Anelastic Limits for Euler Type Systems

Anelastic Limits for Euler Type Systems

Didier Bresch ^∗and GuyM´etivier ^† April 17, 2010

Abstract

In this paper, we present rigorous derivations of anelastic limits for compressible Euler type systems when the Mach (or Froude) number tends to zero. The first and main part is to prove local existence and uniqueness of strong solution together with uniform estimates on a time interval independent of the small parameter. The key new remark is that the systems under consideration can be written in a form where ideas from^MS[MeSc1] can be adapted. The second part of the analysis is to pass to the limit as the parameter tends to zero. In this context, the main problem is to study the averaged effect of fast acoustic waves on the slow incompressible motion. In some cases, the averaged system is completely decoupled from acoustic waves. The first example studied in this paper enters this category: it is a shallow-water type system with topography and the limiting system is the inviscid lake equation (rigid lid approximation). This is similar to the low Mach limit analysis for prepared data, following the usual terminology, where the acoustic wave disappears in a pure pressure term for the limit equation. The decoupling also occurs in infinite domains where the fast acoustic waves are rapidly dispersed at infinity and therefore have no time to interact with the slow motion (see^{Sc,MS, Al}[Sc, MeSc1, Al2]).

In other cases, and this should be expected in general for bounded domains or periodic solutions, this phenomenon does not occur and the acoustic waves leave a nontrivial averaged term in the limit fluid equation, which cannot be incorporated in the pressure term. In this case, the limit system involves a fluid equation, coupled to a nontriv- ial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic en- ergy. This was suspected for the periodic low Mach limit problem for

∗UMR 5127 CNRS, Laboratoire de Math´ematiques, 73376 Le Bourget du lac Cedex, France. Email : didier.bresch@univ-savoie.fr

†IMB, Universit´e de Bordeaux, 33405 Talence cedex, France. Email : Guy.Metivier@math.u-bordeaux.fr

nonisentropic Euler equations in[MeSe2] and proved for finite dimen-^MeSc sional models. The second example treated in this paper, namely Euler type system with heterogeneous barotropic pressure law, is an example where this scenario is rigorously carried out. To the authors’ knowl- edge, this is the first example in the literature where such a coupling is mathematically justified.

Keywords: Compressible Euler equations, heterogeneous media, shallow- water system, rigid-lid approximation, anelastic limit, lake equation, low mach number, low Froude Number.

AMS subject classification: 35Q30, 35B40, 76D05.

1 Introduction

Anelastic limits starting from diffusive systems have been recently stud- ied from a Mathematical point of view for instance in ^BrGiLi[BrGiLi], [Ma1]^Ma1 and ^FeMaNoSt[FeMaNoSt]. These works concern respectively the degenerate vis- cous shallow-water equations with bathymetry and the compressible Navier- Stokes equations with high potential and constant viscosities. In all these papers, the authors consider global weak solutions where the time interval is fixed and, in the ill-prepared case, they prove that, from an energetical point of view, acoustic waves do not interact with the mean velocity field.

In this paper, we consider anelastic limits starting from two compressible Euler-type systems.

I) The first model is the two space dimension shallow-water system with topography, namely:

model1

model1 (1.1)







∂th+ div(hv) = 0,

∂_t(hv) + div(hv⊗v) +h∇(h−h_b) ε² = 0

where v denotes the vertical averaged of the hoziontal velocity field com- ponent, h the height of water and the bathymetry h_b is a given function depending on the space variables, see for instance ^Br[Br],[BrGiLi]. Note the^BrGiLi analysis presented here extends to similar inviscid systems, see for instance

Ma1[Ma1] for corresponding viscous systems.

II) The second modelwe have in mind is the Euler equation with heteroge- neous pressure law

model2

model2 (1.2)







∂tρ+ div(ρv) = 0,

∂_t(ρv) + div(ρv⊗v) +∇(c(x)ρ^γ) ε² = 0

withca given function depending on space variablesx,ρ the density of the fluid andv its velocity.

To prove existence and uniqueness of local strong solution on a time interval which does not depend on the small parameter ε, the main idea is to rewrite such systems under an appropriate form : in Section 2, we prove that after a suitable change of unknows, both system enter the general framework of systems of the following form:

model3

model3 (1.3)







a(∂_tq+v· ∇q) +1

εdiv_xu= 0, b(∂_tm+v· ∇m) +1

ε∇ψ= 0 witha(t, x) andb(t, x) known and positive, and constit

constit (1.4) q = 1

εQ(t, x, εψ), m=µ(t, x)u, v=V(t, x, u, q),

whereQ,µandV are smooth functions of their arguments withQ(t, x,0) = 0,∂_θQ >0 and µ >0. Note thatq is not singular inε and satisfies

constitb

constitb (1.5) q =Q1(t, x, εψ)ψ, with Q1>0.

For such systems, the existence and uniqueness of strong solutions on a time interval independent ofεcan be obtained following the lines of[MeSc1]:^MS first, since the singular term is skew symmetric there are easyL² estimates;

next one commutes the equation with appropriate operators so that the commutators are controlled. This is explained in Section3.^sec3

Concerning the asymptotic limit, we show that the two models (1.1)^model1 and (1.2) yield different analyses. For system (^model2 1.1) the main contribution^model1 of acoustic waves can be written as a gradient and therefore behaves as a pressure term. On the contrary, for system (1.2), the acoustic waves are^model2 strongly coupled to the mean velocity by a term which is not a gradient.

The novelty of this model, compared to usual models studied in the litera- ture, is that in the pressure law,p =cρ^γ, the heterogeneity of the medium is modeled through a functionc(x) which changes from a point to another.

The strong coupling between the acoustic part and the mean field is well known in many models. For instance it occurs for ill-prepared data for the non-isentropic Euler equations, where an additional equation for the entropy is added to the classical Euler system. For unbounded domains, the decou- pling between the acoustics and the mean field is due to the fast dispersion

of the acoustic waves at infinity (see[MeSc1, Al1]). On bounded or periodic^{MS, Al1} domains, acoustic waves still travel very fast but are trapped and their aver- aged effect remains present in the limit. The analysis of this averaging seems to be very difficult in general due to the crossing eigenvalues phenomena, see for instance BrDeGr, MeSc

[BrDeGr, MeSe2]. In ^BrDeGrLi[BrDeGrLi] a semi-formal derivation is given for the nonstationary problem with ill prepared initial data. For System (^model21.2), the fast acoustic waves are governed by a space dependent wave equation, and this induces a nontrivial coupling in the limit. But, this fast wave equation is independent of the solution, in sharp contrast with the nonisentropic Euler’s equation. Using this property, the limit can be car- ried out rigorously. The nonhomogeneity yields an extra non gradient term depending on the waves in the mean momentum equation. The description of the waves dynamics might be of physical and numerical interest.

The paper is organized as follows : in Section 2, we state the main results. In Section 3, we prove the uniform estimates on (u, ψ) which imply existence and uniqueness of strong solutions on a fixed interval of time for the general system. The last section is devoted to the asymptotic analysis when εgo to zero and the rigorous proof of convergence of solutions to solutions of asymptotic models. This section is splitted in four parts: the first one concerns the general problem, the second part (Theorem 2.3) provides a^theoconv framework where we get an asymptotic decoupling between fast and slow scales. Note that System (^model11.1) satisfies the asumptions of this part. The third part concerns dispersion of acoustic waves on R^d which may lead to strong convergence (Theorem2.4) and in the last part we consider averaged^acous acoustics on the torus leading to a limit system involving a fluid equations coupled to a nontrivial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic energy (Theorem 2.6). System (^Theoo 1.2) satisfies the assumptions of^model2 this part and thus it provides an example where this scenario is rigorously carried out. At the end of the paper, we give the coupled limit system corresponding to the heterogenous isentropic Euler system. To the authors’

knowledge, this is the first example in the literature where such a coupling is mathematically justified.

2 Main results.

2.1 Reduction to System (^model31.3)

a) The shallow-water equations. Consider the system (1.1). Using the^model1 mass equation, denoting

ψ= (h−h_b)/ε, q= 1

εln(1 +εψ/h_b), the system (^model11.1) may reads

model11

model11 (2.1)







h_b(∂_tq+v· ∇q) +div(h_bv) ε = 0,

∂tv+v· ∇v+∇ψ ε = 0.

This system is of the form (^model31.3) with

a=hb, b= 1, V =m=u/hb, u=hbv.

b) The heterogeneous Isentropic Euler equations. Consider the sys- tem (^model21.2). Using the mass equation (2.3)₁ and denoting

ψ= γ (γ−1)

(c^1/γρ)^γ−1−1

ε , q= 1

ε(γ−1)ln(1 +ε(γ−1)ψ/γ), the system (^model21.2) reads

model21

model21 (2.2)











c^−1/γ(∂_tq+v· ∇q) +div(c^−1/γv)

ε = 0,

c^−1/γ(∂tv+v· ∇v) +∇ψ ε = 0, This system is of the form (^model31.3) with

a=c^−1/γ, b=c^−1/γ, V =m=c^1/γu, u=c^−1/γv.

2.2 Uniform existence and uniqueness of smooth solution Our first theorem concerns the existence of local strong solution on a time intervall which does not depend on the small parameter. For such purpose we consider the general form (1.3). We work on the domain^model3 D^d which is either the entire space R^d, or a torus T^d, or a mixed of these two types T^d

0 ×R^d−d

0.

mainth Theorem 2.1. There are ε0 > 0 and T0 > 0 which depend only on the H^s(D^d) norm of the initial data, s > ^d₂ + 1, such that for all ε ≤ ε₀ the Cauchy problem for (^model31.3) has a unique solution(u, ψ)∈C⁰([0, T];H^s(D^d)).

In particular, this gives the existence and uniqueness of local strong solution on a time intervall which does not depend on the small parameter εfor Systems (^model11.1) and (1.2).^model2

2.3 Asymptotic limits

In a second part, we look at the asymptotic procedure, lettingεgo to zero, in the ill-prepared case for (1.3) under some assumptions on^model3 a, b, µ, V and Q. We consider different domains, Ω and we investigate three different frameworks.

a) In the first case, assumptions on b and V provide a framework where we get an asymptotic decoupling between fast and slow scales. Note that System (^model11.1) satisfies the asumptions of this part and that this result applies on any domain Ω.

b) Next we work on Ω =R^dwith specific decreasing assumptions at infinity for the coefficients (similar to the one given in[MeSc1]) which provide dis-^MS persion of acoustic waves on R^d. This lead to strong convergence and the decoupling observed in case a) occurs in this case too, but for a different reason.

c) In the last part, we consider averaged acoustics on the torus Ω = T^d. Under some assumptions on a, b, µ, V and Q, we mathematically justify a limit system involving a fluid equations coupled to a nontrivial infinite dimensional system of differential equations which models the energy ex- change between the fluid and some remanent acoustic energy. System (1.2)^model2 satisfies the assumptions of this theorem and thus it provides an example where this scenario is rigorously carried out.

We first prove a result under the following

ass4.1 Assumption 2.2. Suppose that b is constant and the function V has the special form

V(t, x, u, q) =dµu, where dis a constant.

theoconv Theorem 2.3. Under Assumption ^ass4.12.2 and assuming (^initconv4.1), the family of solutionsu^ε (given by Theorem 2.1) converge weakly (in the sense of distri- butions) to the unique solution of

limeq1

limeq1 (2.3) ∂t(µu) +dµu· ∇(µu) +∇π= 0, divu= 0, with initial data(liminitdata

4.11). Moreover,u˜^εconverges strongly touinC⁰([0, T];H_loc^s0 (D^d)) where u˜^ε is defined by (^decomp4.7).

Remark. This theorem applies for system (1.1) and provides the lake equa-^model1 tion (^lake4.16) at the limit.

Next, we split the analysis in two parts:

i)Dispersion of acoustics waves on R^d. Introduce the notation F for func- tionsf on [0, T]×R^dwhich have a limitf(t) asx tends to infinity and such that

coeffatintf

coeffatintf (2.4) |f(t, x)−f(t)| ≤C|x|^−1−δ, |∇_xf(t, x)| ≤C|x|^−2−δ

for some constantC and some δ >0. Let us defineQ₁(t, x) =Q1(t, x,0) =

∂_θQ(t, x,0).

acous Theorem 2.4. Suppose that the coefficients a, b, µ and Q₁ belong to the classF. Then, the family(ψ^ε, u^ε)converges strongly to(0, u)inL²([0, T];H_loc^s0 (R^d)) and the limit equation for u is

(2.5) b(∂_t(µu) +v· ∇(µu)) +∇π = 0, divu= 0 withv=V(t, x, u,0).

Remark. This theorem applies for both systems (1.1) and (^{model1 model2}1.2) provided that the coefficientsh_b orc(x) respectively belong to the class F.

ii) Averaged acoustics on the torus. We make the following assumption.

ass4.4 Assumption 2.5. The reference domain in a torus T^d, the coefficients a, b, µ do not depend on time and are smooth functions of x ∈ T^d, as well as the function Q(x, θ). We further assume that V = d(t, x)u is a linear function ofu.

Theoo Theorem 2.6. Under the Assumption2.5, the limit^ass4.4 usatisfies the equation (4.37)^limequwith initial data (liminitdata

4.11). The additional term I is given by (4.36)^defcalland the α_j’s satisfy the equations (4.38)^alphajwith the initial conditions (4.39).^alphajinit Remark. This theorem applies for system (1.2) and the limit system is given^model2 by (^model234.40), (4.41), (^{defPhi ell}4.42), (²³³4.43) and (^model244.44). Note the strong coupling between mean velocity and acoutic waves.

3 Proof of Theorem 2.1.

sec3

The system being symmetric hyperbolic, solutions are known to exist and to be unique on a small interval of time depending onε. The solutions are continued to a fixed interval, using suitable a-priori estimates for the smooth solutions. Following^MS[MeSc1], we are looking for estimates of

defK

defK (3.1) K := sup

t∈[0,T]

(u, ψ)(t)

H^s, s > d 2 + 1.

Recall the following nonlinear estimates:

Lemma 3.1 (Nonlinear estimates). i) for s > ^d₂, 0 ≤ k_j and k :=

k1+. . .+kp ≤s:

nl1

nl1 (3.2)

u₁. . . u_p

_s−k≤C

u₁. . . u_{p s−k}

1. . .

u_p. . . u_{p s−k}

p

ii) for s >1 +^d₂,1≤kj ≤s and k:=k1+. . .+kp ≤s:

nl2

nl2 (3.3)

u1. . . up

_s−k+p−1 ≤C

u1. . . up

_s−k

1. . .

up. . . up

_s−k

p. Proof. i) well known. For ii) apply (3.2) with^nl1 s−1 > ^d₂ and kj −1≥0 in place ofk_j. Note that k≥p, so that s−k+p−1≤s−1.

The first step is to show that the constant K controls various other derivatives of the unknows which will be present in the analysis of commu- tator estimates.

Lemma 3.2. Smooth solutions of (1.3)^model3satisfy the following estimates eq1.10

eq1.10 (3.4) Ke := sup

t∈[0,T]

s

X

k=0

(ε∂t)^k(u, ψ)(t)

H^s−k ≤C(K),

and for s≤k:

eq1.11

eq1.11 (3.5) sup

t∈[0,T]

(ε∂_t)^k(q, m, v)(t)

H^s−k ≤C(K).e

Here and below, C(K) denotes a constant independent of ε, depending only K, the coefficientsa, b, Q, mv, the dimension dand the indexs

Proof. From the equation

ε∂t(u, ψ) = Φε(t, x, u, ψ)∇(u, ψ) + Ψ_ε(t, x, u, ψ)

with Φε and Ψε uniformly bounded with respect toε≥0. By induction on k, the nonlinear estimates imply

(ε∂t)^k(u, ψ)(t)

_Hs−k ≤C(K).

Next, because Q(t, x,0) = 0, q = ˜Q(t, x, εψ)ψ, so the estimate (3.5) for^eq1.11 k= 0 follows. For k≥1, (ε∂t)^kq is the sum of terms

eq1.12

eq1.12 (3.6) ε^l+p−1(∂_t^l∂_θ^pQ)(t, x, εψ)(ε∂tψ)^k1. . .(ε∂tψ)^kp where

l+p≥1, k_j ≥1, l+k₁+. . .+k_p=k.

Thus (^eq1.113.5) follows from (3.4) and (^eq1.10 3.3).^nl2 Moreover,

eq1.13

eq1.13 (3.7) (ε∂_t)^kq =∂_θQ(t, x, εψ)(ε∂_t)^kψ+r_r

where r_k is a sum of terms (^eq1.123.6) with l+p ≥ 2. In this case, there is (at least) an extra factorεin front of the derivatives and the k_j ≤s−1 for all j. Thus one can apply another full derivative tork and

eq1.14

eq1.14 (3.8)

∂t,xrk

H^s−k ≤C(K).

There estimates form and v are similar and easier . Lemma 3.3. Let (u_k, ψ_k) = (ε∂_t)^k(u, ψ). Then for k≤s:

comm

comm (3.9)







a(∂tq_k+v· ∇q_k) +1

εdivxu_k =f_k, b(∂tmk+v· ∇m_k) +1

ε∇ψ_k=gk

with

qk

qk (3.10) q_k =∂_θQ(t, x, εψ)ψ_k, m_k =µu_k, estcom

estcom (3.11) sup

t∈[0,T]

(f_k, g_k)(t)

L² ≤C(K).e

Proof. By (3.7)^eq1.13

(ε∂_t)^k∂_tq=∂_tq_k+∂_tr_k.

Thusf_k is equal to a∂_tr_k plus a sum of commutators terms ε^l−1∂_t^la(ε∂t)^k−l+1q, ε^l∂_t^l(av) (ε∂t)^k−l∂xq

withl≥1 and the estimate (^estcom3.11) follows. The proof of the estimate forgk

is similar.

This leads to consider the linearized system

lineq

lineq (3.12)







a(∂t(ρψ) +˙ v· ∇(ρψ)) +˙ 1

εdivxu˙ = ˙f , b(∂t(µu) +˙ v· ∇(µu)) +˙ 1

ε∇ψ˙k= ˙g withρ=∂_θQ(t, x, εψ).

Lemma 3.4. There areC0 andC =C(K) such that the solution of (3.12)^lineq satisfies fort≤1:

estL2

estL2 (3.13)

( ˙u,ψ)(t)˙

L² ≤C0(1 +tC(K))

( ˙u,ψ)(0)˙ L²

+C(K) Z _t

0

( ˙f ,g)(t˙ ⁰) L²dt⁰. Proof. The energy is

E(t) = 1 2

√aρψ(t)˙

2 L²+ 1

2

pbµv(t)˙

2 L². Then

d

dtE(t)≤Re f ,˙ ψ˙

L²+ Re ˙g,u˙

L²

+O(kρ∂_ta−a∂_tρk_L^∞ +kρ∂_x(av)−av∂_xρk_L^∞)kψk˙ ²_L2

+O(kµ∂_tb−b∂tµk_L^∞+kµ∂_x(bv)−bv∂xµk_L^∞)kuk˙ ²_L2.

Thus d

dtE(t)≤Re f ,˙ ψ˙

L² + Re ˙g,u˙

L²+C(K)E(t) and

E¹²(t)≤ E¹²(0) +C(K) Z t

0

( ˙f ,g)(t˙ ⁰) L²dt⁰.

Note that εψ(t)

L^∞ ≤ εψ(0)

L^∞+t ε∂tψ

L^∞ ≤ εψ(0)

L^∞+tCK.e Therefore,

estrho

estrho (3.14)

ρ(t)

L^∞ ≤C₀+tC(K) and

( ˙u,ψ)(t)˙

L² ≤C₀(1 +tC(K))E¹²(t)

whereC₀ depends only on theL^∞norm of the initial data foru andψ. The lemma follows.

corestdt Corollary 3.5. There is C₀ which depends only on theH^snorm of the ini- tial data foru andψand there isC(K) such that the(uk, ψk) = (ε∂t)^k(u, ψ) for k≤s, satisfy

estepsdtk

estepsdtk (3.15)

(uk, ψk)(t)

L² ≤C0+tC(K).

Proof. For k ≥ 1, apply the lemma to (3.9).^comm When k = 0, there is a slightly different equation for (u, ψ): writing q= ˜Q(t, x, εψ)ψand consider- ing ˜Q(t, x, εψ) as a known coefficient, yields an equation which is again of the form (3.12) for (u, ψ).^lineq

Next, we estimate the vorticity.

Lemma 3.6. Let ω := curl (bµu). Then h := (∂_t+v∇)ω satisfies for l≤s−1:

(3.16) sup

0≤t≤T

(ε∂t)^lh(t)

H^s−1−l ≤C(K).

Proof.

h= curl (∂tb)µu+v[∇, b]µu

+ [−curl, v∇]bµu.

Corollary 3.7. There is C0 which depends only on the H^s norm of the initial data foruandψand there isC(K)such thatω := curl (bµu)satisfies estvort

estvort (3.17)

(ε∂t)^lω(t)

H^s−1−l ≤C0+tC(K).

where C0 depends only on the H^s norm of the initial data.

Lemma 3.8 (Elliptic estimates).

estdivcurl

estdivcurl (3.18) u

H^k ≤Ck

divu

H^k−1+

curl (bµu)

H^k−1+ u

H^k−1

. Lemma 3.9. For 0≤l≤k≤s,

estrec1

estrec1 (3.19)

(us−k, ψs−k)(t)

H^l≤C₀+ (t+ε)C(K) +C₁

(us−k, ψs−k)(t) H^l−1

where C0 depends only on the H^s norm of the initial data and C1 is inde- pendent of (u, ψ).

Proof. The equation yields

divu=−aε∂_tq−εav∇q, ∇ψ=−bε∂_tm−εbv∇m.

Applying (ε∂_t)^l to these equation, and using (^eq1.133.7), we see that inveq2

inveq2 (3.20) divu_l=−aρψ_l+1−εf_l, ∇ψ_l=−bµu_l+1−εg_l whereρ=∂_θQ(t, x, εψ) and

esterr

esterr (3.21)

(ε∂t)^j(fl, gl)(t)

H^{s−l−j−1} ≤C(K).

Moreover, applying (ε∂_t)^l to curl (bµu) yields epsdtcurl

epsdtcurl (3.22) curl bµul

−(ε∂t)^lω =εhl

whereh_l satisfies estimates similar to (3.21).^esterr

We prove (^estrec13.19) by induction on k. Fork = 0, this is (3.15). Assume^estepsdtk that it is proven at the order k−1. When l = 0, the desired estimate is implied by (3.15). Thus, suppose that 1^estepsdtk ≤l≤k≤s.

Then, from (3.20) and the induction hypothesis we find that^inveq2

divus−k

H^l−1 +

∇ψs−k

H^l−1 ≤C₀+ (t+ε)C(K) With (3.22) and (^epsdtcurl 3.18), we deduce that^estdivcurl

(us−k, ψs−k)(t)

H^l≤C0+ (t+ε)C(K) +

(us−k, ψs−k)(t) H^l−1

that is (3.19)^estrec1

Corollary 3.10. There isC0 which depends only on the H^s norm of initial data and C(K) such that

(3.23)

(u(t), ψ(t)

_Hs ≤C0+ (t+εC(K)).

Corollary 3.11. There are C0, ε0 > 0 and T0 > 0 which depend only on the H^s norm of the initial data, such that for ε≤ε₀ and t≤T₀:

(3.24)

(u(t), ψ(t)

H^s ≤2C₀ Proof. Take T0 ≤ _2C(2C¹

0) and ε0 ≤ _2C(2C¹

0) .

Together with the local existence theorem for symmetric hyperbolic sys- tems, this uniform bound implies Theorem^mainth2.1.

4 Asymptotics

4.1 The general problem

Assume that (u^ε, ψ^ε) is a family of solutions of (1.3) on [0, T^model3 ]×D^s, which sat- isfy the uniform estimates (^defK3.1) (3.4) and (^{eq1.10 eq1.11}3.5) with a fixedK: in particular (ε∂t)^k(u^ε, ψ^ε) and (ε∂t)^k(q^ε, m^ε, v^ε) are uniformly bounded inC⁰([0, T];H^s−k).

We further assume that the initial data converge strongly inH^s: initconv

initconv (4.1) (u^ε_|t=0, ψ^ε_|t=0)→(u0, ψ0) inH^s.

In particular, up to the extraction of a subsequence, we can assume the following weak convergences in the sense of distribution for instance, as ε tends to 0:

weakconv

weakconv (4.2) (u^ε, ψ^ε, q^ε, m^ε, v^ε) * (u, ψ, q, m, v).

Moreover, the constitutive definition ofQand m show that (4.3) q=Q1(t, x,0)ψ, m=µu.

Multiplying the first equation byεand passing to the weak limit shows that incomp

incomp (4.4) divu= 0, ∇ψ= 0.

Consider the curl of the second equation equation of (1.3). It reads^model3 curl

curl (4.5) curl (b(∂t+v^ε· ∇)m^ε) = 0.

The first information we get from it is that ∂tcurl (bµu^ε) = curl (b∂tm^ε) + curl (∂_tb)m^ε

is bounded inC⁰([0, T];H^s−1). Therefore, the vorticityω^ε= curl (bµu^ε) converges strongly

convvort

convvort (4.6) ω^ε→ω= curl (bµu) in C⁰([0, T], H_loc^s0 )

for alls⁰ < s.

To pass to the weak limit in the equation (4.5), we split^curl u^ε into its incompressible and acoustic components, namely we write

decomp

decomp (4.7) u^ε=eu^ε+ 1

b(t, x)µ(t, x)∇G^ε with

decomp2

decomp2 (4.8) G^ε = (∆_{b µ})⁻¹divu^ε, ∆_{b µ} = div( 1 bµ∇).

Here, ∆b µ and (∆b µ)⁻¹ are seen as acting in the space ˙H^±1 or in the space of functions with zero mean on the torus. In particular,

div ˜u^ε= 0, curl (bµ)˜u^ε=ω^ε.

Since divu^ε* 0, ˜u^ε converges weakly to u and the following weak conver- gence holds

convG

convG (4.9) ∇G^ε*0.

In addition, the uniform estimate (4.6) implies that the convergence of ˜^convvort u^ε is strong ang

convtildeu

convtildeu (4.10) ue^ε →u inC⁰([0, T];H_loc^s0 (Ω)).

In particular, the initial data foru is liminitdata

liminitdata (4.11) u_|t=0= ˜u0, whereu₀ = ˜u₀+ _bµ¹∇G₀ as in (4.7).^decomp

We subsitute the splitting (4.7) in the equation (^{decomp curl}4.5). The first term to consider is

b∂tm^ε =b∂t

µue^ε+1 b∇G^ε

=b∂t(µue^ε) +∇∂_tG^ε−∂_tb b ∇G^ε thus

curl b∂_tm^ε

* curl b∂_t(µu) . The next term to consider is

(4.12)

I^ε :=bv^ε· ∇m^ε

=bV

t, x,ue^ε+ 1

bµ∇G^ε,1

εQ(t, x, εψ)

· ∇ µeu^ε+1 b∇G^ε

Hence, passing to the limit in the sense of distributions in the equation (4.5)^curl yields

lim1

lim1 (4.13) curl∂_t(bµu) + curlI = 0 whereI is the weak limit ofI^ε:=v^ε· ∇m^ε, or model6

model6 (4.14) b∂_t(µu) +∇π+I = 0,

for some pressure term ∇π which is in accordance with the divergence free condition divu= 0.

Conditions on V and Qare necessary to compute the limit I.

4.2 Asymptotic decoupling between fast and slow scales The limit of curlI^εis easily computed under the following assumption (^ass4.12.2).

Proof of Theorem2.3.^theoconvDue to the form ofV, the expression ofI^ε simplifies to

I^ε =bdµu^ε· ∇(µu^ε) =d bµue^ε+∇G^ε

· ∇ µue^ε+1 b∇G^ε

.

There is no difficulty in passing to the weak limit in quadratic terms in- volving at least one strongly convergent factor, namely those with ˜u^ε. The remaining term which involves two weakly convergent factors is

d(∇G^ε)· ∇(1

b∇G^ε) = d

2b∇|∇G^ε|². It is an exact gradient, so that

curl (I^ε)* curl bdµu· ∇(µu) . and Theorem ^theoconv2.3 follows.

Shallow-water equations. Note that the model (1.1) satisfies this assumption,^model1 since then

V = u

h_b, µ= 1

h_b b= 1.

Thus analysis above applies to (1.1), and the limit system reads^model1 model12

model12 (4.15)







∂_t(u hb

) + (u hb

)· ∇(u hb

) +∇p= 0, divu= 0,

Recalling thatv=u/h_b, we get exactly the lake equation, namely lake

lake (4.16) ∂_tv+v· ∇v+∇p= 0, div(h_bv) = 0.

This proves the convergence to the inviscid lake equations.

4.3 Dispersion of acoustic waves on R^d The solutions of (1.3) satisfy^model3

modelfastwaves

modelfastwaves (4.17)

(εa∂tQ₁ψ^ε+ divu^ε=εf^ε εb∂_t(µu^ε) +∇ψ^ε=εg^ε.

with Q₁(t, x) = Q1(t, x,0) = ∂_θQ(t, x,0). From the estimates for (u^ε, ψ^ε), we know that f^ε and g^ε are bounded in C⁰([0, T];H^s−1). This systems governs the evolution on small scales of time or order ε.

In this paragraph, we consider solutions on R^d. We sketch the analy- sis of [MeSc1] which proves that the family^MS u^ε converges strongly to u in L²([0, T];H_loc^s0 (R^d)) for s⁰ < s.

Sketch of proof of Theorem 2.4. The system (modelfastwaves

4.17) can be written (4.18) εE₁∂_tE₂U^ε+L(∂_x)U^ε =εF^ε

with (4.19)

U = ψ

u

, E1

a 0 0 b

, E2=

Q₁ 0

0 µ

L(∂x) =

0 divx

∇_x 0

. Following P. G´erard (^Ger[G´er]), see also the introduction in ^Bu[Bu], one in- troduces the microlocal defect measures of subsequences of u^ε. They are measuresM, on Rt×Rτ valued in the space L of trace class operators on L²(R^d). They can be written

M(dt, dτ) =M(t, τ)α(dt, dτ),

whereαis a scalar nonnegative Radon measure andM is an integrable func- tion with respect toαwith values inL. The usual feature of defect measures is that they are supported in the characteristic variety of the equation. In our case, this means that forα-almost all (t, τ),M(t, τ) is valued inH¹(R^d) and

suppM

suppM (4.20) iE₁(t)τ E₂(t) + L(∂_x)

M(t, τ) = 0,

whereE₁(t) and E₂(t) are seen as multiplication operators onL²(R^d).

When τ 6= 0, the kernel of iτ E₁E₂+L(∂_x) is non trivial if and only if τ² is a positive eigenvalue of 1

aQ₁ div 1 bµ∇_x

. When the coefficients belong

to the class F this never occurs (see[RS, H¨^RS,Hor o] and ^MS[MeSc1]), implying that M(t, τ) = 0 for α-almost all tand τ 6= 0. Thus, M is supported inτ = 0 so (4.20) implies that^suppM

L(∂_x)M(t, τ) = 0 or (I−Π(D_x))M(t, τ) = 0, α a.e where Π(Dx) is the orthogonal projector on kerL(∂x):

Π(D_x) ψ

u

=

0

u− ∇(∆⁻¹divu)

.

As a corollary, the microlocal defect measure of (I−Π(Dx))u^εvanishes and, together with the uniform bounds in H^s, this implies that (I−Π(D_x))U^ε converges strongly inL²_loc([0, T]×R^d). Together with the strong convergence of ˜u^ε, this implies the local strong convergence ofu^ε andψ^ε. Since the limit ψ(t, ·)∈H^s(R^d) and∇ψ= 0, the limitψis equal to 0. For further details, we refer the reader to^MS[MeSc1].

4.4 Averaged acoustics on the torus

For the heterogeneous isentropic Euler equations (1.2), the coefficients of the^model2 fast acoustic operator depend only onx, not on time. In this case, the fast evolution is easily analyzed using a spectral decomposition. We first present the averaging method in the extended framework of Assumption (2.5). We^ass4.4 then prove Theorem2.6.^Theoo

Under Assumption (^ass4.42.5), the fast dynamics (modelfastwaves

4.17) reads:

model8

model8 (4.21) εE(x)∂tU^ε+L(∂x)U^ε=εF^ε with

(4.22) E =

aQ

1 0

0 bµ

, L(∂_x) =

0 div_x

∇_x 0

and formF

formF (4.23) F^ε =F₂(t, x, U^ε,∇U^ε) +εFe^ε. withF₂ quadratic in U^ε,∇U^ε:

F2

F2 (4.24) F₂ =

(Q₁)⁻¹∂_θ²Q(x,0)ψdivu−ad u· ∇(Q

1ψ)

−bd u· ∇(µu)

The problem reduces to study interaction of resonant time oscillations linked to the spectral properties of the operator ˜L(x, ∂_x) := E⁻¹(x)L(∂_x).

This operator is self adjoint with respect to the scalar product modscprod

modscprod (4.25)

U, V :=

Z

T^d

E(x)U(x)·V(x)dx.

The fast evolution is governed by the groupE(t) :=e^{−tL(x,∂x)} and the solu- tion of (^model84.21) is

solfast

solfast (4.26) U^ε(t) =E(t

ε)U(0) + Z t

0

E(t−s

ε )E⁻¹F^ε(s)ds.

The filtering method (see e.g. [Sc1]) consists in studying the limit of^Sc1 V^ε(t) :=E(−t

ε)U^ε=U(0) + Z t

0

E(−s

ε)E⁻¹F^ε(s)ds.

Equivalently, V^ε solves inteqV

inteqV (4.27) ∂_tV^ε=E(−t

ε)E⁻¹F^ε.

The group E(t) is unitary in L² for the scalar product (4.25) but it is also^modscprod bounded inH^s(T^d), as a consequence of Theorem ^mainth2.1 applied to the linear equation (^model84.21). In particular, this implies that V^ε and ∂tV^ε are bounded inC⁰([0, T];H^s) and C⁰([0, T];H^s−1) respectively. As a consequence convV Lemma 4.1. Extracting further a subsequence if necessary, V^ε converges

strongly to a limit V in C⁰([0, T];H^s0(T^d)). In particular, asympprof

asympprof (4.28) U^ε(t)∼ E(t

ε)V(t) in C⁰([0, T];H^s0(T^d)).

Next, we analyze the evolution group E(t) using the spectral decompo- sition ofL(x, ∂e x). Its kernel is

(4.29) K= kerLe=n U =

ψ u

: ∇ψ= 0, divu= 0o .

The orthogonal projector Π_K onKis Pi0

Pi0 (4.30) Π_K

ψ u

=

ψ₀ u₀ =u−_bµ¹∇G

,

whereψ0is the average ofψfor the measureaQ₁dxandGsolves div _bµ¹∇G

= divu as in (4.8).^decomp2

The non zero eigenvalues of L(x, ∂e _x) are ±i√

κ_j where the κ_j are the positive eigenvalues of the acoustic wave operator

(4.31) W(aQ

1, bµ)(·) =− 1

aQ₁div 1 bµ∇·

Note thatW(aQ₁, bµ) is self-adjoint inL²(T^d, aQ₁dx). The eigenvectors of Le for the eigenvalue ±i√

κj are

defPhij

defPhij (4.32) Φ±j= 1

√ 2





ψ_j

± 1 ibµ√

κ_j∇ψ_j





whereψj is an eigenvector of W associated to the eigenvalueκj. From now on, we fix an orthonormal basis in L²(T^d, aQ₁dx) made of eigenfunctions (ψ₀, ψ₁. . . , ψ_j. . .) associated to the eigenvalues κ₀ = 0 < κ₁ ≤ κ₂ ≤ . . . of W. Note that κ0 is simple and that ψ0 is a constant. For an integer j ∈ Z\{0} we set λ_j = √

κ_j when j > 0 and λ_j = −√

κ−j when j < 0.

Moreover, Φ_j denotes the function defined in (4.32). The^defPhij {Φ_j} form an orthonormal basis ofK^⊥, the orthogonal ofKfor the scalar product (^modscprod4.25).

Classical properties of the elliptic operator W on the torus imply that κj ≈j^2/d

and thatv belongs to H^s(T^d) if and only if v=X

αjψj with X

j^2s/d|α_j|² <+∞.

Accordingly we expand V^ε using the spectral decomposition of L:e (4.33) V^ε(t) = Π_KV^ε(t) +X

j6=0

α^ε_j(t)Φ_j, α_j(t) =

V^ε(t),Φ_j .

and the series converges inC⁰([0, T];H^s). Therefore Π_KU^ε= Π_KV^ε and : U^ε(t) = Π_KU^ε(t) +X

j6=0

e^−itλj/εα^ε_j(t)Φ_j. The strong convergenceV^ε→V implies the following.

Lemma 4.2. Π_KU^ε converges strongly to Π_KV in C⁰([0, T];H^s0(T^d)) and the α^ε_j converge strongly to α_j :=hV,Φ_ji in C⁰([0, T]).

Thus, asympprofb

asympprofb (4.34) U^ε(t)∼U(t) +X

j6=0

e^−itλj/εα_j(t)Φ_j in C⁰([0, T];H^s0(T^d)).

and in particularU^ε converges weakly to U = Π_KU = Π_KV. The equation (4.27) implies that^inteqV

neweq

neweq (4.35)

(∂_tΠ_KU^ε(t) = Π_KE⁻¹F^ε(t)

∂tα^ε_j(t) =e^itλj/ε

E⁻¹F^ε(t),Φj

.

The limiting equations are obtained by taking the weak limits if the right hand sides. To compute them, we substitute the asymptotic form (4.34) of^asympprofb U^ε in (4.23):^formF

Lemma 4.3. One has F^ε ∼F₂^ε in C⁰([0, T];H^s0−1(T^d)) with F₂^ε=F₂(U,∇U) +X

j6=0

α_je^−tλj/εFf₂(U,∇U; Φ_j,∇Φ_j)

+ X

j6=0,k6=0

α_kα_je^{−t(λj+λk)/ε}Ff₂(Φ_k,∇Φ_k; Φ_j,∇Φ_j),

where Ff₂ is the bilinear form associated to the quadratic termF₂. Therefore, the stationary phase Theorem implies the following.

Lemma 4.4. In the sense of distributions, one has the following weak con- vergence :

Π_K(E⁻¹F^ε)*Π_K E⁻¹(F₂(U,∇U) +I where

defcall

defcall (4.36) I = X

λk+λj=0

α_kα_jFf₂(Φ_k,∇Φ_k; Φ_j,∇Φ_j).

Moreover, e^isλl/ε

E⁻¹F^ε(s),Φl

* X

λj=λl

αj

E⁻¹ F₂(U,∇U; Φj,∇Φ_j) ,Φl

+ X

λj+λk=λl

α_kα_j

E⁻¹F₂(Φ_k,∇Φ_k; Φ_j,∇Φ_j),Φ_l .

With this lemma, we can pass to the limit in the equations (4.35). In^neweq particular, we get that

∂tU = Π_K E⁻¹(F₂(U,∇U) +I .

Tracing back the definition, this gives the condition divu= 0 implies that

∂tψ= 0 and the limitψis not present in the other equations. The equation foru is

limequ

limequ (4.37) bµ∂_tu+bd u· ∇(µu) +I+∇π= 0.

The initial condition for u is still given by (liminitdata

4.11). Summing up, we have proved Theorem2.6 with^Theoo

alphaj

alphaj (4.38)

∂_tα_j = X

λj=λ_l

α_j

E⁻¹ F₂(U,∇U; Φ_j,∇Φ_j) ,Φ_l

+ X

λj+λk=λl

α_kαj

E⁻¹F₂(Φ_k,∇Φ_k; Φj,∇Φ_j),Φ_l .

with the initial conditions alphajinit

alphajinit (4.39) α_j(0) =hU₀,Φ_ji.

Remark 4.5. Generically, that means for general coefficients or for general tori, one expects that the eigenvalues of the wave equations κ_j are simple and that there are no resonances±√

κ_j±√ κ_k±√

κ_l= 0, in which case the formulas above become simpler.

Heterogeneous isentropic Euler equations. The analysis above applies to (1.2). Recall, see Section 2, that for this model^model2

a=c^−1/γ, b=c^−1/γ, µ=c^1/γ, V =du=c^1/γu and that

Q₁(t, x) = 1/γ, ∂_θ²Q(x,0) =−(γ−1)/γ². Using Theorem^Theoo2.6, the limit system (^limequ4.37) reads

model23

model23 (4.40)

(∂_tu+u· ∇(c^1/γu) +I+∇π= 0 divu= 0

whereI is given by (4.36). From (^{defcall F2}4.24),F₂ reads F₂=





−γ−1

γ ψdivu− 1

γ u· ∇ψ

−u· ∇(c^1/γu)





Thus, using that Φj denotes the function defined by

defPhi

defPhi (4.41) Φj = 1

√ 2



 ψj

1 iλj

∇ψ_j





whereψj is an eigenvector of W namely ell

ell (4.42) W(ψj) =−γc^1/γ∆ψj =λ²_jψj

associated to the eigenvalueλ²_j. We get from (4.36)^defcall 233

233 (4.43) I = X

j,`,λk+λj=0

α_kα_j

2|λ_k|² ∇ψ_k· ∇(c^1/γ∇ψ_j) +∇ψ_j · ∇(c^1/γ∇ψ_k) The coefficientsα_j are calculated using (4.38). The first term in right-hand^alphaj side of (^alphaj4.38) is given by

X

λj=λl

αj

E⁻¹ F₂(U,∇U; Φj,∇Φ_j) ,Φl

=− X

λj=λ`

α_j Z

T^d

√1

2iλ_j∇ψ_j·∇(c^1/γu)+u·∇(c^1/γ 1

√2iλ_j∇ψ_j)

·( 1

√2iλ_{`</sub>∇ψ<sub>`})

= X

λj=λ`

αj

2|λ_j|² Z

T^d

∇ψ_j · ∇(c^1/γu) +u· ∇(c^1/γ∇ψ_j)

· ∇ψ_`

The second term in the right-hand side of (4.38) is given by^alphaj X

λj+λk=λl

α_kα_j

E⁻¹F₂(Φ_k,∇Φ_k; Φ_j,∇Φ_j),Φ_l

=− X

λj+λk=λ`

αjα_k Z

T^d

1

√2iλ_j∇ψ_j· ∇(c^1/γ 1

√2iλ_k∇ψ_k)

+ 1

√

2iλ_k∇ψ_k·∇(c^1/γ 1

√ 2iλj

∇ψ_j)

· 1

√

2iλ_{`</sub>∇ψ<sub>`}

− X

λj+λk=λ`

αjαk

Z

T^d

γ−1 γ

ψjdiv( 1

√ 2iλk

∇ψ_k)+ψkdiv( 1

√ 2iλj

∇ψ_j))ψ`

− X

λj+λ_k=λ_`

α_jα_k1 γ

Z

T^d

1

√2iλ_j∇ψ_j· ∇ψ_k+ 1

√2iλ_k∇ψ_k· ∇ψ_j ψ_` Therefore, we conclude that the coefficients α_j are given by the following system of ODE’s

model24

model24 (4.44) ∂tαj = X

λj=λ`

αj

2|λ_j|² Z

T^d

(u· ∇(c^1/γ∇ψ_j) +∇ψ_j · ∇∇(c^1/γu))· ∇ψ_`

− X

λj+λ_k=λ_`

iα_jα_k 2√

2 1 λ_`λ_kλ_j

Z

T^d

(∇ψ_j·∇(c^1/γ∇ψ_k))·∇ψ_{`</sub>+(∇ψ<sub>k</sub>·∇(c<sup>1/γ</sup>∇ψ<sub>j</sub>))·∇ψ<sub>`} .

− X

λj+λk=λ`

i(γ−1)

√2γ² Z

T^d

c^−1/γα_jα_kλ_{`</sub>ψ<sub>j</sub>ψ<sub>k</sub>ψ<sub>`}− i

√2γ Z

T^d

α_jα_k λ_`

λ_jλ_k∇ψ_j·∇ψ_kψ_` .

The first two terms in the right-hand side of (^model244.44) correspond to the part ofF₂ in the momentum equation. The last two terms correspond to part of F₂ in the mass equation.

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