Keywords: dual-frequency laser, solid-state lasers, terahertz, tunable

A two-axis, two-polarization, dual-frequency Ti:Sa laser optimized for high spectral purity CW-THz generation is demonstrated. The laser output power is 50mW. Its mean emission wavelength is optimized around 780nm in order to suit the maximum efficiency of low temperature grown GaAs photomixers. Despite the extremely wide gain bandwidth of Ti:Sa, a proper intracavity filtering design, adapted to this particular laser architecture, enables tunable and singlefrequency operation of each laser mode. The resulting beatnote is shown to be tunable between DC and 1.5 THz by steps of 82GHz. Its linewidth is narrower than 30 kHz without any active stabilization or cavity insolation.

I. INTRODUCTION

ontinuous-wave, tunable THz radiation can be obtained from the beating of two optical fields on a photoconductive antenna [START_REF] Peytavit | Terahertz electromagnetic generation via optical frequency difference[END_REF] [START_REF] Mcintosh | Terahertz photomixing with diode lasers in low-temperature-grown GaAs[END_REF]. The design usually encountered for THz emission on such antennas relies on the beating of two independent lasers [START_REF] Peytavit | Terahertz electromagnetic generation via optical frequency difference[END_REF] [START_REF] Huggard | Generation of millimetre and sub-millimetre waves by photomixing in 1.55 µm wavelength photodiode[END_REF]. Alternative sources has been proposed, such as two longitudinal-mode lasers [START_REF] Hidaka | CW terahertz wave generation by photomixing using a two-longitudinal-mode laser diode[END_REF], or tunable dual-color laser [START_REF] Siebe | A fully tunable dual-color CW Ti:Al2O3 laser[END_REF] [START_REF] Luo | A widely tunable dual-wavelength CW Ti:sapphire laser with collinear output[END_REF]. Semiconductor lasers are compact and low cost, however they suffer from a relatively poor spectral quality, because of the short carrier recombination time and of the Henry factor [START_REF] Osinski | Linewidth broadening factor in semiconductor lasers-An overview[END_REF]. On the contrary, solid state dual-frequency lasers offer a much better spectral purity [START_REF] Czarny | THzdual-frequency Yb 3+ :KGd(WO4)2 laser for continuous wave THz generation through photomixing[END_REF] [START_REF] Alouini | Dual tunable wavelength Er,Yb:glass laser for terahertz beat frequency generation[END_REF]. In a dual-frequency laser, the two optical eigenmodes oscillate with crossed polarizations, and can thus be tuned independently. Furthermore, since the two laser modes oscillate in the same cavity, their frequency fluctuations are strongly correlated and cancel out almost perfectly on the frequency difference. The relevance of this solution was demonstrated in ref [START_REF] Czarny | THzdual-frequency Yb 3+ :KGd(WO4)2 laser for continuous wave THz generation through photomixing[END_REF] at 1.03 µm in a Yb 3+ :KGd(WO 4 ) 2 laser, and in refs [START_REF] Alouini | Dual tunable wavelength Er,Yb:glass laser for terahertz beat frequency generation[END_REF][10] at 1.55 µm, using a

The high resolution spectrometer has been provided by Resolution Spectra System inc within the EU programme NEXPRESSO-Network for EXchange and Prototype Evaluation of photonicS componentS and Optical systems.

The authors are with the Institut de Physique de Rennes, UMR UR1-CNRS 6251, Université of Rennes1 Campus de Beaulieu, 35042 Rennes Cedex,

France.

(e-mail: goulc-hen.loas@univ-rennes1.fr; marco.romanelli@univ-rennes1.fr; mehdi-alouini@univ-rennes1.fr).

Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org.

Er:Yb-doped glass as an active medium. Unfortunately, the most mastered and widespread ps lifetime photoconductive material is LT-GaAs, that requires illumination at optical wavelengths around 800 nm [START_REF] Mcintosh | Terahertz photomixing with diode lasers in low-temperature-grown GaAs[END_REF]. Given its broad emission spectrum [START_REF] Albers | Continuous Wave Tunable Laser Operation of Ti 3+ -Doped Sapphire at 300 K[END_REF], Ti:Al 2 O 3 is a natural candidate for tunable dual-frequency emission at 800 nm. The main difficulty of this approach is the obtainment of single-frequency emission and tunability on both polarization directions because of the extremely wide gain spectral width of Ti:Al 2 O 3 . Indeed, the most common technique to ensure single mode operation of a Ti:Al 2 O 3 laser relies on designing a ring cavity, to get rid of spatial hole burning, in which a Lyot filter and multiple etalons are inserted. Such architecture does not apply for a dual frequency laser which is by essence closed by a linear cavity and which must sustain the oscillation of two crossed polarizations. In the following, we describe how we have designed the laser cavity and the pump geometry in order to obtain dual-frequency and dual-polarization operation. In this work, we report tunable dual-frequency emission from a compact Ti:Sa laser, and demonstrate the high spectral purity of the associated radiofrequency beat note.

II. LASER DESIGN

The dual-frequency laser cavity is represented in Fig. 1. It includes two Ti:Sa crystals as gain media, pumped by a 532 nm single-frequency laser (Coherent, Verdi-12). The cavity is a short plano-concave resonator closed by mirror M 2 , whose radius of curvature is 5 cm. M 1 is a plane mirror coated for maximum reflection (R=99.9%) in the range 740-860 nm and for high transmission (T > 95%) at 532 nm. M 2 is a customized mirror with a maximum reflectivity of 95 % at 780 nm, and a reduced bandwidth of around 20 nm (see Fig. 2). The laser operates near the end of the stability region, it thus gives a free spectral range close to 2.4 GHz with all intracavity components. The cavity is chosen as short as possible, Dual-frequency 780 nm Ti:Sa laser for high spectral purity tunable CW THz generation Goulc'hen Loas, Marco Romanelli, and Mehdi Alouini C that is, around 5 cm in order to facilitate single mode oscillation.

The spectral selectivity of the output coupler M 2 is displayed in Fig. 2, where the amplified spontaneous emission spectrum, obtained when the laser is pumped below threshold, is measured using either a standard mirror (740nm-860nm) or the narrowband M 2 mirror as output couplers. We can note that, when M 2 is used, the bandwidth of the oscillator is drastically reduced to about 10 THz, centered at 780 nm.

AR coated optics are used to focus the pump inside the Ti:Al 2 O 3 crystal. The pump beam waist in the laser crystals was measured to be 40 µm, so that the Rayleigh range is about 7 mm, comparable with the crystal length of 5 mm. Spatial separation inside a portion of the cavity is needed both for independent tuning of each eigenfrequency, and for obtaining a frequency difference larger than the cavity free spectral range [START_REF] Bretenaker | Laser eigenstates in the framework of a spatially generalized Jones matrix formalism[END_REF]. In order to obtain two separated pump beams, a birefringent YVO 4 crystal cut at 45° of its optical axis (Roditi, 5x5x18.36 mm, AR @ 532 nm) is used as a beam displacer. At the crystal end, the two pump beams are spatially separated by 2 mm and orthogonally polarized. Given the fact that the absorption and emission cross-sections of Ti:Sa are anistropic [START_REF] Moulton | Spectroscopic and laser characteristics of Ti:Al2O3[END_REF], we have inserted two separated lasers crystals (Roditi, 4x4x5 mm, doping 0.25 % weight, figure of merit FOM > 150) as active media with their c-axes mutually orthogonals. A second birefringent crystal is thus necessary to recombine the oscillating modes inside the laser cavity. Because of its good transparency, we have chosen a calcite crystal (Moltech, 5x5x18.82 mm, AR @ 800 nm) to compensate the 2-mm beam separation at 780 nm. The crystal length is calculated to optimize the overlap of the ordinary and extraordinary modes with the two pump spots at 780 nm. In principle, the overlap is not perfect for other wavelengths because of dispersion. However, in practice this effect is negligible and does not lead to a degradation of the Gaussian spatial profile of the output beam. For example, a wavelength shift of 3 nm, necessary to obtain a 1.5 THz beat-note, entrains a lateral displacement of the cavity mode waist of less than 0.6 µm. The lasing threshold is obtained for a pump power of 2 W on each arm when all the elements are inserted in the cavity.

We now detail the role of the intracavity elements that are used for spectral selectivity, namely the half-wave plate (HWP) and the étalons.

At first, two thin étalons were inserted on the two separated arms of the laser cavity. These filters are fused silica plates of thickness 25 µm, optical index n=1.47 @ 780 nm, and coated in order to have an intensity reflection coefficient R=50 %. When the étalon is perpendicular to the ordinary propagation axis, the laser oscillation occurs near 782nm and tunability on a few nms is possible. The FSR of the thin étalons is of 8.3 nm or equivalently 4 THz. For other solid-state lasers such as twoaxis Yb 3+ :KGd(WO 4 ) 2 or Er:Glass lasers [START_REF] Czarny | THzdual-frequency Yb 3+ :KGd(WO4)2 laser for continuous wave THz generation through photomixing[END_REF][9], the presence of a rather similar thin étalon is sufficient to ensure stable singlefrequency operation. We found that it was not the case here. In some particular cases, single frequency may be obtained near threshold. Nevertheless, we observed, using a high resolution spectrometer ("Zoom Spectra", Resolution Spectra System Inc) whose resolution is comparable to the free spectral range of the laser cavity, that in general the thin étalon does not provide strong enough selection on adjacent modes of the laser cavity, which are separated by only 2.4 GHz. Thus the "fine" spectral selectivity must be improved. Moreover, even if most of the times one observes a single peak in the optical spectrum for each polarization, under some experimental conditions lasing was observed on two consecutive longitudinal modes of the thin étalon, despite their separation of 8.3 nm. Thus, also the "coarse" spectral filtering provided by M 2 is not sufficient, given the enormous gain bandwidth of Ti:Sa. The situation that is typically encountered is illustrated in Fig. 3. The optical spectrum displays a single peak for each polarization (Fig. 3(a)), but each peak shows a fine structure due to the presence of some (typically 2-3) laser cavity longitudinal modes (Fig. 3(b)). The difficulty in filtering adjacent longitudinal cavity modes is due to the spatial hole burning that is present in a linear, standing-wave laser resonator [START_REF] Mcmackin | Instabilities and chaos in a multimode, standing-wave, cw dye laser[END_REF]. This difficulty is usually circumvented by designing a ring laser cavity [START_REF] Schulz | Single-frequency Ti:Al2O3 ring laser[END_REF], but this cannot be done for dual frequency lasers. Compared to refs [8][9] where the active medium length is 2mm, our longer 5mm long Ti:Sa crystal contributes to enhance spatial hole burning effect.

In order to improve the spectral selectivity, we cannot consider increasing the reflection coefficient of the thin etalon further. Indeed, the reflection coefficient is already quite high, and increasing it further might lead to multiple cavity effects and would induce important losses when the étalon is tilted, thus limiting the oscillator tunability [START_REF] Leeb | Losses introduced by tilting intracavity etalons[END_REF]. Thus, we have chosen to add a thick étalon (1 mm thick YAG plate, n= 1.82, uncoated) on the common path of the two modes. This étalon has a free spectral range (FSR) of 82 GHz. The presence of this filter was found to suppress efficiently adjacent laser cavity modes, thus solving the issue of fine spectral selectivity. Nevertheless, under some conditions two modes of the thin étalons could still be observed, requiring an increase of the coarse selectivity.

In principle, the coarse filtering could be improved by decreasing the thickness of the thin étalon, or by reducing further the bandwidth of the mirror M 2 , but both options turned out to be technically unfeasible. So, we resorted to an additional spectral filtering technique which takes advantage of the presence of a polarization walk-off crystal inside the cavity. To this end, we inserted a HWP in the cavity. The HWP neutral axes are at 45° to the ordinary and extraordinary polarizations, so that the laser polarization remains linear, and its direction is rotated by 90°, if the wavelength is an exact multiple of the HWP optical thickness. On the contrary, for all other wavelengths the polarization becomes elliptical, and this results in intracavity losses when the beam passes through the intracavity calcite crystal. Thus the combination "HWP+birefringent crystal" forces the laser to oscillate preferentially on the HWP resonances. Note that the HWP plate must be placed where the modes are spatially separated, otherwise forked cavity eigenstates are obtained [START_REF] Bretenaker | Laser eigenstates in the framework of a spatially generalized Jones matrix formalism[END_REF]. The losses induced by the HWP plate depend on the plate order, i.e., its thickness. We used an 11 th order plate centered at 780 nm. This introduces sufficient losses on the adjacent modes of the thin étalon to guarantee single frequency operation far from laser threshold.

The combined action of all the spectral filters resulted in stable and tunable single-frequency emission on each laser arm. The total output power is of 50 mW when pumped with 6.4 W. Tunability is obtained by tilting the thin étalons and is analyzed thanks to the high resolution spectrometer (Fig. 4). The frequency difference can be conveniently tuned from a few GHz up to 1.5 THz, corresponding to the frequency bandwidth of LT-GaAs photoconductive materials. Obviously, the frequency difference variations are not continuous because the laser modes "jump" by steps of 82 GHz corresponding to the free spectral range of the YAG plate. In order to detect and characterize the beating between the two laser modes, we have used a 10GHz GaAs photodiode directly connected to an electrical spectrum analyzer. To this end, we have tuned the laser so that the two lasing wavelengths could not be distinguished on the high resolution optical spectrum analyzer. Then, by placing the photodiode after a polarizer, we have recorded the electrical spectrum of the beat-note signal. We note that the beat-note signal amplitude can always be optimized by suitably orienting the polarizer, even if the intensities of the two polarizations are not exactly the same. In particular, this allows compensating for variations of the output power that occur after wavelength tuning.

As illustrated in (Fig. 5(a)) one observes a single peak, thus confirming that only one longitudinal mode for each polarization of the laser cavity (FSR = 2.4 GHz) is present. By zooming this peak, it is possible to measure the linewidth of the beat-note signal (Fig. 5(b)), which is found to be around 30 kHz with a resolution bandwidth (RBW) of 30 kHz and a measurement time of 0.3 s. The drift of the beat-note frequency is measured to be of a few tens of MHz over a minute. We have checked that the laser exhibits the same performances in terms of linewidth and frequency drift whatever its mean wavelength. Despite the relatively high pumping power, the presence of two active crystals in the laser and different filtering levels, these performances are similar to that usually obtained with a less complex two axis dual frequency laser at 1.55 µm [START_REF] Rolland | Narrow Linewidth Tunable Terahertz Radiation By Photomixing Without Servo-Locking[END_REF]. These results are very encouraging since the inherently narrow beatnote that is achieved makes it possible to envisage down conversion and phase-locking to a microwave local oscillator provided that a voltage controlled tunability is implemented to the laser [START_REF] Rolland | Non-linear optoelectronic phase-locked loop for stabilization of optomillimeter waves: towards a narrow linewidth tunable THz source[END_REF].

III. CONCLUSION

We have built a compact THz-beatnote two-frequency Ti:Sa laser oscillating around 780 nm. By a proper cavity design and wavelength filtering, tunable single-frequency operation has been achieved on both polarizations independently, thus providing tunable beat notes spanning from the GHz up to the THz range by steps of 82 GHz. The optical output power is around 50 mW, which should allow us the generation of some µWs in the THz range with state-of-the-art photomixers [START_REF] Peytavit | CW Source Based on Photomixing With Output Power Reaching 1.8 mW at 250 GHz[END_REF]. The linewidth of the beat-note has been measured to be around 30 kHz without any acoustic or thermal insulation. The long term drift of this beatnote is a few tens of MHz over a minute without any stabilization. The next step is to implement to this laser a voltage controlled tunability in order to phase lock the beatnote to a local oscillator after down conversion. Given its operating mean-wavelength around 780 nm, this system provides a valuable alternative to produce high spectral purity CW THz radiation taking advantage of the LT-GaAs photomixing technology. Alternatively, it might be optimized for Coherent Population Trapping in Rubidium.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the laser. λ/2: half wave plate. M1: plane dichroic mirror, M2: 95% output coupler, o: ordinary axis, e: extraordinary axis.

Fig. 2 .

 2 Fig.2. Amplified spontaneous emission spectrum using a standard output coupler (blue) and the customized narrowband M2 mirror (red).

Fig. 3 .

 3 Fig. 3. (a) Optical spectrum of the two-axis laser. With respect to Fig.1, the intracavity HWP and thick étalon are absent. The two peaks correspond to ordinary and extraordinary polarization respectively. (b) Zoom on the first peak, showing that additional filtering is required to get single-frequency emission for each polarization.

Fig. 4 .

 4 Fig.[START_REF] Hidaka | CW terahertz wave generation by photomixing using a two-longitudinal-mode laser diode[END_REF]. Optical spectrum of the two-axis laser as one intracavity thin étalon is tilted while the second one is fixed. Note that each wavelength can be tuned independently. Res 0.01nm.

Fig. 5 .

 5 Fig. 5. Electrical spectrum of the beat-note in the microwave domain. The photocurrent signal is detected by a fast GaAs photodiode. (b) Zoom on the beat-note. RBW 30 kHz, measurement time 0.3 s.

ACKNOWLEDGMENT

The authors thank Cyril Hamel for the mechanical design and optimization. as well as Resolution Spectra System Inc. for providing the "Zoom Spectra" high resolution spectrum analyzer used in this work.


G Bayada
email: guy.bayada@insa-lyon.fr

L Chupin

B Grec

Some theoretical results concerning diphasic flows in thin films

Keywords: Cahn-Hilliard equation, Reynolds equation, Thin flow

We are interested in a model for diphasic fluids in thin flows taking into account both the hydrodynamical and the chemical effects at the interface between the two fluids. A limit problem in thin curved channels is introduced heuristically. It is a system coupling the Reynolds equation and the Cahn-Hilliard equation. We study the mathematical properties of this system, and prove an existence result under some smallness condition on the data.

Introduction

In many applications, the geometry of the flow is anisotropic (i.e. one dimension is small with respect to the others), e.g. in lubrication problems. In the Newtonian case, the flow of a fluid between two close surfaces in relative motion is described by an asymptotic approximation of the Navier-Stokes equations, the Reynolds equation. This equation makes it possible to uncouple the pressure and the velocity. Indeed, in thin films, the pressure is considered to be independent of the direction in which the domain is thin. Thus an equation on the pressure only is obtained, and the velocity can be deduced from the pressure. This approach was introduced by Reynolds, and has been rigorously justified in [START_REF] Bayada | The transition between the Stokes equations and the Reynolds equation: a mathematical proof[END_REF] for the Stokes equation, and generalized afterwards in many works: for the steady-case Navier-Stokes equations [START_REF] Assemien | Inertial effects in the asymptotic behavior of a thin film flow[END_REF], for the unsteady case [START_REF] Bayada | Asymptotic Navier-Stokes equations in a thin moving boundary domain[END_REF], for compressible fluids with the perfect gases law [START_REF] Marusić-Paloka | Rigorous justification of the Reynolds equations for gas lubrication[END_REF]... It is of interest to investigate how this approach can be used for the case of a two fluid flow.

A first diphasic model consists in introducing a variable viscosity η, which is either equal to the viscosity η 1 of one fluid or the viscosity η 2 of the other fluid (that is to say that the fluids are considered to be non-miscible). The behavior of η is described by a transport equation. In that case, when assuming the interface between the two fluids to be the graph of a function, the asymptotic equations corresponding to the thin film approximation can be interpreted as a generalized Buckley-Leverett equation, which governs the behavior of the saturation (i.e. the proportion of one fluid in the mixture) inside the gap, coupled with a generalized Reynolds equation, which governs the behavior of the pressure. These equations are investigated in [START_REF] Paoli | Asymptotic behavior of a two fluid flow in a thin domain: from Stokes equations to Buckley-Leverett equation and Reynolds law[END_REF] without shear effects, and in [START_REF] Bayada | About a generalized Buckley-Leverett equation and lubrication multifluid flow[END_REF], [START_REF] Chupin | Steady state solutions for a lubrication two-fluid flow[END_REF] with shear effects. One of the main disadvantages of the method is that the fluid interface is supposed to be the graph of a function, which hinders for example the formation of bubbles. In addition, this kind of model only takes into account hydrodynamical effects between the two phases, and surface tension effects are neglected.

The second class of models describing diphasic flows, which has been used up to now only for the Navier-Stokes equations, is the class of the so-called diffuse interface models. They take into account chemical properties at the interface between the two fluids, enabling an exchange between the two phases. In this paper, we use a Cahn-Hilliard equation, which involves an interaction potential, enhanced with a transport term. Thus this model describes both the chemical and the hydrodynamical properties of the flow. An order parameter ϕ is introduced, for example the volumic fraction of one phase in the mixture. The surface tension can be taken into account via an additional term depending on ϕ in the Navier-Stokes equations. This kind of model has been studied for the complete Navier-Stokes equations in [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF], and for viscoelastic fluids in [START_REF] Chupin | Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation[END_REF].

In this paper, we consider an asymptotic system (i.e. a thin film approximation) for a diphasic fluid modelled by the Cahn-Hilliard equation. In a similar way as for the Newtonian case, the Navier-Stokes equations are approximated by a modified Reynolds equation, in which the viscosity is not constant anymore. We study the Reynolds/Cahn-Hilliard system, and prove the existence and the regularity of a weak solution under a smallness assumption on the initial data and the geometry.

Let us describe briefly the main steps of the mathematical analysis. First, we study the Reynolds equation and investigate the regularity of the pressure and the velocity as functions of the order parameter. Next, we prove the existence of a solution to the system Reynolds/Cahn-Hilliard, by using a Galerkin process, which consists in introducing finite dimension approximations of ϕ. After obtaining a priori estimates for these approximations, we conclude that they converge to a solution of the system Reynolds/Cahn-Hilliard.

This paper is organized as follows. In Section 2, we introduce the twodimensional model for a diphasic fluid in a thin film, which consists of a generalized Reynolds equation and of a diffuse-interface model (the Cahn-Hilliard equation). In Section 3, we state the main theorem, and give the main steps and difficulties of the proof. In Section 4, we deal with the Reynolds equation, and obtain some existence and regularity result on the velocity field and the pres-sure. In Section 5, we first introduce some specific results on trace estimates and Poincaré inequalities. They are used in the rest of the section for obtaining a priori estimates for the Cahn-Hilliard equation. At last, convergence results are deduced from these estimates, and allow to conclude the proof of the main theorem. Section 6 presents some preliminary numerical results obtained with this model in order to highlight the features of the model.

Modelling a diphasic fluid in a thin film

In this section, we will first present how a fluid is described in a thin domain by the Reynolds equation. Next, we introduce the hydrodynamical Cahn-Hilliard model for any fluid. Lastly, we combine both aspects and state the model of a diphasic fluid in a thin domain. We introduce the physical domain Ω Ω " px, zq P R 2 , 0 ă x ă L, 0 ă z ă hpxq ( .

The thin film approximation for an incompressible fluid leads to the following equations (see [START_REF] Bayada | The transition between the Stokes equations and the Reynolds equation: a mathematical proof[END_REF]), describing the behavior of the pressure p and the velocity field u " pu, vq, η being the viscosity of the fluid.

B z pη B z uq " B xp, B z p " 0, B xu `Bz v " 0.
In these equations, the thin film assumption leads to the decoupling of the pressure and the velocity, as well as the simplification of the equations.

We will see that it is possible to prove an existence theorem assuming a small size condition on the physical domain Ω (see Theorem 3.3). In order to understand the dependance of the solution with respect to the domain Ω, we rescale the spatial variable px, zq using a dilatation coefficient λ. More precisely, we suppose that the domain is small and can be written as Ω " tpx, zq P R 2 , 0 ă λx ă λL, 0 ă λz ă λhpxqu, and we rewrite the system using the following change of variable and domain

λx Ñ x, λz Ñ z, Ω Ñ Ω " tpx, zq P R 2 , 0 ă x ă L, 0 ă z ă hpxqu. (2)
We assume that there exists three constants ph m , h M , h 1 M q P R 3 `such that the function h P C 2 pRq (see Fig. 1) satisfies

@x P r0, Ls, 0 ă h m ď hpxq ď h M and |h 1 pxq| ď h 1 M , (3)
and h 1 pLq " 0 as well as Dε ą 0 such that @x P r0, εs, h 1 pxq " h 2 pxq " 0, Observe that the regularity of h ensures that the domain Ω defined by (2) satisfies the segment property and cone property (see [2, § 4.2 and 4.3]).

x z u " ps, 0q The Reynolds equation now writes

u " p0, 0q Q Ω hpxq 0 L ε
B z pη B z uq " λ B x p, B z p " 0, B x u `Bz v " 0. ( 4
)
We choose boundary conditions on u suitable for lubrication applications: Dirichlet boundary conditions are imposed on the velocity on tz " 0u and tz " hpxqu in order to model shear effects. The boundary conditions are written: @x P r0, Ls upx, 0q " s and upx, hpxqq " vpx, 0q " vpx, hpxqq " 0.

Without loss of generality, the constant shear velocity s is supposed to be positive. For the lateral part of the boundary, it has been showed in [START_REF] Bayada | The transition between the Stokes equations and the Reynolds equation: a mathematical proof[END_REF] that only the input flow Q " ż hp0q 0 up0, ξq dξ needs to be prescribed. Observe that according to the divergence-free condition and the boundary conditions on u, this flow is constant on any "vertical" section of the domain:

B x ˜ż hpxq 0 upx, ξqdξ ¸" h 1 pxqupx, hpxqq loooooooomoooooooon "0 `ż hpxq 0 B x upx, ξqdξ " ´ż hpxq 0 B ξ vpx, ξqdξ " ´vpx, hpxqq `vpx, 0q " 0, thus Q " λ ż hpxq 0 upx, ξqdξ, @x P p0, Lq. (6)
Remark 2.1. We use the Reynolds equation to describe the behavior of the fluid. This equation is an approximation of the (Navier)-Stokes system for thin domains (in which the height is much smaller than the length). The anisotropy of the physical domain is therefore taken into account in this step. Further, the equation is written down in (4) in a rescaled form in the domain Ω (with length and height of the same order of magnitude). No assumption on the shape is needed for the domain Ω. As we already stated, the parameter λ will allow us to control the smallness of the physical domain.

Modelling one fluid in a thin domain

The usual procedure [START_REF] Bayada | The transition between the Stokes equations and the Reynolds equation: a mathematical proof[END_REF] is to integrate twice the first equation of (4) with respect to z, make use of the boundary conditions [START_REF] Boyer | Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model[END_REF] and of the fact that B z p " 0. This allows us to express u as a function of p:

upx, zq " zpz ´hpxqq 2η λ B x ppxq `s ˆ1 ´z hpxq ˙. (7)
Then, putting this expression in the divergence-free equation leads to the Reynolds equation:

λ B x ˆh3 12η B x p ˙" sB x ˆh 2 ˙. (8)
A first boundary condition on p is deduced from the ones on u. In fact, the choice of the input flow Q corresponds to a Neumann condition for p at x " 0. This condition can be determined as a function of Q by

Q " λ ż hp0q 0 up0, ξqdξ " ´λ2 B x pp0q hp0q 3 12η `λshp0q 2 .
Let us denote w :" 12ηpλshp0q{2 ´Qq λ 2 hp0q 3 " B x pp0q.

Moreover, the solution p of ( 8) with the Neumann boundary condition B x pp0q " w is defined up to a constant. We can thus choose ppLq " 0 to gain a well-defined pressure p. It is to be noticed that once p is computed from [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF], then [START_REF] Bayada | About a generalized Buckley-Leverett equation and lubrication multifluid flow[END_REF] allows us to compute u, while the other component v of the velocity field is obtained by: vpx, zq " ´ż z 0 B x upx, ξq dξ.

Modelling a mixture

Since we want to study the mixture of two fluids, we introduce an order parameter ϕ describing the volumic fraction of one fluid in the flow. All physical parameters can be written as functions of ϕ, in particular the viscosity η. We assume that the function η satisfies η P C 1 pRq such that

Dpη m , η M , η 1 M q P R 3 ; @ϕ P R, 0 ă η m ď ηpϕq ď η M and η 1 pϕq ď η 1 M . (9)
A possible explicit form of the viscosity is given in the following Remark:

Remark 2.2. For ϕ P r´1, 1s, we can use a specific realistic law as a function of the viscosities of the two fluids η 1 and η 2 (see [START_REF] Boyer | A theoretical and numerical model for the study of incompressible mixture flows[END_REF] or [START_REF] Onuki | Phase transitions of fluids in shear flow[END_REF]):

1 ηpϕq " 1 `ϕ 2η 1 `1 ´ϕ 2η 2 for ϕ P r´1, 1s, (10)
so that ϕ " 1 and ϕ " ´1 correspond respectively to the fluids of viscosity η 1 and η 2 only. However, we will not always be able to prove mathematically that ϕ remains in the interval r´1, 1s (see [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF]).

The effects of a possible variation of the density in the mixture will not be taken into account in this paper. Therefore, the density of the mixture is assumed to be constant (i.e. the two densities of the two incompressible phases ρ 1 and ρ 2 are supposed to be equal). Let us notice that due to the loss of the local conservation equation for the density, the non-homogeneous case ρ 1 ‰ ρ 2 induces further difficulties (see [START_REF] Boyer | Nonhomogeneous Cahn-Hilliard fluids[END_REF]).

We choose the Cahn-Hilliard equation in order to describe the evolution of ϕ. This equation consists of both a transport term, taking the mechanical effects into account, and a diffusive term modelling the chemical effects. The Cahn-Hilliard equation is written in the rescaled domain Ω:

λ B t ϕ `u ¨∇ϕ ´1 λ Pe div pBpϕq∇µq " 0, (11)
µ " ´α2 λ 2 ∆ϕ `F 1 pϕq. ( 12
)
Recall that the constant λ is a rescaling constant allowing us to follow the dependance on the domain size. The variable µ is the chemical potential, Bpϕq is called mobility, Pe is the Péclet number, α is a non-dimensional parameter measuring the thickness of the diffuse interface, and the function F is called Cahn-Hilliard potential. Physical considerations show that F must have a double-well structure, each of the wells representing one of the two fluids. A rational choice for F is given by a logarithmic form (for more details, we refer to [START_REF] Doi | Dynamics of domains and textures[END_REF] or [START_REF] Guton | Phase transitions and critical phenomena[END_REF])

F pξq " 1 ´ξ2 `c pp1 `ξq logp1 `ξq `p1 ´ξq logp1 ´ξqq , for some constant 0 ă c ă 1, or its polynomial approximation

F pξq " p1 ´c1 ξ 2 q 2 ,
where c 1 is another constant. These physically realistic potentials share several mathematical properties. In the following, we prove mathematical results for potentials F having these properties:

• The function F is supposed to be regular (e.g. of class C 2 pRq).

• Since F is a physical potential, it is bounded from below. Moreover, only the derivative of F occurs in the equations, therefore the addition of a constant does not change the equations. It is thus realistic to make the following assumption:

DF 0 ą 0 ; @ξ P R F pξq ě F 0 . (13)
• The convexity of the potential corresponds to the stability of the mixture. Usual potentials contain some stable and unstable regions (see for example Figure 2). In order to include such cases, we impose:

DF 5 ě 0 ; @ξ P R F 2 pξq ě ´F5 . (14)
• Moreover, the following hypothesis on the growth of the potential is imposed:

DF 1 , F 2 ą 0 Dr ą 1 ; @ξ P R |F 1 pξq| ď F 1 |ξ| r `F2 and |F 2 pξq| ď F 1 |ξ| r´1 `F2 . ( 15
)
This hypothesis is satisfied for any polynomial function.

• At last, we state a generalization of the convexity:

@γ P R DF 3 pγq ą 0, F 4 pγq ě 0 ;

@ξ P R pξ ´γqF 1 pξq ě F 3 pγqF pξq ´F4 pγq. (16)
ϕ F pϕq These assumptions are satisfied by a function of the form F pϕq " ϕ 4 4 ´ϕ2 2 `F0 (as in Figure 2), which can be used as a model case. As far as the mobility B is concerned, it is supposed to be regular B P C 2 pRq, positive, and bounded from above and from below:

DpB m , B M q P R 2 ; @ξ P R 0 ă B m ď Bpξq ď B M . (17)
Let us mention that other types of functions B can be considered, in particular the degenerate case Bpξq " p1 ´ξ2 q σ , with σ ě 0, which has been studied in [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF] and in [START_REF] Elliott | On the Cahn-Hilliard equation with degenerate mobility[END_REF], but introduces further mathematical difficulties. Equations ( 11)-( 12) must be equipped with boundary conditions on ϕ and µ.

We are interested here in injection phenomena, which arise for example in lubrication or polymer injection problems. To this end, it is important to control the composition of the input. Thus we use Dirichlet boundary conditions on some part of the boundary, namely where the fluid is supplied. For the other part of the boundary, classical Neumann boundary conditions for both ϕ and µ are considered. Let us observe that in previous works ( [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF] and [START_REF] Chupin | Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation[END_REF]) Neumann boundary conditions were imposed on the whole boundary.

x z Let us define (see Figure 3)

Γ l Γ r Γ 0 Γ b Γ t Ω hpxq 0 L
Γ l " tp0, zq P R 2 , 0 ă z ă hp0qu, Γ r " tpL, zq P R 2 , 0 ă z ă hpLqu, Γ b " tpx, 0q P R 2 , 0 ă x ă Lu, Γ t " tpx, zq P R 2 , z " hpxqu, Γ 0 " tpx, zq P BΩ, x ą 0u.
Thus, the boundary conditions are written, denoting n the exterior normal to the domain, as follows:

ϕ| Γ l " ϕ l , µ| Γ l " 0 and Bϕ Bn ˇˇΓ 0 " 0, Bµ Bn ˇˇΓ 0 " 0, (18)
for some given boundary value ϕ l defined on Γ l , satisfying the following hypothesis:

Hypothesis 2.3. We assume that ϕ l P H 7{2 pΓ l q satisfies

ϕ 1 l p0q " ϕ 1 l php0qq " 0, |ϕ 1 l | L 2 pΓ l q ă ε. ( 19
)
for some small ε ą 0 depending on all the data. We will explain further how ε is determined (see Proposition 5.12).

Finally, let us define the initial condition: ϕ| t"0 " ϕ 0 P H 3 pΩq, where ϕ 0 is supposed to be satisfying the same boundary conditions as ϕ. Compatibility conditions also imply that µ 0 defined by µ 0 " ´α2 λ 2 ∆ϕ 0 `F 1 pϕ 0 q satisfies the same boundary conditions as µ.

Modelling a mixture in thin films

A diphasic flow in a thin domain is described by a modified Reynolds system of the form (4), where the viscosity η is not constant anymore but depends on the order parameter ϕ. Because of the non-constant viscosity, the coefficients in the Reynolds equation (which depend on η) depend on ϕ. Let us introduce the following expressions that will be useful in the following:

We use the fact that u is divergence-free and the boundary conditions in order to write ż hpxq 0 B x upx, zq dz " B x ˜ż hpxq 0 upx, zq dz ¸" 0.

After integrating [START_REF] Marusić-Paloka | Rigorous justification of the Reynolds equations for gas lubrication[END_REF], we obtain λ B x ´r dpxqB x p pxq ¯" sB x pr epxqq ,

where the coefficients r d and r e are given by [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF]. Therefore the whole system (Reynolds/ Cahn-Hilliard) is written:

$ ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' % λ B x p r d B x pq " s B x r e (26a) u " ˜b ´a r b r a ¸λB x p `s ´1 ´a r a ¯(26b) vp¨, zq " ´ż z 0 B x up¨, ξqdξ (26c)
λ B t ϕ `u B x ϕ `v B z ϕ ´1 λ Pe divpBpϕq∇µq " 0 (26d) µ " ´α2 λ 2 ∆ϕ `F 1 pϕq. (26e)
The coefficients a, b, r a, r b, r d, r e are explicit functions of ϕ (given by ( 20)-( 21)). The functions B, F are also known functions of ϕ. The quantities Pe, α are physical constants. The boundary conditions on ϕ and µ are given in [START_REF] Girault | Finite element approximation of the Navier-Stokes equations[END_REF]. Let us notice that equations (26b)-(26c) imply that the following boundary conditions are satisfied for u: upx, 0q " s, upx, hpxqq " vpx, 0q " vpx, hpxqq " 0,

λ

ż hp0q 0 up0, ξq dξ " Q. ( 28
)
As far as the pressure p is concerned, we impose an homogeneous Dirichlet boundary condition at x " L and a Neumann boundary condition at x " 0, which is given as a function of Q and s. These boundary conditions are written: (i) For the usual Sobolev spaces, we denote by | ¨|p the L p -norm in Ω, and by } ¨}s the H s -norm in Ω. We also introduce ~¨~2 which contains the second-order derivatives:

w :" B x pp0q " Q ´sλ ˜hp0q ´1 r ap0q
~¨~2 2 " |B z ¨|2 2 `|B 2 x ¨|2 2 `|B 2 xz ¨|2 2 `|B 2 z ¨|2 2 .
(ii) Let us define the following function spaces:

X pΩq " tf P H 1 pΩq X L 8 pΩq, B z f P H 1 pΩqu, Φ 1 0 " tφ P Dp Ωq, φ| Γ l " 0u H 1 pΩq
.

We introduce the weak form of ( 26).

Problem 3.2. Let ϕ l P H 7{2 pΓ l q, and ϕ 0 P H 3 pΩq satisfying (18), and T ą 0. Find pp, u, v, ϕ, µq such that -the following regularity is satisfied:

p P L 8 p0, T ; H 2 p0, Lqq, u P L 8 p0, T ; X pΩqq, v P L 8 p0, T ; L 2 pΩqq, ϕ P L 8 p0, T ; H 1 pΩqq X L 2 loc p0, T ; H 3 pΩqq X C 0 pp0, T q; H 1 pΩqq, µ P L 2 loc p0, T ; Φ 1 0 q.

10

-the velocity field u ϕ " pu, vq is given as a function of ϕ by (26a), (26b), (26c) equipped with the boundary conditions ( 27), ( 28) and (29).

-for any ψ P Φ 1 0 ,

λ ż Ω B t ϕ ψ `żΩ 1 λ Pe Bpϕq∇µ ∇ψ `żΩ u ϕ ¨∇ϕψ " 0, ( 30
) with µ " ´α2 λ 2 ∆ϕ `F 1 pϕq. (31)
-the initial condition ϕ| t"0 " ϕ 0 is satisfied as well as the boundary conditions [START_REF] Girault | Finite element approximation of the Navier-Stokes equations[END_REF] for ϕ.

The following sections are dedicated to the proof of the main theorem:

Theorem 3.3. Let T ą 0, ϕ l satisfying Hypothesis (2.
3), ϕ 0 P H 3 pΩq satisfying (18), F and η satisfying the assumptions stated in Section 2.2. If λ is small enough then there exists a solution pp, u, v, ϕ, µq to Problem 3.2.

Sketch of the proof

We present here the sketch of the proof of the main theorem. All the details and computations are given in Sections 4 and 5. The proof is divided into two main parts, since the Reynolds equation and the Cahn-Hilliard are treated separately.

Step 1. As far as the Reynolds equation is concerned, we prove the following proposition: Proposition 3.4. Assume that the viscosity η satisfies (9). For any ϕ P H 1 pΩq, the Reynolds equation (26a) equipped with the boundary conditions (29) admits a unique solution which satisfies

B x p P H 1 p0, Lq.
The velocity field u " pu, vq given as a function of p by (26b)-(26c) satisfies u P H 1 pΩq X L 8 pΩq and v P L 2 pΩq, with B z v P L 2 pΩq.

Moreover, we have the following estimates

|u| 8 ď C and |v| 2 ď C}ϕ} 1 , (32)
where the constant C does not depend on the scaling defined by λ.

Let us sketch the main steps of the proof of Proposition 3.4:

• The Reynolds equation can be solved explicitly, so that p is given as a function of the coefficients r d and r e (given as functions of ϕ by ( 21)): recalling definition (29) of w, we can integrate the Reynolds equation once and obtain λ r d B x p " s r e `λ r dp0q w ´s r ep0q,

where the coefficients r dp0q and r ep0q only depend on ϕ l and are thus known. If r d does not vanish, we compute formally B x p, and then p using the boundary condition ppLq " 0. In order to obtain estimates on the pressure, we have to prove that the coefficients r d and r e are regular enough (see Lemma 4.1), and that r dpϕq is greater than a strictly positive constant (i.e. the operator B x pd B x ¨q must be coercive, see Lemma 4.2).

• As far as the velocity is concerned, u is given by

u " λ f B x p `g,
where the coefficients are given by f " ´b ´r b r a a ¯and g " ´1 ´a r a ¯s (and a, b, r a, r b are defined in ( 20)). It is enough to prove the regularity of f and g in order to deduce the needed estimate on u from the estimate on B x p (see Lemma 4.3).

• The velocity v is given by vpx, zq " ´ż z 0 B x upx, ξq dξ, and the regularity of v follows from the regularity of u (see Lemma 4.4).

Step 2. As far as the Cahn-Hilliard equation is concerned, we proceed as in the earlier works on Cahn-Hilliard equation (e.g. [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF]), and we apply the Galerkin method in order to prove the existence of a solution to the system (30)-(31). This process consists in building approximate solutions pϕ n , µ n q in finite dimension (see Section 5.2), for which the existence follows from the Cauchy-Lipschitz theorem. For these approximate solutions pϕ n , µ n q, we prove the following proposition (see Section 5.5):

Proposition 3.5. For all 0 ď t ď T , let Yptq " α 2 2λ 2 |∇ϕ n ptq| 2 2 `żΩ F pϕ n ptqq, Zptq " α 2 2λ 2 |∇ϕ n ptq| 2 2 `|∇µ n ptq| 2 2 `|∆ϕ n ptq| 2 2 `żΩ F pϕ n ptqq.
Then the following estimate is satisfied:

Y 1 ptq `C1 Zptq ď f pYptqqZptq `C2 ,
where C 1 , C 2 are positive constants, and f : R Ñ R is a continuous function satisfying f p0q " 0.

Let us emphasize the main features of the proof:

• Although estimates on the Cahn-Hilliard equation are similar to the ones in [START_REF] Boyer | Mathematical study of multi-phase flow under shear through order parameter formulation[END_REF] or [START_REF] Chupin | Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind[END_REF], they involve supplementary terms due to the different boundary conditions: because of the non-homogeneous Dirichlet condition on ϕ n on the left-hand side of the domain (fluid injection), the conservation of the quantity of each fluid is not satisfied anymore (in the sense that the mean value mpϕ n q " 1 |Ω| ż Ω ϕ n is not constant with respect to time). For example, since mpϕ n q is not constant, we cannot apply classical inequalities on ϕ n ´mpϕ n q, such as the Poincaré inequality, and we have to work with the boundary value of ϕ n on the left-hand side of the domain (see Sections 5.3 and 5.4).

• In order to control the boundary and source terms with the ones on the left-hand side of the estimate, we have to work in adequate function spaces and choose in a suitable way the coefficients in front of each term. This is obtained only by imposing a smallness assumption on λ which depends on all other data of the problem.

From Proposition 3.5, we deduce the convergence of the linear terms. However, it is not enough to conclude the convergence of the nonlinear terms and the initial condition. To this end, we need more regularity on ϕ n and will prove the following proposition: Proposition 3.6. There exists C ą 0 such that for any T ą 0:

}ϕ n } L 2 p0,T ;H 3 pΩqq ď CT `C, › › › › dϕ n dt › › › › L 2 p0,T ;H ´1 pΩqq ď CT `C.
This proposition allows us to deduce the convergence of all terms in adequate function spaces, using classical compacity results from [START_REF] Simon | Compact sets in the space L p p0, T ; Bq[END_REF].

About the Reynolds equation

The letter C will then denote any constant depending on physical parameters (s, Q, η M , η m , Pe, α, F 1 , F 2 , F 3 , F 4 , B m , L, hpxq,...), but independent of the unknowns (u, p, ϕ, µ) and of λ. c). We will present the details of the proof for the case i " 1. The same computations can be used to obtain the regularity results for i " 0, i " 2. Let bpx, zq "

ż z 0 ξ ηpϕpx, ξqq dξ.
Let us prove that b P X pΩq for any ϕ P H 1 pΩq.

Ź First we prove that b P L 2 pΩq : for any px, zq P Ω, we have bpx, zq 2 "

ˆż z 0 ξ ηpϕpx, ξqq dξ ˙2 ď ´1 η m ż z 0 ξdξ ¯2 ď Cz 4 .
After integrating with respect to z and x, we get

ż L 0 ż hpxq 0 bpx, zq 2 dz dx ď C.
Ź Next, we show that b P H 1 pΩq and B z b P H 1 pΩq:

-On one hand, B x bpx, zq " ´ż z 0 ξη 1 pϕpx, ξqq ηpϕpx, ξqq 2 B x ϕpx, ξq dξ, with B x ϕ P L 2 pΩq. Let px, zq P Ω. Using hypothesis (9), we compute

|B x bpx, zq| 2 " ˜ż z 0 ξη 1 pϕpx, ξqq ηpϕpx, ξqq B x ϕpx, ξqdξ ¸2 ď η 1 M 2 η 2 m ż z 0 ξ 2 dξ ż z 0 |B x ϕpx, ξq| 2 dξ ď Cz 3 ż hpxq 0 |B x ϕpx, ξq| 2 dξ.
After integrating with respect to z and then with respect to x, we get

ż hpxq 0 |B x bpx, yq| 2 dy ď C ż hpxq 0 |B x ϕpx, ξq| 2 dξ, |B x b| 2 2 " ż L 0 ż hpxq 0 |B x bpx, yq| 2 dy dx ď C|B x ϕ| 2 2 ă 8.
It follows that B x b P L 2 pΩq.

-On the other hand, B z bpx, zq " z{ηpϕpx, zqq P H 1 pΩq, since ϕ P H 1 pΩq and using [START_REF] Boyer | Nonhomogeneous Cahn-Hilliard fluids[END_REF]. By definition of b, we know that bpx, 0q " 0, @x P r0, Ls. Therefore, the usual trace theorem for the Sobolev space H 1 pΩq implies that

|bpx, zq| 2 ď z ż z 0 pB ξ bpx, ξqq 2 dξ ď h M ż hpxq 0 pB ξ bpx, ξqq 2 dξ " h M |B z b| 2 L 2 p0,hpxqq ď C}B z b} 2 H 1{2 p0,hpxqq ď C}B z b} 2 1 , thus |b| 2 8 ď C}B z b} 2 1 ă 8.
It remains to prove the regularity of r a, r b, r c, r d, r e.

Ź For the coefficients of the form r apxq " apx, hpxqq, r bpxq " bpx, hpxqq, r cpxq " cpx, hpxqq, H 1 -regularity can be obtained using the same procedure as in the first part of the proof.

Ź For r d and r e, the key point of the proof is to observe that H 1 p0, Lq (which is embedded in L 8 p0, Lq) is an algebra: pf, gq P H 1 p0, Lq 2 ñ f g P H 1 p0, Lq. , and using the fact that r a, r b, r c belong to H 1 p0, Lq, we need to show that 1{r a remains bounded. Since η ď η M , we have r apxq "

ż hpxq 0 1 ηpϕpx, ξqq dξ ě h m η M i.e. 1 r a ď C. (34)
From the regularity of r a, r b, r c, from the algebra structure and from (34), we deduce that In order to prove the assertion, it suffices to prove that there exists δ ą 0 such that

r d P H 1 p0, Lq,
˜ż h 0 z 2 η dz ¸˜ż h 0 1 η dz ¸´˜ż h 0 z η dz ¸2 ě δ ˜ż h 0 1 η dz ¸.
Let us denote by P the following polynomial

P : ν Þ Ñ ż hpxq 0 ˜z a ηpϕpx, zqq `ν a ηpϕpx, zqq ¸2 dz " ż hpxq 0 z 2 ηpϕpx, zqq `ν2 ηpϕpx, zqq `2zν ηpϕpx, zqq dz.
From ( 9), we get

P pνq ě 1 η M ż hpxq 0 ´z2 `2zν `ν2 ¯dz " 1 3η M phpxq 3 `3hpxq 2 ν `3hpxqν 2 q.
A simple study of the right-hand side polynomial proves that @ν P R, @x P p0, Lq, hpxq 2 `3hpxqν `3ν

P pνq ´hpxq 3 12η M " ν 2 ż h 0 1 η `2ν ż h 0 z η `ż h 0 z 2 η ´hpxq 3 12η M is negative: 4 ˜ż hpxq 0 zdz ηpϕpx, zqq ¸2´4 ˜ż hpxq 0 dz ηpϕpx, zqq ¸«˜ż hpxq 0 z 2 dz ηpϕpx, zqq ¸´hpxq 3 12η M ff ď 0,
that is to say

˜ż h 0 z 2 η dz ¸˜ż h 0 1 η dz ¸´˜ż h 0 z η dz ¸2 ě h 3 m 12η M ˜ż h 0 1 η dz ¸, i.e. r d ě h 3 m 12η M ą 0.
The two previous lemmas 4.1 (regularity of the coefficients) and 4.2 (coercivity of the operator) with formula (33) imply that B x p P H 1 p0, Lq, thus p P H 2 p0, Lq.

and B x p is given by (33), thus:

λ|B x p| 8 ď 1 min xPp0,Lq r dpxq ´s|r e| 8 `λ| r dp0q||w| `s|r ep0q| ¯. (37)
Let us obtain estimates for these coefficients.

Ź Using the boundedness hypothesis on η, and applying the Cauchy-Schwarz inequality and the fact that @x P p0, Lq, hpxq ď h M , we can write for all px, zq P Ω apx, zq "

ż z 0 dξ ηpϕpx, ξqq ď h M η m ď C, thus |a| 8 , |r a| 8 ď C. (38)
Ź Similar computations for b, c and r b, r c give

|b| 8 , | r b| 8 ď C, |c| 8 , |r c| 8 ď C. (39)
Ź Recalling definition [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF] of r e, and using (34), it follows from (39):

|r e| 8 " |b| 8 min xPp0,Lq r apxq ď C. (40)
Ź We compute also from (29)

λ|w| ď C. (41)
Thus, using all these estimates in (37), we get

λ|B x p| 8 ď C, (42)
and combined with (36) and obvious estimates for a, r a, b, r b, we obtain the needed estimate:

|u| 8 ď C.
Lemma 4.4. Assume that the viscosity η satisfies (9). Assume ϕ P H 1 pΩq.

Then the vertical velocity v given by (26c) satisfies

|v| 2 ď C}ϕ} 1 .
Proof. The regularity of v follows from the regularity of u, equation (26c) and the regularity of the coefficients (Lemma 4.1):

vpx, zq " ´ż z 0 B x upx, ξqdξ.
From the Cauchy-Schwarz inequality, we deduce that

|v| 2 ď h M |B x u| 2 . (43)
Let us introduce the coefficients f " b´a r b r a and g " 1´a r a

, so that u " λf B x p`sg.

Therefore |B x u| 2 ď λ|B x f | 2 |B x p| 8 `λ|f | 8 |B 2 x p| 2 `s|B x g| 2 , (44)
and B 2 x p is given by taking the derivative of (33) with respect to x:

λ|B 2 x p| 2 ď 1 min xPp0,Lq r dpxq ´s|B x r e| 2 `λ|B x r d| 2 |B x p| 8 ¯. ( 45
)
Let us obtain estimates for each coefficient separately:

Ź We have |f | 8 ď | r b| 8 `C|a| 8 | r b| 8 . (46)
Ź It remains to obtain estimates of the derivatives of the coefficients with respect to x. We can compute B x a " ż y 0 η 1 pϕq ηpϕq 2 B x ϕ, and the Cauchy-Schwarz inequality yields

|B x a| 2 2 ď η 1 M 2 η 4 m ż Ω ˆż y 0 B x ϕpx, zq dz ˙2 ď C ż Ω ż y 0 |B x ϕ| 2 ď C}ϕ} 2 1 , (47)
and similar estimates for all the other coefficients:

|B x a| 2 , |B x r a| 2 ď C}ϕ} 1 , |B x b| 2 , |B x r b| 2 ď C}ϕ} 1 , |B x c| 2 , |B x r c| 2 ď C}ϕ} 1 . (48)
Ź Let us write

B x ´a r a ¯" B x a r a ´a B x r a r a 2 .
From (34), we know that r a ě h m η M . This estimate combined with (38) and

(48) suffices to prove that

ˇˇBx ´a r a ¯ˇˇ2 ď C}ϕ} 1 , (49)
and

ˇˇˇˇB x ˜r b r a ¸ˇˇˇˇ2 ď C}ϕ} 1 . ( 50
) Ź Since B x d " B x c ´Bx r b r b r a ´r bB x ˜r b r a ¸, B x e " B x ˜r b r a ¸, B x f " B x b ´Bx a r b r a ´aB x ˜r b r a ¸, B x g " B x ´a r a ¯, (51)
it follows, using (48), ( 49), ( 50) in (51), that

|B x r d| 2 ď C}ϕ} 1 , |B x r e| 2 ď C}ϕ} 1 , |B x f | 2 ď C}ϕ} 1 , |B x g| 2 ď C}ϕ} 1 . (52)
Putting (35), ( 52), (37) in ( 45) and (44), we deduce an estimate for each of the three terms in (44):

Ź The first term is estimated by:

λ|B x f | 2 |B x p| 8 ď C}ϕ} 1 .
Ź For the second term, we have:

|f | 8 δ ´s|B x r e| 2 `λ|B x r d| 2 |B x p| 8 ¯ď C}ϕ} 1 .
Ź The third term |B x g| 2 is exactly estimate (52).

Therefore, using (43) and these three estimates for |B x u| 2 , we obtain:

|v| 2 ď C|B x u| 2 ,
which proves the lemma.

Remark 4.5. Observe that it is not straightforward to prove that v P L 8 pΩq if ϕ only lies in H 1 pΩq. We get easily |v| 8 ď C|B x u| 8 , however the H 1 -regularity of ϕ is not sufficient to conclude. We need a lift operator for the boundary value ϕ l of the order parameter ϕ.

Lemma 5.1. Let ϕ l P H 7{2 pΓ l q satisfy Hypothesis 2.3. There exists φl P H 7{2 pΩq such that the following conditions are satisfied

φl | Γ l " ϕ l , ∇ φl | Γ0 ¨n " 0, α 2 λ 2 ∆ φl | Γ l " F 1 pϕ l q, ∇∆ φl | Γ0 ¨n " 0.
Proof. For any px, zq P Ω, let us define φl px, zq " χpxqϕ l ´hp0qz hpxq ¯`F, where F is the solution of the following problem:

$ ' ' & ' ' % ∆F " λ 2 α 2 ψpxqF 1 ´ϕl ´hp0qz hpxq ¯¯´h p0q hpxq χpxqϕ 2 l ´hp0qz hpxq ¯in Ω, F| Γ l " 0, ∇F| Γ0 ¨n " 0,
and the functions χ and ψ are smooth functions satisfying the following conditions:

χp0q " 1, χ 1 p0q " 0, χ 2 p0q " 0, ψp0q " 1, @x P rε, Ls χpxq " χ 1 pxq " χ 2 pxq " χ 3 pxq " 0, @x P rε, Ls ψpxq " ψ 1 pxq " 0.

By regularity of the Laplacian [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], it follows immediately that F P H 7{2 pΩq, thus φl P H 7{2 pΩq. Since h 1 pxq " 0 for x P r0, εs, the two last conditions imply that χh 1 is identically zero, and so are the other functions Let us check that this function φl satisfies the claimed conditions:

χ 1 h 1 , χ 2 h 1 , χ 3 h 1 , ψh 1 , ψ 1 h 1 and χh 2 . x L ε 1 hpxq χ ψ
• On Γ l , F is zero, and since χp0q " 1, φl has the right value.

• On Γ 0 , we know that ∇F| Γ0 ¨n " 0, and we have to treat separately the three different boundaries for the other term:

-On Γ b , B y φl px, 0q " χpxq hp0q hpxq ϕ 1 l p0q " 0 by [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. -On Γ t , h 1 pxqB x φl px, hpxqq ´By φl px, hpxqq " h 1 pxqχ 1 pxqϕ l php0qq χpxq h 1 pxq 2 hp0q hpxq ϕ 1 l php0qq´χpxq hp0q hpxq ϕ 1 l php0qq " 0 by [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] and using that h 1 χ 1 is identically zero.

-On Γ r , all the terms of B x φl pL, yq contain either χpLq or χ 1 pLq, which are both equal to zero.

Let us compute the Laplacian of φl . In order to improve the readibility, we denote Y " hp0qy hpxq :

∆ φl "χ 2 pxqϕ l pY q ´2 h 1 pxqY hpxq χpxqχ 1 pxqϕ 1 l pY q `h1 pxq 2 Y 2 hpxq 2 χpxqϕ 2 l pY q ´h2 pxqh 2 pxq ´2h 1 pxq 2 hpxq hpxq 3 Y χ 1 pxqϕ 1 l pY q ` hp0q hpxq χpxqϕ 2 l pY q `λ2 α 2 ψpxqF 1 pϕ l pY qq ´hp0q hpxq χpxqϕ 2 l pY q.

• We can compute the Laplacian on Γ l : ∆ φl p0, yq " λ 2 α 2 F 1 pϕ l pyqq, since χ 1 p0q " χ 2 p0q " 0, h 1 p0q " 0 and ψp0q " 1.

• For Γ 0 , we treat again each boundary separately:

-On Γ b , we have to compute B y ∆ φl at px, 0q. Using that y " 0 and ϕ 1 l p0q " 0, we obtain that B y ∆ φl px, 0q " 0. -On Γ t , we compute h 1 pxqB x ∆ φl ´By ∆ φl at px, hpxqq. The terms in h 1 are multiplied by either χ, χ 1 , χ 2 , χ 3 , ψ, or ψ 1 , and are therefore identically zero. For the other terms, we use the same arguments and that ϕ 1 l php0qq " 0 to conclude that the normal derivative of ∆ φl is zero on Γ t .

-On Γ r , we observe that χpLq " χ 1 pLq " χ 2 pLq " χ 3 pLq " ψpLq " ψ 1 pLq " 0, thus B x ∆ φl pL, yq " 0.

Useful inequalities Sobolev embeddings.

Let us recall the Poincaré inequality and usual Sobolev embeddings.

Proposition 5.2 (Poincaré inequality).

Let Ω Ă R 2 defined by (2). For any f P H 1 pΩq such that f | Γi " 0 on one of the three parts Γ l , Γ b , Γ r of the boundary,

|f | 2 ď C|∇f | 2 . ( 53
)
Proposition 5.3 (Sobolev embeddings). Let Ω Ă R 2 defined by (2). Then for any 2 ď q ă `8, we have H 1 0 pΩq ãÑ L q pΩq. More precisely, for any f P H 1 pΩq with f | Γi " 0 on one of the three parts Γ l , Γ b , Γ r of the boundary, we have

|f | q ď C}f } 1 . (54)
Equivalence of norms. (see [START_REF] Dauge | Elliptic boundary value problems on corner domains, Smoothness and asymptotics of solutions[END_REF] for a proof) Proposition 5.4. Let f P H 2 pΩq such that f | Γi " 0 on one of the three parts Γ l , Γ b , Γ r of the boundary. We have

}f } 2 ď C|∆f | 2 . ( 55
)
Trace estimates. (see [START_REF] Adams | Sobolev spaces[END_REF] for a proof) Proposition 5.5. For any f P H 1 pΩq such that f | Γi " 0 on one of the three parts Γ l , Γ b , Γ r of the boundary, we have

|f | L 2 pΓ l q ď C|∇f | 2 .
Corollary 5.6. For ϕ P H 2 pΩq satisfying the boundary conditions (18), we can apply this proposition to B x ϕ, since B x ϕ| Γr " 0, and deduce that

|B x ϕ| L 2 pΓ l q ď C|∇B x ϕ| 2 ,
and if we combine this relation with Proposition 5.4, we obtain

|B x ϕ| L 2 pΓ l q ď C|∆ϕ| 2 . (56)

Galerkin approximations

Let us build Galerkin approximations of ϕ and µ. Since H 1 pΩq is a separable Hilbert space, there exists an Hilbertian basis pψ i q iě1 of H 1 pΩq. The functions ψ i can be chosen to be eigenfunctions of the Laplacian ´∆ with the boundary conditions

Bψ i Bn | Γ0 " 0, ψ i | Γ l " 0,
and we denote by λ i the corresponding eigenvalues. As far as the regularity of the functions ψ i is concerned, we have ψ i P H 2 pΩq (this result can be deduced from [START_REF] Dauge | Elliptic boundary value problems on corner domains, Smoothness and asymptotics of solutions[END_REF]). We define Ψ n " Spanpψ 1 , ¨¨¨, ψ n q, and P Ψn the orthogonal projector on Ψ n in L 2 pΩq. As a projector, P Ψn satisfies:

pP Ψn f, gq " pf, P Ψn gq, @pf, gq P L 2 pΩq 2 , ( 57
)
where p¨, ¨q denotes the scalar product in L 2 pΩq.

Recalling that φl P H 7{2 pΩq is a lifting of the boundary condition ϕ l defined in Lemma 5.1, we consider the following approximation of ϕ:

ϕ n ptq " n ÿ i"1 β i ptqψ i `φ l ,
where β i are unknown functions to be determined. In this setting, ϕ n p0q ´φ l is the orthogonal projection of ϕ 0 ´φ l on Ψ n . Let us introduce the following auxiliary function a, which will be useful in order to define µ n :

Proposition 5.7. There exists a P H 1 pΩq such that a| Γ l " F 1 pϕ l q, ∇a ¨n| Γ0 " 0.

Proof. Let us define a by apx, zq " F 1 ´ϕl ´hp0qz hpxq ¯¯. We check that a satisfies the claimed conditions.

• On Γ l , ap0, zq " F 1 pϕ l pzqq.

• On Γ b , B z apx, 0q " ´hp0q hpxq ϕ 1 l p0qF 2 pϕ l p0qq " 0 by [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF].

• On Γ t , the normal derivative is written h 1 pxqB x apx, hpxqq ´Bz apx, hpxqq.

The two terms are again equal to zero thanks to [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF].

• On Γ r , B x apL, zq " ´h1 pLqhp0qz hpLq 2 ϕ 1 l ´hp0qz hpxq ¯F 2 ´ϕl ´hp0qz
hpxq ¯¯, which is also zero since h 1 pLq " 0. Taking (30)-(31) into account, let us define pϕ n , µ n q as the solution of the following weak problem:

Problem 5.8. Find ϕ n " n ř i"1 β i ptqψ i `φ l and µ n such that λ ż Ω B t ϕ n ψ `żΩ 1 λ Pe
Bpϕ n q∇µ n ∇ψ `żΩ u ϕn ¨∇ϕ n ψ " 0, @ψ P Ψ n , (58)

µ n " ´α2 λ 2 ∆ϕ n `a `PΨn pF 1 pϕ n q ´aq, (59)
with the initial condition ϕ n | t"0 " ϕ 0 and the boundary conditions

µ n | Γ l " 0, ϕ n | Γ l " ϕ l , ∇µ n ¨n| Γ0 " ∇ϕ n ¨n| Γ0 " 0, (60)
and where u ϕn is defined for each ϕ n by the formulas (22)-( 23) and (25).

This problem can indeed be obtained from ( 26) because the boundary term vanishes, as proved in the following proposition. Proposition 5.9. Let pϕ n , µ n q solution of Problem 5.8. Then the boundary term coming from the integration by parts cancels:

ż Γ Bpϕ n q∇µ n ¨n ψ " 0.

Proof.

Ź On Γ 0 , we can compute ∇µ n ¨n| Γ0 , using that the functions ψ i are eigenfunctions of ´∆: ∇µ n ¨n| Γ0 " ´α2 λ 2 ∇∆ϕ n ¨n| Γ0 `∇a ¨n| Γ0 loooomoooon "0 by Def. 5.7

`∇P Ψn pF 1 pϕ n q ´aq ¨n| Γ0 looooooooooooooomooooooooooooooon "0, since PΨ n pF 1 pϕnq´aqPΨn " ´α2 λ 2 ∇ ˜n ÿ

i"1

β i λ i ψ i ¸¨n| Γ0 `∇∆ φl ¨n| Γ0 loooooomoooooon
"0 by Lem. 5.1

.

Since ψ i P Ψ n for any i ď n, we have ∇ψ i ¨n| Γ0 " 0, we deduce ∇µ n ¨n| Γ0 " 0.

Ź On Γ l , the boundary term is also equal to zero, since ψ P Ψ n , and thus vanishes on Γ l . Observe that the weak formulation (58)-( 59) is well-defined since ψ i P H 1 0 pΩq implies that µ n P H 1 pΩq. Indeed, the functions ψ i are eigenfunctions of ´∆, thus the regularity follows from definition (59).

Remark 5.10. Observe that the chosen approximation (59) of µ satisfies the same boundary conditions as µ, because of the definition of φl in Lemma 5.1. Moreover, if it converges, it is towards µ " ´α2 λ 2 ∆ϕ`F 1 pϕq, since P Ψn converges towards the identity. Indeed, F 1 pϕ n q ´a satisfies the right boundary conditions in Φ 1 0 (by construction of a, see Proposition 5.7):

Ź F 1 pϕ n q ´a " 0 on Γ l , Ź ∇pF 1 pϕ n q ´aq ¨n " 0 on Γ 0 .

Lemma 5.11. For n P N, there exist t n ą 0 and pβ i q 1ďiďn P C 1 p0, t n q such that ϕ n ptq "

n ř i"1 β i ptqψ i `φ l is a solution of Problem 5.8.
Proof. Replacing ϕ n by its expression as a function of β i , the system (58)-(59) becomes:

λ n ÿ i"1 β 1 i ptq ż Ω ψ i ψ`ż Ω 1 λ Pe B ˜n ÿ i"1 β i ptqψ i `φ l ¸∇µ n ∇ψ `n ÿ i"1 β i ptq ż Ω u t n ř i"1 βiptqψi`φ l u ¨∇ψ i ψ " 0, @ψ P Ψ n , µ n " ´α2 λ 2 n ÿ i"1 β i ptqλ i ψ i `a `PΨn F 1 ˜n ÿ i"1 β i ptqψ i `φ l ´a¸.
This formulation is an ordinary differential equation on pβ i q 1ďiďn . The functions B and F 1 are of class C 1 on R. Moreover, the function u as a function of ϕ n given by (26b)-(26c)-( 26a) is also C 1 on R `(with respect to time): indeed, u ϕn is given as a combination of coefficients of the form ż z 0 ξ{ηpϕ n px, ξqqdξ, and the function η is C 1 by assumption [START_REF] Boyer | Nonhomogeneous Cahn-Hilliard fluids[END_REF]. The second component of the velocity v is given as a function of u, and is also C 1 on R `. Therefore, the Cauchy-Lipschitz theorem ensures the existence of a unique solution pβ i q 1ďiďn on a time interval r0, t n q.

Last, let us introduce another auxiliary function b, which is another lifting of the boundary condition ϕ l and will be used to apply Poincaré inequality: Proposition 5.12. There exists b P H 2 pΩq such that for some small ε ą 0 that will be determined later,

b| Γ l " ϕ l , ∇b ¨n| Γ0 " 0, |B x b| 2 ă ε.
Proof. Let us define b by bpx, zq " ϕ l ´hp0qz hpxq ¯. Let us check that b satisfies the claimed conditions. The first ones are the sames as in Proposition 5.7, and are satisfied in the same way:

• On Γ l , bp0, zq " ϕ l pzq.

• On Γ b , B z bpx, 0q " hp0q hpxq ϕ 1 l p0q " 0 by [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF].

• On Γ t , the normal derivative is written h 1 pxqB x bpx, hpxqq ´Bz bpx, hpxqq.

The two terms are again equal to zero thanks to [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF].

• On Γ r , B x bpL, zq " ´h1 pLqhp0qz hpLq 2 ϕ 1 l ´hp0qz hpxq ¯, which is also zero since h 1 pLq " 0.

Last, we observe that

|B x b| 2 2 " ż Ω h 1 pxq 2 hp0q 2 z 2 hpxq 4 ˇˇˇϕ 1 l ˆhp0qz hpxq ˙ˇˇˇ2 dxdz ď C|ϕ 1 l | 2 L 2 p0,1q ,
and thus by ( 3) and [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], this term can be arbitrarily small. Therefore, in order to ensure the smallness of |B x b| 2 , we have to choose ε sufficiently small. Therefore, this determines the smallness assumption on ε in [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] in Hypothesis 2.3.

Equation on ϕ

Let us now focus on obtaining estimates of ϕ n , µ n in appropriate function spaces. In the sequel, we drop the subscripts n for the sake of readability, and we write ϕ, µ instead of ϕ n , µ n . Lemma 5.13. For ϕ and µ solutions of (58)-( 60), the following applies: .

λ d dt ˜α2 2λ 2 |∇ϕ| 2 2 `żΩ F pϕq ¸`´B m λ Pe ´1¯| ∇µ| 2 2 ď C ˜´|u| 2 8 `|v|
‹ For A 1 , we use integration by parts:

A 1 " ´α2 λ 2 ż Ω B t ϕ∆ϕ `żΩ B t ϕ a " α 2 2λ 2 d dt |∇ϕ| 2 2 ´α2 λ 2 ż Γ B t ϕ ∇ϕ ¨n `żΩ B t ϕ a.
The boundary condition ∇ψ i ¨n| Γ0 " 0, and the fact that ϕ l is independent of t allow us to treat the boundary term:

´α2 λ 2 ż Γ B t ϕ lo omo on " 0 on Γ l ∇ϕ ¨n loomoon " 0 on Γ0 " 0, thus A 1 " α 2 2λ 2 d dt |∇ϕ| 2 2 `żΩ B t ϕ a. (63)
‹ For the second term, observe that from the time-independency of φl and ψ i P Ψ n , it yields

P Ψn B t ϕ " P Ψn ˜n ÿ i"1 β 1 i ptqψ i ¸" n ÿ i"1 β 1 i ptqψ i " B t ϕ. ( 64
)
Now, we use property ( 57) and (64):

A 2 " pB t ϕ, P Ψn pF 1 pϕq´aqq " pP Ψn B t ϕ, F 1 pϕq´aq " pB t ϕ, F 1 pϕq´aq.

Thus, A 2 can be expressed as a time derivative plus a second term which will cancel with the last term in (63):

A 2 " ż Ω B t ϕF 1 pϕq ´żΩ B t ϕ a " d dt ż Ω F pϕq ´żΩ B t ϕ a. ( 65
)
Ź The B-term is trivially estimated using that Bpϕq ě B m (see ( 17)):

B " 1 λ Pe ż Ω Bpϕq|∇µ| 2 ě B m λ Pe |∇µ| 2 2 . ( 66
)
Ź For the D-term, we split it into two terms:

D " ż Ω uB x ϕµ loooomoooon "D1
`żΩ vB y ϕµ loooomoooon "D2

.

‹ We use Poincaré inequality (53) and Young's inequality

D 1 " ż Ω uB x ϕµ ď |u| 8 |B x ϕ| 2 |µ| 2 ď C|u| 8 |B x ϕ| 2 |∇µ| 2 ď 1 2 |∇µ| 2 2 `C|u| 2 8 |B x ϕ| 2 2 .
Now, observe that B x ϕ is zero on Γ r , and thus the Poincaré inequality yields

|B x ϕ| 2 2 ď L|B 2 x ϕ| 2 2 ď C|∆ϕ| 2 2 .
Combining these two estimates, we obtain

D 1 ď 1 2 |∇µ| 2 2 `C|u| 2 8 |∆ϕ| 2 2 . (67)
‹ For D 2 , we apply Hölder's inequality with two exponents q and q 1 strictly greater than 2 such that 1 q `1 q 1 " 1 2 and the Sobolev inequality (54) for |µ| q 1 with the Poincaré inequality (53):

D 2 " ż Ω vB y ϕµ ď |v| 2 |B y ϕ| q |µ| q 1 ď C|v| 2 |B y ϕ| q }µ} 1 ď C|v| 2 |B y ϕ| q |∇µ| 2 .
Now, we use (54) for |B y ϕ| q , and Young's inequality

D 2 ď C|v| 2 }B y ϕ} 1 |∇µ| 2 ď 1 2 |∇µ| 2 2 `C|v| 2 2 }B y ϕ} 2 1 .
It remains to apply (55) with a function equals to zero on Γ l . This is done using b defined in Definition 5.12. Since }B y pϕ ´bq} 1 ď C|∆pϕ ´bq| 2 , we have

D 2 ď 1 2 |∇µ| 2 2 `C|v| 2 2 p|∆ϕ| 2 2 `~b~2 2 q. (68)
Putting ( 63), ( 65), (66), (67), ( 68) into (62), and rearranging terms, we get

λ d dt ˜α2 2λ 2 |∇ϕ| 2 2 `żΩ F pϕq ¸`´B m λ Pe ´1¯| ∇µ| 2 2 ď C ˜´|u| 2 8 `|v| 2 2 ¯|∆ϕ| 2 2 `|v| 2 2 ~b~2 2 ¸. (69)
This proves inequality (61).

Equation on µ

Lemma 5.14. For ϕ and µ solutions of (58)-(60), the following inequality applies:

α 2 λ 2 |∇ϕ| 2 2 `F3 p0q ż Ω F pϕq ď C λ 2 |∆ϕ| 2 2 `1 2 |∇µ| 2 2 `C|∇ϕ| 2r 2 `C|∇ϕ| 2 2 `Tl . ( 70
)
where T l is independent of ϕ, µ and of time t, and is given by:

T l " C `1 `}b} 2 1 `| φl | 2 2 `}b} 2r 1 ˘`|a| 2 | φl | 2 `C ˆ1 λ 2 `1˙} b} 2 1 .
Proof.

If
As before, let us treat each term separately.

Ź For B, we use integration by parts, and obtain:

B " α 2 λ 2 |∇ϕ| 2 2 ´α2 λ 2 ż Γ ϕ∇ϕ ¨n loooooooomoooooooon ":B1 `żΩ a ϕ lo omo on p‹q . (72)
Observe that since ∇ϕ ¨n| Γ0 " 0, the boundary term B 1 is zero on ΓzΓ l . Using (56) and Young's inequality, it follows:

|B 1 | " α 2 λ 2 ˇˇˇż Γ l ϕ l B x ϕ ˇˇˇď α 2 λ 2 |ϕ l | L 2 pΓ l q |B x ϕ| L 2 pΓ l q ď C λ 2 |ϕ l | L 2 pΓ l q |∆ϕ| 2 ď C λ 2 p|∆ϕ| 2 2 `}b} 2 1 q, (73)
where we used b as a lift of ϕ l .

Ź For the D-term, let us use the projector property (57) and the fact that ϕ´φ l P Ψ n (i.e. P Ψn pϕ´φ l q " ϕ´φ l , and thus P Ψn ϕ " ϕ´pId´P Ψn q φl ):

D " pP Ψn pF 1 pϕq ´aq, ϕq " pF 1 pϕq ´a, P Ψn ϕq " pF 1 pϕq, ϕq loooomoooon ":D1

´pa, ϕq loomoon ´p‹q ´pF 1 pϕq, pId ´PΨn q φl q loooooooooooooomoooooooooooooon ":D2

`pa, pId ´PΨn q φl q looooooooooomooooooooooon ":D3

.

The term ´p‹q cancels with the one in (72). Hypothesis [START_REF] Doi | Dynamics of domains and textures[END_REF] with γ " 0 yields

D 1 " ż Ω F 1 pϕq ϕ ě ż Ω F 3 p0qF pϕq ´F4 p0q|Ω| ě ż Ω F 3 p0qF pϕq ´C. (74)
As far as D 2 is concerned, we use the fact that Id ´PΨn is a projector, thus its operator norm (in L 2 pΩq) is equal to 1. We also use the property [START_REF] Dauge | Elliptic boundary value problems on corner domains, Smoothness and asymptotics of solutions[END_REF] for |F 1 pϕq| and (54) for |ϕ| r 2r to obtain (if r ą 1):

|D 2 | " |pF 1 pϕq, pId ´PΨn q φl q| ď | φl | 2 |F 1 pϕq| 2 ď C| φl | 2 pF 1 |ϕ| r 2r `F2 |Ω|q ď C| φl | 2 p}ϕ} r 1 `1q ď C| φl | 2 ´|∇ϕ| r 2 `}b} r 1 `1ď C ´| φl | 2 2 `|∇ϕ| 2r 2 `}b} 2r 1 `1¯, (75)
where in the third line, we used the lifting b of the boundary condition ϕ l defined in Definition 5.12 to apply Poincaré inequality. Observe that we proved the following estimate on F 1 pϕq, which will be used in the following:

|F 1 pϕq| 2 2 ď C ´|∇ϕ| 2r 2 `}b} 2r 1 `1¯. (76)
Last, we use again the fact that the operator norm of Id ´PΨn is equal to 1, and write

D 3 ď |a| 2 | φl | 2 . (77)
Ź For the A-term, Young's inequality combined with the Poincaré inequality for ϕ (using b as a lifting of ϕ l ) and (53) for µ yields:

A " ż Ω µ ϕ ď |µ| 2 |ϕ| 2 ď C|∇µ| 2 pC|∇ϕ| 2 `}b} 1 q ď C|∇µ| 2 p|∇ϕ| 2 `}b} 1 q ď 1 2 |∇µ| 2 2 `C `|∇ϕ| 2 2 `}b} 2 1 ˘. (78)
Putting ( 72)-( 78) in (71), and rearranging terms, it follows:

α 2 λ 2 |∇ϕ| 2 2 `F3 p0q ż Ω F pϕq ď C λ 2 |∆ϕ| 2 2 `1 2 |∇µ| 2 2 `C|∇ϕ| 2r 2 `C|∇ϕ| 2 2 `C ´|ϕ l | 2 L 2 pΓ l q `| φl | 2 2 `}b} 2r 1 ¯`|a| 2 | φl | 2 `C ˆ1 `1 λ 2 ˙}b} 2 1 `C,
which is the inequality (70) we claimed.

Lemma 5.15. For ϕ and µ solutions of (58)-(60), the following estimate applies for any θ, κ ą 0:

p α 2 λ 2 ´3κq|∆ϕ| 2 2 ď 1 2 |∇µ| 2 2 `1 2 |∇ϕ| 2 2 `C κ |∇ϕ| 2r 2 `Sl , (79)
where S l is independent of ϕ, µ and of time t, and is given by:

S l " C κ `}b} 2r 1 `1˘`1 κ |a| 2 2 .
Proof. Multiplying (59) by ´∆ϕ and integrating by parts, we get α 2 λ 2 |∆ϕ| 2 2 " ´pµ, ∆ϕq loooomoooon ":A `żΩ P Ψn pF 1 pϕq ´aq ∆ϕ looooooooooooomooooooooooooon ":B `pa, ∆ϕq loomoon

":D . (80)
Ź We treat the D-term with Young's inequality with some constant κ ą 0:

D " pa, ∆ϕq ď 1 κ |a| 2 2 `κ|∆ϕ| 2 2 . (81)
Ź For the B-term, we use the projector property (57) and Young's inequality to obtain the following estimate:

B " pP Ψn pF 1 pϕq ´aq, ∆ϕq ď F 1 pϕq| 2 |∆ϕ| 2 `|a| 2 |∆ϕ| 2 ď 1 κ |a| 2 2 `κ|∆ϕ| 2 2 `κ|∆ϕ| 2 2 `C κ |F 1 pϕq| 2 2 .
Then we can use (76) to deduce that

B ď 2κ|∆ϕ| 2 2 `1 κ |a| 2 2 `C κ ´|∇ϕ| 2r 2 `}b} 2r 1 `1¯. (82)
Ź As far as the A-term is concerned, it is computed by integration by parts:

A " ´pµ, ∆ϕq " Thanks to Young's inequality, we have

A " ´p∇µ, ∇ϕq ď 1 2 |∇µ| 2 2 `1 2 |∇ϕ| 2 2 . (83)
Finally, we use (81)-( 83) in (80) to obtain

p α 2 λ 2 ´3κq|∆ϕ| 2 2 ď 1 2 |∇µ| 2 2 `1 2 |∇ϕ| 2 2 `C κ |∇ϕ| 2r 2 `C κ `}b} 2r 1 `1˘`1 κ |a| 2 2 .
This concludes the proof.

Convergence results

A priori estimates

Let us sum (61), (70) and c 3 ˆ(79), where c 3 is a positive constant that will be determined in the sequel. We obtain

λ d dt ˆα2 2λ 2 |∇ϕ| 2 2 `żΩ F pϕq ˙`´B m λ Pe ´3 2 ´c3 2 ¯|∇µ| 2 2 `´α 2 λ 2 ´C ´c3 2 ¯|∇ϕ| 2 2 `´c 3 p α 2 λ 2 ´3κq ´C λ 2 ¯|∆ϕ| 2 2 `F3 p0q ż Ω F pϕq ď C ´p|u| 2 8 `|v| 2 2 q|∆ϕ| 2 2 `|v| 2 2 ~b~2 2 ¯`´C `c3 C κ ¯|∇ϕ| 2r 2 `c3 S l `Tl . (84)
To control the right hand side member of (84) we recall that we proved in (32) that |u| 8 ď C, |v| 2 ď C}ϕ} 1 .

We apply the Poincaré inequality choosing b as a lift for ϕ to gain

|u| 2 8 ď C, |v| 2 2 ď C|∇ϕ| 2 2 `C}b} 2 1 . (85)
Estimate (84) becomes

λ d dt ˆα2 2λ 2 |∇ϕ| 2 2 `żΩ F pϕq ˙`´B m λ Pe ´3 2 ´c3 2 ¯|∇µ| 2 2 `´α 2 λ 2 ´C ´c3 2 ¯|∇ϕ| 2 2 `´c 3 p α 2 λ 2 ´3κq ´C λ 2 ´C¯| ∆ϕ| 2 2 `F3 p0q ż Ω F pϕq ď C|∇ϕ| 2 2 |∆ϕ| 2 2 `C´1 `c3 κ ¯|∇ϕ| 2r 2 `C `C λ 2 `C c 3 κ . (86)
In order to ensure

B m λ Pe ´3 2 ´c3 2 ě B m 2λ Pe , α 2 λ 2 ´C ´c3 2 ě α 2 2λ 2 , c 3 p α 2 λ 2 ´3κq ´C λ 2 ´C ě c 3 α 2 2λ 2 ,
we will choose c 3 , λ such that

3 2 `c3 2 ď B m 2λ Pe , C `c3 2 ď α 2 2λ 2 , 3κc 3 `C λ 2 `C ď c 3 α 2 2λ 2 .
We choose c 3 with c 3 α 2 large enough such that the third condition can be rewritten as

3κc 3 `C ď ˆc3 α 2 2 ´Cl ooooooomooooooon ą0 1 λ 2 .
Newt, choosing λ ą 0 small enough ensures the required inequalities. Estimate (86) becomes

λ d dt ˆα2 2λ 2 |∇ϕ| 2 2 `żΩ F pϕq ˙`B m 2λ Pe |∇µ| 2 2 `α2 2λ 2 |∇ϕ| 2 2 `c3 α 2 2λ 2 |∆ϕ| 2 2 `F3 p0q ż Ω F pϕq ď C|∇ϕ| 2 2 |∆ϕ| 2 2 `C´1 `c3 κ ¯|∇ϕ| 2r 2 `C `C λ 2 `C c 3 κ . (87)
Let us define for all t ě 0,

Yptq " α 2 2λ 2 |∇ϕptq| 2 2 `żΩ F pϕptqq, Zptq " α 2 2λ 2 |∇ϕptq| 2 2 `|∇µptq| 2 2 `|∆ϕptq| 2 2
`żΩ F pϕptqq, so that 0 ď Yptq ď Zptq, since F ą 0 (by assumption ( 13)).

Lemma 5.16. There exists strictly positive constants C 1 , C 2 and f : R Ñ R an increasing continuous function satisfying f p0q " 0 satisfying • C 1 ą 0;

• there exists M ą 0 such that

‹ f pM q ă C 1 {2; ‹ C 2 ă M C 1 {2
. such that the a priori estimate (87) can be rewritten in the following form:

Y 1 ptq `C1 Zptq ď f pYptqqZptq `C2 . ( 88
)
Proof. In order to rewrite (87) as the inequality (88), we have to set apart the linear terms (with respect to Z) and the nonlinear terms (which will appear in f pYqZ). Defining

C 1 :" 1 λ min " B m 2λ Pe , 1, c 3 α 2 2λ 2 , F 3 p0q * ą 0 and C 2 :" C λ ´1 `1 λ 2 `c3 κ ¯ą 0,
we rewrite (87) as

Y 1 ptq `C1 Zptq ď f pYptqqZptq `C2 .
We can also give explicitely the form of f , which is given, up to a multiplicative constant, by

f pxq " C 2λ 2 α 2 x `C´1 `c3 κ ¯ˆ2λ 2 α 2 ˙2r x r´1 .
For r ą 1, it is always possible to find M ą 0 such that f pM q ă C 1 {2.

It remains to impose that the right-hand side is controlled by C 1 , i.e. that C 2 ă M C 1 {2. This is achieved by imposing some smallness conditions on λ.

Indeed, if λ " 0 then we have C 1 " 1`F3p0q λ and C 2 " C λ ´1 `c3 κ ¯.
It is then possible to find M ą 0 satisfying the desired property, since the two constants are of the same order in λ. This concludes the proof.

From now on, let us come back to the notations with the subscripts n introduced in Section 5.2, denoting the Galerkin approximations. The proof of the main theorem consists in showing that t n " `8 for any n ě 1, and that ϕ n converges in appropriate function spaces. Lemma 5.17. For any n P N, under a smallness assumption on λ and Hypothesis 2.3, there exists C ą 0 such that for any T ą 0, }ϕ n } L 8 pR `;H 1 pΩqq ď C, }ϕ n } L 2 p0,T ;H 2 pΩqq ď CT, }µ n } L 2 p0,T ;Φ 1 0 q ď CT. (89) Proof. Let n P N, T ą 0. The assumptions are enough to apply Lemma 5.16 with Proposition 7.1 (given in Appendix) which implies that Y n P L 8 p0, T q with a bound independent of T , and Z n P L 1 p0, T q with a bound depending on T . From this, we deduce several results on ϕ n , µ n :

• The quantity ∇ϕ n is bounded in L 8 p0, 8; L 2 pΩqq, uniformly with respect to n.

• The quantities ∇µ n , ∇ϕ n and ∆ϕ n are bounded in L 2 loc p0, 8; L 2 pΩqq, uniformly with respect to n.

• Furthermore, applying the Poincaré inequality to ϕ n allows us to control the whole H 1 pΩq-norm by the L 2 -norm of the gradient.

• As far as the H 2 -norm of ϕ n is concerned, we know by Proposition 5.4 that it is equivalent to the L 2 -norm of the Laplacian, and thus controlling |∆ϕ n | 2 is enough to control the whole H 2 pΩq-norm.

• For µ n , the Poincaré inequality (53) also allows us to control the H 1 -norm by the L 2 -norm of the gradient.

From these arguments, we conclude that there exists C ą 0 such that for any T ą 0, estimate (89) is satisfied.

Let us observe that the first estimate of (89) is enough to show that the time interval p0, t n q on which the functions ϕ n exist is p0, `8q.

Estimates (89) are not enough to conclude for the convergence of the nonlinear terms and of the initial condition ϕ n p0q. Therefore, some more regularity on ϕ n and B t ϕ n will be proved in the next subsection. We also note that the value of the scaling λ is now fixed: the constants C which appear from now can depend on λ.

5.5.2. H 3 -estimate for ϕ Lemma 5.18. For any n P N, under a smallness assumption on λ, there exists C ą 0 such that for any T ą 0

}ϕ n } L 2 p0,T ;H 3 pΩqq ď CT `C. ( 90
)
Proof. We compute the gradient of (59):

α 2 λ 2 ∇∆ϕ n ´∇a " ∇P Ψn pF 1 pϕ n q ´aq looooooooooomooooooooooon ":A ´∇µ n . (91)
Ź Let us prove that |A| 2 2 ď |∇F 1 pϕ n q| 2 2 . The difficulty here is to switch the two operators ∇¨and P Ψn ¨. We have by integration by parts

|A| 2 2 "
ż Ω ∇P Ψn pF 1 pϕ n q ´aq ¨∇P Ψn pF 1 pϕ n q ´aq " ´żΩ ∆P Ψn pF 1 pϕ n q ´aqP Ψn pF 1 pϕ n q ´aq `(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ż Γ ∇P Ψn pF 1 pϕ n q ´aq ¨n P Ψn pF 1 pϕ n q ´aq, where the boundary term on Γ cancels since P Ψn pF 1 pϕ n q´aq P Ψ n . Let us denote Φ 1 Q F 1 pϕ n q ´a "

8 ř i"1 γ i ψ i . We have P Ψn pF 1 pϕ n q ´aq " n ř i"1 γ i ψ i .
Thus, we can compute

|A| 2 2 " ´żΩ n ÿ i"1 λ i γ i ψ i n ÿ i"1 γ i ψ i ,
and since the ψ i are orthogonal, we have

|A| 2 2 " ´n ÿ i"1 pλ i γ i ψ i , γ i ψ i q " ´n ÿ i"1 p∆γ i ψ i , γ i ψ i q " n ÿ i"1 p∇γ i ψ i , ∇γ i ψ i q
" pP Ψn ∇pF 1 pϕ n q ´aq, P Ψn ∇pF 1 pϕ n q ´aqq " |P Ψn ∇pF 1 pϕ n q ´aq| 2 2 ď |∇pF 1 pϕ n q ´aq| 2 2 ď |∇F 1 pϕ n q| 2 2 `|∇a| 2 2 , since the operator norm of P Ψn is equal to 1.

Ź It follows from hypothesis [START_REF] Dauge | Elliptic boundary value problems on corner domains, Smoothness and asymptotics of solutions[END_REF] on F that:

|∇F 1 pϕ n q| 2 2 ď ż Ω pF 1 |ϕ n | r´1 `F2 q 2 |∇ϕ n | 2 ď Cp|∇ϕ n | 2 2 `|ϕ r´1 n ∇ϕ n | 2 2 q.
Since r ą 1, the Hölder inequality implies

|∇F 1 pϕ n q| 2 2 ď Cp|∇ϕ n | 2 2 `ˆż Ω |ϕ 2pr´1q n | q ˙1{q ˆżΩ |∇ϕ n | 2q 1 ˙1{q 1 q " Cp|∇ϕ n | 2 2 `|ϕ n | 2pr´1q 2pr´1qq |∇ϕ n | 2 2q 1 q, with 1
q `1 q 1 " 1, for any q ą 1. Let q " 1 r ´1 . Then 2pr ´1qq ě 2, thus H 1 pΩq ãÑ L 2pr´1qq pΩq and 2q 1 ě 2, thus H 1 pΩq ãÑ L 2q 1 pΩq. We finally obtain

|A| 2 2 ď Cp|∇ϕ n | 2 2 `}ϕ n } r´1 1 }ϕ n } 2 2 q `α2 |∇∆ φl | 2 2 |∇F 1 pϕ n q| 2 2 `|∇a| 2 2 , (92)
Ź At last, taking the L 2 -norm of (91), it follows from (92) that

α 2 λ 2 |∇∆ϕ n | 2 2 ď Cp|∇µ n | 2 2 `|∇ϕ n | 2 2 `}ϕ n } r´1 1 }ϕ n } 2 2 q `|∇a| 2 2 ,
This estimate combined with (89) and the regularity of φl (Lemma 5.1) allows us to conclude that estimate (90) is satisfied.

5.5.3. Time derivative estimate for ϕ Lemma 5.19. For any n P N, under a smallness assumption on λ, there exists C ą 0 such that for any T ą 0,

› › › › dϕ n dt › › › › L 2 p0,T ;H ´1pΩqq ď CT `C. (93)
Proof. We introduce the dual operator P Ψn of P Ψn . Equation ( 58) can be rewritten in the following form: pλB t ϕ n , P Ψn χq`pu ϕn ¨∇ϕ n , P Ψn χq´1 λ Pe pdivpBpϕ n q∇µ n q, P Ψn χq " 0, @χ P Φ 1 0 , which becomes

λ dϕ n dt " ´PΨ n ´uϕn B x ϕ n `vϕn B z ϕ n ´1 λ Pe divpBpϕ n q∇µ n q ¯.
Let us treat each term separately:

Ź By Proposition 3.4, we have

u ϕn P L 8 p0, T ; H 1 pΩqq, v ϕn P L 8 p0, T ; L 2 pΩqq.
Moreover, previous estimate (90) implies that ϕ n belongs to L 2 p0, T ; H 3 pΩqq. By a classical result on the multiplicative algebra structure of the Sobolev spaces proved e.g. in [START_REF] Hörmander | Lectures on nonlinear hyperbolic differential equations[END_REF], we deduce that

u ϕn B x ϕ n P L 2 p0, T ; H 1 pΩqq, v ϕn B z ϕ n P L 2 p0, T ; L 2 pΩqq,
with the following estimate:

}u ϕn B x ϕ n } L 2 p0,T ;H 1 q `}v ϕn B z ϕ n } L 2 p0,T ;L 2 q ď C `}u ϕn } L 8 p0,T ;H 1 q `}v ϕn } L 2 p0,T ;L 2 q `}ϕ n } L 2 p0,T ;H 3 q ˘. Ź Furthermore, since B ď B m : } divpBpϕ n q∇µ n q} H ´1 ď B m |∇µ n | 2 .
It follows the claimed estimate (93).

Final convergence results

It is now possible to prove the main theorem 3.3, re-stated here for the sake of readibility:

Theorem. Let T ą 0, ϕ l satisfying Hypothesis (2.3), ϕ 0 P H 3 pΩq satisfying (18), F satisfy the assumptions stated in Section 2.2. Under a smallness assumption on λ, there exists a solution pp, u, ϕ, µq of Problem 3.2.

Proof. From the previous Lemmas 5.17, 5.18 and 5.19 (i.e. estimates (89), ( 90), ( 93)), we obtain the following convergence results (up to a subsequence):

ϕ n á ϕ in L 8 pR `; H 1 pΩqq ˚-weak, ϕ n á ϕ in L 2 loc pR `; H 3 pΩqq weak, µ n á µ in L 2 loc pR `; Φ 1 0 q weak, dϕ n dt á dϕ dt in L 2 loc pR `; H ´1pΩqq weak.
Moreover, Proposition 3.4 combined with the previous global convergence result on ϕ implies the following convergence results (up to a subsequence):

u n á u in L 8 pR `; X pΩqq ˚-weak, v n á v in L 8 pR `; L 2 pΩqq ˚-weak, p n á p in L 8 pR `; H 2 p0, Lqq ˚-weak.
Therefore, from the convergences of ϕ n , we deduce

ϕ n Ñ ϕ in L 2 loc pR `; H 2 pΩqq strong.
Furthermore, by a classical embedding result due to [START_REF] Simon | Compact sets in the space L p p0, T ; Bq[END_REF], we deduce from (90) and (93) that for any T ą 0

ϕ n Ñ ϕ in C 0 pr0, T q; L 2 pΩqq strong, ϕ n á ϕ in C 0 pr0, T q; H 1 pΩqq weak.
Therefore, we can conclude for the convergence of the nonlinear terms:

• Since ϕ n converges strongly in C 0 pr0, T q; L 2 pΩqq X L 2 loc pR `; H 2 pΩqq, the nonlinear terms Bpϕ n q and F 1 pϕ n q converge strongly in C 0 pr0, T q; L 2 pΩqq.

• As far as the convection term u ϕn ¨∇ϕ n is concerned, we know from Lemmas 4.3 and 4.4 that u ϕn is bounded in L 8 pR `; L 2 pΩqq. From the strong convergence of ∇ϕ n in L 2 loc pR `; L 2 pΩqq, we conclude the convergence of u ϕn ¨∇ϕ n .

Lastly, we deduce from the last convergence result that ϕ n p0q converges weakly to ϕp0q in H 1 pΩq, and thus ϕp0q " ϕ 0 because P Ψn converges to the identity for the strong topology of operators. For the boundary conditions on ϕ, the previous convergence result in H 3 pΩq also allows us to conclude that both the Dirichlet (on Γ l ) and the Neumann condition (on Γ 0 ) pass to the limit for ϕ n . Using again the convergence of P Ψn and the fact that ψ i satisfies the homogeneous Dirichlet and Neumann boundary conditions, we deduce that ϕ satisfies (60). For µ, we know that µ n converges weakly to µ in Φ 1 0 . It remains to prove that the functions u ϕ , ϕ and µ satisfy (58), (59). Let ρ P D 1 pR `q, and let N ą 1. For any n ě N , ϕ n satisfies (58) with ψ " µ N . We multiply this equation by ρptq and integrate by parts. From the convergence results stated above, we can pass to the limit in this equation. The limit equation obtained is fulfilled for any N ě 1, and any ρ P D 1 pR `q, thus we conclude from the density of Spanpψ i q iě1 in H 1 pΩq that u ϕ , ϕ and µ satisfy (58), where u ϕ is defined by the formulas ( 22)-( 23) and [START_REF] Simon | Compact sets in the space L p p0, T ; Bq[END_REF].

Lastly, since P Ψn converges to the identity for the strong topology of operators (see Remark 5.10), the dominated convergence theorem allows us to conclude that ϕ and µ also satisfy (59).

Numerical illustration

In this section, we present some preliminary numerical results solving system [START_REF] Sarrazin | Experimental and numerical study of droplets hydrodynamics in microchannels[END_REF], in order to show some features of the model. Let us emphasize that in contrary to other bifluid models, this model does not assume that the interface between the two fluids is a graph, and therefore allows more general configurations, such as drops.

The equations are discretized in a standard way by finite differences. In order to deal with the fact that the domain is not rectangular, we rescaled the equations to work in the rescaled domain Ω rescaled " tpx, yq, x P p0, Lq, y P p0, 1qu. In order to preserve a maximal principle on ϕ, we use the same flux limiters for the Cahn-Hilliard equation as in [START_REF] Boyer | Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model[END_REF]. The boundary conditions are treated by introducing artificial variables in fictive cells on the boundary of the domain.

Influence of the different viscosities

Viscosity is widely used for fluid characterization, and allows us to model different types of behavior for the fluids, even for Newtonian ones (which is the framework of this study). It is of interest to compare the results obtained in both scenarios, when a drop of a less viscous fluid is immersed in a more viscous one, or when a drop of a more viscous fluid is immersed in a less viscous one. Indeed, the results can vary in a qualitative way.

In order to focus on the influence of the viscosity, we use a simple domain of constant thickness h " 1, and we neglect the shear effects by choosing the shear velocity s " 0. The test cases are carried out with the parameter α related to the thickness of the interface chosen equal to α " 0.015, with an input flow Q " 0.5. The time step δt is adapted from the C.F.L. condition, with δt ď 0.01. Thus, we model a situation in which the flow "pushes" the drop in the other fluid, from the left hand side to the right.

• If we want to model for example a drop of oil in water, we choose η 2 {η 1 " 80. We obtain the results presented in Figure 5. We observe that a viscous drop is not really deformed when immersed in a less viscous fluid.

- • On the other hand, choosing η 2 {η 1 " 1{80, we model a drop of water in oil. The results are given in 6. On the contrary to the previous case, the drop is strongly deformed.

- Of course, these numerical results could be enhanced with a model taking the surface tension into account.

Drop transport applications

Another example which allows to validate the program corresponds to the observation of recirculations inside a drop. Indeed, numerical and experimental works [START_REF] Colin | Ecoulement de gouttes dans des microcanaux: simulations numériques et expériences[END_REF], [START_REF] Sarrazin | Experimental and numerical study of droplets hydrodynamics in microchannels[END_REF] have showed that due to the blending dynamics, recirculations are observed.

If we compute the relative velocity, we observe recirculations inside the drops, as in Figure 7. To this end, we define a mean value of the velocity ū, for example It is of interest to note that this asymptotic model, which is in fact a very simple one when comparing to the whole Navier-Stokes system coupled with the Cahn-Hilliard equation, allows us nevertheless to observe very fine phenomena, such as recirculations inside a drop. (94)

Assume that • f is an increasing continuous function such that f p0q " 0,

• C 1 ą 0,

• there exists M ą 0 such that

f pM q ă C 1 2 and C 2 ă M C 1 2 .
If Yp0q ă M , then there exists a constant C such that }Y} L 8 p0,T q ď M.

Moreover, we have }Z} L 1 p0,T q ď CT `C.

Proof. Suppose that there exists 0 ă T ˚ă T , such that YpT ˚q " M and Y 1 pT ˚q ą 0. Then, evaluating (94) at T ˚, and using the hypothesis on C 2 , we get 0 ă Y 1 pT ˚q ď ZpT ˚qpf pM q ´C1 q `C2 ď ´C1 2 ZpT ˚q `C2 ď C 1 2 pM ´ZpT ˚qq.

But since M " YpT ˚q ď ZpT ˚q, we have M ´ZpT ˚q ď 0, which leads to a contradiction.

The regularity of Z follows by integrating (94) over p0, T q, and using the regularity of Y:

C 1 2 }Zptq} L 1 p0,T q ď YpT q `C1 2 }Zptq} L 1 p0,T q ď Yp0q `C2 T ď M `C2 T, which is written }Zptq} L 1 p0,T q ď CT `C.

Figure 1 :

 1 Figure 1: Domain Ω and boundary conditions on the velocity

Figure 2 :

 2 Figure 2: Possible shape of the potential F pϕq

Figure 3 :

 3 Figure 3: Domain Ω and notations for the boundary

Notations 3 . 1 .

 31 Let us define some notations and function spaces.

4. 1 .

 1 Regularity of the coefficients Lemma 4.1. Assume that the viscosity η satisfies (9). If ϕ P H 1 pΩq, the coefficients defined in (20), (21) have the following regularity: a, b, c P X pΩq, r a, r b, r c, r d, r e P H 1 p0, Lq. Proof. Assume ϕ P H 1 pΩq. The terms a, b, c are of the form ż z 0 ξ i {ηpϕpx, ξqq dξ, for i " 0, 1, 2 (see definition (20) of a, b,

Ź

  Next we show that b P L 8 pΩq: since B z b P L 2 pΩq, we can write bpx, zq " bpx, 0q `ż z 0 B ξ bpx, ξq dξ.

Recalling the definitions r d " ˜r c ´r b 2

 2

Remark 4 . 6 . 5 .

 465 Since (26a)-(26b)-(26c) are steady-state equations, the constants in the previous estimates are also independent of time, so that the L 8 pΩqand L 2 pΩq-estimates of Lemma 4.3 and 4.4 can also be written in L 8 p0, T ; L 8 pΩqq and L 8 p0, T ; L 2 pΩqq for any T ą 0. About the Cahn-Hilliard equation 5.1. Useful results and inequalities 5.1.1. Boundary conditions and lift operator

Figure 4 :

 4 Figure 4: Possible shapes of function χ and ψ

Figure 5 :

 5 Figure 5: A drop of oil (in yellow) in water (in dark blue)

Figure 6 :

 6 Figure 6: A drop of water (in dark blue) in oil (in yellow)

Figure 7 :

 7 Figure 7: Recirculations in a drop and shape of the drop

1 .

 1 Let T ą 0. Let Y and Z be two functions in C 1 pr0, T sq, such that there exists three real constants C 1 , C 2 and a function f : R Ñ R satisfying Y 1 `C1 Z ď f pYqZ `C2 , 0 ď Y ď Z on r0, T s.

  r e P H 1 p0, Lq.

	Proof. By definition (21), r dpxq can be written in the form:
	r dpxq " r cpxq	´r bpxq 2 r apxq	"	ż hpxq 0	z 2 ηpx, zq	dz	´˜ż hpxq 0 ż hpxq 0	¸2 dz dz ηpx, zq z ηpx, zq 1	.
	4.2. Coercivity of the operator
	Lemma 4.2. Assume that the viscosity η satisfies (9). Let r d be defined by (21).
	It satisfies the following estimate:
		@x P p0, Lq, r dpxq ě δ :"	h 3 m 12η M	ą 0.	(35)

  4.3. Estimates of |u| 8 and |v| 2 Lemma 4.3. Assume that the viscosity η satisfies (9). Assume ϕ P H 1 pΩq. The horizontal velocity u given by (26b) satisfies |u| 8 ď C. Proof. The regularity of u follows from the regularity of p, equation (26b) and the regularity of the coefficients (Lemma 4.1):

	u " pb	´ar b r a	qλ B x p `sp1	´a r a	q P XpΩq.
							ż z
	Moreover, we know that u is a combination of coefficients of the form	ξ{ηpϕqdξ.
	Indeed						0
	|u| 8 ď ¨|b| 8	`|a| 8 | r b| 8 min xPp0,Lq r apxq	'λ|B x p| 8	`s ¨1	`|a| 8 min xPp0,Lq r apxq	',

  Proof. Let us take ψ " µ P Ψ n in the weak formulation (58).

							Using Definition
	(59) for µ, we get
		ż	2			ż
	λ	Ω looooooooooooooooooooooooomooooooooooooooooooooooooon B t ϕp´α λ 2 ∆ϕ `a `PΨn pF 1 pϕq ´aqq	looooooooooomooooooooooon Ω `1 λ Pe Bpϕq|∇µ| 2	"	looooooomooooooon ´żΩ u ¨∇ϕµ	.
				":A		":B	":D
							(62)
	Let us obtain estimates for each term A, B, D:
		Ź The A-term is composed of two parts:
				ż
			A "	´α2 λ 2 loooooooooooooooomoooooooooooooooon B t ϕ∆ϕ `żΩ B t ϕ a Ω	`żΩ loooooooooooooomoooooooooooooon B t ϕP Ψn pF 1 pϕq ´aqq
				":A1		":A2
							¸.	(61)
						2 2 ¯|∆ϕ| 2 2 `|v| 2 2 ~b~2 2

  we multiply (59) by ϕ, we get `pP Ψn pF 1 pϕq ´aq, ϕq looooooooooomooooooooooon

		2
	pµ, ϕq lo omo on	" p´α λ 2 ∆ϕ `a, ϕq looooooooomooooooooon
	":A	":B

":D

.

  Position de la frontiere au temps t 0.000 pour a = 1 et b= 1

	1.006									1.000
	0.894									0.778
	0.781									0.556
	0.669									0.333
	0.556									0.111
	0.444									-0.111
	0.331									-0.333
	0.219									-0.556
	0.106									-0.778
	-0.006
	-0.006	0.106	0.219	0.331	0.444	0.556	0.669	0.781	0.894	1.006
		.

Acknowledgments

The authors want to thank the reviewers for their relevant and helpful remarks and comments, which helped to improve the paper.