Layered incompressible fluid flow equations in the limit of low Mach number and strong stratification

Physica D 237 (2008) 1466–1487 www.elsevier.com/locate/physd

Layered incompressible fluid flow equations in the limit of low Mach number and strong stratification Eduard Feireisl a , Anton´ın Novotn´y b,∗ , Hana Petzeltov´a a a Mathematical Institute AS CR, Zitn´ ˇ a 25, 115 67 Praha 1, Czech Republic b IMATH, Universit´e du Sud Toulon-Var, BP 132, 839 57 La Garde, France

Available online 25 March 2008

Abstract We investigate the low Mach number regimes of the complete Navier–Stokes–Fourier system which describes an evolution of a viscous, heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows. This investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill-prepared initial data. The physically expected limiting equations as the Mach and Froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification. We give a rigorous mathematical proof of this result. c 2008 Elsevier B.V. All rights reserved.

PACS: 83.2; 47.4 Keywords: Navier–Stokes–Fourier system; Low Mach number regimes; Stratified fluids

1. Introduction Stratified fluids whose densities, sound speed and other parameters are functions of a single depth coordinate occur widely in nature. Several so-called mesoscale regimes in the atmosphere modeling involve fluid flows of strong stable stratification but weak rotation. Numerous observations, numerical experiments as well as purely theoretical results to explain this situation have been recently surveyed by Majda [28, Chapter 6]. In particular, several simplified systems of dynamic equations that describe the fluid motion with strong stratification was proposed. The basic system of equations to be discussed in this paper reads as follows: • Hydrostatic balance: ∇x p(%, ˜ ϑ) = %∇ ˜ x F,

%˜ = %(x ˜ 3 ), F = F(x3 );

(1.1)

• Anelastic Navier–Stokes system:   1 ∂t (%v) ˜ + divx (%u ˜ ⊗ v) + %∇ ˜ H Π = µ(ϑ) ∆v + ∇H divx u , 3   1 ∂t (%u ˜ 3 ) + divx (%uu ˜ 3 ) + %∂ ˜ x3 Π = µ(ϑ) ∆u 3 + ∂x3 divx u + ∂x3 %˜ χ 3 divx (%u) ˜ = 0, u = (u 1 , u 2 , u 3 ),

(1.2) (1.3) (1.4)

v = (u 1 , u 2 , 0);

∗ Corresponding author. Tel.: +33 0 4 94142625; fax: +33 0 4 94142633.

E-mail address: [email protected] (A. Novotn´y). c 2008 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2008.03.027

E. Feireisl et al. / Physica D 237 (2008) 1466–1487

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• Diagnostic equation for the vertical component of the velocity: 1 ∂F u3 = χ , d ∂ x3 where d denotes a positive constant related to radiation cooling in the atmosphere, cf. (1.21). −

(1.5)

Eq. (1.1) determines the static distribution of the density %˜ provided the fluid is subjected to a conservative driving force with a potential F. The motion of the fluid is described by the Eulerian velocity u = (u 1 , u 2 , u 3 ), with v = (u 1 , u 2 , 0) and u 3 denoting its horizontal and vertical component, respectively. Analogously the subscript H denotes differential operators acting on the horizontal independent variables x1 , x2 , for example ∇H = (∂x1 , ∂x2 ). The function p representing the thermodynamic pressure in the hydrostatic balance equation (1.1) as well as the viscosity coefficient µ in (1.2) and (1.3) depends on the equilibrium temperature ϑ which is assumed to be constant. The symbol Π in (1.2) and (1.3) denotes the mean normal stress (the pressure in an incompressible flow). Note that in the “classical” anelastic model equations proposed by Lipps and Hemler [27], Ogura and Phillips [35], the quantity χ in (1.3) is directly related to variations of the (potential) temperature, while, in the present setting, χ and the vertical component of the velocity are interrelated through the diagnostic Eq. (1.5). If the pressure satisfies Boyle’s law p = %ϑ, and F = −x3 represents the gravitational potential, relation (1.5) reads χ = d1 u 3 giving rise to a “friction” term in Eq. %˜ (1.3) of the form − ϑd u3. In the hypothetical case χ = 0, the flow is layered horizontally so that the vertical velocity vanishes, while the stacking of the layers is governed by hydrostatic balance (1.1), and the viscosity induces transfer of horizontal momentum between the layers through vertical variations of the horizontal velocity (see Majda [28, Chapter 6.2] for further discussion as well as relevant reference material). For the sake of simplicity, we consider the periodic boundary conditions with respect to the horizontal variables (x1 , x2 ), more specifically, we take the underlying spatial domain Ω = T 2 × (0, 1), where T 2 = {(0, 1)|{0,1} }2 is a two-dimensional torus. In addition, we impose the free slip boundary conditions for the velocity: u · n|∂ Ω = 0,

(Tn) × n|∂ Ω = 0,

(1.6)

where the symbol   2 T = µ(ϑ) ∇x u + ∇xt u − divx uI , 3

(1.7)

denotes the viscous stress tensor, and n stands for the outer normal vector. Last but not least, Eqs. (1.2)–(1.4) are endowed with the initial conditions u(0, ·) = u0 (·),

where div(%u ˜ 0 ) = 0.

(1.8)

In the following we shall use the standard notation: L p (Ω ), W k, p (Ω ) (L p (Ω ; R s ), W k, p (Ω ; R s )), 1 ≤ p ≤ ∞, (k, s) ∈ N2 for the Lebesgue respectively, Sobolev spaces of scalar (R s -vector) valued functions on Ω ; L q (0, T ; L p (Ω )), L q (0, T ; W k, p (Ω )) (L q (0, T ; L p (Ω ; R s )), L q (0, T ; W k, p (Ω ; R s ))), 1 ≤ q ≤ ∞ for the corresponding Bochner spaces; D(G) for the space of C ∞ functions with the compact support in G and D0 (G) for the corresponding space of distributions. With these basic notations at hand, we are now able to introduce the notion of weak solutions to problem (1.1)–(1.8) in the spirit of Leray [22]. In the weak formulation we require the following: • Eqs. (1.1) and (1.5) have to be satisfied a.e. in (0, T ) × Ω . • The velocity field u ∈ L 2 (0, T ; W 1,2 (Ω ; R 3 )) has zero normal trace on ∂Ω = T 2 × ({x3 = 0} ∪ {x3 = 1}) and the anelastic approximation (1.2)–(1.4) endowed with (1.6) and (1.8) is verified in the weak sense, namely   Z TZ Z TZ 1 ˜ t ϕE + %u ˜ ⊗ u : ∇x ϕ) E dxdt = µ(ϑ) ∇x u : ∇x ϕE + divx u divx ϕE dxdt (%u∂ 3 0 Ω 0 Ω Z TZ Z − χ ∂x3 %ϕ ˜ 3 dxdt − %u ˜ 0 ϕ(0, E ·)dx (1.9) 0

Ω

Ω

for any test function ϕE ∈ D([0, T ); D(Ω ; R 3 )),

ϕE · n|∂ Ω = 0,

divx (%˜ ϕ) E = 0.

(1.10)

System (1.1)–(1.5) may be viewed as an asymptotic limit of the Boussinesq approximation, where the Froude number Fr is small (see Klein [20], Majda [28, Chapter 6.2]). On the other hand, the Boussinesq system itself can be derived as a singular limit of the full Navier–Stokes–Fourier system provided the Mach number Ma tends to zero (see Zeytounian [40,41], among others). Pursuing the program originated in [11,13] we will show that system (1.1)–(1.5) can be obtained as a singular limit of the full

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Navier–Stokes–Fourier system provided both the Mach number Ma and the Froude number Fr are proportional to a small parameter ε. The result holds true for any ill-prepared initial data and on an arbitrary time interval (0, T ) (see Theorem 3.1). The Navier–Stokes–Fourier system constitutes, in the framework of continuum fluid mechanics, a closed set of equations governing the time evolution of a viscous, compressible, and heat conducting fluid (see for instance Gallavotti [16]). Under the scaling t ≈ ε2 t, x ≈ εx, u ≈ εu, the basic laws of classical mechanics can be written in the form: • Mass conservation: ∂t % + divx (%u) = 0.

(1.11)

• Balance of momentum: ∂t (%u) + divx (%u ⊗ u) +

1 1 ∇x p = divx S + 2 %∇x F. ε2 ε

(1.12)

• Entropy production: q 1 % = σ + 2 Q, ∂t (%s) + divx (%su) + divx ϑ ε ϑ   1 q · ∇x u σ = ε 2 S : ∇x u − . ϑ ϑ • Total energy balance:  Z  2 Z Z ε 1 τ E(τ ) = %|u|2 + %e − %F (t)dx = E 0 + 2 %Qdx. 2 ε 0 Ω Ω

(1.13) (1.14)

(1.15)

• Gibbs’ relation: ϑ Ds = De + p D

  1 . %

(1.16)

Similarly to (1.6), system (1.11)–(1.16) is considered on the spatial domain Ω = T 2 × (0, 1), and supplemented with the conservative boundary conditions u · n|∂ Ω = 0,

(Sn) × n|∂ Ω = 0,

q · n|∂ Ω = 0,

(1.17) (1.18)

where q denotes the heat flux. Here the fluid density % = %(t, x), the Eulerian velocity u = u(t, x), and the absolute temperature ϑ = ϑ(t, x) are the state variables, while the pressure p = p(%, ϑ), the specific internal energy e = e(%, ϑ), and the specific entropy s = s(%, ϑ) are determined through a set of suitable constitutive relations satisfying (1.16). Analogously as in (1.7), the viscous stress tensor S is given through Newton’s rheological law   2 t S = µ ∇x u + ∇x u − divx uI + ηdivx uI, (1.19) 3 while the heat flux q obeys Fourier’s law q = −κ∇x ϑ,

(1.20)

where the transport coefficients µ, η, κ, and ϑ depend effectively on the absolute temperature ϑ. The source term Q appearing in (1.13)–(1.15) represents the heating rate per unit mass. With regard to possible applications to geophysical flows, Q is related to radiative cooling, latent heat release from condensation, or various chemical reactions as the case may be. We shall assume that the physical system under consideration admits an equilibrium temperature ϑ > 0, accordingly ∂Q (ϑ) < 0, ∂ϑ where the latter stipulation is necessary for the corresponding rest state to be stable. Under these circumstances, a simple ansatz for Q reads Q(ϑ) = 0,

Q = d(ϑ − ϑ),

d > 0.

(1.21)

Such a Q can be interpreted as radiation cooling in models of atmospheric flows. In the meteorological models, d1 is usually called radiation relaxation time scale. For further discussion on the relevance of the heat source/sink terms in atmospheric modeling see Chapter 8 in Zeytounian [39]).

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In terms of the standard terminology, the scaling parameter ε in (1.11)–(1.15) corresponds to the regime Ma = Fr = ε, and Da = ε12 , where Ma, Fr, and Da denote the Mach, Froude, and Damkoehler number, respectively (see Klein et al. [21]). Accordingly, system (1.11)–(1.15) is supplemented with the initial data:   (1)  %(0, ·) = %0,ε = %˜ + ε%0,ε , Z (1.22) (1) (1)  %0,ε dx = 0, {%0,ε }ε>0 bounded in L ∞ (Ω ); Ω

u(0, ·) = u0,ε , u0,ε → u0 weakly in L 2 (Ω ; R 3 );   (1)  ϑ(0, ·) = ϑ0,ε = ϑ + εϑ0,ε , Z (1) (1)  ϑ0,ε dx = 0, {ϑ0,ε }ε>0 bounded in L ∞ (Ω ),

(1.23) (1.24)

Ω

where %˜ = %(x ˜ 3 ), ϑ = const > 0 satisfy the hydrostatic balance equation (1.1). Under certain technical hypotheses specified in Section 2 below, system (1.11)–(1.21), supplemented with the initial conditions (1.22)–(1.24), admits a weak solution {%ε , uε , ϑε } on an arbitrary time interval (0, T ) (see [9,10,12,15]). The main objective of the present study is to show that %ε → %, ˜ uε → u,

and

ϑε → ϑ

in a certain sense as ε → 0,

where the trio {%, ˜ u, ϑ} represents a solution to problem (1.1)–(1.5). There is a vast amount of literature devoted to the scaling analysis of problems arising in mathematical fluid mechanics. The reader may consult the pioneering papers of Ebin [8], Klainerman and Majda [19], Schochet [36], the monographs by Majda [28], Zeytounian [41], the survey studies of Danchin [5], Klein et al. [21], Masmoudi [29], Schochet [37], or the recent contributions to the theory by Alazard [1,2], Bresch et al. [3], Danchin [4], Desjardins et al. [6], Hagstrom and Lorenz [17], Hoff [18], Lions and Masmoudi [24,25], Metivier and Schochet [31,32], among others. In terms of the standard terminology introduced in the above-mentioned studies, the main difficulties to be dealt with in the present paper are characterized as follows: • The asymptotic density distribution determined by the function %˜ depends on the vertical coordinate. This is related to the fact that the driving force in Eq. (1.12) is proportional to ε−2 . • The initial data are “ill-prepared” meaning the initial velocity fields are arbitrarily large, while the initial distribution of the density and the temperature approaches the equilibrium state (%, ˜ ϑ) at a rate proportional to the Mach number. • Convergence towards the target problem is established on an arbitrary (large) time interval (0, T ). As already observed by Lions and Masmoudi [24], Schochet [36] in the context of isentropic gas dynamics, the principal difficulty when dealing with the ill-prepared data is to control the time oscillations of the gradient component of the velocity field associated to the fast acoustic waves. The acoustic waves owe their existence to the presence of compressibility in the fluid, and they have no counterpart in the limit system. If the temperature fluctuations are neglected, the time evolution of the acoustic waves in a stratified medium is governed by a wave equation with the wave speed depending effectively on the vertical coordinate. The relevant mathematical theory was developed by Wilcox [38] and subsequently used in [11] in order to study the low Mach number limit of a stratified isentropic fluid flow (see also recent work of Masmoudi [30] for similar results). The situation becomes even more complicated, when the influence of the temperature fluctuations must be taken into account. In this case, the resulting acoustic equation turns out to be a rather complicated system interrelating the density changes to the entropy fluctuations. Fortunately, the presence of the heat source term Q provides a stabilizing effect forcing the “thermal” waves to fall away at a rate proportional to the Mach number as soon as the pressure p obeys classical Boyle’s law p = %ϑ. However, the need of sufficiently strong uniform estimates as well as the underlying existence theory require the pressure to be a coercive function of the density in the spirit of [10,12]. Consequently, the constitutive equations interrelating p, e, and s must undergo additional structural scalings specified in Section 2. The paper is organized as follows. Section 2 is a brief introduction to the mathematical theory of the Navier–Stokes–Fourier system. In particular, we review the available existence results, together with a list of relevant structural hypotheses imposed on the constitutive equations. Section 3 contains the main result formulated in Theorem 3.1. The rest of the paper is devoted to the proof of Theorem 3.1. Section 4 derives uniform estimates on the family of solutions to the Navier–Stokes–Fourier system independent of the singular parameter ε. In addition, each quantity is shown to admit a decomposition into an “essential” and “residual” component, where the former is uniformly bounded while the latter vanishes in the asymptotic limit for ε → 0. Section 5 focuses on the convergence %ε → %, ˜ ϑε → ϑ. At this stage, we identify the resulting equations to be satisfied by the limit {%, u, ϑ} leaving open only the problem of convergence of the convective terms %ε uε ⊗ uε . Section 6 forms the heart of the paper, focusing on the analysis of acoustic waves. On the basis of spectral analysis of the corresponding wave operator, we eliminate the influence of the entropy fluctuations by means of the stabilizing effect provided by the heat source term Q. Finally, the proof of the main result is completed in Section 7.

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2. Preliminaries 2.1. Structural hypotheses The technical restrictions imposed on the thermodynamic quantities p, s, e as well as the transport coefficients µ, η, κ, and d can be viewed as a compromise between the mathematical need of suitable a priori estimates, required by the available existence theory, and physically admissible hypotheses dictated by the nature of the phenomena to be captured by the model. Accordingly, the pressure p is taken in the form   a % 1 5 F R F pεR = ε ϑ 4 , (2.1) p = pε (%, ϑ) = pε (%, ϑ) + pε (ϑ), pε = ϑ 2 P ε 3 , ε 3 ϑ2 where the function P satisfies P(Y ) = Y

P ∈ C 1 [0, ∞), 0 < cv ≤

5 3 P(Y ) −

P 0 (Y )Y

Y

for 0 ≤ Y ≤ 1,

P 0 (Y ) > 0

for all Y > 0,

for all Y > 0,

≤ cv

(2.3)

P(Y )

= p∞ > 0. 5 Y3 Note that the state Eq. (2.1) approaches the perfect gas law p = %ϑ provided ε → 0. In order to comply with Gibbs’ relation (1.16), the specific internal energy e can be taken in the form lim

(2.2)

Y →∞

e = eε (%, ϑ) = eεF (%, ϑ) + eεR (%, ϑ),

(2.4)

(2.5)

where eεF (%, ϑ) =

  5 % 31ϑ2 P ε 3 , 2ε % ϑ2

eεR (%, ϑ) = εa

ϑ4 . %

(2.6)

The quantities pεR , eεR represent the effect of radiation. Assumption (2.3) implies that the specific heat at constant volume cv 5 is strictly positive and bounded from above. Condition (2.4) says that p F (%, ϑ) is equivalent to p∞ % 3 for the large values of the 3 quantity %/ϑ 2 (i.e. the fluid behaves like a Fermi gas in these extreme regimes), while (2.1) means that the fluid obeys the perfect 3 gas law in the standard regimes when %/ϑ 2 is bounded. This is true, in particular, for the range of densities and temperatures in the physics of atmosphere and suits well to the applications we have in mind. Note also that pε , eε meet the standard thermodynamic stability hypotheses ∂ pε (%, ϑ) > 0, ∂%

cv =

∂eε (%, ϑ) > 0. ∂ϑ

(2.7)

The reader may consult [10,12] and the references cited therein to learn more about the physical background of (2.1) and (2.6) as well as the associated technical restrictions (2.2)–(2.4). In accordance with (1.16), the specific entropy reads s = sε (%, ϑ) = sεF (%, ϑ) + sεR (%, ϑ),

(2.8)

where   % sεF (%, ϑ) = S ε 3 − S(ε), ϑ2

4 ϑ3 sεR (%, ϑ) = ε a , 3 %

(2.9)

with 3 35 P(Y ) − P 0 (Y )Y . (2.10) 2 Y2 The transport coefficients µ, η, and κ are continuously differentiable functions of the absolute temperature satisfying technical restrictions S 0 (Y ) = −

0 < µ(1 + ϑ) ≤ µ(ϑ) ≤ µ(1 + ϑ), 0 < κ(1 + ϑ 3 ) ≤ κ(ϑ) ≤ κ(1 + ϑ 3 ) for any ϑ > 0.

η ≡ 0,

(2.11) (2.12)

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2.2. Global-in-time weak solutions In the framework of weak solutions in the spirit of Leray’s original paper [22], all differential operators appearing in (1.11)–(1.15) are to be understood in the sense of distributions. Accordingly, we shall say that the trio (% ≥ 0, u, ϑ > 0), 5

% ∈ L ∞ (0, T ; , L 3 (Ω )),

u ∈ L 2 (0, T ; W 1,2 (Ω ; R 3 )),

5

%u ∈ L ∞ (0, T ; L 4 (Ω )),

ϑ ∈ L 2 (0, T ; W 1,2 (Ω )) ∩ L ∞ (0, T ; L 4 (Ω )) is a global-in-time weak solution to the problems (1.11)–(1.24) and (2.1)–(2.12) if the following holds: • Following Diperna and Lions [7] we require that the couple of functions %, u represents a renormalized solution of the continuity equation (1.11), namely the integral identity Z Z TZ %0 B(%0 )ϕ(0, ·)dx (2.13) (%B(%)∂t ϕ + %B(%)u · ∇x ϕ − b(%)divx uϕ) dxdt = − 0

Ω

Ω

holds for any test function ϕ ∈ D([0, T ) × Ω ) and any b such that Z % b(z) b ∈ L ∞ ∩ C[0, ∞), B(%) = B(1) + dz. z2 1 Here and in what follows, we anticipate implicitly that all (non-linear) quantities are integrable on (0, T ) × Ω . • Similarly, a weak formulation of the momentum Eq. (1.12), namely  Z TZ  1 %u · ∂t ϕE + %u ⊗ u : ∇x ϕE + 2 p divx ϕE dxdt ε 0 Ω  Z TZ  Z 1 = S : ∇x ϕE − 2 %∇x F · ϕE dxdt − %0 u0 · ϕ(0, E ·)dx ε 0 Ω Ω is verified for any test function ϕE ∈ D([0, T ); D(Ω ; R 3 )) satisfying ϕE · n|∂ Ω = 0. • As in [12], the entropy balance equations (1.13) and (1.14) is replaced by the integral inequality   Z TZ Z TZ h i q 1 q · ∇x ϑ 1 % 2 · ∇x ϕ dxdt + ε S : ∇x u − + 2 Qϕdxdt %s∂t ϕ + %su · ∇x ϕ + ϑ ϑ ε ϑ 0 Ω ϑ 0 Ω Z ≤− %0 s(%0 , ϑ0 )ϕ(0, ·)dx

(2.14)

(2.15)

Ω

for any test function ϕ ∈ D([0, T ) × Ω ), ϕ ≥ 0. The key feature of this approach is to assert that the “genuine” entropy production σ associated to a weak solution is in fact a non-negative measure satisfying   1 q · ∇x ϑ 2 σ ≥ ε S : ∇x u − (2.16) ϑ ϑ in place of (1.14), whereas (2.15) can be written in the form Z TZ h q i %s∂t ϕ + %su · ∇x ϕ + · ∇x ϕ dxdt + hσ, ϕi ϑ 0 Ω Z TZ Z 1 % Qϕdxdt − %0 s(%0 , ϑ0 )ϕ(0, ·)dx. =− 2ϑ ε 0 Ω Ω • Finally, in accordance with (1.15), the total energy E satisfies  Z  2 ε 2 E(τ ) = %|u| (τ ) + %e(%, ϑ)(τ ) − %(τ )F dx 2 Ω  Z  2 Z Z 1 τ ε 2 %0 |u0 | + %0 e(%0 , ϑ0 ) − %0 F dx + 2 = %Qdxdt 2 ε 0 Ω Ω

(2.17)

for a.a. τ ∈ (0, T ).

(2.18)

Due to (2.17) and (2.18), it is a routine matter to check that for any smooth weak solution relation (2.16) reduces to (1.14) and Eq. (1.13) holds (see Remark 2.3 in [12]).

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3. Main result Having collected the necessary preliminary material we are ready to state the main result of the present paper. Theorem 3.1. Let Ω = T 2 × (0, 1). Assume that the thermodynamic functions p = pε , e = eε , s = sε are given through (2.1)– (2.10), while the fluxes S, q satisfy (1.19) and (1.20), where the transport coefficients µ, η, and κ obey (2.11) and (2.12). Let the heat source term Q be given by (1.21), and let F satisfy F = F(x3 ) = −x3 . Finally, let {%ε , uε , ϑε }ε>0 be a family of weak solutions to the Navier–Stokes–Fourier system (1.11)–(1.21) on (0, T ) × Ω in the sense specified in Section 2.2 emanating from the initial data {%0,ε , u0,ε , ϑ0,ε }ε>0 satisfying (1.14)–(1.24), where %, ˜ ϑ solve (1.1). Then, at least for a suitable subsequence, 5

%ε → %˜

in L ∞ (0, T ; L 3 (Ω )),

ϑε → ϑ

in L (0, T ; W 2

1,2

(3.1)

(Ω ; R )),

uε → u weakly in L (0, T ; W 2

3

1,2

(3.2)

(Ω ; R )), 3

(3.3)

and ϑε − ϑ →χ ε2

in D0 ((0, T ) × Ω ), χ ∈ L 2 (Q T ),

(3.4)

where (%, ˜ ϑ, u, χ ) is a weak solution of problem (1.1)–(1.8), supplemented with the initial data %u(0, ˜ ·) = weak lim H%˜ [%u ˜ 0,ε ], ε→0

(3.5)

where H%˜ denotes a weighted Helmholtz projection introduced in Section 6. The existence of global-in-time weak solutions to the Navier–Stokes–Fourier system (1.11)–(1.21) can be shown by the methods introduced in [12,15]. The presence of the extra terms associated to the heat source Q requires only minor modifications. The rest of the paper is devoted to the proof of Theorem 3.1. 4. Uniform estimates The first step in the proof of Theorem 3.1 consists in deriving uniform estimates for the sequence {%ε , uε , ϑε , }ε>0 independent of ε → 0. To begin with, making use of a suitable family of spatially homogeneous test functions and taking b = 0 in (2.13), we deduce that the total mass is a constant of motion, specifically, Z Z %ε (t)dx = %dx ˜ for all t ≥ 0. (4.1) Ω

Ω

4.1. Dissipation equality It was observed in [13] and later on exploited in [11] that the dissipation equality plays an essential role in the analysis of low Mach number flows. While in [13] the equilibrium density %˜ is constant, and while in [11] the underlying flow is isentropic, we investigate in the present paper the general situation of heat conducting non-isentropic flow with the space-dependent equilibrium density. In the present setting, the dissipation inequality follows from (2.17) and (2.18) and reads:  Z  2 Z Z ε 1 τ %ε %ε |uε |2 + %ε eε (%ε , ϑε ) − ϑ%ε sε (%ε , ϑε ) − %ε F (τ )dx + ϑσε [[0, τ ] × Ω ] + 2 Q(ϑ − ϑε )dxdt 2 ε 0 Ω ϑε Ω  Z  2 ε 2 = (4.2) %ε,0 |uε,0 | + %ε,0 eε (%ε,0 , ϑε,0 ) − ϑ%ε,0 sε (%ε,0 , ϑε,0 ) − %ε,0 F dx 2 Ω for a.a. τ ∈ (0, ∞). Most of the uniform estimates that our analysis leans on can be deduced from it. Indeed, in accordance with hypotheses (2.5) and (2.8),   4 3 4 %eε (%, ϑ) − ϑ%sε (%, ϑ) = Hϑ,ε (%, ϑ) + εa ϑ − ϑϑ , 3

(4.3)

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where we have set Hϑ,ε (%, ϑ) = %eεF (%, ϑ) − %ϑsεF (%, ϑ).

(4.4)

Since ∂2 1 ∂ F HΘ ,ε (%, Θ) = p (%, Θ), 2 % ∂% ε ∂%

(4.5)

while ∂ ∂ HΘ ,ε (%, ϑ) = %(ϑ − Θ) sεF (%, ϑ); (4.6) ∂ϑ ∂ϑ we can use the thermodynamic stability hypotheses (2.7) in order to see that for any fixed Θ > 0 the function HΘ ,ε enjoys two remarkable properties: • % 7→ HΘ ,ε (%, Θ) is strictly convex, • ϑ→ 7 HΘ ,ε (%, ϑ) is decreasing for ϑ < Θ and increasing whenever ϑ > Θ. On the other hand, as %, ˜ ϑ satisfy the hydrostatic balance Eq. (1.1), we deduce from (4.5) that there is a constant k such that ∂ H (%, ˜ ϑ) = F + k ∂% ϑ,ε

in Ω .

Consequently, by virtue of the fact that %ε − %˜ is a function of zero mean, relation (4.2) can be written in the form  Z 
1 1 %ε |uε |2 + 2 Hϑ,ε (%ε , ϑε ) − (%ε − %)∂ ˜ % Hϑ,ε (%, ˜ ϑ) − Hϑ,ε (%, ˜ ϑ) (τ )dx ε Ω 2   Z Z Z %ε 1 4 1 4 1 1 τ a ϑε4 − ϑε3 ϑ + ϑ (τ )dx + 2 ϑσε [[0, τ ] × Ω ] + 4 Q(ϑ − ϑε )dxdt + 3 3 ε ε 0 Ω ϑε Ω ε  Z 
1 1 = ˜ % Hϑ,ε (%, ˜ ϑ) − Hϑ,ε (%, ˜ ϑ) dx %0,ε |u0,ε |2 + 2 Hϑ,ε (%0,ε , ϑ0,ε ) − (%0,ε − %)∂ ε Ω 2   Z 1 4 3 1 4 4 + a ϑ0,ε − ϑ0,ε ϑ + ϑ dx. 3 3 Ω ε

(4.7)

(4.8)

In accordance with (4.6), we have ∂ Hϑ,ε (%, ˜ ϑ)

= 0; ∂ϑ whence, by virtue of hypotheses (1.22)–(1.24) imposed on the initial distribution of %0,ε , u0,ε , and ϑ0,ε , the right-hand side of (4.8) is bounded uniformly with respect to ε → 0. As for the expression on the left-hand side of (4.8), we claim the following assertion. Lemma 4.1. Let pεF , eεF , sεF be determined through (2.1)–(2.10), where the function P satisfies hypotheses (2.2)–(2.4). Given 0 < % ≤ %˜ ≤ %, ϑ > 0, there exists a constant c = c(%, %, ϑ) such that Hϑ,ε (%, ϑ) − (% − %)∂ ˜ % Hϑ,ε (%, ˜ ϑ) − Hϑ,ε (%, ˜ ϑ)  2 2 |% − %| ˜ + |ϑ − ϑ| if %/2 < % < 2% and ϑ/2 < ϑ < 2ϑ, ≥ c(%, %, ϑ) × F %eε (%, ϑ) + %ϑ|sεF (%, ϑ)| otherwise.

(4.9)

Proof. Given 0 < Θ1 < Θ2 consider an auxiliary function HΘ ,ε (%, ϑ) defined by (4.4) for Θ ∈ [Θ1 , Θ2 ]. In accordance with (4.5) and (4.6), we have HΘ ,ε (%, ϑ) ≥ HΘ ,ε (%, Θ) ≥ HΘ ,ε (%Θ , Θ),

(4.10)

where %Θ is uniquely determined through the relation ∂ HΘ ,ε (%Θ , Θ) = 0. ∂%

(4.11)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

On the other hand, in agreement with hypothesis (2.2), 3

HΘ ,ε (%, Θ) =

3 Θ2 3 %Θ + %Θ log(%) − %Θ log(Θ) provided % ≤ . 2 2 ε 3/2

Consequently, if ε ∈ (0, Θ1 /%), %Θ can be identified as the unique solution of the equation 3 5 log(Θ) − , 2 2 and it is independent of ε. Moreover, log(%Θ ) =

HΘ ,ε (%, ϑ) ≥ HΘ ,ε (%Θ , Θ) ≥ −c(Θ1 , Θ2 )

for all %, ϑ ≥ 0

(4.12)

uniformly for ε → 0. Furthermore, since HΘ ,ε (%, Θ) is convex in % and since (4.11) holds true, we get Z % Z z 1 ∂ F p (τ, Θ) dτ dz, HΘ ,ε (%, ϑ) ≥ HΘ ,ε (%, Θ) = HΘ ,ε (%Θ , Θ) + τ ∂% ε %Θ %Θ

(4.13)

where we have used (4.5). By virtue of hypotheses (2.2)–(2.4), inf P 0 (Y ) > 0;

Y ≥0

whence, seeing that   τ 1 ∂ F Θ 0 p (τ, Θ) = P ε 3 , τ ∂% ε τ Θ2 we deduce from (4.12) and (4.13) Θ % log(%) − c0 (Θ1 , Θ2 ) 2 It follows from (4.12) that HΘ ,ε (%, ϑ) ≥

for all %, ϑ ≥ 0.

(4.14)

ϑ F %s (%, ϑ) ≥ −c1 (ϑ), 2 ε

(4.15)

H2ϑ,ε (%, ϑ) = %eεF (%, ϑ) − 2ϑ%sεF (%, ϑ) ≥ −c2 (ϑ).

(4.16)

Hϑ/2,ε (%, ϑ) = %eεF (%, ϑ) − and, by the same token,

After a simple manipulation, relations (4.15) and (4.16) give rise to   Hϑ,ε (%, ϑ) ≥ c3 (ϑ) %eεF (%, ϑ) + %ϑ|sεF (%, ϑ)| − c4 (ϑ), with c3 (ϑ) > 0.

(4.17)

Combining (4.14) and (4.17) we infer that Hϑ,ε (%, ϑ) − (% − %)∂ ˜ % Hϑ,ε (%, ˜ ϑ) − Hϑ,ε (%, ˜ ϑ)   ≥ c5 (ϑ) %eεF (%, ϑ) + %ϑ|sεF (%, ϑ)| − c6 (%, %, ϑ),

with c5 (ϑ) > 0.

(4.18)

Finally, by virtue of (4.5) and (4.6), n inf Hϑ,ε (%, ϑ) − (% − %)∂ ˜ % Hϑ,ε (%, ˜ ϑ) − Hϑ,ε (%, ˜ ϑ) | o (%, ϑ) ∈ [0, ∞)2 \{%/2 < % < 2%, ϑ/2 < ϑ < 2ϑ} = c(%, %, ϑ) > 0.

(4.19)

Relations (4.18) and (4.19), together with (4.5) and (4.6) yield (4.9). Setting % = inf %˜ ≤ sup %˜ = % Ω

Ω

we introduce the following notation: Mess = {(%, ϑ) | %/2 < % < 2%, ϑ/2 < ϑ < 2ϑ},



E. Feireisl et al. / Physica D 237 (2008) 1466–1487

1475

Mres = [0, ∞) × [0, ∞) \ Mess . Similarly, for a measurable function w we write w = [w]ess + [w]res , where [w]ess = w char{(t, x) ∈ (0, T ) × Ω | (%ε (t, x), ϑε (t, x)) ∈ Mess }, [w]res = w char{(t, x) ∈ (0, T ) × Ω | (%ε (t, x), ϑε (t, x)) ∈ Mres } with char B denoting the characteristic function of the set B. Note that, unlike the sets Mess , Mres , the components [w]ess , [w]res depend on the values of %ε , ϑε , thus on ε. The subscript ess, meaning essential, stands for the dominating component, while res, meaning residual, denotes the part vanishing in the asymptotic limit for ε → 0. It follows from (4.5) and (4.6) that min

(%,ϑ)∈Mres

Hϑ,ε (%, ϑ) ≥

min

(%,ϑ)∈∂ Mess

Hϑ,ε (%, ϑ) > 0,

(4.20)

where the latter quantity is independent of ε for ε small enough. Now, we are ready to establish a set of uniform estimates that follow directly from dissipation Eq. (4.8). To begin with, we see that ess sup k%ε u2ε k L 1 (Ω ) ≤ c.

(4.21)

t∈(0,T )

Next we observe that (4.9) and (4.20) imply ess sup meas{x ∈ Ω | (%ε (t, x), ϑε (t, x)) ∈ Mres } ≤ cε 2 . t∈(0,T )

It follows from (4.9) that

 

%ε − %˜

ess sup ≤c

ε t∈(0,T ) ess L 2 (Ω ) and, by the same token,

" #

ϑ −ϑ

ε ess sup ε t∈(0,T )





≤ c,

(4.22)

(4.23)

(4.24)

ess L 2 (Ω )

where the bounds are uniform with respect to ε → 0. Furthermore, we get kσε kM+ [0,T ]×Ω ≤ cε 2 ,

(4.25)

in particular, by virtue of (2.16),



2 t

sup ∇x uε + ∇x uε − divx uε I ≤ c,

2 3 ε>0 L (0,T ;L 2 (Ω ;R 3×3 )) !



∇ x ϑε

∇x log(ϑε )

sup + ≤ c,

ε 2

2 ε ε>0 L (0,T ;L 2 (Ω ;R 3 )) L (0,T ;L 2 (Ω ;R 3 ))

(4.26)

(4.27)

where we have used the constitutive relations (1.19) and (1.20), together with the structural hypotheses (2.11) and (2.12). Using in (4.26) Korn’s inequality and then both in (4.26) and (4.27) the Poincar´e type inequality   Z kξ k L 2 (Ω ) ≤ c k∇ξ k L 2 (Ω ) + |ξ |dx , M

where M ⊂ Ω , |M| ≥ M > 0 and c = c(M), we arrive at sup kuε k L 2 (0,T ;W 1,2 (Ω ;R 3 )) ≤ c, ε>0

(4.28)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487



ϑ − ϑ ε

sup 

ε ε>0

L 2 (0,T ;W 1,2 (Ω ;R 3 ))



log ϑ − log ϑ ε

+

ε

  ≤ c.

(4.29)

L 2 (0,T ;W 1,2 (Ω ))

Last but not least, the last term on the left-hand side of (4.8) yields Z TZ %ε |ϑε − ϑ|2 dxdt ≤ cε 4 , 0 Ω ϑε

(4.30)

where we have used the constitutive relation (1.21). Next, the “radiation” part in (4.8) gives rise to ess sup kϑε − ϑk4L 4 (Ω ) ≤ cε. t∈(0,T )

(4.31)

Finally, we derive estimates on the residual part of the density that follow from (4.8) and (4.9). To begin with, it follows from hypotheses (2.2)–(2.4) that 5

P(Y ) ≥ c(Y + Y 3 ), c > 0,

for all Y ≥ 0.

Consequently, by using (2.6), we get Z 5 4 3 [%ε ]res ess sup dx ≤ cε 3 , t∈(0,T ) Ω

(4.32)

(4.33)

and, similarly, Z ess sup

t∈(0,T ) Ω

[%ε ϑε ]res dx ≤ cε2 .

(4.34)

In addition, by virtue of (4.14) and (4.22), Z ess sup [%ε log(%ε )]res dx ≤ cε2 .

(4.35)

Similarly, we get a uniform estimate on the residual component of entropy Z [%ε |sεF (%ε , ϑε )|]res dx ≤ cε2 . ess sup

(4.36)

t∈(0,T ) Ω

t∈(0,T ) Ω

Finally, combining (4.21) with (4.23) and (4.33), namely ess sup k%ε k t∈(0,T )

5

L 3 (Ω )

≤ c,

we conclude ess sup k%ε uε k t∈(0,T )

5

L 4 (Ω ;R 3 )

≤ c.

(4.37)

4.2. Refined pressure estimates We adapt the technique developed in [14] in order to obtain an additional piece of information concerning integrability of the pressure. This step is necessary in order to control the quantity pε − %ε ϑε during the limit process. Note that our approach is based on the fact that the state equation for the pressure approaches asymptotically the classical perfect gas law p = %ϑ. The principal idea is to take the quantities   Z 1 ϕ(t, E x) = ψ(t)∇x ∆−1 b(% ) − b(% )dx , ψ ∈ D(0, T ), ε ε n |Ω | Ω as test functions in the variational formulation of the momentum Eq. (2.14). Here the symbol ∆−1 n stands for the inverse of the Laplace operator on Ω supplemented with the homogeneous Neumann boundary conditions on ∂Ω . Note that the time derivative of the quantity b(%ε ) can be expressed by means of the renormalized equation (2.13). A local variant of this approach was used by Lions [23] in order to establish a refined a priori estimates on the barotropic pressure necessary for the existence theory. A global version has been introduced in [14] and in [33], where ∇x ∆−1 was replaced

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

by the so-called Bogovskii operator (see [34] about more details concerning the use of the Bogovskii operator in the dynamics of compressible fluids). In both cases using ϕE as a test function in (2.14) leads to Z Z Z T Z Z 1 T 1 b(%ε )dxdt p (% , ϑ )dx p (% , ϑ )b(% )dxdt = ψ ψ ε ε ε ε ε ε ε ε2 0 ε 2 |Ω | 0 Ω Ω Ω   Z Z Z 1 T 1 −1 b(%ε )dx dxdt + Iε , (4.38) − 2 %ε ∇x F · ∇x ∆n b(%ε ) − ψ |Ω | Ω ε 0 Ω where T

Z Iε =

Z Ω

0

T

Z + 0

T

Z + 0

E dt − (Sε : ∇x ϕE − %ε uε ⊗ uε : ∇x ϕdx)

T

Z

∂t ψ

Z

0

  Z 1 %ε uε · ∇x ∆−1 b(% ) − b(% )dx dxdt ε ε n |Ω | Ω Ω

Z

%ε uε · ∇x ∆−1 n divx (b(%ε )uε )dxdt Ω   Z Z 1 −1 0 0 ψ %ε uε · ∇x ∆n (%ε b (%ε ) − b(%ε ))divx uε − (b(%ε ) − b (%ε )%ε )divx uε dx dxdt. |Ω | Ω Ω ψ

The last two terms are coming from Eq. (2.13) written in the form ∂t b(%ε ) = −div(b(%ε )uε ) − (%ε b0 (%ε ) − b(%ε ))divx uε

in D0 ((0, T ) × Ω ).

(4.39)

Utilizing estimates (4.23)–(4.37) established in the preceding section one can show, exactly as in [14], that all integrals included in Iε are bounded uniformly for ε → 0 and that (2.13) implies (4.39) as soon as |b(%)| ≤ c%β , |%b0 (%)| ≤ c%β

with β > 0 sufficiently small.

(4.40)

In order to comply with (4.40), we take b ∈ C ∞ [0, ∞) such that  0 for 0 ≤ % ≤ 2%, b(%) = ∈ [0, %β ] for 2% < % ≤ 3%,  β % if % > 3%,

(4.41)

with β > 0 sufficiently small to be specified later. In particular, b(%ε ) = b([%ε ]res ); whence, by virtue of (4.35), Z b(%ε )dx ≤ cε2 . ess sup

(4.42)

t∈(0,T ) Ω

Furthermore, we get   Z Z 1 1 −1 b(%ε )dx dx %ε ∇x F · ∇x ∆n b(%ε ) − |Ω | Ω ε2 Ω    Z  Z 1 %ε − %˜ 1 −1 ∇x F · ∇x ∆n b(%ε ) − b(%ε )dx dx = ε Ω ε |Ω | Ω ess   Z  Z 1 %ε − %˜ −1 ∇x F · ∇x ∆n b(%ε ) − b(%ε )dx dx + |Ω | Ω ε2 Ω res   Z Z 1 1 + 2 b(%ε )dx dx, %˜ ϑ b(%ε ) − |Ω | Ω ε Ω where we have exploited the fact that %, ˜ ϑ solve the diagnostic equation (1.1). Furthermore, using the standard elliptic estimates for ∆n we deduce Z     Z %ε − %˜ 1 1 −1 ∇x F · ∇x ∆n b(%ε ) − b(%ε )dx dx ε Ω ε |Ω | Ω ess



 Z

%ε − %˜

c

b(%ε ) − 1

ess sup ≤ ess sup b(% )dx ε



ε ε |Ω | 2 t∈(0,T )

ess L (Ω )

t∈(0,T )

Ω

(4.43)

6

L 5 (Ω )

,

(4.44)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

Z     Z 1 %ε − %˜ −1 b(% ) − b(% )dx dx ∇ F · ∇ ∆ ε ε x x n |Ω | Ω ε2 Ω res



 Z

%ε − %˜



b(%ε ) − 1

≤ ess sup b(% )dx ess sup ε



2 |Ω | ε 1 t∈(0,T )

t∈(0,T )

res L (Ω )

Ω

L 4 (Ω )

,

(4.45)

while the last integral on the right-hand side of (4.43) is bounded in view of (4.42). Finally, it is easy to check that, by virtue of (4.35), Z Z Z 6β 6 5 5 |b(%ε )| dx ≤ [%ε ]res dx ≤ [%ε log(%ε )]res dx ≤ cε 2 Ω

Ω

Ω

as soon as β ≤ 5/6. Thus we conclude, combining estimates (4.38)–(4.45), that Z TZ pε (%ε , ϑε )b(%ε )dxdt ≤ ε 2 c,

(4.46)

Ω

0

with b given by (4.41), where β ∈ (0, 65 ]. 5. Convergence, part I Taking advantage of the uniform bounds established in the preceding section, we pass to the asymptotic limit for ε → 0 in the field Eqs. (2.13)–(2.17). 5.1. Equation of continuity With estimates (4.23), (4.28), (4.33) and (4.37) at hand we have %ε → %˜

5

in L ∞ (0, T ; L 3 (Ω )),

uε → u weakly in L (0, T ; W 2

(5.1) 1,2

(Ω ; R )), 3

(5.2)

5 4

%ε uε → %u ˜ weakly- ∗ in L ∞ (0, T ; L (Ω ; R 3 )).

(5.3)

It is thus easy to let ε → 0 in (2.13) with b = 0 in order to obtain divx (%u) ˜ = %˜ divH v + ∂x3 (%u ˜ 3 ) = 0,

(5.4)

where we have set v = (u 1 , u 2 , 0). 5.2. Momentum equation To begin with, we examine the pressure term  5    2 ϑ % 4 ε ε pε (%ε , ϑε ) = %ε ϑε +  P ε 3  − %ε ϑε  + εa(ϑε4 − ϑ ). ε 2 ϑε Our aim is to show that pε ≈ %ε ϑε in the asymptotic limit ε → 0. This will be done by means of the following two steps: • Writing 4

4

4

ϑε4 − ϑ = [ϑε4 − ϑ ]ess + [ϑε4 − ϑ ]res and using (4.30), we obtain 4

k[ϑε4 − ϑ ]ess k L 2 ((0,T )×Ω ) ≤ cε 2 .

(5.5)

As far as the residual component is concerned, we have Z TZ Z TZ 4 |[ϑε4 − ϑ ]res |dxdt ≤ c |ϑε − ϑ|([ϑε ]3res + [ϑ]3res )|dxdt 0

Ω

0

Ω

≤ kϑε − ϑk L 2 (0,T ;L 4 (Ω )) ess sup

t∈(0,T )

where the last inequality follows from (4.22), (4.29) and (4.31).



k[ϑε ]3res k

4

L 3 (Ω )

+ k[ϑ]3res k

 4

L 3 (Ω )

7

≤ cε 4 ,

(5.6)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

Thus we infer, making use of (5.5) and (5.6), that

ϑ4 − ϑ4 3

ε

≤ cε 4 .



1 ε

(5.7)

L ((0,T )×Ω )

• The next and more delicate task is to show that    5  1  ϑε2  %ε  P ε 3 − %ε ϑε  → 0 in L 1 ((0, T ) × Ω ) for ε → 0. ε ε2 ϑε2

(5.8)

To this end, observe first that hypothesis (2.2) yields  5      Z T Z 52 Z 2 ϑε ϑ % % ε ε ε ) (    P ε 3  − %ε ϑε  dxdt 2 2 ε P ε 3 − %ε ϑε dxdt = ε 0 Ω ϑε ≤ε 3 %ε3 ϑε2 ϑε2 res  5    Z ϑε2 % ε  ( ) dxdt    = P ε 3 − %ε ϑε 2 2 ε 2 ϑε ≤ε 3 %ε3 , %ε ≤M ϑε res  5    Z ϑε2 % ε ( ) dxdt.     + 2 2 ε P ε 3 − %ε ϑε 3 2 ϑε ≤ε 3 %ε , %ε >M ϑε res It follows from hypotheses (2.2) and (2.4), together with (4.22), that  5    Z TZ Z ϑε2 8 2 %ε  ( )    3 1res dx ≤ c2 (M)ε 3 . 2 2 dxdt ≤ c1 (M)ε ε P ε 3 − %ε ϑε 3 0 Ω ϑε ≤ε 3 %ε , %ε ≤M ϑε2 res

(5.9)

On the other hand, by virtue of (4.46) and hypotheses (2.2) and (2.4), we have   5 Z Z 2 2 5 +β % ϑ ε ε P ε 3  %εβ dxdt ≤ cε 2 ; ε 3 %ε3 dxdt ≤ {%ε >M} {%ε >M} ε ϑε2 therefore Z

 5    ϑε2 % ( )   ε  2 2 ε P ε 3 − %ε ϑε ϑε ≤ε 3 %ε3 , %ε >M ϑε2

res

Z 2 5 c2 dxdt ≤ c1 ε 3 %ε3 dxdt ≤ β . M {%ε >M}

Consequently, we infer that   5   a 1  ϑε2  %ε  4 P ε 3 − %ε ϑε  + (ϑε4 − ϑ ) → 0 2 ε ε ε ϑε2

in L 1 ((0, T ) × Ω )

(5.10)

for ε → 0.

(5.11)

Now, in accordance with (1.1), we can write Z Z Z Z Z Z 1 T 1 T %ε 1 T ϑ div ϕ E + % ∇ F · ϕ) E dxdt = ϑ div ( % ˜ ϕ)dxdt E + %ε (ϑε − ϑ)divx ϕdxdt. E (% ε ε x ε x x ε2 0 Ω ε 2 0 Ω %˜ ε2 0 Ω

(5.12)

Indeed, this identity is the rigorous formulation of the chain of evident formal equalities %ε ∇x (%ε ϑε ) − %ε ∇x F = ∇x (%ε (ϑε − ϑ)) + ϑ∇x %ε − ϑ ∇x %˜ = ∇x (%ε (ϑε − ϑ)) + ϑ %∇ ˜ x %˜



%ε %˜



.

Now, we shall prove that the latter term in (5.12) gives rise to the friction force ∂x3 %˜ χ (see (1.3)), where χ verifies (1.5). In order to show this statement, some preliminary considerations are needed. To begin with, writing " # " # ϑε − ϑ ϑε − ϑ ϑε − ϑ = + ε2 ε2 ε2 ess

res

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

we easily deduce from (4.30) that "

ϑε − ϑ ε2

# →χ

weakly in L 2 ((0, T ) × Ω ).

(5.13)

ess

On the other hand, # " ϑε − ϑ %˜ ε2

%˜ − %ε = ε 

res

"

 res

ϑε − ϑ ε

# + [ %ε ϑε ]res p

r

res

%ε ϑε

ϑε − ϑ ε2

! ,

where, in accordance with (4.30) and (4.34), p [ %ε ϑε ]res

r

%ε ϑε

ϑε − ϑ ε2

! →0

in L 1 ((0, T ) × Ω ) for ε → 0.

(5.14)

Moreover, combining (4.22), (4.23), (4.29) and (4.33) with H¨older’s inequality we get

 " #

%˜ − %  ϑε − ϑ ε

ε ε res





14 30

≤ c1 ε

res L 1 ((0,T )×Ω )

" #

 

ϑ −ϑ

%˜ − %ε

ε



5 ε ε res L ∞ (0,T ;L 3 (Ω ))





7

≤ c2 ε 15 . (5.15)

res L 2 (0,T ;L 6 (Ω ))

Relations (5.14) and (5.15) give rise to "

ϑε − ϑ ε2

# →0

in L 1 ((0, T ) × Ω ) as ε → 0.

(5.16)

res

Furthermore, by means of (5.13) and (5.16), the last integral in (5.12) can be treated as follows: 1 ε2

T

Z

Z Ω

0

E = %ε (ϑε − ϑ)divx ϕdxdt

T

Z

"

Z Ω

0

%ε

ϑε − ϑ ε2

# divx ϕdxdt E +

T

Z 0

ess

# "   ϑε − ϑ %ε res ε2 Ω

Z

divx ϕdxdt, E

(5.17)

res

where, in accordance with (5.13), Z

T

"

Z Ω

0

%ε

ϑε − ϑ ε2

# divx ϕdxdt E →

T

Z

Z

0

ess

Ω

%χ ˜ divx ϕdxdt, E

(5.18)

as ε → 0

(5.19)

while, exactly as in (5.14), Z 0

T

Z

" #   ϑε − ϑ %ε res ε2 Ω

divx ϕdxdt E →0 res

for any fixed test function ϕ. E With relations (5.12)–(5.19) at hand, it is not difficult to let ε → 0 in (2.14) in order to obtain   Z TZ Z TZ
1 %u∂ ˜ t ϕE + %u ⊗ u : ∇x ϕE dxdt = µ(ϑ) ∇x u : ∇x ϕE + divx u divx ϕE dxdt 3 0 Ω 0 Ω Z TZ Z − χ ∂x3 %ϕ ˜ 3 dxdt − %u ˜ 0 ϕ(0, E ·)dx 0

Ω

(5.20)

Ω

for any test function ϕE ∈ D([0, T ); D(Ω ; R 3 )),

ϕE · n|∂ Ω = 0,

divx (%˜ ϕ) E = 0.

(5.21)

In (5.20), the symbol %u ⊗ u stands for a weak L 1 −limit of {%ε uε ⊗ uε }ε>0 . Clearly, relations (5.20) and (5.21), together with the anelastic constraint (5.4), represent a suitable variational formulation of (1.2)–(1.4) as soon as we can show that

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487 T

Z 0

Z Ω

%u ⊗ u : ∇x ϕdxdt E =

T

Z

Z Ω

0

%u ⊗ u : ∇x ϕdxdt E

(5.22)

for any test function satisfying (5.21). This will be the objective of the last two sections. 5.3. Entropy balance Our ultimate goal in the section is to let ε → 0 in entropy balance equation (2.17) in order to recover (1.5). To this end, observe first that %ε s(%ε , ϑε ) = [%ε s(%ε , ϑε )]ess + [%ε s(%ε , ϑε )]res ,

(5.23)

where, in view of (4.36) and (5.6), [%ε s(%ε , ϑε )]res → 0

in L 1 ((0, T ) × Ω ),

(5.24)

while, by virtue of (4.23) and (4.24), and hypothesis (2.2),   3 [%ε s(%ε , ϑε )]ess → %˜ log(ϑ) − log(%) ˜ in L q ((0, T ) × Ω ) for any q ≥ 1. 2

(5.25)

Similarly, by means of (4.25), hσε , ϕi → 0

for any fixed ϕ ∈ D((0, T ) × Ω ).

Furthermore, Z TZ 0

%ε ϑ − ϑε ϕdxdt = ε2 Ω ϑε

T

Z 0

%ε Ω ϑε

Z

"

(5.26)

ϑ − ϑε ε2

# ϕdxdt +

T

Z 0

ess

%ε Ω ϑε

Z

"

ϑ − ϑε ε2

# ϕdxdt,

(5.27)

res

where, in accordance with (4.23), (4.24) and (5.13), # " Z TZ Z TZ %ε ϑ − ϑε −1 χ ϕdxdt, ϕdxdt → − %ϑ ˜ ε2 0 Ω ϑε 0 Ω

(5.28)

ess

while, considering separately the cases ϑε ≤ # " Z TZ %ε ϑ − ϑε ϕdxdt → 0. ε2 0 Ω ϑε

ϑ 2

and ϑε >

ϑ 2,

by virtue of (4.29), (4.30) and (4.33), (5.29)

res

Writing qε κ(ϑε ) = −ε ϑε ϑε



∇ x ϑε ε

 (5.30)

one can use (4.29) and (4.31) in order to conclude that Z TZ qε · ∇x ϕ dxdt → 0 as ε → 0 ϑε 0 Ω

(5.31)

for any fixed ϕ ∈ D((0, T ) × Ω ). Finally, the convective term in (2.17) gives rise to Z TZ Z TZ %ε s(%ε , ϑε )uε · ∇x ϕdxdt = [%ε s(%ε , ϑε )]ess uε · ∇x ϕdxdt 0

Ω

Ω

0

T

Z

Z

+ 0

where, in accordance with (5.2) and (5.25), Z TZ Z [%ε s(%ε , ϑε )]ess uε · ∇x ϕdxdt → − 0

Ω

for any fixed ϕ ∈ D((0, T ) × Ω ).

Ω

T 0

[%ε s(%ε , ϑε )]res uε · ∇x ϕdxdt,

Z Ω

%˜ log(%)u ˜ · ∇x ϕdxdt

(5.32)

(5.33)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

On the other hand, we claim that Z TZ [%ε s(%ε , ϑε )]res uε · ∇x ϕdxdt → 0. 0

(5.34)

Ω

Indeed, in view of (4.22), it is enough to show that k%ε s(%ε , ϑε )uε k L q ((0,T )×Ω ) ≤ c

for a certain q > 1.

(5.35)

In order to see (5.35), one can use (4.28), (4.31), and (4.37) to deduce that 4 %ε s R (%ε , ϑε )uε = ε aϑε3 uε 3

12

is bounded in L 2 (0, T ; L 11 (Ω )).

(5.36)

On the other hand, it follows from hypotheses (2.3) and (2.10) that |S(εY ) − S(ε)| ≤ c| log(Y )|

for all Y > 0;

whence, due to (2.9) |%ε sεF (%ε , ϑε )| ≤ c%ε (| log(%ε )| + | log(ϑε )|),

(5.37)

where, by virtue of (4.23) and (4.33), ess sup k%ε | log(%ε )|k L q (Ω ) ≤ c t∈(0,T )

for any 1 ≤ q <

5 . 3

(5.38)

Furthermore, it follows from (4.29) and (4.37) that {%ε log(ϑε )uε }ε>0

30

is bounded in L 2 (0, T ; L 29 (Ω ; R 3 )).

(5.39)

Combining (5.35)–(5.39) we obtain (5.34). Summing up the previous estimates, one can take ε → 0 in (2.17). Taking into account (5.4) and its renormalized counterpart, namely div(%˜ log %u) ˜ + %divu ˜ = 0, we obtain (1.5). 6. Analysis of the acoustic waves The regularization technique via the spectral analysis and the Fourier series used in Sections 6 and 7 are reminiscent to Lions and Masmoudi [24], where the similar analysis is performed in the “simple” case of the constant equilibrium density %. ˜ Another available regularization technique based on mollifiers was proposed by Lions and Masmoudi in [26] and applied to system (6.2) and (6.3) with constant coefficient %. ˜ Its applicability to the case of space-dependent %˜ would be worth investigating. Following [11] we introduce a weighted Helmholtz decomposition: H%˜ [v] = v − %∇ ˜ x Φ,

H⊥ ˜ x Φ, %˜ [v] = %∇

(6.1)

where Φ is the unique solution of the Neumann problem divx (%∇ ˜ x Φ) = divx v

in Ω ,

Z ∇x Φ · n|∂ Ω = 0,

Φdx = 0. Ω

Since %˜ is smooth and strictly positive on Ω , it can be shown, by means of the standard elliptic theory, that H%˜ is a bounded linear 1,q operator on Wn (Ω ; R 3 ) and on L q (Ω ; R 3 ) for any 1 < q < ∞ provided, in the latter case, the quantity divx v is identified with a 1,q linear form on Wn (Ω ; R 3 ). Here, the subscript n stands for the subspace of W 1,q (Ω ; R 3 ) consisting of functions with zero normal trace. The time evolution of the acoustic waves is governed by a hyperbolic system that can be deduced from (2.13) and (2.14), specifically,   Z TZ  Z η εrε ηt + %Q ˜ ε · ∇x dxdt = − εr0,ε η(0, ·)dx (6.2) %˜ 0 Ω Ω

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

for all η ∈ D([0, T ) × Ω ), Z Z TZ
εQ0,ε · ϕ(0, E ·)dx εQε · ∂t ϕE + ϑrε divx ϕE dxdt = − Ω 0 Ω     Z TZ  ϕE ϕE − %ε [uε ⊗ uε ] : ∇x dxdt +ε Sε : ∇x %˜ %˜ 0 Ω   Z Z
1 T ϕE + (%ε ϑε − pε ) + %ε (ϑ − ϑε ) divx dxdt ε 0 Ω %˜

(6.3)

for all ϕE ∈ D([0, T ); D(Ω ; R 3 )), ϕE · n|∂ Ω = 0, where we have set Qε =

%ε uε , %˜

Q0,ε =

%0,ε u0,ε , %˜

rε =

%ε − %˜ , ε%˜

r0,ε =

%0,ε − %˜ . ε %˜

(6.4)

System (6.2) and (6.3) represents a variational formulation of a wave equation with a vertically dependent speed of propagation ε∂t rε + (1/%) ˜ divx (%Q ˜ ε ) = 0, ε∂t Qε + ϑ∇x rε = Gε studied in detail by Wilcox [38]. In the latter equation we have set Gε = (ε/%) ˜ (divx Sε − divx (%ε uε ⊗ uε )) +


11 ∇x (%ε (ϑ − ϑε )) + (%ε ϑε − pε ) . ε %˜

Note that the last two integrals on the right-hand side of (6.3) are small of order ε, more specifically, by virtue of (4.21), (4.28) and (4.31), together with (5.11), (5.13) and (5.14),       Z Z Z TZ 
ϕE ϕE ϕE 1 T (%ε ϑε − pε ) + %ε (ϑ − ϑε ) divx ε Sε : ∇x − %ε [uε ⊗ uε ] : ∇x dxdt + dxdt % ˜ % ˜ ε %˜ 0 Ω 0 Ω   Z TZ ϕE =ε dxdt, (6.5) (Aε + Bε ) : ∇x %˜ 0 Ω where Aε → 0

in L 1 (0, T ; L 1 (Ω ; R 3×3 )),

sup kBε k L 2 (0,T ;L 1 (Ω ;R 3×3 )) ≤ c.

(6.6)

ε>0

The “elliptic” part of (6.2) and (6.3) gives rise to the following eigenvalue problem:   ω %∇ ˜ x = λU, ϑdivx U = λω in Ω , U · n|∂ Ω = 0, %˜ or, equivalently,      ω ω −divx %∇ ˜ x = Λ%˜ %˜ %˜

in Ω ,

∇x ω · n|∂ Ω = 0,

(6.7)

λ2 = −Λϑ.

(6.8) ∞,m

j The elliptic problem introduced in (6.8) admits a complete system of real eigenfunctions {ω j,m } j=0,m=1 , together with the

∞,m

j corresponding family of real eigenvalues {Λ j,m } j=0,m=1 such that

m 0 = 1,

Λ0,1 = 0,

ω0,1 = %, ˜

0 < Λ1,1 = · · · = Λ1,m 1 (= Λ1 ) < Λ2,1 = · · · = Λ2,m 2 (= Λ2 ) < · · · ,

(6.9)

∞,m

j where the functions {ω j,m } j=0,m=1 form an orthonormal basis of the weighted Hilbert space L 21/%˜ (Ω ),

Z Ω

ω j,` ωk,m



dx %˜



= δ j,k δ`,m .

The symbol m j in (6.9) denotes multiplicity of the eigenvalue Λ j (cf. Chapter 3 in Wilcox [38]). Thus all solutions of the original problem (6.7) can be written in the form q λ j = i Λ j ϑ,

(6.10)

(6.11)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

and U j,m = i( Λ j )

−1

p

%∇ ˜ x



ω j,m %˜



for j = 1, . . . , m, and m = 1, . . . , m j .

(6.12)

The eigenspace associated with Λ0,1 = 0 coincides with the space of divergence-less functions L 2div,1/%˜ (Ω ; R 3 ) = {v ∈ L 21/%˜ (Ω ; R 3 ) | divx v = 0}. The weighted Hilbert space L 21/%˜ (Ω ; R 3 ) admits an orthogonal decomposition of the form L 21/%˜ (Ω ;

R )= 3

L 2div,1/%˜ (Ω ;

R )⊕ 3

n

∞,m j span{iU j,m } j=1,m=1

o L 21/%˜ (Ω ;R 3 )

,

(6.13) ∞,m

j with the corresponding projectors represented by H%˜ and H⊥ %˜ introduced in (6.1). Accordingly, the functions {iU j,m } j=1,m=1 form

2 3 an orthonormal basis on the Hilbert space H⊥ %˜ [L 1/%˜ (Ω ; R )]. Taking ϕ = ψ(t)ω j,m in (6.2), and ϕ = ψ(t)U j,m in (6.3), we obtain a system of equations p   ε∂t [rε ] j,m + i pΛ j [Qε ] j,m = 0, on (0, T ), ε∂t [Qε ] j,m + i Λ j [rε ] j,m = ε Aεj,m + ε Bεj,m

for j = 1, 2, . . ., m = 1, . . . m j , where we have set Z Z [rε ] j,m = rε ω j,m dx, [Qε ] j,m = Qε · U j,m dx. Ω

(6.14)

(6.15)

Ω

Note that, by virtue of (6.6), Aεj,m → 0

in L 1 (0, T ) for ε → 0,

(6.16)

{Bεj,m }ε>0

are bounded in L 2 (0, T ).

(6.17)

while

7. Convergence, part II In accordance with what have been already achieved in Section 5, the proof of Theorem 3.1 is complete as soon as we show (5.22), specifically,     Z TZ Z TZ ϕE ϕE %u ˜ ⊗ u : ∇x dxdt → dxdt (7.1) %ε [uε ⊗ uε ] : ∇x %˜ %˜ 0 Ω 0 Ω for any ϕE satisfying ϕE ∈ D(0, T ; D(Ω ; R 3 )),

ϕE · n|∂ Ω = 0,

divx ϕE = 0

in Ω .

(7.2)

The first step in the proof of (7.1) is to observe that H%˜ [%ε uε ] → H%˜ [%u] ˜ = %u ˜ in L 1 ((0, T ; L 1 (Ω ; R 3 ))).

(7.3)

Indeed, taking ϕE =

ψ H%˜ [%φ], ˜ %˜

ψ ∈ D(0, T ),

φ ∈ C ∞ (Ω ; R 3 ),

φ · n|∂ Ω = 0

as a test function in the momentum balance (2.14), and using relations (5.11), (5.12), (5.18) and (5.19), we infer that the mappings Z Z dx %ε uε (t) · H%˜ [%φ] ˜ H%˜ [%ε uε ] · φdx t ∈ [0, t] 7→ = %˜ Ω Ω are precompact in C[0, T ]; whence, in agreement with (4.37), 5

H%˜ [%ε uε ] → H%˜ [%u] ˜ = %u ˜ in Cweak ([0, T ]; L 4 (Ω ; R 3 )).

(7.4)

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

Consequently, using (5.2) we get  Z T Z Z TZ dx H%˜ [%ε uε ] · H%˜ [%u ˜ ε] H%˜ [%ε uε ] · uε dxdt dt = %˜ 0 Ω 0 Ω Z Z TZ H%˜ [%u] ˜ · udxdt = → 0

Ω

T

Z Ω

0

2 dx

%˜ |u| 2

%˜

 dt.

(7.5)

Due to (5.1), relation (7.5) implies that   Z T Z Z T Z 2 dx 2 dx |H%˜ [%u ˜ ε ]| |%u| ˜ dt → dt, %˜ %˜ 0 Ω 0 Ω or, equivalently, H%˜ [%u ˜ ε ] → %u ˜ in L 2 (0, T ; L 2 (Ω ; R 3 )),

(7.6)

which, by the same token, gives rise to (7.3). More easily, due to (5.3) and (5.4), H⊥ %˜ [%ε uε ] → 0

5

weakly- ∗ in L ∞ (0, T ; L 4 (Ω ; R 3 )).

(7.7)

Keeping (7.3) in mind we observe easily that showing (7.1) reduces to   Z TZ ϕE dx ⊥ dt → 0 as ε → 0 H⊥ [ %Q ˜ ] ⊗ H [ %u ˜ ] : ∇ ε x ε %˜ %˜ %˜ %˜ 0 Ω

(7.8)

for all ϕE satisfying (7.2), where Qε is the quantity introduced in (6.4). Indeed %ε uε ⊗ uε =

1 1 1 H%˜ [%Q ˜ ε ] ⊗ (%u ˜ ε ) + H⊥ [%Q ˜ ε ] ⊗ H%˜ [%u ˜ ε ] + H⊥ [%Q ˜ ε ] ⊗ H⊥ ˜ ε ], %˜ [%u %˜ %˜ %˜ %˜ %˜

where, by virtue of (5.2), (7.3), (7.6) and (7.7)  H%˜ [%Q ˜ ε ] ⊗ uε → %u ˜ ⊗ u, weakly in L 1 (0, T ; L 1 (Ω ; R 3×3 )). H⊥ [ %Q ˜ ] ⊗ H [ %u ˜ ] → 0 ε % ˜ ε %˜ Now we set n o H⊥ [ %Z] ˜ %˜

M

mj X

X

=

[Z] j,m U j,m ,

{ j;0<Λ j ≤M} m=1

where, similarly to (6.15), we have introduced the Fourier coefficients Z Z · U j,m dx [Z] j,m = Ω

L 1 (Ω ;

for any Z ∈ R 3 ). We have h i hn o h n o ii ⊥ ⊥ ⊥ ⊥ H⊥ [ %Q ˜ ] ⊗ H [ %u ˜ ] = H [ %Q ˜ ] + H [ %Q ˜ ] − H [ %Q ˜ ] ε ε ε ε ε %˜ %˜ %˜ %˜ %˜ M M hn o h n o ii ⊥ ⊥ ⊥ ⊗ H%˜ [%u ˜ ε] + H%˜ [%u ˜ ε ] − H%˜ [%u ˜ ε] , M

(7.9)

M

where n o ⊥ H⊥ [ %Q ˜ ] − H [ %Q ˜ ] ε ε %˜ %˜

M

n o ⊥ = H⊥ [(% − %)u ˜ ] − H [(% − %)u ˜ ] ε ε ε ε %˜ %˜

M

n o ⊥ + H⊥ [ %u ˜ ] − H [ %u ˜ ] . ε ε %˜ %˜ M

By virtue of the uniform estimates (4.23), (4.28) and (4.33) n o ⊥ H⊥ [(% − %)u ˜ ] − H [(% − %)u ˜ ] → 0 in L 1 (0, T ; L 1 (Ω ; R 3 )). ε ε ε ε %˜ %˜ M

On the other hand, using (6.10), together with Parseval’s identity on the Hilbert space L 21/%˜ (Ω ), we get kdivx (%u ˜ ε )k2L 2 (Ω ) 1/%˜

=

mj ∞ X X j=1 m=1

Λ j [uε ]2j,m ;

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E. Feireisl et al. / Physica D 237 (2008) 1466–1487

whence

n o 2

⊥

˜ ε ] − H⊥ [ %u ˜ ]

H%˜ [%u

2 ε %˜

M L 1/%˜ (Ω )

and we are allowed to conclude that n o ⊥ H⊥ [ %u ˜ ] − H [ %u ˜ ] →0 ε ε %˜ %˜ M

mj X

X

=

[uε ]2j,m ≤

{ j;Λ j >M} m=1

1 kdivx (%u ˜ ε )k2L 2 (Ω ) M 1/%˜

in L 2 (0, T ; L 21 (Ω ; R 3 )) as M → ∞ uniformly in ε. %˜

In the light of the previous arguments, the proof of (7.8) reduces to showing that   Z TZ n o n o ϕE dx ⊥ ⊥ H%˜ [%Q ˜ ε] ⊗ H%˜ [%u ˜ ε] : ∇x dt → 0 M M %˜ %˜ 0 Ω or, equivalently, T

Z

Z n Ω

0

o H⊥ [ %Q ˜ ] ε %˜

M

n o ⊗ H⊥ [ %Q ˜ ] ε %˜

M

: ∇x

  ϕE dx dt → 0 %˜ %˜

(7.10)

for all ϕE satisfying (7.2) and for any fixed M. In order to see (7.10), we first observe, using (6.12), that     Z TZ Z TZ n o n o ϕE ϕE dx ⊥ ( %∇ ˜ Ψ ⊗ ∇ Ψ ) : ∇ H⊥ [ %Q ˜ ] ⊗ H [ %Q ˜ ] : ∇ dt = dxdt, x ε x ε x x ε ε %˜ %˜ M M % ˜ % ˜ %˜ 0 Ω 0 Ω where Ψε =

  mj X X [Qε ] j,m ω j,m p . %˜ Λj j≤M m=1

Integrating by parts and using the fact that ϕE is a solenoidal function, we get     Z TZ Z TZ ϕE ϕE dxdt = − divx (%∇ ˜ x Ψε ) ∇x Ψε · dxdt, (%∇ ˜ x Ψε ⊗ ∇x Ψε ) : ∇x %˜ %˜ 0 Ω 0 Ω where, by virtue of (6.8), −divx (%∇ ˜ x Ψε ) =

mj X X p

Λ j [Qε ] j,m ω j,m .

j≤M m=1

Making use of the fact that the quantities [Qε ] j,m satisfy the acoustic equation (6.14) we have T

Z − 0

  Z TZ X X mj ω j,m ϕE divx (%∇ ˜ x Ψε ) ∇x Ψε · dxdt = iε ∂t [rε ] j,m ∇x Ψε · ϕE dxdt %˜ %˜ Ω 0 Ω j≤M m=1 Z TZ X X mj
ω j,m [rε ] j,m ∇x Ψε · ∂t ϕE dxdt = iε ˜ 0 Ω j≤M m=1 % Z TZ X X mj ω j,m − iε [rε ] j,m ∂t ∇x Ψε · ϕE dxdt. ˜ 0 Ω j≤M m=1 %

Z

Consequently, in order to complete the proof of (7.1), it is enough to show that Z Z mj T X X ω j,m [rε ] j,m ∂t ∇x Ψε · ϕE dxdt ≤ c. 0 Ω j≤M m=1 %˜ To this end, it follows from the acoustic equation (6.14) that  X X    m mj ω j,m ω j,m −i X Xj ε ε ∂t ∇x Ψε = [rε ] j,m ∇x + (A j,m + B j,m )∇x . ε j≤M m=1 %˜ %˜ j≤M m=1

(7.11)

E. Feireisl et al. / Physica D 237 (2008) 1466–1487

1487

Thus, as ϕE is solenoidal, # " X X #   Z T Z "X X mj mj ω j,m ω j,m [rε ] j,m [rε ] j,m ∇x · ϕE dxdt = 0; %˜ %˜ 0 Ω j≤M m=1 j≤M m=1 whence (7.11) follows from (6.16), (6.17). Having established the relation (7.1) we have completed the proof of Theorem 3.1. Acknowledgements The work of E. Feireisl was supported by Grant 201/05/0164 of GA CR in the general framework of research programmes supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503 and partially by the Universit´e du Sud Toulon-Var. The work of A. Novotn´y was partially supported by the Neˇcas Center for Mathematical Modeling (LC06052) financed by M+SMT. The work of H. Petzeltov´a was supported by Grant 201/05/0164 of GA CR in the general framework of research programmes supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503. References [1] T. Alazard, Low Mach number flows, and combustion, SIAM J. Math. Anal. 38 (4) (2006) 1186–1213. 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