D Loum

C Tatkeu

I Dayoub

M Heddebaut

J-M Rouvaen

New Passenger Services in Railway Stations Based on Mode Group Diversity Multiplexing Optical Fiber Communications

Keywords: Optical Fiber, Passenger Services, Railway Station, Mode Group Diversity Multiplexing

Nowadays, because of its wide bandwidth and high communication capability, the optical fiber is more and more used for high data rate transmission of information in railway environments. Conventionally, only one service is sent over the fiber at a time. However, many different services can be simultaneously conveyed in railway stations such as passenger information service, cellular phone, Wi-Fi… The objective of the work proposed in this paper is to demonstrate the potential benefits of transmitting radio signals over fiber in a railway environment. The main idea is to exploit the full capacity of the fiber by transmitting multiple services using the same fiber. Since, different services are operating in different frequency bands; we propose a new multiplexing technique called -Mode Group Diversity Multiplexing-(MGDM) to ensure the transmission of multiple services using the same fiber, without additional infrastructure. There are numerous advantages of the proposed technique e.g. faster and reliable data exchange, high resolution video surveillance capability, high data rate information exchange in railway stations. We present, in this paper, the physical characteristics of optical fibers, performance of MGDM multiplexing technique, and the influence of the laser excitation conditions at the entrance of the fiber on the performances of the system.

Introduction

The development of cable networks with optical fibers has an impact on a large number of applications in the civil and military domains. Optical fibers are known for their large bandwidth and their high data rate, superior to those offered by other technologies. However, this bandwidth is not yet fully utilized by the currently available techniques. Generally a single service is transmitted by fiber, while several services could be combined within the same optical fiber. At present, single-mode (SMF-Single-Mode Fiber) and multi-mode (MMF-Multi-Mode Fiber) optical fibers equip more and more buildings, in particular, shopping malls, places and sites of public transports sites such as stations, airports, highways, ports or multimodal platforms. In these places, fibers connections are often used as means of extension of radio systems in particular between buildings. Furthermore, 90 % of the new buildings in Europe are at present equipped with multi-modes optical fibers MMF [START_REF] Dayoub | Wireless Systems Radio-optic demonstrator for distributed antenna system indoor wireless applications using low-cost VCSELs[END_REF]. To enable multiple services, (multi-frequency bands operation), we propose the exploitation of a new multiplexing technique called MGDM -Mode Group Diversity Multiplexing, which is a technique based on the transmission and the spatial reception of data. It consists of using 2 or 3 lasers sources simultaneously inside the same fiber. Our main objective is to provide multiple services using the same fiber, without additional infrastructure. In this paper we present, in the first part, the characteristics of various types of optical fibers, before highlighting the advantages of the MGDM multiplexing technique. We also study the influence of the axial delay (or offset) and of the spot size at the entrance of the fiber. We evaluate the transmission performance in terms of communication Bit Error Rate (BER). This evaluation study is performed by simulation using the 'VPI TM Systems' tools. We operate at three conventional wavelengths commonly used for radio transmission over fiber (850 nm, 1300 nm and 1550 nm). We consider fiber optics lengths from 10 m to 3 km. After several simulations, we show that the optimal "spot size" is around 8 µm and that up to three different services can be implemented efficiently in a single optical fiber. Considering these different services, we then discuss the 1-D and 2-D OCDMA codes used for the multiplexing techniques. In this paper, we concentrate on the 2-D codes which are associated to the MGDM technique. Finally, we conclude and give directions for future research.

Characteristics of optical fibers

Optical fiber is constituted of a very fine glass or plastic thread which possesses the property of guiding light. It serves the transmissions of ground and oceanic data and offers a data rate far superior than coaxial cables. It supports a "wide band" network through which can be carried out various types of services (television, telephony, video conferencing…). There are various types of optical fiber: single-mode, multi-mode and micro-structured fiber.

The optical fiber presents many advantages: It offers wide bandwidth of the Terahertz order, and enables data rate of several Gigabits per second. It offers a high level of safety during the transmissions, with low attenuations. It also has a long lasting of life, which requires few interventions according to maintenance. It provides a thermal insensitivity, a high degree of immunity against electromagnetic radiations and jamming. However, opticsal fiber possesses some disadvantages such as difficulties of adaptation with the opto-electronic transducers during installation. It requires precise mechanical alignment and specially adapted connectors. Moreover, the initial installation cost is high compared to conventional networks and it is hard to find adequately trained personnel. Table 1 presents characteristics of various types of optical fibers, and Figure 1 presents an example of a micro-structured optical fiber. In spite of the high level of performance, signals which cross across the optical fiber undergo attenuation. Various losses like the absorption, the modal dispersal, the imperfection of the connections, the presence of the curvatures and the micro-curvatures inside the fiber are usually represented by Rayleigh distributions. The attenuation of the signal depends essentially on the nature of the fiber. The single-mode fiber at wavelength 1550 nm (or a frequency of 193.5 THz) has attenuation of about 0.2 dB / km, at 1300 nm (or a frequency of 230.7 THz) attenuation is about 0.4 dB / km and the multi-mode fiber of wavelength at 850 nm (or a frequency of 352.9 THz) has attenuation of about 3 dB / km.

In Table 2, we present the advantages and disadvantages of various types of optical fiber: singlemode, multi-mode and micro-structured. Having presented the characteristics of various types of fibers, in the following paragraph, we develop the multi-users services aspects intended for railway stations applications. For this purpose, we present and evaluate the performances of multiplexing techniques capable of implementing multiple access and managing multi-users.

Mode Group Diversity Multiplexing (MGDM)

The Mode Group Diversity Multiplexing (MGDM) technique is based on the spatial transmission and reception of data [START_REF] Awad | RoF & Multi Services In Single-MMF LAN Using Mode Group Diversity Multiplexing[END_REF]. It consists of sending packet information corresponding to various users. It is an optical technique which aims at creating independent communication channels on a multi-mode optical fiber, using subsets of propagating modes. By associating a service with a propagated mode, the MGDM can offer same features as multiplexing using wavelength with a lower cost. This technique is depicted in following Figure 2. At one extremity of the optical fibre, several laser beams noted "ei" carrying different passenger services, feed the transmission line. They are characterized by a spot size radius, noted "w", and an offset radius in the plane xOy. At the opposite extremity, the energy corresponding to these different laser beams is concentrated in disks of radius "ri" where the different passenger services signals can be simply recovered. To efficiently manage MGDM multiplexing technique, the generation of certain number of optical codes and the choice of multi-mode fiber is necessary. This approach provides several distinct, independent propagation channels in the optical fiber. Associated with these different propagation channels, we can also encode the digital signals on each of these channels.

Different optical codes used

The principle of data encoding not only ensures secure transmission, but also allows several users or services to communicate on the same transmission medium.

There are several types of optical codes, including the 1-D, 2-D and 3-D codes. Among the 1-D codes there are OOC-Optical Orthogonal Code, PC-Prime Code and EPC-Extended Prime Code (which is derived from PC code).

In the following paragraphs, we explain how these codes are constructed and generated.

 OOC codes

OOC codes are composed of unipolar sequences characterized by four parameters (L, W, a h , c h ) where L is the length of the sequence, W is the weight of the code corresponding to the number of chips equal to "1", a h and c h are respectively the constraints for autocorrelation and cross-correlation.

The formula providing the available number of users is:

         ) 1 ( 1 W W L N ooc [2]
Table 3 provides Maximum autocorrelation is the weight of the code and the cross-correlation is between 0 and 1.

 PC codes

The generation of PC codes performed using Galois's formula GF (p) of a prime number p such as:

      ) mod ( . 1 , , 1 , 0 ) ( ), ( : , , , ) ( ) 1 ( 1 0 p j i s p p GF p GF j et i with s s s j S ij p i i i p i         (1)
The following formula is used:

       otherwise 0 ) ( , ) ( if 1 ) ( p GF j i jp j S n n C p i p i (2)
Using equations ( 1) and ( 2) we present in Table 4 an example of PC code. For N=5 users we obtain a code of length L = 25, with weight of code W = 5. In Fig. 4, for the PC code, maximum autocorrelation is the weight of the code but autocorrelation values are duplicated. The cross-correlation is situated between 0 and 1.

 2-D codes

The 2D codes enable the spreading of the data in both domains simultaneously, which allows to decrease the constraints on the length of the codes (L, F), on the number of available codes N as well as improving the performances of the system in term of Bit Error Rate (BER) for a given number of active users. Figure 5 presents an example of 2D system in both time and frequency domains. A family of 2D codes used simultaneously in the temporal and spectral domains is characterized by the parameters (L x F, W, ha, hc), where W is the weight of the matrices of code, ha and hc are the values of autocorrelation and cross-correlation defined by:

               1 0 1 0 * max L i F j j i j i a c c h                  1 0 1 0 * max L i F j j i j i c d c h   with: c and d are two matrices of a family of code,   1 , 0 and  j i j i d c ,    . 1 , 0 , 1 , 0     F j L i  represent the asynchronous gap between two codes,   1 , 0   F  et  is a sum modulo-F.
The objective is to generate families of codes which allow to obtain a large number of sequences with optimal properties of autocorrelation and cross-correlation (ha=hc=1) in order to limit the impact of Multiple Access Interference -IAM.

-Example of FFH code

The following figure 6 presents the position of chips and the evolution of autocorrelation and crosscorrelation functions of 2-D code FFH-Fast Frequency Hopping.

We have represented in Table 5 an example of FFH code. We obtain, for N=20 users, a code of length L = 25 and weight of code W = 5. The autocorrelation of FFH code is best because we obtain a single peak corresponding to the maximum of correlation. The values of cross-correlation are between 0 and 1. We conclude that 2D codes are quite interesting to use because they increase the number of available users.

FFH code N=p(p-1) L=

Simulations results

The following simulation results are obtained by using the VPI TM systems software. VPI TM Systems is a simulation software for optical transmission channel, which can be used to develop and support sophisticated design tools for photonic devices including network components.

It supports modeling several types of optical fiber by using several opto-electronic components. In our case, it takes into account the majority of the optical fiber parameters.

We study the evolution of the Bit Error Rate (BER) with respect to the emitted power for three typical wavelengths used for radio transmission over fiber (850 nm, 1300 nm and 1550 nm), by using fiber lengths ranging from 10 m to 3 km and different "spot size" of 3 µm and 8 µm.

In figure 7 we have an example of a simulated channel of optical transmission with wavelength of 850 nm and spot sizes 3 µm and 8 µm. It is better to use fiber length lower than1 km because the gain obtained is about 7 dB compared to optical fiber of 3 km. This type of application could be used inside railway stations.

The variation of spot size permits to define the number of laser source that could be sent according to the diameter of optical fiber. In practical it is possible to send simultaneously up to 3 lasers sources inside the same core of optical fiber and to manage up to 3 different canals. These permit transmission of much more information at the same time.

Conclusion and perspectives

We presented the physical characteristics of different types of optical fibers, namely SMF, MMF and micro-structured fibers. We develop the MGDM technique for the multiple users and we give some examples of optical codes necessary for this technique. We presented the results of our simulations obtained through VPI TM Systems software. These results lead us to conclude that the best results are obtained for spot size of 8 µm. With a spot size of 8 µm, up to 3 lasers sources may be sent. This shows the potential benefits of transmitting radio signals over fiber in a railway environment. In the future work we will include a third dimension of simulation for augmenting the capacity and number of users to be sent.

Figure 1 Fig. 1 :

 11 Figure 1 presents an example of a micro-structured optical fiber.

Fig. 2 :

 2 Fig. 2: Mode Group Diversity Multiplexing (MGDM) technique implementation.

3 :

 3 Fig.3: a) Position of chips, b) autocorrelation and c) cross-correlation of OOC code.

Table 4 :

 4 Fig.4: a) Position of chips, b) autocorrelation and c) cross-correlation of PC code.

Fig. 5

 5 Fig. 5: 2-D OCDMA, Time and Wavelength.

  Example generation of FFH code for 20 users.

  Fig.6: a) Position of chips, b) autocorrelation and c) cross-correlation of FFH code.

Fig. 7

 7 Fig. 7 Evolution of the BER vs. emitted power for spot size w= 3 μm and w= 8 μm, Wavelength = 850 nm.

Table 1 :

 1 Characteristics of various types of optical fibers.

	Type of optical fiber	Single-mode	Multi-mode
	Nature	Graded-index fiber	Step-index fiber	Graded-index fiber	Step-index fiber
	Diameter core / cladding	10 µm/125 µm	50-62,5 µm/125 µm	200 µm/300 µm
	Bandwidth	10 GHz/ km	100-1000 MHz/ km	25 MHz / km
	Wavelength /	1300 nm / 230.7 THz	850 nm / 352.9 THz
	Frequency	1550 nm / 193.5 THz	1300 nm / 230.7 THz

Air duct Silica core Air duct Silica core d 137µm



Air duct diameter: d ~ 1-2µm

Distance between 2 air duct:~ 1

-3µm Micro-structured fiber

	Types of optical fiber	Advantages	Disadvantages
	Single-mode	Very large bandwidth. No distortion.	Very high price
	Multi-mode with	Low price	High loss and distortion of the
	step-index	Ease of implementation	signal
	Multi-mode with graded-index	Reasonable bandwidth Good quality of transmission	Complexity of implementation
	Micro-structured	Large bandwidth Low distortion	Difficulties of manufacturing. Limited development.

Table 2 :

 2 Comparison of the various types of optical fiber: advantages and disadvantages.

Code sequence

Cross-correlation of 2D PC/PC code


Eduard Feireisl

Antonín Novotný

Multiple scales and singular limits for compressible rotating fluids with general initial data

We study the singular limit of a rotating compressible fluid described by a scaled barotropic Navier-Stokes system, where the Rossby number = ε, the Mach number = ε m , the Reynolds number = ε -α , and the Froude number = ε n are proportional to a small parameter ε → 0. The inviscid planar Euler system is identified as the limit problem. The proof is based on the application of the method of relative entropies and careful analysis of oscillatory integrals describing the propagation of Rossby-acoustic waves.

Introduction

The basic system of equations considered in this paper are the continuity equation and the momentum equation describing the time evolution of the mass density ̺ = ̺(t, x) and the (relative) velocity u = u(t, x) of a compressible, rotating fluid:

∂ t ̺ + div x (̺u) = 0, (1.1) ∂ t (̺u) + div x (̺u ⊗ u) + 1 ε ̺(f × u) + 1 ε 2m ∇ x p(̺) = ε α div x S(∇ x u) + 1 ε 2n ̺∇ x G, (1.2)
S(∇ x u) = µ ∇ x u + ∇ t x u -2 3 div x uI + ηdiv x uI, µ > 0, η ≥ 0.

(1.

3

)
The fluid is confined to an infinite slab Ω = R 2 × (0, 1), (1.4) where it satisfies the slip condition

u • n| ∂Ω = [S(∇ x u) • n] tan | ∂Ω = 0 (1.5)
imposed on the horizontal boundary.

The model (1.1 -1.5) may be viewed as a crude approximation (f -plane model) of the Earth atmosphere in a plane tangent to the Earth at a certain latitude, see Vallis [34, Chapter 2, Section 2.3]. Accordingly, the gravitational force is taken parallel to the vertical projection of the rotation axis:

f = [0, 0, 1], ∇ x G = [0, 0, -1],

The momentum equation (1.2) contains a small parameter ε related to different characteristic numbers resulting from the scale analysis: Rossby number = ε, Mach number = ε m , Reynolds number = ε -α , Froude number = ε n , see Klein [START_REF] Klein | Scale-dependent models for atmospheric flows[END_REF]. We consider the singular limit problem for ε ց 0 in the multiscale regime:

m 2 > n ≥ 1, α > 0 (1.6)
for the ill-prepared initial data

̺(0, •) = ̺ 0,ε = ̺ε + ε m ̺ (1) ε , u(0, •) = u 0 , (1.7)
where ̺ε is a solution to the static problem

∇ x p(̺ ε ) = ε 2(m-n) ̺ε ∇ x G in Ω. (1.8)
In particular, since n ≥ 1, the centrifugal force, parallel to the vertical axis, is dominated by gravitation (cf. Durran [START_REF] Durran | Is the Coriolis force really responsible for the inertial oscillation ?[END_REF]).

Formally, it is not difficult to identify the limit problem. Indeed fast rotation is expected to eliminate the vertical motion, the vanishing viscosity (high Reynolds number) makes the limit system inviscid (hyperbolic), while the low Mach number regime drives the fluid to incompressibility. The limit problem is therefore expected to be the incompressible Euler system for the planar velocity field

v = [v 1 , v 2 ], ∂ t v + div x (v ⊗ v) + ∇ x Π = 0, div x v = 0 in (0, T ) × R 2 .
(1.9)

Our main goal is to put these formal arguments on rigorous grounds. The phenomena discussed above have been investigated by many authors. The fact that highly rotating fluids become planar (two-dimensional), and, accordingly, fast rotation has a regularizing effect, was observed by Babin, Mahalov, and Nicolaenko [START_REF] Babin | Global regularity of 3D rotating Navier-Stokes equations for resonant domains[END_REF], [START_REF] Babin | 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity[END_REF], Bresch, Desjardins, and Gerard-Varet, [START_REF] Bresch | Rotating fluids in a cylinder[END_REF], Chemin et al. [START_REF] Chemin | Mathematical geophysics[END_REF], among others. The inviscid limit is a well studied and partially still open challenging problem, see Clopeau, Mikelic, Robert [START_REF] Clopeau | On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions[END_REF], Kato [START_REF] Kato | Remarks on the zero viscosity limit for nonstationary Navier-Stokes flows with boundary[END_REF], Masmoudi [START_REF] Masmoudi | The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary[END_REF], [START_REF] Masmoudi | Incompressible, inviscid limit of the compressible Navier-Stokes system[END_REF], [START_REF] Masmoudi | Examples of singular limits in hydrodynamics[END_REF], Sammartino and Caflisch [START_REF] Sammartino | Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space I, Existence for Euler and Prandtl equations[END_REF], [START_REF] Sammartino | Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space II, Construction of the Navier-Stokes solution[END_REF], Swann [START_REF] Swann | The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R 3[END_REF], Temam and Wang [START_REF] Temam | On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity[END_REF], [START_REF] Temam | Boundary layers associated with incompressible NavierStokes equations: The noncharacteristic boundary case[END_REF], to name only a few. The low Mach number limits were proposed in the pioneering papers by Ebin [START_REF] Ebin | The motion of slightly compressible fluids viewed as a motion with strong constraining force[END_REF], and Klainerman and Majda [START_REF] Klainerman | Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids[END_REF], and later reexamined in the context of weak solutions by Lions and Masmoudi [START_REF] Lions | Incompressible limit for a viscous compressible fluid[END_REF], see also the survey by Danchin [6], Gallagher [START_REF] Gallagher | Résultats récents sur la limite incompressible[END_REF], and Schochet [START_REF] Schochet | The mathematical theory of low Mach number flows. M2ANMath[END_REF]. To the best of our knowledge, the simultaneous effects of these three mechanisms has not yet been treated in the literature.

The present paper may be viewed as complementary to our previous study [START_REF] Feireisl | Scale interactions in compressible rotating fluids[END_REF], where we examined the "single-scale" limit corresponding to n = 0, m = 1, α > 0.

(1.10)

Although this problem looks formally very similar to the present setting, the methods employed as well as the limit system are different, cf. [START_REF] Feireisl | Scale interactions in compressible rotating fluids[END_REF]. The central issue to be discussed is the behavior of the oscillatory part of solutions to the scaled system. These are described in the case (1.10) by a system of linear equations with constant coefficients, while the situation (1.6) gives rise to a problem with coefficients depending on the scaling parameter ε.

Similarly to [START_REF] Feireisl | Scale interactions in compressible rotating fluids[END_REF], our approach is based on the concept of finite energy weak solutions satisfying the relative entropy inequality identified in [START_REF] Feireisl | Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids[END_REF], see Section 2. After collecting the necessary preliminary material, we state the main result in Section 3. Section 4 reviews the basic estimates, independent of ε ց 0, for solutions of the family of scaled problems. Section 5 is the heart of the paper. We establish decay estimates on the oscillatory part of solutions to the scaled problems by means of careful analysis of certain oscillatory integrals. Here, inspired by the analysis of Guo, Peng, and Wang [START_REF] Guo | Decay estimates for a class of wave equations[END_REF], we make use of frequency cut-off operators and estimates based on van Corput's lemma. The proof of convergence towards the limit system is completed in Section 6.

Preliminaries, weak solutions, relative entropy inequality

We suppose that the pressure p ∈ C[0, ∞) ∩ C 3 (0, ∞) is a given function of the density enjoying the following property

p(0) = 0, p ′ (̺) > 0 for all ̺ > 0, lim ̺→∞ p ′ (̺) ̺ γ-1 , γ > 3 2
.

(2.1)

In addition, without loss of generality, we assume that p is "normalized" setting

p ′ (1) = 1. (2.2)
Finally, we introduce the pressure potential H,

H(̺) = ̺ ̺ 1 p(z) z 2 dz, (2.3)
noting that

H ′′ (̺) = p ′ (̺) ̺ , H ′′ (1) = 1.

Static solutions

As already mentioned above, the static solutions ̺ε solve the problem (1.8), specifically, we take

H ′ (̺ ε ) = ε 2(m-n) G + H ′ (1)
, where G(x) = -x 3 ;

(2.4) whence ̺ε (x) = ̺ε (x 3 ), sup

x 3 ∈[0,1] |̺ ε (x 3 ) -1| ≤ cε 2(m-n) . (2.5)
As indicated by our choice of the initial data (1.7), the solutions of the evolutionary problem (1.1 -1.3), (1.5), (1.7) satisfy far field conditions in the form

̺ → ̺ε , u → 0 as |x| → ∞.
(2.6)

Finite energy weak solutions

We say that [̺, u] is a finite energy weak solution of the problem (1.1 -1.3), (1.5), (1.7), (2.6) on the space-time cylinder (0, T ) × Ω if the following holds:

• Regularity. The functions ̺, u belong to the class

̺ ≥ 0, (̺ -̺ε ) ∈ L ∞ (0, T ; L 2 + L γ (Ω)), u ∈ L 2 (0, T ; W 1,2 (Ω; R 3 )), u • n = u 3 | ∂Ω = 0. (2.7)
• Equations. The equation of continuity (1.1) and the momentum equation (1.2) are replaced by integral identities

T 0 Ω (̺∂ t ϕ + ̺u • ∇ x ϕ) dx dt = - Ω ̺ 0,ε ϕ(0, •) dx (2.8)
for any ϕ ∈ C ∞ c ([0, T ) × Ω), and

T 0 Ω ̺ ε u ε • ∂ t ϕ + (̺u ⊗ u) : ∇ x ϕ - 1 ε ̺(f × u)•ϕ + 1 ε 2m p(̺)div x ϕ dx dt (2.9) = T 0 Ω ε α S(∇ x u) : ∇ x ϕ - 1 ε 2n ̺∇ x G • ϕ dx dt - Ω ̺ 0,ε u 0,ε • ϕ(0, •) dx for any ϕ ∈ C ∞ c ([0, T ) × Ω; R 3 ), ϕ • n| ∂Ω = 0.
• Energy. The energy inequality

Ω 1 2 ̺|u| 2 + 1 ε 2m (H(̺) -H ′ (̺ ε )(̺ -̺ε ) -H(̺ ε )) (τ, •) dx + ε α τ 0 Ω S(∇ x u) : ∇ x u dx dt (2.10) ≤ Ω 1 2 ̺ 0,ε |u 0,ε | 2 + 1 ε 2m (H(̺ 0,ε ) -H ′ (̺ ε )(̺ 0,ε -̺ε ) -H(̺ ε )) dx holds for a.a. τ ∈ (0, T ).
Note that the existence theory in the class of finite energy weak solutions was developed by Lions [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF] and later extended in [START_REF] Feireisl | On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids[END_REF] to the sofar "critical" adiabatic exponent γ > 3 2 .

Relative entropy

For future analysis, it is convenient to replace the energy inequality (2.10) by the relative entropy inequality containing more transparent piece of information on the asymptotic behavior of solutions for ε → 0. To this end, we introduce the relative entropy functional

E ε ̺, u r, U = Ω 1 2 ̺|u -U| 2 + 1 ε 2m H(̺) -H ′ (r)(̺ -r) -H(r) dx, (2.11)
cf. [START_REF] Feireisl | Relative entropies, suitable weak solutions, and weakstrong uniqueness for the compressible Navier-Stokes system[END_REF], [START_REF] Feireisl | Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids[END_REF], Germain [START_REF] Germain | Weak-strong uniqueness for the isentropic compressible Navier-Stokes system[END_REF]. It can be shown, see [START_REF] Feireisl | Relative entropies, suitable weak solutions, and weakstrong uniqueness for the compressible Navier-Stokes system[END_REF], that any finite energy weak solution [̺, u] specified in Section 2.2 satisfies the relative entropy inequality:

E ε ̺, u r, U (τ ) + ε α τ 0 Ω S(∇ x u) -S(∇ x U) : ∇ x u -∇ x U dx dt ≤ (2.
12)

E ε ̺ 0,ε , u 0,ε r(0, •), U(0, •) + τ 0 Ω ̺ (∂ t U + u • ∇ x U) • (U -u) dx dt +ε α τ 0 Ω S(∇ x U) : ∇ x (U -u) dx dt + 1 ε τ 0 Ω ̺(f × u) • (U -u) dx dt + 1 ε 2m τ 0 Ω (r -̺)∂ t H ′ (r) + ∇ x H ′ (r) -H ′ (̺ ε ) • (rU -̺u) dx dt - 1 ε 2m τ 0 Ω div x U p(̺) -p(r) dx dt- 1 ε 2n τ 0 Ω (̺ -r)∇ x G • U dx dt
for all sufficiently smooth "test functions" r, U satisfying

U • n| ∂Ω = 0, r > 0, U, (r -̺ε ) compactly supported in Ω.
Note the assumptions concerning the decay and regularity can be relaxed to basically any couple [r, U] for which (2.12) makes sense via the standard density argument, see [START_REF] Feireisl | Relative entropies, suitable weak solutions, and weakstrong uniqueness for the compressible Navier-Stokes system[END_REF].

3 Main result . Similarly, we use the symbols ∇ h , div h to denote the differential operators acting on the horizontal variables only. Finally, the symbol H denotes the Helmholtz projection onto the space of solenoidal functions in Ω, while H h is the Helmholtz projection in R 2 . Let v 0 ∈ W k,2 (R 2 ; R 2 ), k ≥ 3, divv 0 = 0, be given. It is well known (see for instance Kato and Lai [START_REF] Kato | Nonlinear evolution equations and the Euler flow[END_REF]) that the Euler system (1.9) supplemented with the initial data

v(0) = v 0 admits a regular solution (v, Π), unique in the class v ∈ C([0, T ]; W k,2 (R 2 ; R 2 )), ∂ t v ∈ C([0, T ]; W k-1,2 (R 2 ; R 2 )), Π ∈ C([0, T ]; W k,2 (R 2 )).
We are ready to formulate our main result.

Theorem 3.1 Let the pressure p = p(̺) satisfy the hypotheses (2.1), (2.2). Suppose that the exponents α, m, n are given such that

α > 0, m 2 > n ≥ 1.
Let the initial data [̺ 0,ε , u 0,ε ] be given by (1.7), where the stationary states ̺ε satisfy (2.4),

̺ (1) 0,ε L 2 ∩L ∞ (Ω) ≤ c, ̺ (1)
0,ε → ̺ (1) 0 in L 2 (Ω), u 0,ε → u 0 in L 2 (Ω; R 3 ), (3.1)
with ̺

(1)

0 ∈ W k-1,2 (Ω), u 0 ∈ W k,2 (Ω; R 3 ) for a certain k ≥ 3. (3.2) Let [̺ ε , u ε ] be a finite energy weak solution of the problem (1.1 -1.3), (1.5), (1.7), (2.6) in the space-time cylinder (0, T ) × Ω.
Then ess sup

t∈(0,T ) ̺ ε (t, •) -̺ε (L 2 +L γ )(Ω) ≤ ε m c (3.3) √ ̺ ε u ε → v      weakly-(*) in L ∞ (0, T ; L 2 (Ω; R 3 )), strongly in L 1 loc ((0, T ) × Ω; R 3 ), (3.4
)

where v = [v h , 0]
is the unique solution of the Euler system (1.9), with the initial data

v(0, •) = H h 1 0 u 0 (x h , x 3 ) dx 3 .
The rest of the paper is devoted to the proof of Theorem 3.1.

Uniform bounds

We start with the nowadays standard estimates that follow directly from the energy inequality (2.10). These are conveniently formulated in terms of the essential and residual components of a measurable function h,

h = h ess + h res , h ess = χ(̺ ε )h, χ ∈ C ∞ c (0, ∞), 0 ≤ χ ≤ 1, χ = 1 in an open interval contaning 1, h res = (1 -χ(̺ ε ))h.
Since the initial data are given by (1.7), where the functions ̺

(1) 0,ε , u 0,ε satisfy the hypotheses (3.1), (3.2), the integral on the right-hand side of (2.2) is bounded uniformly for ε ց 0. As the stationary states ̺ε are chosen to satisfy (2.5), we deduce the following bounds independent of ε:

ess sup t∈(0,T ) √ ̺ ε u ε L 2 (Ω;R 3 ) ≤ c, (4.1)
ess sup

t∈(0,T ) ̺ ε -̺ε ε m ess L 2 (Ω) ≤ c, (4.2)
ess sup

t∈(0,T ) [̺ ε ] res γ L γ (Ω) + ess sup t∈(0,T ) [1] res L 1 (Ω) ≤ ε 2m c, (4.3)
and

ε α T 0 Ω ∇ x u ε + ∇ x u t ε - 2 3 div x u ε I 2 dx dt ≤ c, (4.4)
cf. [9, Section 2]. Obviously, the estimates (4.1), (4.2) yield (3.3), which, combined with (2.5) gives rise to

̺ ε → 1 in L ∞ (0, T ; L q loc (Ω)) for any 1 ≤ q < γ. (4.5)
Moreover, combining (4.5) with (4.1 -4.3) we obtain

√ ̺ ε u ε → u weakly-(*) in L ∞ (0, T ; L 2 (Ω; R 3 )), (4.6)
and

̺ ε u ε → u weakly-(*) in L ∞ (0, T ; L 2 + L 2γ/(γ+1) (Ω; R 3 )), (4.7)
passing to suitable subsequences as the case may be. Finally, we may let ε → 0 in the equation of continuity to deduce that

div x u = 0, u • n| ∂Ω = 0 in the sense of distributions in (0, T ) × Ω. (4.8)

Decay estimates and oscillatory integrals

With our convention (2.2), the equation describing the oscillatory part of solutions reads

ε m ∂ t s + div x V = 0, (5.1)
ε m ∂ t V + ωf × V + ∇ x s = 0, ω = ε m-1 , V • n| ∂Ω = 0, (5.2)
cf. [START_REF] Feireisl | Scale interactions in compressible rotating fluids[END_REF]. Re-scaling in the time we arrive at

∂ t s + div x V = 0, (5.3)
∂ t V + ωb × V + ∇ x s = 0, V • n| ∂Ω = 0, (5.4)
with the operator

B(ω) : s V → div x V ωf × V + ∇ x s .
The operator B is skew symmetric in the space L 2 (Ω) × L 2 (Ω; R 3 ), with the domain of definition

D[B(ω)] = [r, V] r ∈ W 1,2 (Ω), V ∈ L 2 (Ω; R 3 ), div x V ∈ L 2 (Ω), V • n = V 3 | ∂Ω = 0 ,
and the kernel

N (B(ω)) = [q, v] q = q(x h ), q ∈ W 1,2 (R 2 ), v = [v h (x h ), 0], div h v h = 0, ωf × v + ∇ x q = 0 .
Let P(ω) denote the projection

P(ω) : L 2 (Ω) × L 2 (Ω; R 3 ) → N (B(ω)).
Exactly as in [11, Section 4.1.1] we can show that

P(ω)[r, U] = [q, v] if -∆ h q+ω 2 q = ω 1 0 curl h U h dx 3 +ω 2 1 0 r dx 3 in R 2 , v = [v 1 , v 2 , 0], ωv 1 = -∂ x 2 q, ωv 2 = ∂ x 1 q. (5.5)

Spectral analysis

Thanks to our special choice of the geometry of the spatial domain Ω, we may reformulate the problem (5.3), (5.4) in terms of the Fourier variables. To this end, we observe, exactly as in [START_REF] Feireisl | Scale interactions in compressible rotating fluids[END_REF], that the underlying spatial domain Ω may be equivalently replaced by

Ω = R 2 × T 1 ,
where

T 1 = [-1, 1] {-1,1}
is a "flat" sphere, and where s, V h were extended as even functions of the vertical variable x 3 , while V 3 was extended as odd in x 3 . In other words, all quantities are understood as 2-periodic functions in the vertical x 3 variable. Accordingly, for each function g ∈ L 2 ( Ω), we introduce its Fourier representation

ĝ(ξ, k), ξ = [ξ 1 , ξ 2 ] ∈ R 2 , k ∈ Z, where ĝ(ξ, k) = 1 √ 2 1 -1 R 2 exp (-iξ • x h ) g(x h , x 3 ) exp (-ikx 3 ) dx h dx 3 .
We have

g(x h , x 3 ) = k∈Z F -1 ξ→x h [ĝ(ξ, k)] exp (ikx 3 ) ,
where the symbol F x h →ξ denotes the standard Fourier transform on R 2 . Thus the problem (5.3), (5.4) can be written in the form

d dt       ŝ(t, ξ, k) V1 (t, ξ, k) V2 (t, ξ, k) V3 (t, ξ, k)       + i      0 ξ 1 ξ 2 k ξ 1 0 ωi 0 ξ 2 -ωi 0 0 k 0 0 0            ŝ(t, ξ, k) V1 (t, ξ, k) V2 (t, ξ, k) V3 (t, ξ, k)       = 0, ŝ(0, ξ, k) V(0, ξ, k) = ŝ0 (ξ, k) V0 (ξ, k) ; (5.6)
with the Hermitian matrix

A(ξ, k, ω) =      0 ξ 1 ξ 2 k ξ 1 0 ωi 0 ξ 2 -ωi 0 0 k 0 0 0      .
Of course, solutions of (5.6) depend also on the parameter ω = ε m-1 .

Spectral properties of the matrix A

After a bit tedious but straightforward manipulation (see [START_REF] Feireisl | A singular limit for compressible rotating fluids[END_REF]), we can check that A(ξ, k, ω) possesses four eigenvalues

λ 1 (|ξ| 2 , k, ω) = ω 2 +|ξ| 2 +k 2 + √ (ω 2 +|ξ| 2 +k 2 ) 2 -4ω 2 k 2 2 1/2 , λ 2 (|ξ| 2 , k, ω) = -λ 1 (|ξ| 2 , k, ω), λ 3 (|ξ| 2 , k, ω) = ω 2 +|ξ| 2 +k 2 - √ (ω 2 +|ξ| 2 +k 2 ) 2 -4ω 2 k 2 2 1/2 , λ 4 (|ξ| 2 , k, ω) = -λ 3 (|ξ| 2 , k, ω).
(5.7)

Note that λ 3 (|ξ| 2 , 0, ω) = λ 4 (|ξ| 2 , 0, ω) = 0 are the zero eigenvalues corresponding to the non-void kernel of the matrix Athe Fourier image of the null-space of the operator B(ω), see [START_REF] Feireisl | A singular limit for compressible rotating fluids[END_REF]. As for the eigenvectors [q, v 1 , v 2 , v 3 ], we have

ξ 1 v 1 + ξ 2 v 2 + kv 3 = λq, ξ 1 q + iωv 2 = λv 1 , ξ 2 q -iωv 1 = λv 2 , kq = λv 3 , (5.8)
from which we immediately deduce

v 1 = µ(λξ 1 + iωξ 2 ), v 2 = µ(λξ 2 -iωξ 1 ), µλ|ξ| 2 = λq -kv 3 , , kq = λv 3 , (5.9)
where µ is a free parameter that is fixed to normalize the norm of the eigenvector to be one.

Eigenvectors

We denote by E = E(ξ, k, ω) = [q, v 1 , v 2 , v 3 ] the normalized eigenvectors. Our goal is to show that diagonalizing matrices Q, Q T , formed by the eigenvectors, are L p -multipliers in the ξ variable restricted to compact subsets of R 2 \ {0}. This amounts to showing that sup ω∈(0,1)

max 0<a≤|ξ|≤b<∞ ∇ A ξ E j (ξ, k, ω) ≤ c = c(A, a, b, k), j = 1, 2, 3, 4, k ∈ Z (5.10) with any multi-index A = (A 1 , A 2 ).
We distinguish two cases:

Case λ = 0:

In this case, we necessarily have k = 0 and λ = ±λ 3 , and the orthonormal basis of eigenvectors can be taken in the form

E 1 = µ -iω, -ξ 2 , ξ 1 , 0 , µ = |ξ| 2 + ω 2 -1/2 , E 2 = [0, 0, 0, 1].
Clearly (5.10) holds.

Case λ = 0 :

We find that

v 1 = µ(λξ 1 + iωξ 2 ), v 2 = µ(λξ 2 -iωξ 1 ), λ 2 -k 2 λ 2 q = µ(ξ 2 1 + ξ 2 2 ), v 3 = k λ q.
Thus, the corresponding normalized eigenvector has the form

E = µ λ 2 |ξ| 2 λ 2 -k 2 , λξ 1 + iωξ 2 , λξ 2 -iωξ 1 , kλ|ξ| 2 λ 2 -k 2 , with µ = λ 4 + |k| 2 λ 2 (λ 2 -k 2 ) 2 |ξ| 4 + (λ 2 + ω 2 )|ξ| 2 -1/2
.

We consider first the case λ = ±λ 1 . We check that

λ 2 1 ≥ |ξ| 2 /2, (ω 2 + |ξ| 2 + k 2 ) 2 -4ω 2 k 2 ≥ |ξ| 4 , (5.11)
λ 2 1 -k 2 = ω 2 -k 2 + |ξ| 2 + (ω 2 -k 2 ) 2 + |ξ| 2 (|ξ| 2 + 2(ω 2 + k 2 )) 2 ≥ |ξ| 2 2 .
Consequently (5.10) is satisfied. Finally, if 0 = λ = ±λ 3 , we note that

λ 3 (|ξ| 2 , k, ω) = ω|k| 1 λ 1 (|ξ| 2 , k, ω) , λ 2 3 -k 2 = k 2 (ω 2 -λ 2 1 ) λ 2 1 ,
(5.12)

λ 2 1 -ω 2 = -ω 2 + |ξ| 2 + |k| 2 + (ω 2 + |ξ| 2 + k 2 ) 2 -4k 2 ω 2 2 ≥ |ξ| 2 /2.
Using identity (5.12) 1 , we write E in terms of λ 1 and verify (5.10) employing this explicit formula and estimates (5.11), (5.12).

Frequency cut-off

As we shall see below, it will be convenient to approximate the initial data for the problem (5.3), (5.4) by a frequency truncation represented by a function ψ ∈ C ∞ c (0, ∞). Accordingly, solutions of the problem (5.3), (5.4) will be, in a certain way, composed of the quantities

Z(t, x h , k, ω) = F -1
ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) ĥ(ξ) , j = 1, 2, 3, 4, k ∈ Z, ω ∈ (0, 1), (5.13) where ĥ stands for the Fourier transform of the "initial data". Our goal will be to derive suitable dispersive estimates for the mapping h → Z.

We start with the L 1 -L ∞ decay estimates. To this end, write

Z(t, •, k, ω) L ∞ (R 2 ) ≤ F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) L ∞ (R 2 ) h L 1 (R 2 ) . (5.14)
Consequently,

F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) (x h ) (5.15) = π √ 2 2π 0 ∞ 0 exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) exp i|ξ||x h | sin θ |ξ| d|ξ|dθ = π √ 2 ∞ 0 exp ± iλ j (r 2 , k, ω)t ψ(r)rJ 0 (r|x h |) dr,
where the symbols J m , m = 0, 1, . . . denote the Bessel functions, cf. Guo, Peng, and Wang [START_REF] Guo | Decay estimates for a class of wave equations[END_REF].

Finally, performing a simple change of variables, we get

F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) (x h ) = π √ 2 2 ∞ 0 exp ± iλ j (z, k, ω)t ψ( √ z)J 0 ( √ z|x h |) dz.
(5.16)

Decay estimates

Supposing λ j = 0 we derive the desired decay estimates. To this end, we use van Corput's lemma, see Stein [30, Chapter 8.1.2, Proposition 2 and Corollary]:

Lemma 5.1 Let Λ = Λ(z) be a smooth function away from the origin,

∂ z Λ(z) monotone, |∂ z Λ(z)| ≥ Λ 0 > 0 for all z ∈ [a, b], 0 < a < b < ∞. Let Φ be a smooth function on [a, b]. Then b a exp (iΛ(z)t) Φ(z) dz ≤ c 1 tΛ 0 |Φ(b)| + b a |∂ z Φ(z)| dz ,
where c is an absolute constant independent of Λ and Φ.

Going back to the oscillatory integral (5.16), we distinguish two cases.

Case |x h | > t β

Using the decay properties of J 0 and the fact that ψ is compactly supported away from zero, we get

F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) (x h ) ≤ c(ψ)t -β/2 whenever |x h | > t β .
(5.17)

5.3.2 Case |x h | ≤ t β
The idea is to use van Corput's lemma. Let [a, b] be a closed interval, a > 0, containing the support of ψ( √ z). In accordance with the hypotheses of Lemma 5.1, we have to verify that

• |∂ z λ j (z, k, ω)| ≥ Λ 0 (a, b, ω, k) > 0 for all z ∈ [a, b], ω ∈ (0, 1); • ∂ z Λ j is monotone on [a, b].
If the two conditions are satisfied, we get

F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) (x h ) ≤ c(ψ) 1 Λ 0 (a, b, k, ω)t 1 + t β/2 for |x h | ≤ t β , (5.18)
where again we have used the properties of the Bessel functions, namely,

J ′ 0 (z) = -J 1 (z).
Now, our goal is to verify the hypotheses of Lemma 5.1. We have

∂ z λ 1 (z, k, ω) = 1 2 1 λ 1 (z, k, ω) β 1 (z, k, ω), β 1 (z, k, ω) = 1 2   1 + ω 2 + z + k 2 (ω 2 + z + k 2 ) 2 -4ω 2 k 2   ,
and, similarly,

∂ z λ 3 (z, k, ω) = 1 2 1 λ 3 (z, k, ω) β 3 (z, k, ω), β 3 (z, k, ω) = 1 2   1 - ω 2 + z + k 2 (ω 2 + z + k 2 ) 2 -4ω 2 k 2   , k = 0.
Furthermore,

∂ z β 1 (z, k, ω) = - 2ω 2 k 2 [(ω 2 + z + k 2 ) 2 -4ω 2 k 2 ] 3/2 , ∂ z β 3 (z, k, ω) = 2ω 2 k 2 [(ω 2 + z + k 2 ) 2 -4ω 2 k 2 ] 3/2 .
Summing up the previous relations, we conclude that

∂ z λ 1 (z, k, ω) is a decreasing functions of z, while ∂ z λ 3 (z, k, ω
) is an increasing functions of z for k = 0.

Consequently, we deduce that

∂ z λ 1 (z, k, ω) ≥ ∂ z Λ 1 (b, k, ω) ≥ Λ(ψ, k) for z ∈ [a, b], ω ∈ (0, 1). (5.19)
Finally, we have

∂ z λ 3 (z, k, ω) < 0, |∂ z λ 3 (z, k, ω)| ≥ |∂ z λ 3 (b, k, ω)|,
where

|∂ z λ 3 (b, k, ω)| ≥ 1 2   ω 2 + b + k 2 (ω 2 + b + k 2 ) 2 -4ω 2 k 2 -1     ω 2 + b + k 2 -(ω 2 + b + k 2 ) 2 -4ω 2 k 2 2   -1/2 = √ 2ω|k| (ω 2 + b + k 2 ) 2 -4ω 2 k 2 -1/2 ω 2 + b + k 2 + (ω 2 + b + k 2 ) 2 -4ω 2 k 2 -1/2
≥ c(ψ, k)ω for k = 0, ω ∈ (0, 1).

Thus, reviewing (5.17), (5.18) we may infer that

F -1 ξ→x h exp ± iλ j (|ξ| 2 , k, ω)t ψ(|ξ|) L ∞ (R 2 )
≤ c(ψ, k) max 1 ωt 1-β/2 ; 1 t β/2 , t > 0, (5.20) as soon as λ j = 0, which gives rise to the decay estimates

Z(t, •, k, ω) L ∞ (R 2 ) ≤ c(ψ, k) max 1 ωt 1-β/2 ; 1 t β/2 h L 1 (R 2 ) .
(5.21)

Next, seeing that the mapping h → exp iλ j (ξ, k, ω)t h is an isometry on L 2 (R 2 ), we deduce

Z(t, •, k, ω) L 2 (R 2 ) ≤ c h L 2 (R 2 ) (5.22)
Finally, interpolating (5.21) and (5.22), we obtain the

L p -L q estimates Z(t, •, k, ω) L p (R 2 ) ≤ c(ψ, p, k) max 1 ωt 1-β/2 ; 1 t β/2 1-2 p h L p ′ (R 2 ) for p ≥ 2, 1 p + 1 p ′ = 1, λ j = 0. (5.23)
Keeping in mind that ω scales like ε m-1 while the time t is proportional to ε -m we observe that taking 0 < β < 2 m yields the effective decay of Z ε = Z(t/ε m , k, ω) on any compact subinterval of (0, T ]. In particular, the optimal choice β = 1/m gives rise to

Z t ε m , •, k, ω L p (R 2 ) ≤ c ε 1 2 -1 p max 1 t 1-1/2m ; 1 t 1/2m 1-2 p h L p ′ (R 2 ) , p ≥ 2, λ j = 0, t ∈ (0, T ].
(5.24)

Convergence

In this final part, we complete the proof of Theorem 3.1. The basic idea is to use the relative entropy inequality (2.12) for a suitable choice of test functions r and U.

Initial data decomposition

We start be introducing suitable smoothing operators imposed on the initial data. Taking a family of smooth functions

ψ δ ∈ C ∞ c (0, ∞), 0 ≤ ψ δ ≤ 1, ψ δ ր 1 as δ → 0, and φ δ = φ δ (x h ) ∈ C ∞ c (R 2 ), 0 ≤ φ δ ≤ 1, φ δ ր 1 as δ → 0, we introduce ̺ (1) 0 δ (x h , x 3 ) = 1 √ 2 |k|≤1/δ F -1 ξ→x h ψ δ (|ξ|) ̺ (1) 0 φ δ (ξ, k) exp (-ikx 3 ) , (6.1)
and, similarly,

[u 0,j ] δ (x h , x 3 ) = 1 √ 2 |k|≤1/δ F -1 ξ→x h ψ δ (|ξ|) (u 0,j φ δ )(ξ, k) exp (-ikx 3 ) , j = 1, 2, 3. (6.2)
Now, we write the initial data in the form

̺ (1) 0 δ = s 0,ε,δ + q 0,ε,δ , where -∆ h q 0,ε,δ + ω 2 q 0,δ = ω 1 0 curl h [[u 0 ] h ] δ dx 3 + ω 2 1 0 ̺ (1) 0 δ dx 3 (6.3) [u 0 ] δ = V 0,ε,δ + v 0,ε,δ , with ω[v 0,ε,δ ] 1 = -∂ x 2 q 0,ε,δ , ω[v 0,ε,δ ] 2 = ∂ x 1 q 0,ε,δ . (6.4)
Finally, we choose the functions r, U in the relative entropy inequality as

r = r ε,δ = ̺ε + ε m (q ε,δ + s ε,δ ), U = U ε,δ = v ε,δ + V ε,δ , (6.5)
where [s ε,δ , V ε,δ ] is the unique solution of the acoustic-Rossby system (5.1), (5.2), emanating from the initial data

s ε,δ (0, •) = s 0,ε,δ , V ε,δ (0, •) = V 0,ε,δ , while the functions q ε,δ , v ε,δ are interrelated through ωf × v ε,δ + ∇ x q ε,δ = 0, (6.6)
where q ε,δ is the unique solution of the problem

∂ t ∆ h q ε,δ -ω 2 q ε,δ + 1 ω ∇ ⊥ h q ε,δ • ∇ x ∆ h q ε,δ -ω 2 q ε,δ = 0, q ε,δ (0, •) = q 0,ε,δ . (6.7)

Decay of the oscillatory component

First we claim that, in view of the dispersive estimates (5.24) (with 0 < β < 2/m), and the properties of the eigenvectors of the matrix A, discussed in detail in Section 5.1.2, we get

s ε,δ → 0 in L p (0, T ; W l,q (Ω)), V ε,δ → 0 in L p (0, T ; W l,q (Ω)) as ε → 0 (6.8)
for any fixed δ > 0, 1 ≤ p < ∞, 2 < q ≤ ∞ and l = 0, 1, . . . We note that

   s ε,δ (t, x h , k) V ε,δ (t, x h , k    = F -1 ξ→x h                Q T (ξ, ω, k)                exp iλ 1 (ξ, ω, k) t ε m , 0, 0, 0 0, exp iλ 2 (ξ, ω, k) t ε m , 0, 0 0, 0, exp iλ 3 (ξ, ω, k) t ε m , 0 0, 0, 0, exp iλ 4 (ξ, ω, k) t ε m                Q(ξ, ω, k)ψ(|ξ|) ĥ0 (ξ)                = F -1 ξ→x h                ψ(|ξ|)Q T (ξ, ω, k)                exp iλ 1 (ξ, ω, k) t ε m , 0, 0, 0 0, exp iλ 2 (ξ, ω, k) t ε m , 0, 0 0, 0, exp iλ 3 (ξ, ω, k) t ε m , 0 0, 0, 0, exp iλ 4 (ξ, ω, k) t ε m                ψ(|ξ|)Q(ξ, ω, k)ψ(|ξ|) ĥ0 (ξ)                , ω = ε m-1 ,
where ψ ∈ C ∞ c (0, ∞) has been chosen so that ψ ψ = ψ. In accordance with Section 5.1.2, the quantities ψQ, ψQ T are L p -Fourier multipliers with norm independent of ω. Thus the desired decay estimates (6.8) follow from (5.24).

Convergence of the non-oscillatory component

We introduce the scaled function qε,δ = q ε,δ /ω and observe that

∂ t ∆ h qε,δ -ω 2 qε,δ + ∇ ⊥ h qε,δ • ∇ h ∆ h qε,δ -ω 2 qε,δ , q ε,δ (0) = q 0,ε,δ , (6.9)
∆ h q0,ε,δ -ω 2 q0,δ = 1 0 curl h [[u 0 ] h ] δ dx 3 + ω 1 0 ̺ (1) 0 δ dx 3 .
Since the initial data are regular, we may use the result of Oliver [START_REF] Oliver | Classical solutions for a generalized Euler equation in two dimensions[END_REF]Theorem 3] to deduce that

{∆q ε,δ + ε 2 qε,δ } ε>0 is bounded in C r ([0, T ]; W l,2 (Ω)), r ≥ 0, l = 0, 1, . . . (6.10)
Moreover, as ∆q ε,δ satisfies that transport equation (6.9) with the initial data in L p (R 2 ), we get

{∆q ε,δ + ω 2 qε,δ } ε>0 is bounded in L ∞ ([0, T ]; L p (Ω)) for any 1 < p < ∞. (6.11)
Next, we recall the "energy estimates" that can be obtained multiplying (6.9) on qε,δ and integrating by parts:

R 2 |∇ x qε,δ | 2 + ω 2 |q ε,δ | 2 (τ, •) dx = R 2 |∇ x q0,ε,δ | 2 + ω 2 |q 0,ε,δ | 2 dx. (6.12) Note that R 2 ∇ ⊥ h qε,δ • ∇ h ∆ h qε,δ qε,δ dx = - R 2 ∇ ⊥ h qε,δ • ∇ h qε,δ ∆ h qε,δ dx = 0. Since v ε,δ = ∇ ⊥ h qε,δ , we get {v ε,δ } ε>0 is bounded in C r ([0, T ]; W l,2 (R 2 )), r ≥ 0, l = 0, 1, . . . (6.13)
Finally, we compute

∂ t qε,δ = (∆ h -ω 2 ) -1 [∇ h (v ε,δ curlv ε,δ )] ,
where, furthermore,

vcurl h v = v 1 (∂ x 1 v 2 -∂ x 2 v 1 ) ; v 2 (∂ x 1 v 2 -∂ x 2 v 1 ) , with v 1 ∂ x 1 v 2 = ∂ x 1 (v 1 v 2 ) + 1 2 ∂ x 2 v 2 2 , v 2 ∂ x 2 v 1 = ∂ x 2 (v 1 v 2 ) + 1 2 ∂ x 1 v 2 1 .
Consequently, we may infer that {∂ t qε,δ } ε>0 is bounded in C r ([0, T ]; W l,q (R 2 )), r ≥ 0, q > 1, l = 0, 1, . . . (6.14) All the above estimates may depend on δ but are uniform with respect to ε ց 0.

In view of the above estimates, it is easy to pass to the limit for ε → 0 in order to get

v ε,δ → v δ , ∂ t v ε,δ → ∂ t v δ weakly-(*) in L ∞ (0, T ; W l,2 (R 2 )), l = 0, 1, . . . , (6.15)
where

v δ , ∂ t v δ ∈ C([0, T ]; W l,2 (R 2 )), l = 0, 1, 2, . . . . (6.16)
We have, in particular,

v ε,δ → v δ in L q (0, T ; L q loc (Ω)), 1 ≤ q < ∞;
whence, by virtue of (6.9),

∂ t curl h v δ + v δ • ∇ h curl h v δ = 0. Seeing that curl h div h (v δ ⊗ v δ ) = curl h (v δ • ∇ h v δ ) = v δ • ∇ h curl h v δ ,
we deduce existence of Π δ ∈ C([0, T ]; W l,2 (R 2 )), l = 0, 1, 2, . . . , (6.17) where the couple (v δ , Π δ ) is the unique solution of the Euler system (1.9), emanating from the initial data

v δ (0, •) = 1 0 H h [[u 0 ] δ ] dx 3 . (6.18)

Relative entropy inequality

We return to the relative entropy inequality (2.12), where ̺ = ̺ ε , u = u ε and the test functions r and U are given by (6.5). In what follows, we examine step by step all terms on the right-hand side of (2.12) and perform the limits; first for ε → 0, then for δ → 0.

Initial data

We have

E ε ̺ 0,ε , u 0,ε r(0, •), U(0, •) (6.19) = Ω 1 2 ̺ 0,ε |u 0,ε -[u 0 ] δ | 2 dx + Ω 1 ε 2m H 1 + ε m ̺ (1) 0,ε -H ′ (1 + ε m [̺ (1) 0 ] δ ) (̺ (1) 0,ε -[̺ (1) 0 ] δ -H(1 + ε m [̺ (1) 0 ] δ ) dx ≤ c u 0,ε -[u 0 ] δ 2 L 2 (Ω;R 3 ) + ̺ (1) 0,ε -[̺ (1) 0 ] δ 2 L 2 (Ω;R 3 ) → c u 0 -[u 0 ] δ 2 L 2 (Ω;R 3 ) + ̺ (1) 0 -[̺ (1) 0 ] δ 2 L 2 (Ω;R 3 ) as ε → 0.
The most left quantity obviously tends to zero as δ ց 0.

Viscosity

We write

ε α τ 0 Ω S(∇ x U ε,δ ) : ∇ x (U ε,δ -u ε ) dx dt ≤ ε α c 1 (δ) τ 0 Ω |∇ x (U ε,δ -u ε )| dx dt, (6.20)
and, by Korn's inequality,

ε α τ 0 Ω |∇ x (U ε,δ -u ε )| dx dt ≤ ε α 2 τ 0 Ω (S ε (∇ x u ε ) -S ε (∇ x U ε,δ ) : ∇ x (U ε,δ -u ε ) dx dt + c 2 ε α .

Forcing term

We have

1 ε 2m (r ε,δ -̺ ε )∂ t H ′ (r ε,δ ) + ∇ x H ′ (r ε,δ ) -H ′ (̺ ε ) • (r ε,δ U ε,δ -̺ ε u ε ) (6.21) + 1 ε 2m div x U ε,δ p(r ε,δ ) -p(̺ ε ) + 1 ε 2n (r ε,δ -̺ ε )∇ x G • U ε,δ = 1 ε 2m p(r ε,δ ) -p ′ (r ε,δ )(r ε,δ -̺ ε ) -p(̺ ε ) div x U ε,δ + 1 ε 2m (r ε,δ -̺ ε )H ′′ (r ε,δ ) ∂ t r ε,δ + div x (r ε,δ U ε,δ ) + 1 ε 2m ̺ ε ∇ x H ′ (r ε,δ ) • (U ε,δ -u ε )-∇ x H ′ (̺ ε ) • (r ε,δ U ε,δ -̺ ε u ε ) + 1 ε 2n (r ε,δ -̺ ε )∇ x G • U ε,δ , where ∂ t r ε,δ + div x (r ε,δ U ε,δ ) = (̺ ε -1)div x V ε,δ + ∇ x ̺ε • U ε,δ + (6.22) ε m ∂ t q ε,δ + ε m div x ((s ε,δ + q ε,δ )(v ε,δ + V ε,δ )) .
and

̺ ε ∇ x H ′ (r ε,δ ) • (U ε,δ -u ε ) -∇ x H ′ (̺ ε ) • (r ε,δ U ε,δ -̺ ε u ε ) (6.23) = ̺ ε ∇ x H ′ (r ε,δ ) -H ′′ (̺ ε )(r ε,δ -̺ε ) -H ′ (̺ ε ) • (U ε,δ -u ε ) +̺ ε ∇ x H ′′ (̺ ε ) • (U ε,δ -u ε )(r ε,δ -̺ε ) + ̺ ε ε m H ′′ (̺ ε ) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ) +(̺ ε -r ε,δ )∇ x H ′ (̺ ε ) • U ε,δ .
Here we have used the fact that the term

̺ ε ∇ x H ′ (̺ ε ) • (U ε,δ -u ε ) in the expansion of ̺ ε ∇ x H ′ (r ε,δ ) • (U ε,δ -u ε ) cancels with the same term in the expansion of ∇ x H ′ (̺ ε ) • (r ε,δ U ε,δ -̺ ε u ε ).
Now we use formula (6.21) with the second and third terms at the right hand side expressed through formulas (6.22) and (6.23): we keep the terms

1 ε m ̺ ε H ′′ (̺ ε ) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ) and 1 ε m (r ε,δ -̺ ε )∇ x H ′ (r ε,δ ) [∂ t q ε,δ + div x ((s ε,δ + q ε,δ )(v ε,δ + V ε,δ
))] as they are, and estimate the decay of all remaining terms as ε → 0. We observe that, thanks to (6.9), (6.12), sup t∈(0,T )

q ε,δ (t, •) L 2 (R 2 ) ≤ c, sup t∈(0,T ) ∆q ε,δ (t, •) L 2 (R 2 ) ≤ cε m-1 , sup t∈(0,T ) ∂ t q ε,δ (t, •) L 2 (R 2 ) ≤ cε m-1 .
(6.24) Now, we use (2.4-2.5), (4.1-4.4), (6.5), (6.8), (6.13), (6.24) to deduce the following estimates:

1 ε 2n τ 0 Ω (r ε,δ -̺ ε )∇ x G • U ε,δ dxdt ≤ cε m-2n , (6.25) 1 ε 2m τ 0 Ω p(r ε,δ ) -p ′ (r ε,δ )(r ε,δ -̺ ε ) -p(̺ ε ) div x U ε,δ dxdt (6.26) = 1 ε 2m τ 0 Ω p(r ε,δ ) -p ′ (r ε,δ )(r ε,δ -̺ ε ) -p(̺ ε ) res div x V ε,δ dxdt + 1 ε 2m τ 0 Ω p(r ε,δ ) -p ′ (r ε,δ )(r ε,δ -̺ ε ) -p(̺ ε ) ess div x V ε,δ dxdt = h(ε, δ), 1 ε 2m τ 0 Ω (r ε,δ -̺ ε )H ′′ (r ε,δ ) (̺ ε -1)div x V ε,δ + ∇ x ̺ε • U ε,δ dxdt ≤ cε m-2n , (6.27) 1 ε 2m τ 0 Ω ̺ ε ∇ x H ′ (r ε,δ ) -H ′′ (̺ ε )(r ε,δ -̺ε ) -H ′ (̺ ε ) • (U ε,δ -u ε ) dxdt = h(ε, δ), (6.28) 1 ε 2m Ω ̺ ε ∇ x H ′′ (̺ ε ) • (U ε,δ -u ε )(r ε,δ -̺ε ) dxdt + τ 0 Ω (̺ ε -r ε,δ )∇ x H ′ (̺ ε ) • U ε,δ dxdt ≤ cε m-2n .
(6.29) Here and hereafter, h(ε, δ) denotes a generic function having the property h(ε, δ) → h(δ) as ε → 0, h(δ) → 0 as δ → 0. (6.30)

Note that the dispersive decay estimates (6.8) play a crucial role in the analysis.

Taking into account (6. [START_REF] Klainerman | Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids[END_REF]), (6.20), using the identity (6.21) with the third and fourth terms expressed through (6.22-6.23), and employing the asymptotic behavior from formulas (6.25-6.29), we may rewrite the relative entropy inequality (2.12) in the form

E ε ̺ ε , u ε r ε,δ , U ε,δ (τ ) ≤ τ 0 Ω ̺ ε (∂ t U ε,δ + u ε • ∇ x U ε,δ ) • (U ε,δ -u ε ) dx dt (6.31) + 1 ε τ 0 Ω ̺ ε (f × u ε ) • (U ε,δ -u ε ) dx dt + 1 ε m τ 0 Ω (r ε,δ -̺ ε )H ′′ (r ε,δ ) ∂ t q ε,δ + div x ((s ε,δ + q ε,δ )U ε,δ ) dx dt + 1 ε m τ 0 Ω
̺ ε H ′′ (̺ ε ) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δu ε ) dx dt + h(ε, δ).

Coriolis force

We may write ̺ ε H ′′ (̺ ε ) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δu ε ) (6.32)

= ̺ ε (H ′′ (̺ ε ) -H ′′ (1)) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ) + ̺ ε H ′′ (1) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ),
where, by the same reasoning as in estimates (6.25-6.29),

1 ε 2m τ 0 Ω ̺ ε (H ′′ (̺ ε ) -H ′′ (1)) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ) dxdt ≤ cε m-2n
Recalling our convention H ′′ (1) = 1 we get for the second term in (6.32),

1 ε m ̺ ε H ′′ (1) ∇ x s ε,δ + ∇ x q ε,δ • (U ε,δ -u ε ) = ̺ ε -∂ t V ε,δ - 1 ε f × V ε,δ - 1 ε f × v ε,δ • (U ε,δ -u ε ) = -∂ t V ε,δ - 1 ε (f × U ε,δ ) • u ε ,
where the last term cancels out with the Coriolis force term

1 ε f × u ε • (U ε,δ -u ε ) = 1 ε (f × u ε ) • U ε,δ
. Thanks to formula (6.32) the relative entropy inequality (6.31) reduces to

E ε ̺ ε , u ε r ε,δ , U ε,δ (τ ) ≤ τ 0 Ω ̺ ε (∂ t v ε,δ + u ε • ∇ x U ε,δ ) • (U ε,δ -u ε ) dx dt (6.33) + 1 ε m τ 0 Ω
(r ε,δ -̺ ε )H ′′ (r ε,δ ) ∂ t q ε,δ + div x ((s ε,δ + q ε,δ )U ε,δ ) dx dt + h(ε, δ).

Estimating the remaining terms

We observe that

τ 0 Ω ̺ ε (∂ t v ε,δ + u ε • ∇ x U ε,δ ) • (U ε,δ -u ε ) dx dt = τ 0 Ω ̺ ε (v ε,δ + U ε,δ • ∇ x U ε,δ ) • (U ε,δ -u ε ) dx dt - τ 0 Ω ̺ ε (U ε,δ -u ε ) • ∇ x U ε,δ • (U ε,δ -u ε ) dx dt,
where the terms in the first expression at the right hand side containing the quantities s ε,δ , V ε,δ tend to 0 in the limit lim δ→0 lim ε→0 thanks to dispersive estimates (6.8). Consequently, employing (2.5), we obtain

τ 0 Ω ̺ ε (∂ t v ε,δ + u ε • ∇ x U ε,δ ) • (U ε,δ -u ε ) dx dt ≤ τ 0 Ω (∂ t v ε,δ + v ε,δ • ∇ x v ε,δ ) • (v ε,δ -u ε ) dx dt + c τ 0 E ε ̺ ε , u ε r ε,δ , U ε,δ dt + h(ε, δ).
Similarly, we deduce 1 ε m τ 0 Ω (r ε,δ -̺ ε )H ′′ (r ε,δ ) ∂ t q ε,δ + div x ((s ε,δ + q ε,δ )U ε,δ ) dx dt = τ 0 Ω q ε,δ ∂ t q ε,δ dx dt + h(ε, δ),

where we have used (4.1-4.3),(6.6), (6.24). Now, we employ the energy equality (6.12) to observe that

τ 0 Ω v ε,δ • ∂ t v ε,δ + q δ,ε • ∂ t q ε,δ dx dt = 0. Consequently, τ 0 Ω (∂ t v ε,δ + v ε,δ • ∇ x v ε,δ ) • (v ε,δ -u ε ) dxdt + τ 0 Ω q ε,δ ∂ t q ε,δ dx dt = - τ 0 Ω (∂ t v ε,δ + v ε,δ • ∇ x v ε,δ ) • u ε dxdt,
where we have used the identity τ 0 Ω v ε,δ • ∇ x v ε,δ • v ε,δ dxdt = 0. We remark that

τ 0 Ω (∂ t v ε,δ + v ε,δ • ∇ x v ε,δ ) • u ε dx dt → τ 0 Ω (∂ t v δ + v δ • ∇ x v δ ) • u dx dt = τ 0 Ω ∇ x Π δ • u dx dt = 0,
where we have used (4.7-4.8), (6.15) and equations (1.9) with (v, Π) replaced by (v δ , Π δ ). Consequently, the entropy inequality (6.33) takes the form

E ε ̺ ε , u ε r ε,δ , U ε,δ (τ ) ≤ h(ε, δ),
where the function h satisfies (6.30). This finishes the proof of Theorem 3.1

For a vector field b = [b 1

 1 , b 2 , b 2 ], we introduce the horizontal component b h = [b 1 , b 2 ] writing b = [b h , b 3 ]

* Eduard Feireisl acknowledges the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.


S Kračmar

Š Nečasová

A Novotný

The motion of a compressible viscous fluid around rotating body

We consider the motion of compressible viscous fluids around a rotating rigid obstacle when the velocity at infinity is non zero and parallel to the axis of rotation. We prove the existence of weak solution.

Viscous compressible fluid in a rotating frame

We consider a rigid body S with boundary Γ such that Ω = R 3 \ S is an exterior domain. We assume that the body is rotating around the x 3 -axis with the angular velocity ω = (0, 0, |ω|). The position of the body at time t > 0 is given by the formula S(t) = {y = O(t)x; x ∈ S}, S(0) = S

(1.1)

where

O(t) =   cos |ω|t -sin |ω|t 0 sin |ω|t cos |ω|t 0 0 0 1   . (1.2)
We shall investigate the motion of viscous compressible fluid around the obstacle S(t).

The exterior domain Ω(t) occupied by the viscous compressible fluid is time dependent:

Ω(t) = {y = O(t)x; x ∈ Ω} Γ(t) = ∂Ω(t) = {y = O(t)x; x ∈ Γ}. (1.
3)

The fluid in Ω(t) is governed by the compressible Navier-Stokes equations. The unknowns, σ, v -density and velocity -solve the nonlinear system ∂ t (σv) + div (v ⊗ σv) + ∇p(σ) -div S(∇v) = 0 Ω(t), t ∈ (0, ∞)

∂ t σ + div (σv) = 0 in Ω(t), t ∈ (0, ∞) v(t, y) = ω × y on Γ(t), t ∈ (0, ∞) σ(t, y) → ϱ ∞ , v(t, y) → a ∞ as |y| → ∞, (1.4)
completed with the initial conditions σ(0, y) = σ 0 (y), v(0, y) = v 0 (y), y ∈ Ω.

. Here and hereafter, S is the viscous stress tensor -a linear function of ∇ x v specified later, see (2.1), p is the pressure -a given function of the density, see (2.2), a ∞ ∈ R 3 is a constant vector denoting a (prescribed) velocity of the flow at infinity and ϱ ∞ > 0 is a (prescribed) positive constant density at infinity. We omit the action of external forces for the sake of simplicity.

We shall transform the time dependent domain Ω(t) to the fixed domain Ω by employing the change of variables x = O(t) T y,

and introducing new functions u(t, x) = O(t) T v(t, y), ρ(t, x) = σ(t, y).

(1.6)

It is easy to verify that

O(-t) = O T (t) = O -1 (t), O -1 Ȯ = |ω|   0 -1 0 1 0 0 0 0 0   , O -1 Ȯ x = ω × x, Ȯ-1 O x = -ω × x.
Moreover, the operations of inversion of the matrix O(t) and of the derivative with respect to t applied on O(t) are commuting. Using these facts, we easily find that the first term of the left-hand side in (1.4) 1 reads in the new coordinates:

∂ t (σv)(t, y) = ∂ ∂t ( (O(t) ρu) ( t, O -1 (t)y )) = Ȯ(t)ρu (( t, O -1 (t)y ) + O(t) ∂ t (ρu) ( t, O -1 (t)y ) + O(t)∇ x (ρu) ( t, O -1 (t)y )) Ȯ-1 (t)y = O(O -1 Ȯ)ρu (t, x) + O ∂ t (ρu) (t, x) + O ∇ x (ρu) (t, x)) Ȯ-1 Ox = O [ ∂ t (ρu)-(ω×x)•∇ x (ρu)+ρ(ω×u) ] = O [ ∂ t (ρu)-div (ρ(ω × x) ⊗ u)+ρ(ω×u) ] ,
where the following standard conventions have been used

∇ x u = [ a ij ] i,j=1,2,3 , a ij = ∂u i ∂x j , (z × A) k = 3 ∑ j=1 z j A j,k , z = [z 1 , z 2 , z 3 ], A = [ A jk ] j,k=1 ,2,3
,

div x (z ⊗ w) = ∇ x • (z ⊗ w), z ⊗ w = [ z i w k ] i,k=1,2,3
.

Analogously, div y (σv ⊗ v) = O div x (ρu ⊗ u)
and div y S(∇v)(t, y) = O div x S(∇u).

Finally, we observe that the non homogenous Dirichlet boundary conditions (1.4) 3 transform to u = ω × x, on the fixed boundary ∂S and the conditions at infinity remain unchanged. Consequently, after the change of variables (1.5) equations (1.4) transform to

∂ t (ρu) + div (ρ (u -ω × x) ⊗ u) + ρω × u + ∇p(ρ) = div S(∇u) in (0, T ) × Ω, (1.7) ∂ t ρ + div (ρ(u -ω × x)) = 0 in (0, T ) × Ω, (1.8) u = ω × x on (0, T ) × ∂Ω, (1.9)
u(t, x) → a ∞ ∈ R 3 , ρ(t, x) → ρ ∞ > 0 as |x| → ∞ (1.10) ρ(0, x) = ρ 0 (x), (ρu)(0, x) = q 0 (x) = ϱ 0 u 0 (x). (1.11)
This is a classical formulation of equations for unknown functions (ρ, u)(t, x), (t, x) ∈ (0, T ) × Ω that is equivalent to the formulation (1.4). We shall deal with this problem in the following sections. We refer for more details concerning the above change of variables to Galdi [14, Chapter 1], where the calculations are performed in the case of the (incompressible) Navier-Stokes equations.

Although the problem of the moving and rotating body in a compressible fluid is of obvious practical importance, to the best of our knowledge, there is no mathematical literature about the subject. Our goal in this paper is to investigate this problem with the large initial data and in the long time run, in the context of weak solutions in the spirit introduced to the theory of incompressible Navier-Stokes equations by Leray [START_REF] Leray | Sur le mouvement d'un liquide visqueux emplissant l'espace[END_REF] and generalized later to compressible fluids in monographs by Lions [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF], Feireisl [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF] and in [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF], [START_REF] Novotný | Introduction to the Mathematical Theory of Compressible Flow[END_REF]. Notice, that the case without rotation has been studied in [START_REF] Novo | The compressible Navier-Stokes equations with influx-outflux boundary conditions[END_REF], [START_REF] Novotný | Introduction to the Mathematical Theory of Compressible Flow[END_REF] and the case with zero velocity at infinity has been investigated in [START_REF] Sýkora | Motion of a compressible fluid in a time dependent domain[END_REF].

The "incompressible" analogue of this evolution problem is investigated in Hishida [START_REF] Hishida | An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle[END_REF]. The reader can also consult Hieber and coauthors [START_REF] Geissert | L p theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle[END_REF], and references quoted therein for the related results. The steady version of the incompressible case is investigated in Galdi and co-authors [START_REF] Galdi | An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady -state problems[END_REF][START_REF] Galdi | Steady-state Navier-Stokes flows past a rotating body: Leray solutions are physically reasonable[END_REF][START_REF] Galdi | Further results on steady-state flow of a Navier-Stokes liquid around a rigid body. Existence of the wake[END_REF], Farwig et al. see [START_REF] Farwig | An L q -analysis of viscous fluid flow past a rotating obstacle[END_REF][START_REF] Farwig | L q -theory of a singular "winding" integral operator arising from fluid dynamics[END_REF][START_REF] Farwig | A weighted L q approach to Oseen flow around a rotating body[END_REF]. Variational approach was used in Deuring et al. [START_REF] Deuring | On pointwise decay of linearized stationary incompressible viscous flow around rotating and translating bodies[END_REF] and in [START_REF] Kračmar | Anisotropic L 2 -estimates of weak solutions to the stationary Oseen-type equations in 3D-exterior domain for a rotating body[END_REF].

It is well known from the works of P.L. Lions [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF], Feireisl [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF] (see also [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF], [START_REF] Novotný | Introduction to the Mathematical Theory of Compressible Flow[END_REF]), that the estimates available in the compressible models do not allow to pass to the limit in the non-linear terms in the equations. Instead, one has to use the methods of compensated compactness (div-curl lemma, effective viscous flux, various commutator lemmas, oscillations defect measure, renormalized solutions of the transport equation) and to search strong convergence within the structure of equations. We show, that after the transformation to a fixed domain one can adapt these techniques to the equations in the rotating frame. Although this choice is natural in the present context, there are many situations (e.g. domains with oscillating boundaries), where it is more convenient to work in the moving domain. We refer to [START_REF] Feireisl | Incompressible limits of fluids excited by moving boundaries[END_REF][START_REF] Feireisl | Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains[END_REF] and references quoted there, for relevant results.

Finally, notice that another situation, including Coriolis (and possibly a centrifugal force), when the fluid sticks or slips on the boundary of the body in the rotating frame and when the velocity at infinity tends to zero, enters into the framework of the existence results investigated in [START_REF] Jesslé | Navier-Stokes-Fourier system, weak solutions, relative entropy, weak-strong uniqueness[END_REF]. This context is typical of the models in the physics of atmosphere; it has been therefore broadly investigated in the context of singular limits, see e.g. Bresch, Chemin, Desjardins, Feireisl, Gallagher, Grenier, Gérard-Varet, Novotny and others, [START_REF] Bresch | Rotating fluids in a cylinder[END_REF][START_REF] Chemin | Mathematical geophysics[END_REF][START_REF] Feireisl | Low Mach number limits of compressible rotating fluids[END_REF][START_REF] Feireisl | A singular limit for compressible rotating fluids[END_REF][START_REF] Feireisl | Novotný ıMulti-scale Analysis of Compressible Viscous and Rotating Fluids[END_REF].

The paper is organized as follows: In the first Section, we introduce the weak formulation and state the main result. The weak solutions are first established on large balls with help of invading domains via a penalization method described in Section 4. In order to achieve the non zero prescribed velocity at infinity, the penalization uses an auxiliary vector field u ∞ , that is constructed in Section 3.The limit process on large balls with the penalization parameter tending to infinity is performed in Section 5. Finally, the proof of the existence result is completed in Section 6, where the limit with the radius of large balls tending to infinity is established. The interesting feature in this limit process appears in the renormalized continuity equation, where the test functions have to be chosen in such a way in order to eliminate the possible contributions of the terms containing ω × x, (that explode at infinity) to the oscillations in the density sequence. This issue is handled in the last part of Section 6.

Setting of the problem, weak solutions

Before introducing the notion of weak solutions of the problem (1.7-1.10) and in order to determine the appropriate functional spaces, we shall specify the form of the viscous stress tensor and the pressure. Following the standard existence theory in the situation without rotation, we deal with the Newtonian fluids, where

S(∇ x u) = µ T(∇ x u) + η div x u I, T(∇ x u) = ∇ x u + ∇ T x u - 2 3 div x u I, (2.1)
with µ > 0, η ≥ 0. Moreover, we assume that the pressure is a given function of the density that satisfies hypotheses

p ∈ C[0, ∞) ∩ C 1 (0, ∞), p(0) = 0, p ′ (ϱ) > 0 for all ϱ > 0, lim ϱ→∞ p ′ (ϱ) ϱ γ-1 = a > 0 (2.2)
for a certain γ > 3/2. Notice that this assumption covers at least one physically reasonable case, namely isentropic flows of a monoatomic gas. In view of these assumptions, the weak formulation of problem (1.7-1.10) reads:

Definition 1

We shall call a couple (ρ, u) a renormalized weak solution of problem (1.7-2.2) if:

(i) Functions (ϱ, u) are such that ρ ≥ 0, ρ -ϱ ∞ ∈ L ∞ (0, T ; L γ + L 2 (Ω)), ϱ(u -a ∞ ) 2 ∈ L ∞ (0, T ; L 1 (Ω)), u -a ∞ ∈ L 2 (0, T ; W 1,2 (Ω)), u(t)| ∂Ω = ω × x for a.a. t ∈ (0, T ). (2.3) (ii) Density ϱ ∈ C weak ([0, T ]; L γ (K))
for any compact K ⊂ Ω and the equation of continuity (1.8) is satisfied in the weak sense,

∫ Ω ϱφdx τ 0 = ∫ τ 0 ∫ Ω ϱ∂ t φ + ϱ(u -ω × x) • ∇ x φ dx dt, (2.4)
for all τ ∈ [0, T ] and any test function φ ∈ C ∞ c ([0, T ] × Ω), as well as in the renormalized weak sense

∫ Ω b(ϱ)φdx τ 0 = ∫ τ 0 ∫ Ω b(ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dx dt + ∫ τ 0 ∫ Ω ( b(ϱ) -ϱb ′ (ϱ) ) divu φ dx dt (2.5) for any τ ∈ [0, T ], for any φ ∈ C 1 c ([0, T ] × Ω) and for all b ∈ C c [0, ∞) ∩ C 1 c (0, ∞) (2.6)
Here and in what follows

∫ Ω gφdx τ 0 = ∫ Ω g(τ, x)φ(τ, x)dx - ∫ Ω g 0 (x)φ(0, x)dx.
(iii) The linear momentum ϱu ∈ C weak ([0, T ], L 2γ/(γ+1) (K)) for any compact K ⊂ Ω and the momentum equation (1.7) is satisfied in the weak sense

∫ Ω ϱu • φdx τ 0 + ∫ τ 0 ∫ Ω S(∇ x u) : ∇ x φ dx dt = ∫ τ 0 ∫ Ω ( ϱu • ∂ t φ + ϱ(u -ω × x) ⊗ u : ∇ x φ -ϱ(ω × u) • φ + p(ϱ) div x φ ) dxdt (2.7)
for all τ ∈ [0, T ] and for any test function φ ∈ C ∞ c ([0, T ] × Ω). Notice that the class (2.6) of the renormalizing functions b can be enlarged to

b ∈ C[0, ∞) ∩ C 1 (0, ∞), z b ′ (z) ∈ L ∞ (0, 1), b(z)/z 5γ/6 , (zb ′ -b)(z)/z γ/2 ∈ L ∞ (1, ∞)
(2.8) by using the Lebesgue dominated convergence theorem.

Here and hereafter, we denote

H(ρ) = ρ ∫ ρ 1 p(s) s 2 ds, (2.9) E(ϱ|ϱ ∞ ) = H(ϱ) -H ′ (ϱ ∞ )(ϱ -ϱ ∞ ) -H(ϱ ∞ ).
(2.10)

We are now in a position to state the main existence result.

Theorem (Existence of weak solutions) Assume that ρ ∞ > 0 and

Ω = R 3 \ S,
where S is a bounded Lipschitz domain. Suppose that the vector ω is parallel to the vector a ∞ , and that the viscous stress tensor S and the pressure p satisfy (2.1), (2.2). Assume that

ϱ 0 ≥ 0, E(ρ 0 |ϱ ∞ ) ∈ L 1 (Ω) ϱ 0 (u 0 -a ∞ ) 2 ∈ L 1 (Ω). (2.11)
Then the problem (1.7-1.11) admits at least one renormalized weak solution.

An auxiliary result

Let us denote the open ball with radius r centered at the origin by B r . Suppose without loss of generality that S ⊂ B 1/2 and B 1 ⊂ B R . To begin, we claim that there exists a smooth vector field u ∞ ∈ C ∞ c (R 3 ) such that:

u ∞ =    ω × x in S supp u ∞ ⊂ B 2R a ∞ in B (3/2)R \ B 1 (3.1) div u ∞ = 0 in S ∪ (R 3 \ B 1 ) (3.2)
Let us consider non-increasing functions

Φ 1 , Φ 2 , Ψ R ∈ C ∞ c ([0, +∞)) as follows: 0 ≤ Φ 1 ≤ 1, Φ 1 = 1 in [ 0, 5/9) , Φ 1 = 0 in [2/3, +∞) , 0 ≤ Φ 2 ≤ 1, Φ 2 = 1 in [ 0, 7/9) , Φ 2 = 0 in [ 8/9, +∞) , 0 ≤ Ψ R ≤ 1, Ψ R = 1 in [ 0, (3/2)R)) , Ψ R = 0 in [2R -1/4, +∞) . Let us set ã∞ (x) = a ∞ (1 -Φ 2 (|x|))Ψ R (|x|).
Using the classical Stokes formula it is easy to check that

∫ B 2R-1/4 \B (3/2)R div ã∞ (x) dx = 0.
So, from Lemma 3.1 (see further) we get that there exists a vector field b

∞ ∈ W 1,p 0 (B 2R-1/4 \ B (3/2)R ), p ∈ (1, +∞) and div b ∞ (x) = div ã∞ (x) for x ∈ B 2R-1/4 \ B (3/2)R .
It is easy to see that divergence of the expression (ω

× x) Φ 1 (|x|) ∈ C ∞ c (R 3 ) is zero. Now, let extend b ∞ (•) by zero in R 3 and set u ∞ (x) = (ω × x) Φ 1 (|x|) + m ε * (ã ∞ -b ∞ )(x),
where 0 < ε < 1/8 is fixed and m ε (•) is the standard mollifier. In the previous reasoning, we have used the auxiliary result concerning the so called Bogovskiǐ operator (see e.g. Galdi [START_REF] Galdi | An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady -state problems[END_REF]): B G :

{ F ∈ L q (G); ∫ G F dx = 0 } -→ W 1,q 0 ( G; R N )
is a bounded linear operator. More precisely, for any q ∈ (1, ∞),

∥∇B G (F)∥ L q (G;R N ×N ) ≤ c(G, q)∥F∥ L q (G) ; 2. the function v = B G (F) solves the problem div v = F in G, v| ∂G = 0; 3. if, moreover, F can be written in the form F = div Q for a certain Q ∈ L p (G, R N ) satisfying Q • n| ∂G = 0
where n stands for the outward unit normal vector to ∂G, then it holds

∥B G (F)∥ L p (G;R N ) ≤ c(G, p)∥Q∥ L p (G) .

Penalized problem

In what follows, we denote

V = B 2R
and extend ϱ 0 by 0 and u 0 by ω × x in S. We shall solve on (0, T ) × V the following penalized problem

∂ t (ρu) + div ((ρ(u -ω × x) ⊗ u)) + ρω × u -div S(∇u) +∇p(ρ) + k • 1 {(V \B R )∪S} (u -u ∞ ) = 0    (4.1) ∂ t ρ + div (ρ(u -ω × x)) = 0. (4.2)
u| ∂V = 0, (4.3) (iii) Momentum equation is satisfied in the weak sense ,

ϱ(0) = ϱ 0 , ϱu(0) = ϱ 0 u 0 . ( 4
(i) ρ ≥ 0, ρ -ϱ ∞ ∈ L ∞ (0, T ; [L γ + L 2 ](V )), ϱ(u -u ∞ ) 2 ∈ L ∞ (0, T ; L 1 (V )), u -u ∞ ∈ L 2 (0, T ; W 1,2 0 (V ; R 3 )).
∫ τ 0 ∫ Ω ( ϱu•∂ t φ+ϱ(u-ω×x)⊗u : ∇ x φ-ϱ(ω×u)•φ-k•1 {(V \B R )∪S} (u-u ∞ )•φ (4.6) +p(ϱ)div x φ ) dx dt = ∫ Ω ϱuφ dx τ 0 + ∫ τ 0 ∫ Ω S(∇ x u) : ∇ x φ dx dt for all τ ∈ [0, T ] and for any test function φ ∈ C ∞ c ([0, T ] × V ).
(iv) The following energy inequality is satisfied

∫ V ( 1 2 ϱ|u -u ∞ | 2 + E(ϱ|ϱ ∞ ) ) dx τ 0 + ∫ τ 0 ∫ V ( S(∇ x (u -u ∞ )) : ∇ x (u -u ∞ ) + k1 (V \B R )∪S |u -u ∞ | 2 ) dx dt ≤ - ∫ τ 0 ∫ V p(ρ)div u ∞ + ∫ τ 0 ∫ V ρ(u -ω × x) • ∇ x u ∞ • (u ∞ -u) dx dt - ∫ τ 0 ∫ V ϱ(ω × u ∞ ) • (u -u ∞ ) dx dt - ∫ τ 0 ∫ V S(∇ x u ∞ ) : ∇ x (u -u ∞ ) dx dt (4.7)
for a. a. τ ∈ (0, T ).

Under assumptions (2.11) with u ∞ defined in formulas (3.1-3.2), problem (4.1-4.2) admits at least one weak solution (ϱ k , u k ). This result can be proved similarly as in Theorems 7.7, 7.79 [25, Sections 7.1.2, 7.12.6].

Limit k → ∞

We deduce from estimate (4.7) the following estimates, with c = c(R) a generic constant that may depend on R but is independent of k:

ess sup t∈(0,T ) ∫ V ρ k |u k -u ∞ | 2 dx ≤ c(R) (5.1) ess sup t∈(0,T ) ∫ V E(ρ k |ρ ∞ )dx ≤ c(R) (5.2)
∥u k ∥ L 2 (0,T ;W 1,2 (V ;R 3 )) ≤ c(R) (5.3) ∥u k -u ∞ ∥ L 2 ((0,T )×((V \B R )∪S);R 3 ) ≤ c(R) √ k (5.4)
When deducing (5.3) we have used the Korn-Poincaré type inequality in the form

∥S (∇(u k -u ∞ ))∥ L 2 (V ;R 3×3 ) ≥ c∥u k -u ∞ ∥ W 1,2 0 (V ;R 3×3 ) , (5.5)
see Theorem 10.16 in [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]. Finally, introducing for any

h ∈ L 1 ((0, T ) × V ) [h] ess = 1 {|ρ-ρ∞|≤ρ∞/2} • h, [h] res = h -h ess, (5.6)
and taking into account

E(ρ|ρ ∞ ) ≥ c ( |ρ -ρ ∞ | 2 • 1 {ρ∞/2≤ρ≤2ρ∞} + (1 + ρ γ ) • 1 {ρ<ρ∞/2 or ρ>2ρ∞} )
(see e.g. [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]) we get from (5.2):

ess sup t∈(0,T ) ∥[ρ k -ρ ∞ ] ess ∥ L 2 (V ) ≤ c(R), ess sup t∈(0,T ) ∥[ρ k ] res ∥ L γ (V ) ≤ c(R). (5.7)
On the basis of these estimates we get the existence of a subsequence (ϱ k , u k ) and a couple (ϱ, u) such that

ϱ k -ϱ ∞ ⇀ ϱ -ϱ ∞ in L ∞ (0, T ; [L 2 + L γ ](V )) u k ⇀ u in L 2 (0, T ; W 1,2 (V, R 3 )),
where ϱ ≥ 0. It can be proved via quite recent but nowadays classical methods (including effective viscous flux, oscillations defect measures, renormalized solutions of continuity equation) [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF], [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF], [START_REF] Novotný | Introduction to the Mathematical Theory of Compressible Flow[END_REF], [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF] that the couple (ϱ, u) has the following properties 1 : (i) ϱ ∈ C weak (0, T ; L γ (V )) and the continuity equation is satisfied in the weak sense

∫ V ϱφdx τ 0 = ∫ τ 0 ∫ V ϱ∂ t φ + ϱ(u -ω × x) • ∇ x φdx dt (5.8)
for all τ ∈ [0, T ] and any φ ∈ C ∞ c ([0, T ] × V ). Similarly, the renormalized continuity equation is satisfied in the form:

∫ V b(ϱ)φdx τ 0 + ∫ τ 0 ∫ V ( ϱb ′ (ϱ) -b(ϱ) ) divuφdxdt = ∫ τ 0 ∫ V b(ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dxdt (5.9)
for all τ ∈ [0, T ], for any b belonging to the class (2.6) and for any

φ ∈ C 1 c ([0, T ] × V ).
(ii) Since u -ω × x = 0 a.e. in (0, T ) × S, equation (5.8) yields, in particular,

∫ T 0 η ′ (t) ∫ S ϱφ(x)dxdt = 0
for all η ∈ C 1 (0, T ) and for all φ ∈ C 1 c (S). This means that for a.a. (t, x) ∈ (0, T ) × S, ϱ(t, x) = ϱ 0 (x) = 0.

With this information at hand, formula (5.8) yields, in particular,

∫ Ω R ϱφdx τ 0 = ∫ τ 0 ∫ Ω R ϱ ( ∂ t φ + (u -ω × x) • ∇ x φ ) dx dt (5.10)
for all τ ∈ [0, T ] and for any test function

φ ∈ C ∞ c ([0, T ] × (B R \ S)).
Here and in what follows, we denote Ω R = Ω ∩ B R = B R \ S. Similarly, formula (5.9) yields

∫ Ω R b(ϱ)φdx τ 0 = ∫ τ 0 ∫ Ω R ( b(ϱ) -ϱb ′ (ϱ) ) divuφdxdt ∫ τ 0 ∫ Ω R b(ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dxdt (5.11)
for all τ ∈ [0, T ], for any b belonging to the class (2.6) and

φ ∈ C 1 c ([0, T ] × (B R \ S)).
(iii) There holds ϱu ∈ C weak (0, T ; L 2γ/γ+1 (V ; R 3 )) and

∫ T 0 ∫ Ω R ( ϱu • ∂ t φ + ϱ(u -ω × x) ⊗ u : ∇ x φ -ϱ(ω × u) • φ + p(ϱ)div x φ ) dx dt (5.12) = ∫ Ω R ϱuφdx τ 0 + ∫ T 0 ∫ Ω R S(∇ x u) : ∇ x φdx dt for any τ ∈ [0, T ] and all test functions φ ∈ C ∞ c ([0, T ] × Ω R ). Moreover, u = u ∞ a.e. in (0, T ) × [(V \ B R ) ∪ S].
(5.13) (iv) The following energy inequality is satisfied 6 Limit R → ∞

∫ Ω R ( 1 2 ϱ|u -u ∞ | 2 + E(ϱ|ϱ ∞ ) ) dx τ 0 + ∫ τ 0 ∫ Ω R (S(∇ x (u -u ∞ )) : ∇ x (u -u ∞ )dxdt ≤ - ∫ τ 0 ∫ B 1 \S p(ρ)divu ∞ - ∫ τ 0 ∫ B 1 \S ρ(u -ω × x) • ∇ x u ∞ • (u -u ∞ )dxdt - ∫ τ 0 ∫ B 1 \S ϱ(ω × u ∞ ) • (u -u ∞ ) dx dt - ∫ τ 0 ∫ B 1 \S S(∇ x u ∞ ) : ∇ x (u -u ∞ )dxdt (5.

Weak limits

We set

U ∞ = { u ∞ in B 1 , a ∞ in R 3 \ B 1 } . (6.1)
Here and hereafter we extend ρ by ρ ∞ outside B R (and denote the new function by ϱ), and u by U ∞ outside B R and denote the new function again by u. Using (5.1-5.7) we can deduce from (5.14) estimates:

∫ R 3 [1] res dx ≤ c (6.2) ess sup t∈(0,T ) ∥[ρ R -ρ ∞ ] ess ∥ L 2 (R 3 ) ≤ c (6.3) ess sup t∈(0,T ) ∥[ρ R ] res ∥ L γ (R 3 ) ≤ c (6.4) ess sup t∈(0,T ) ∥ϱ R |u R -U ∞ | 2 ∥ L 1 (R 3 ) = 0. (6.5) ∥u R -U ∞ ∥ L 2 ((0,T )×W 1,2 (R 3 ;R 3 )) ≤ c (6.6)
Hereafter, we describe in a concise form the nowadays classical argument including effective viscous flux identity, oscillations defect measures and renormalized continuity equation leading to the proof of the main theorem. It is to be noticed that the main issue is to show the strong convergence of the density sequence.

We use in momentum equation (5.12) (written with

(ϱ R , ϑ R , u R ) on Ω R ) with the test function φ = B Ωn (ϱ ν R ) ∈ L ∞ (0, T ; W 1,γ/ν 0 (Ω n ; R 3 )) ∩ L 2 (0, T ; L p (Ω n )), ν γ = 1 2 = 1 p
, where ν > 0 is sufficiently small and n ∈ N is fixed. After some technical calculations similar to [11, Section 2.2.5, Appendix], where one uses the Bogovskii lemma 3.1, one derives that there exists ν > 0 such that for all R ≥ n,

∫ T 0 ∫ Ωn ϱ γ+ν R dxdt ≤ c(n). (6.7)
Proceeding as in [25, Section 7.12.6] we deduce from estimates (6.2-6.7), by means of a diagonalisation procedure, existence of a couple (ϱ, u) such that

ϱ R -ϱ ∞ ⇀ * ϱ -ϱ ∞ in L ∞ (0, T ; L γ + L 2 (Ω)), u R -U ∞ ⇀ u -U ∞ in L 2 (0, T ; W 1,2 (Ω; R 3 )), p(ϱ R ) ⇀ p(ϱ) in L q ((0, T ) × Ω n ) with some q > 1, ∀n ∈ N, (6.8)
as R → ∞. Here and hereafter g(ϱ, u) denotes L 1 weak limit of the sequence g(ϱ R , u R ).

Moreover, using continuity equation (5.10), renormalized continuity equation (5.9) and momentum equation (5.12) together with estimate (6.3-6.5) (all written with (ϱ R , u R ) on Ω R )), we get, after employing the Arzela-Ascoli compactness argument and a density argument,

ϱ R → ϱ in C weak ([0, T ]; L γ (Ω n )) and in L 2 (0, T ; W -1,2 (Ω n )), ∀n ∈ N, h(ϱ R ) → h(ϱ) in C weak ([0, T ]; L q (Ω n )), q ∈ [1, ∞) and in L 2 (0, T ; W -1,2 (Ω n )), ∀n ∈ N, where h belongs to (2.6), ϱ R u R → ϱu in C weak ([0, T ]; L 2γ/(γ+1) (Ω n ; R 3 )) and in L 2 (0, T ; W -1,2 (Ω n , R 3 )), ∀n ∈ N, ϱ R u R ⊗ u R ⇀ ϱu ⊗ u in L 2 (0, T ; L 6γ/(4γ+3) (Ω n , R 3×3 )), ∀n ∈ N.
(6.9) In all three cases, the strong L 2 W -1,2 convergence follows from the previous C weak L q convergence and the compact embedding L q (G) ֒→֒→ W -1,2 (G), q > 6/5, where G is a bounded domain.

At this stage, we may pass to the limit R → ∞ in equations (5.12), (5.10) and (5.9) that are written with (ϱ R , ϑ R , u R ) on Ω R . We get

∫ Ω ϱφ dx τ 0 = ∫ τ 0 ∫ Ω ( ϱ∂ t φ + ϱ(u -ω × x) • ∇ x φ ) dx dt (6.10)
for all τ ∈ [0, T ] and for any φ ∈ C 1 c ([0, T ] × Ω);

∫ Ω ϱu • φ dx τ 0 (6.11) = ∫ τ 0 ∫ Ω ( ϱu • ∂ t φ + ϱ(u -ω × x) ⊗ u : ∇ x φ + p(ϱ)div x φ -S(∇ x u) : ∇ x φ ) dx dt
for all τ ∈ [0, T ] and for any φ ∈ C 1 c ([0, T ] × Ω; R 3 );

∫ τ 0 ∫ Ω ϱL k (ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dxdt (6.12) = ∫ τ 0 ∫ Ω T k (ϱ)divuφdxdt + ∫ Ω ϱL k (ϱ)φdx τ 0 and ∫ τ 0 ∫ Ω T k (ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dxdt (6.13) = ∫ τ 0 ∫ Ω (ϱT ′ k (ϱ) -T k (ϱ))divuφdxdt + ∫ Ω T k (ϱ)φdx τ 0 ,
where τ ∈ [0, T ] and φ ∈ C 1 c ([0, T ) × Ω). In the above formulas, we have taken

T k (z) = kT (z/k), L k (z) = ∫ z 1 T k (w)
w 2 dw, ∀z ≥ 0, T (z) = min{z, k}.

6.2 Strong convergence of densities

T k (ϱ)div x u -T k (ϱ)div x u = 1 4 3 µ + η ( p(ϱ)T k (ϱ) -p(ϱ) T k (ϱ) ) . (6.14)
To get this identity, one needs to subtract the limit R → ∞ of the momentum equation (5.13

) on Ω R with (ϱ R , u R )) tested with φ = ζ∇∆ -1 [T k (ϱ R )ζ], ζ ∈ C ∞ ((0, T ) × Ω n ; R 3
) from the limiting momentum equation (6.11) with the test function

φ = ζ∇∆ -1 [T k (ϱ)ζ],
where n is fixed. Here and hereafter we denote (∇∆ -1 )

j (v) = -F -1 [ iξ j |ξ| 2 F(v)(ξ) ] (that is a continuous operator from L p (R 3 ) to {v ∈ L 3p/(3-p) (R 3 ; R 3 ), ∇v ∈ L p (R 3 ; R 3×3 )}, 1 < p < 3 and R the Riesz operator, (R[v]) ij = (∇ ⊗ ∇∆ -1 ) ij v = F -1 [ ξ i ξ j |ξ| 2 F(v)(ξ)
] with F the Fourier transform. It is important to recall that R is a continuous operator from L p (R 3 ) → L p (R 3 , R 3×3 ) for any 1 < p < ∞. Since S is linear, this procedure yields

lim R→∞ ∫ T 0 ∫ Ω ζ(t, x) ( p(ϱ R )T k (ϱ R ) -S(∇ x u R ) : R[ζT k (ϱ R )] ) dxdt = ∫ T 0 ∫ Ω ζ(t, x) ( p(ϱ) T k (ϱ) -S(∇ x u) : R[ζT k (ϱ)] ) dxdt lim R→∞ ∫ T 0 ∫ Ω (u R -ω × x) • ( ζT k (ϱ R )R[ζϱ R u R ] -ζϱ R u R R[ζT k (ϱ R )] ) dxdt - ∫ T 0 ∫ Ω (u R -ω × x) • ( ζT k (ϱ)R[ζϱu] -ζϱuR[ζT k (ϱ)] ) dxdt, (6.15)
where

(R[z]) j = ∑ 3 k=1 R jk [z k ].
In fact, other (lower order) terms that should eventually appear at the right hand side are zero by virtue of a standard compactness argument.

At this stage, we need a div-curl type lemma in the form [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]Theorem 10.27], stating Lemma 6.1.

Let U R ⇀ U in L p (R 3 ; R 3 ), v R ⇀ v in L q (R 3 ) as R → ∞, where 1 p + 1 q = 1 r < 1. Then v R R[U R ] -R[v R ]U R ⇀ vR[U] -R[v]U in L r (R 3 ; R 3 ).
Combining this lemma with the convergence established in the first two lines of (6.9), we get

( ζT k (ϱ R )R[ζϱ R u R ] -ζϱ R R[ζT k (ϱ R )]u R ) (t) → ( ζT k (ϱ)R[ζϱu] -ζϱR[ζT k (ϱ)]u ) (t)
weakly in L r (R 3 ; R 3 ) with some r > 6/5 for all t ∈ [0, T ] as R →∞; whence by the compact imbedding L r (Ω n ) ֒→֒→ W -1,2 (Ω n ) and the Lebesgue dominated convergence theorem used over (0, T ) we conclude that the r.h.s. of (6.15) is equal to 0. This yields the effective viscous flux identity (6.14).

Oscillations defect measure and renormalized continuity equation

Due to (2.2), p(ϱ) = dϱ γ + p m (ϱ), for some d > 0, where ∂ ϱ p m (ϱ) ≥ 0. (6.16)

We have

d lim sup R→0 ∫ T 0 ∫ Ω ζ|T k (ϱ R ) -T k (ϱ)| γ+1 dxdt ≤ d lim sup R→∞ ∫ T 0 ∫ Ω ζ ( (T k (ϱ R ) -T k (ϱ) ) (ϱ γ R -ϱ γ )dxdt ≤ (6.17) ∫ T 0 ∫ Ω ζ ( ϱ γ T k (ϱ) -ϱ γ T k (ϱ) ) dxdt ≤ ∫ T 0 ∫ Ω ζ ( p(ϱ) T k (ϱ) -p(ϱ) T k (ϱ) ) dxdt, where ζ ∈ C ∞ c ((0, T ) × Ω), ζ ≥ 0.
The first inequality is an algebraic one, to derive the second one, we have used convexity of ϱ → ϱ γ and concavity of ϱ → T k (ϱ). Finally, to derive the third one, we have employed the standard relation between the weak limits of monotone functions that reads

p m (ϱ)T k (ϱ) -p m (ϱ) T k (ϱ) ≥ 0, (6.18)
cf. e.g. [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]Theorem 10.19].

The right hand side of inequality (6.17) can be calculated from the effective viscous flux identity (6.14); using (6.6) we find that (6.19) where the expression at the left hand side is called oscillations defect measure, see Feireisl [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF].

∫ T 0 ∫ Ωn |T k (ϱ R ) -T k (ϱ)| γ+1 dxdt ≤ c(n)∥T k (ϱ R ) -T k (ϱ)∥ L 2 ((0,T )×Ωn) ≤ ∥T k (ϱ R ) -T k (ϱ)∥ (γ-1)/(2γ) L 1 ((0,T )×Ωn) ∥T k (ϱ R ) -T k (ϱ)∥ (γ+1)/(2γ) L γ+1 ((0,T )×Ωn) ; whence osc γ+1 [ϱ R → ϱ]((0, T )×Ω n ) ≡ sup k>0 lim sup R→∞ ∫ T 0 ∫ Ωn |T k (ϱ R )-T k (ϱ)| γ+1 dxdt ≤ c(n),
On the other hand, relation (6.19) implies that the limit quantities ϱ, u satisfy the renormalized equation of continuity (2.5), see Feireisl [START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF], Let osc q [ϱ R → ϱ](G) < ∞ (6.20)

for 1 q < 1 -1 r , where (ϱ R , u R ) solves the renormalized continuity equation (2.5) (with Ω replaced by G). Then the limit functions ϱ, u solve (2.5) with (Ω replaced by G) as well.

Strong convergence of density sequence

We deduce using Lemma 6.2 that

∫ τ 0 ∫ Ω ϱL k (ϱ) ( ∂ t φ + (u -ω × x) • ∇ x φ ) dxdt (6.21) = ∫ τ 0 ∫ Ω T k (ϱ)divuφdxdt+ ∫ Ω ϱL k (ϱ)φdx τ 0 ,
where L k (ϱ) is defined in (6.13), τ ∈ [0, T ] and φ ∈ C ∞ c ([0, T ] × Ω). Reasoning as in (6.18) we verify that p(ϱ)T k (ϱ) -p(ϱ) T k (ϱ) ≥ 0; (6.22) whence subtracting (6.12), (6.21) and noticing that (ϱ log ϱ -ϱ log ϱ)(0) = 0, we deduce after letting k → ∞, ∫ Ω ( ϱ log ϱ -ϱ log ϱ

) (τ )Φ R (x) (6.23) ≤ ∫ τ 0 ∫ Ω ( ϱ log ϱ -ϱ ∞ log ϱ ∞ ) (u -ω × x) • ∇ x Φ R (x)dxdt - ∫ τ 0 ∫ Ω ( ϱ log ϱ -ϱ ∞ log ϱ ∞ ) (u -ω × x) • ∇ x Φ R (x)dxdt.
Here, we have used (6.22), and we have chosen Φ R (x) = η(

x 3 /R α )ϕ(|x ′ |/R), R > 1, α > 0, x ′ = (x 1 , x 2 ), η ∈ C ∞ c (R), ϕ ∈ C ∞ c (R), 0 ≤ η, ϕ ≤ 1, ϕ(|x ′ |) =    ϕ(|x ′ |) = 1 if |x ′ | ≤ 1 ϕ(|x ′ |) = 0 if |x ′ | ≥ 2    , η(s) =    η(s) = 1 if |s| ≤ 1 η(s) = 0 if |s| ≥ 2    .
It is easy to check that ∥∇ x Φ R ∥ L ∞ (R 3 ;R 3 ) → 0 as R → ∞, ∥∂ x 3 Φ R ∥ q L q (R 3 ) ≤ cR 2+α-αq , 1 ≤ q < ∞, and that (ω × x) • ∇ x Φ R (x) = 0.

Consequently, ∫

Ω ( ϱ log ϱ -ϱ log ϱ

) (τ )Φ R (x) (6.24) ≤ ∫ τ 0 ∫ Ω ϱ log ϱ -ϱ ∞ log ϱ ∞ U ∞ ∂ x 3 Φ R (x) dxdt - ∫ τ 0 ∫ Ω ϱ log ϱ -ϱ ∞ log ϱ ∞ |U ∞ | ∂ x 3 Φ R (x) dxdt, + ∫ τ 0 ∫ Ω ( ϱ log ϱ -ϱ ∞ log ϱ ∞ ) (u -U ∞ ) • ∇ x Φ R (x)dxdt

Lemma 3 . 1 .G

 31 Let G ⊂ R N be a bounded domain with Lipschitz boundary. Then there exists a linear operator B G = (B ) with the following properties:1.

. 4 )

 4 Definition 4.1. A couple (ϱ, u) is a renormalized finite energy weak solution of the penalized problem (4.1-4.4) if

(4. 5 )

 5 (ii) Equation of continuity(1.8) is satisfied in the weak sense, see(2.4) with Ω replaced by V for any test function φ ∈ C ∞ c ([0, T ] × V ), as well as in the renormalized weak sense, see(2.5) with Ω replaced by V for any b as in (2.5) and any φ ∈ C 1 c ([0, T )×V ).

  [START_REF] Galdi | On the motion of a rigid body in a viscous liquid: A mathematical analysis with applications[END_REF] for a. a. τ ∈ (0, T ).When deriving(5.14) we have used the facts that: 1) ω and u ∞ are parallel if x ∈ B3 2 R \ B 1 and 2) properties (3.1), (3.2) and (5.13).

Lemma 6 . 2 (

 62 Renormalized continuity equation). Let G ⊂ R 3 be open set and letϱ R ⇀ ϱ in L 1 ((0, T ) × G), u R ⇀ u in L r ((0, T ) × G; R 3 ), ∇u R ⇀ ∇u in L r ((0, T ) × G; R 3×3 ), r > 1.

We shall give more details on this procedure when treating the limit R → ∞ in the next section

* S. Kracmar acknowledges the support of the Czech Science Foundation (GA ČR) project P201/11/1304 † S. Nečasová acknowledges the support of the Czech Science Foundation (GA ČR) project P201/11/1304 in the general framework of RVO 67985840 ‡ The work was supported by the MODTERCOM project within the APEX programme of the region Provence-Alpes-Côte d'Azur, France

Due to uniform bounds (6.2-6.6), the right hand side of this inequality tends to 0 as R → ∞, provided we chose α > 2. Consequently,

It is to be noticed that the couple (ϱ, u) satisfies the energy inequality in the form

25) for a. a. τ ∈ (0, T ), where U ∞ is defined in (6.1).