in introduction. Next, we present some results executed on a real domain: the dam of Calacuccia in Corsica. In this case, the eigenfunctions are obtained with the finite elements Modulef software.

2.1. Comparison between the NSCI model and the SW model

We have shown in Section 1.1 that the following problem, solved with the Galerkin method, had a fixed point and we propose in this section a numerical resolution method of the following equations

α(ρ_n,m, ψ) + (div (ρ_n,mv_n,m), ψ) = (h_n,m, ψ), ∀ψ∈H¹(Ω)_m, (47) (h_n,mu_n,m, ϕ) + (ρ_n,mv_n,m· ∇vn,m, ϕ) +µA(u_n,m, ϕ) +a

∇(ρn,m)², ϕ

= (ρ_n,mf, ϕ) + (g_n,m, ϕ),∀ϕ∈ Vn. (48) Notice that we seta=¹₂ in (48), in order to execute the comparison test described in the next section.

We use the global Galerkin method to approximateu_nm andρ_n,mwhich are searched under the form

u_n,m= n

i=1

a_i(t)ϕ_i(x) =

n1

i=1

c_i(t)U p_i(x) +

n2

i=1

d_i(t)U q_i(x), (49)

ρ_n,m= m l=1

b_l(t)p_l(x) (50)

whereU p_i=^√∇p_λⁱ

i andU q_i =^Curlõi^qi are solutions of problem (B) presented in introduction.

We describe the numerical resolution method on a time step at time t. Replacing ψ in (47) by p_k, the continuity equation leads to

b^j+1_k (t)−∆t n i=1

m l=1

a^j_i(t)b^j+1_l (t)

Ωp_lϕ_i∇p_k dx=b_k(t−∆t), (51) where j represents the implicit iterations on a time step. Next, the equation (48) leads to a linear system of the form

1

∆t n i=1

m l=1

b_l(t−∆t)a^j+1_i (t)

Ωp_lϕ_iϕ_k dx+µΥ_iδ_ik

= 1

∆t m l=1

n i=1

b_l(t−∆t)a_i(t−∆t)

Ωp_lϕ_iϕ_k dx +

m l=1

n i=1

n r=1

a^j_i(t)a^j_r(t)b^j+1_l (t) 1

2

Ωϕ_iϕ_rp_ldivϕ_k dx+1 2

Ωϕ_iϕ_r∇p_lϕ_k dx−

Ωp_lcurlϕ_iα(ϕ_r)ϕ_k dx

+1 2

m l=1

m s=1

b^j+1_l (t)b^j+1_s (t)

Ω

p_lp_sdivϕ_k dx+ m l=1

b^j+1_l (t)

Ω

p_lfⁿϕ_k dx (52)

where Υ_i represents the eigenvaluesλ_i orµ_i respectively if ϕ_i is equal toU p_i or U q_i. When we start the method (thenj = 0), we set

b^j=0_l (t) =b_l(t−∆t), l= 1, . . . , m, a^j=0_i (t) =a_i(t−∆t), i= 1, . . . , n.

Solving (51) and (52), we obtain b¹_l and a¹_i. Next, we apply a fixed point method for j ≥ 2 by computing b^j_l and a^j_i, and we repeat these computations until we get a certain convergence criterion on the sequences,

where is fixed,

b^j+1_l (t)−b^j_l(t)

L²(Ω)≤ , a^j+1_i (t)−a^j_i(t)

L²(Ω)² ≤ . 2.1.2. Comparison test with a shallow water model

In order to validate our approximate method, we notice that using (1) we can rewrite (2) as follows ρ∂u

∂t +ρu· ∇u−µ∆u+1

2∇(ρ²) =ρf (53)

and dividing byρ(noticed that if ρ₀>0 thenρ >0), we obtain

∂u

∂t +u· ∇u−µ

ρ∆u+∇ρ=f. (54)

As we have analyzed and solved the following shallow water problem [7]

∂ρ

∂t + div (uρ) = 0, (55)

∂u

∂t +u· ∇u−ν∆u+∇ρ=f, (56)

u·n= 0, curlu= 0 on Σ, (57)

u(t= 0) =u₀(x), ρ(t= 0) =ρ₀(x), ρ₀(x)>0, (58) it is natural to compare the solution of (1, 2), approximated by (47, 48) at each time step, to the solution of the previous shallow water equations in which we setν= _ρ(t)^µ whereρ(t) is the solution obtained with (47, 48) at time t.

2.1.3. Numerical results

1. Tests on a square with a constant wind.

Taking a wind f under the form of a gradient, an obvious asymptotic solution of NSCI model, with u₀= 0 andρ₀=K >0, is

¯

u= 0 and the solution of 1

2∇ρ¯²= ¯ρf.

So, iff = (1,0) =∇φ(x) =∇(x+C) for instance, then

¯

ρ=φ(x) +C (59)

whereC =_meas(Ω)¹

Ωρ₀(x)−φ(x) dx. This example is interesting because we verify that the behaviour of the approximate solution is near the exact solution whent−→ ∞(notice that we have not proved that ( ¯ρ,u) is the asymptotic solution of (1, 2)). We have represented in Figures 1 and 2 the variation¯ ρ ofρ around its mean value, for both models. We obtain some linear fields, which verify (59).

We represent in Figure 3 the evolution of Ec(u_n) = ¹₂||u_n||²_L2(Ω)² = ¹_2n

i=1a_i(t)² for both models (which corresponds to kinetic energy if ρ is constant but we employ this term by abuse). This figure shows that the velocity initialized to zero, oscillates and goes to zero whent goes to∞.

0 1 0

1

−0.392

−0.392

−0.317−0.317−0.317 −0.242−0.242 −0.168−0.168−0.168 −0.0932−0.0932−0.0932 −0.0186−0.0186 0.0560.0560.056 0.1310.131 0.2050.2050.205 0.280.280.28 0.3540.354

0.429

0.429

0.429

Figure 1. Variation ofρfor NSCI model.

0 1

0 1

−0.392

−0.392 −0.317−0.317−0.317 −0.242−0.242 −0.168−0.168−0.168 −0.0932−0.0932−0.0932 −0.0186−0.0186 0.05590.05590.0559 0.1310.131 0.2050.2050.205 0.280.280.28 0.3540.354

0.429

0.429

0.429

Figure 2. Variation of ρ for SW model.

0 10

0 0.001 0.002 0.003 0.004 0.005 0.006 0.006813

0 10

0 0.001 0.002 0.003 0.004 0.005 0.006 0.006813

NSCI model SW model

Figure 3. Kinetic energy.

Next, we have represented in Figure 4 the evolution of ||∇ρ−f||_L²_(Ω)² for both models (notice that the curves are quasi-identical) in order to verify that this difference goes to zero.

2. Tests on a “real” domain: the dam of Calacuccia (Corsica).

The Figure 5 represents the bathymetry of the dam of Calacuccia whose the maximal depth is approxi- mately fifty meters. The tests are computed with a west dominant wind characteristic of this geographical area (Fig. 6). The results obtained for both models are quasi identical and for the velocity fields, we observe a circulation in the direction of the wind close to the coasts (where the height of water is low) and

0 10

−0.9653 0 0.7424

0 10

−0.9653 0 0.7424

NSCI model SW model

Figure 4. Evolution of||∇ρ−f||_L²_(Ω)².

0 10 15

0 1 2 3 4 5 6 7 8 9

8.46

8.46 8.46

8.46

12.3

12.3

12.3 12.3

12.3 12.3

16.2

16.2

16.2 16.2

16.2

16.2 20 20

20

20

20

20 23.9

23.9 23.9

23.9

23.9

27.8 27.8

27.8

27.8

27.8

31.6 31.6 31.6

31.6

35.5 35.5

35.5

35.539.3

39.3 39.3

43.247

Figure 5. Smoothed bathymetry.

a recirculation by the center of the domain where the depth is most important (Figs. 7 and 8). The dif- ference observed on the maximal velocity is probably due to the truncating level of the Galerkin method.

Indeed, for these computations, we have used the 30 first eigenvectors for each (N) and (D) problem and on a complex domain, contrary to the square case, we need a higher number of modes for better representing the solution. More precision is more expensive and this is mainly due to the nonlinear terms.

We remark in the Figures 9 and 10 that the surface elevation is only influenced by the wind.