Keywords:

Algorithm to compute quantiles of 2D (and 3D) distributions.

• Well-founded statistics on top of them.

• Useful for multi-objective optimization problems.

Expected RunTime ECDF [HAB + 16] q 0=q |f(x)-f(x*)|=q* ^qq* 0 q* r … q t q t P(q*=q) P q t P P(q*<q) q t P Expected Quality q t P Expected RunTime

QT Empirical Attainment Function Levelsets [LIS14]

q 0=q |f(x)-f(x*)|=q* ^qq* 0 q* r … q t q t P(q*=q) P q t P P(q*<q) q t P Expected Quality q t P Expected RunTime q t q 0=q |f(x)-f(x*)|=q* ^qq* 0 q* r … q t q t P(q*=q) P q t P P(q*<q) q t P Expected Quality q t P Expected RunTime q t q t P QT Empirical Attainment Function Quality-Time Empirical Attainment Function

• EQ-and ERT-ECDF are trivialy computed:

• fix a target quality (resp. time),

• traverse all runs across time (resp. quality),

• compute the ratio of better-than-target.

• QT-EAF requires a more complex algorithm [GdFF02].

q t P E R T -E C D F E Q -E C D F Q T -E A F le v e ls e t

Part 2

Quantiles on joint distributions

Closeness

• A set of optimization trajectories forms a closed-set.

• Bounded by: • Which are essentially equivalent to quantiles.

• optimal solution, q ∈ [0, bound[, • time budget, t ∈ [1, budget[, • P ∈ [0,
• Because the sample is finite, there is a finite number of levelsets.

• At most r level sets for r input sets.

• (Assuming minimization on both axis).

• "Peel" level sets.

• Sweep one axis in increasing values,

• sweep the other in decreasing values.

• Essentially computes incremental Pareto-optimal archives.

• O(m log m + nm), m points in n runs (asymptotically optimal). • Output surfaces instead of level sets.

• C++ implementation.

• Within IOHexperimenter https://iohprofiler.github. io/IOHexp/.

• May be ported if needed.

Part 4

Statistics

Examples

• Orthogonal partial section statistics:

• Area Under curves (EQ-ECDF and ERT-ECDF).

• Attainment surface (EAF).

• Global statistics:

• Volume under the EAF ≈ sum/mean-like.

• Volume under a levelset ≈ quantile-like.

• Volume under a subset of levelsets (scaling approximation). • We want to optimize both quality and time.

• Covariance [GdFFH01]. q t P E R T -E C D F E Q -E C D F Q T -E A F le v e ls e t
• Because we have no clue about budget or target in advance.

• Maximize an average aggregate?

No.

• Maximize volume under the EAF. • Well-founded statistics on top of them.

• Useful for multi-objective optimization problems.

• Example: automated tuning of stochastic optimization solvers.

Expected

  Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10 Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10

  Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10

•

  Computes level sets of the distribution [GdFFH01].

• 20 |

 20 Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10 Algorithm 3 dimensions • O(n 2 m log m), m points in n runs, • a logarithmic factor worse than the upper bound.

  Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10 Example of use Automated design • Consider the automated design of an optimization solver as a (meta) bi-ojbective optimization problem [Dre03].

30|•

  Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10 Conclusion Recall Algorithm to compute quantiles of 2D (and 3D) distributions.

31|•

  Johann Dreo | Quantile-like measures on multi-dimensional distributions of closed sets | 2021-06-10 Conclusion Perspective Quantify the time/memory/loss compromises.• Impact of the statistic choice on the optimization sub-problem.• Use in multi-objective problems.Introduction• Optimization = find the optimumx * minimizing an objective function f

Statistics & Probability Letters, 57(2):179-182, April 2002.

Pros and cons

• If all level sets are computed, EAF is the true function (given the sample).

• EAH is an approximation (which converges with the discretization). . . • . . . but can be computed on log-log scales, for better resolution.

• EAH scales better regarding the number of points. . . • . . . but require more memory.

Why?

• Multi-objective problems:

• either Pareto-optimal approaches (heavy on user),

• either objectives aggregation (not good math properties).

• How to aggregate randomized observations?

Statistics for bi-objective problems Why?

• Multi-objective problems:

• either Pareto-optimal approaches (heavy on user),

• either objectives aggregation (not good math properties).

• How to aggregate randomized observations?

• EA[FH] is a way to aggregate Pareto-optimal fronts.

• One can compute statistics on it.

Part 6

Supplementary material

Bibliography

Johann Dreo.

Adaptation de la métaheuristique des colonies de fourmis pour l'optimisation difficile en variables continues. Application en génie biologique et médical.

Theses, Université Paris XII Val de Marne, December 2003.

Johann Dreo.

Using performance fronts for parameter setting of stochastic metaheuristics.

In Franz Rothlauf, editor, Genetic and Evolutionary Computation Conference, GECCO 2009, Proceedings, Montreal, Québec, Canada, July 8-12, 2009, Companion Material, pages 2197-2200. ACM, 2009. Viviane Grunert da Fonseca and Carlos M. Fonseca.

A link between the multivariate cumulative distribution function and the hitting function for random closed sets.

Terminal distribution

• The output distribution of a randomized search heuristic is the probability to reach a given quality target:

IMG

Temporal convergence

• Because any solver (should) be quasi-ergodic:

• then it (should) converge in a finite time:

• Thus the probability of attaining the target should not decrease over time:

• Hence the trajectory in objective space of a run is monotonic.

Formal concepts

Pareto Optimality

• The trajectory in objective space being monotonic, all its points are non-dominated, and the set is Pareto-optimal.

Formal concepts

QT-EAF

• Given a set of non-dominated sets:


domain: phys spi

text:
Pierre Bourdon

Rodwane Chtouki

Laurent Lombard

Anasthase Liméry

Julien Le Goüet

François Gustave

Hermance Jacqmin

Didier Goular

Christophe Planchat

Anne Durécu

Coherent Beam Combining of Lasers: Toward Wavelength Versatility and Long-Range Operation Compliance

Keywords: Fiber laser, Coherent Beam Combining, nonlinear optics, Optical Parametric Oscillator, phase control, turbulence mitigation

Coherent beam combining (CBC) by active phase control is an efficient technique to power scale fiber laser sources emitting in the near-infrared, between 1 and 2 µm, up to the multi-kilowatt level. Interestingly, it has been demonstrated by our team that CBC could also be used to power scale mid-infrared sources, frequency converters, generating a wavelength between 3 and 5 µm. We present our latest results on coherent combining of continuous-wave highefficiency mid-infrared sources: optical parametric oscillators (OPOs) and detail the difficulties encountered to achieve this combining, as well as the main limitations to efficient operation of CBC in this case.

In a second part of this talk, we also present recent results on coherent combining of seven 1.5-µm fiber lasers through active phase control, using frequency-tagging, and operating efficiently on a remote target. A testbed has been designed to combine these 7 lasers on a remote surface, with phase-locking operating through analysis of the optical signal backscattered by the target, in a so-called target-in-the-loop (TIL) experiment. In such TIL configuration, CBC mitigates both laser-amplification-induced and atmospheric turbulence-induced phase fluctuations simultaneously. CBC demonstrated proper operation outdoors, on a target located up to 1 km from the laser and the results from this experimental campaign will be described.

INTRODUCTION

Coherent beam combining (CBC) techniques involving active phase control of the laser emitters have demonstrated their potential to power scale continuous-wave [START_REF] Shekel | 16kW single mode CW laser with dynamic beam for material processing[END_REF] and pulsed [START_REF] Lombard | Coherent beam combination of narrow-linewidth 1.5 μm fiber amplifiers in a long-pulse regime[END_REF] fiber lasers through coherent addition of the power emitted by multiple separate amplifiers. Using all-fiber components, very compact configurations of coherently combined fiber amplifiers can be designed, with demonstrated overall emitted power exceeding the kilowatt level [START_REF] Shekel | 16kW single mode CW laser with dynamic beam for material processing[END_REF].

However, high power fiber lasers are only available within a finite set of wavelengths, mainly amongst 1 µm, 1.5 µm and 2 µm. Hopefully, nonlinear crystals offer the capability to convert these laser lines to access higher wavelengths in the midinfrared bands, for instance around 4 µm. But as with lasers, power scaling of frequency converters is limited, essentially by the damage threshold of the nonlinear medium. Coherent combining techniques could be useful to power scale frequency converters beyond this damage threshold limit. Unfortunately, combination techniques are difficult to implement in this case, as active phase control requires fast phase modulators at the midinfrared wavelength emitted by the converter. Such components are not exactly "off-the-shelf" and are not as practical nor as fast as the all-fiber modulators available at more standard wavelengths.

Taking advantage of the phase-matching relations involved in any frequency converting process, one can indirectly control the converted wave phase through control of the pump wave phase. When this pump wave is delivered by fiber amplifiers, such a frequency-converter coherent combining configuration can be achieved using standard off-the-shelf all-fiber phase modulators. * pierre.bourdon@onera.fr; phone (+33) 1 80 38 63 82; fax (+33) 1 80 38 63 45; www.onera.fr Our team successfully applied this approach to perform efficient coherent combination of second harmonic generators and difference frequency generators [START_REF] Odier | Coherent combining of second-harmonic generators by active phase control of the fundamental waves[END_REF], [START_REF] Odier | Coherent combining of fiber-laser-pumped 3.4-µm frequency converters[END_REF], [START_REF] Odier | Coherent combining of mid-infrared difference frequency generators[END_REF]. In a first part of this paper, we present the challenges of extending this indirect phase control technique to midinfrared OPO coherent combining, as well as the main limitations to efficient operation of CBC in this case. An experimental test of OPO CBC is also detailed. CBC has been demonstrated for a large number of lasers [START_REF] Bourderionnet | Collective coherent phase combining of 64 fibers[END_REF], [START_REF] Chang | First experimental demonstration of coherent beam combining of more than 100 beams[END_REF], but many realizations use the fact that the laser beams are spatially separated at the output of the lasers, before overlapping after propagation and becoming undistinguishable. While the laser beams are still spatially distinct, it's feasible to measure the phase of each laser and to control this phase in real time to achieve phase-locking. However, for some applications, such process cannot be applied as phase measurement has to be done once the laser beams have already begun to overlap and to interfere.

That's the case of target-in-the-loop CBC where one wants to phase-lock the lasers on a remote target, maximizing the power density deposited at long range. In this case, phase measurement at the output of the lasers is useless, as beam propagation induces additional phase shifts that are not accounted for in this measurement. It's necessary to find some way of driving the laser phases, benefiting from the information available in the optical signal backscattered by the remote target.

One approach is to try and maximize the intensity of this backscattered signal, as maximum power density deposited on the target corresponds to a maximum of this backscattered signal too. It's been done, for instance, very early on, during the first experiments of TIL-CBC on a glint target [START_REF] Pearson | Coherent optical adaptive techniques: design and performance of an 18-element visible multidither COAT system[END_REF], and, later on, up to 7 km using a stochastic parallel gradient descent algorithm [START_REF] Weyrauch | Experimental demonstration of coherent beam combining over a 7 km propagation path[END_REF], a classical approach to maximize optical intensity by step-by-step optical wavefront correction.

In 2009, our team demonstrated a more practical and promising approach that can be performed using frequency-tagging for CBC phase control [START_REF] Jolivet | Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence[END_REF]. Frequency-tagging each optical channel at a specific frequency to assess the phase fluctuations to be compensated for is an efficient technique for CBC, also known as LOCSET [START_REF] Shay | First experimental demonstration of self-synchronous phase locking of an optical array[END_REF]. Frequency-tagging has the advantage of "engraving" specific information on the signal emitted by each optical channel through low-depth modulation. This tagging process enables to easily retrieve the phase information from each channel, within the complex interference signal generated once the laser beams have overlapped. Moreover, this phase information retrieval can also be achieved using the backscattered signal from a remote target. We also demonstrated that a simple evolution of the configuration of detection used for direct CBC could transform a standard CBC system into a fully operational TIL-CBC device [START_REF] Jolivet | Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence[END_REF].

The tests we performed in 2009 were in the laboratory with artificially generated atmospheric turbulence. In a second part of this paper, we present the work done recently to build a testbed coherently combining seven 1.5-µm fiber lasers through active phase control, using frequency-tagging. We also present the first results obtained on TIL-CBC operating this testbed. These results are obtained propagating the laser beams through the atmosphere, hence through real atmospheric turbulence aver a distance ranging up to 1 km.

COHERENT COMBINING OF OPTICAL PARAMETRIC OSCILLATORS

Principle and experimental setup for CBC of parametric amplifiers

Coherent beam combining of non-degenerate three wave mixing nonlinear processes was first demonstrated on devices that didn't require a cavity to operate: difference frequency generators. But the theoretical bases are identical to those of OPO CBC, as both experiments combine parametric amplifiers. To achieve CBC of DFGs or OPOs, phase control of one of the converted waves, the signal or the idler waves, has to be performed. Indirect phase control benefits from the phase-matching condition.

The electric field corresponding to each nonlinearly coupled wave can be written as:

(1) with = , or p and the complex amplitude of the pulsation wave. If we assume that the pump wave is not depleted when passing through the crystal, which is true on the first millimetres of the nonlinear crystal where parametric amplification is still weak, the coupled-amplitude equations describing difference frequency generation are:

, = - + . (2)
where . Propagation equation of the idler wave is:

(3)

with

. A similar equation can be written for the signal wave.

Maximum power conversion is achieved for perfect phase-matching condition, i.e. ∆k = 0.

This system of equations can be solved, in the case of an unseeded idler wave Ai(0) = 0, into:

These solutions correspond to a signal field that keeps its initial phase-offset, and is simply amplified by the interaction, while the idler wave phase-offset depends on both the pump wave's and the signal wave's.

(5-a)

Similarly, in the QPM case, high conversion efficiency can be achieved in a periodically poled crystal. The relation between the phases of the pump, signal and idler waves can be obtained propagating the waves in the crystal, domain by domain and step by step:

(5-b)

Due to this phase relation, controlling the pump wave phase gives complete control of the idler wave phase. For instance, in the QPM case, a ϕp0 phase shift induced on the pump wave will compensate for an idler wave phase variation of ϕp0 -ϕs0, where ϕs0 is the initial phase of the signal wave at the entrance of the nonlinear crystal.

Consequently, phase control of DFG and OPO converters is achievable, using all-fiber electro-optic phase modulators to control the pump wave.

We demonstrated experimentally the feasibility of such indirect phase control combining two DFG modules pumped by 1.064-µm Yb:fiber lasers delivering up to 15 W power each (see Fig. 1).

To fix and maintain the signal wavelength and phase, it is necessary to seed the DFGs with a signal seed at 1553 nm from a laser diode amplified by a commercial erbium-doped fiber amplifier split in two. Using this 1553-nm seeder, we are able to keep both DFG channels at a common and fixed signal wavelength.

The DFG nonlinear crystals are PPLN crystal from Covesion and their length is L = 20 mm. With a 1064-nm pump wavelength and a seeded signal at 1553 nm, quasi-phase matching in the PPLN crystal generates an idler wave at 3.4 µm. The pump beams are focused down to 100 µm waist in the PPLN crystals.

Coherent combining of these DFG modules is achieved using the idler waves interference signal to close the feedback loop (see Fig. 2). Time-averaged combining efficiency is excellent: the residual phase error is λ/28 rms for the idler wave. This experiment is the proof that CBC of continuous wave DFG using indirect phase control is feasible, hence proving that OPO CBC is also feasible as the nonlinear process involved in both cases is the same: parametric amplification.

The main difference between SHG CBC and DFG CBC is that simultaneous phase locking that has been proved feasible for SHG combination [START_REF] Odier | Coherent combining of second-harmonic generators by active phase control of the fundamental waves[END_REF], cannot be achieved so easily when combining non-degenerate three-wave mixing frequency

⎩ ⎪ ⎨ ⎪ ⎧ = !" # $$ # % & * ∆ & = !" & # $$ & # % * ∆ ∆) = ) p -) s -) i # & # = - * -. / % / # & -= !" # $$ # ⎩ ⎪ ⎨ ⎪ ⎧ = 0 123 4 --& / % / # & = -& % 0
converters [START_REF] Odier | Coherent combining of mid-infrared difference frequency generators[END_REF]. Even if seeded, the optical path difference between the two signal channels induces a phase difference that is not compensated for, and the signal waves are not phase-locked simultaneously when the idler waves are.

Experimental tests of OPO CBC, challenges and limitations

First step of experimentally combining OPOs was to design an efficient continuous-wave OPO cavity. Different cavities were tested, ring cavities and linear ones, and the best results were obtained with the linear cavity presented in Fig. 3 that offered the lowest emission threshold. The cavity is 20-cm long and is formed between two concave mirrors with 50-mm and 130-mm curvature radius respectively. Both are high reflection coated for the signal wavelength at 1.5 µm (Rs > 97 %) and the output coupler is high transmission at the idler midinfrared wavelength. Both mirrors offer a low reflection at the pump wavelength to minimize the pump losses.

With this linear cavity, the OPO threshold was around 7 -8 W and the OPO delivered a few hundreds of milliwatts of idler power at 3.8-µm wavelength as can be seen in Fig. 4. Using a wavemeter and finely tuning finely the temperature of the second NL crystal, we were able to bring the second signal wavelength very close to the first one, sufficiently close to allow for fine adjustment of the second signal wavelength to the other signal wavelength value using cavity length tuning with a piezo-electric micro-positioner.

Unfortunately, even with a singly-resonant OPO that should provide smooth tuning of the signal wavelength with the cavity length, we observed thermally induced longitudinal mode-hops that made it impossible to implement proper feedback on the signal wavelength through cavity length control.

Parasitic Fabry-Perot effect due to residual reflectivity of the cavity mirrors at the pump wavelength were also observed and induced detrimental power fluctuations even after limiting their impact as much as possible.

Coherent combining of cw OPOs couldn't be achieved and would require a higher level of stabilization of the OPO cavities than the one available on the experimental setup, as well as a choice of mirrors specifically designed for the OPO cavity with higher transmittance for the pump wavelength.

TARGET-IN-THE-LOOP COHERENT COMBINING OF 7 FIBER LASERS

The laser testbed for CBC and TIL-CBC

The laser configuration is a standard 7-channel master oscillator power amplifier (MOPA) configuration at 1.5 µm wavelength. A single-frequency low power (40 mW) master oscillator (MO) is split and amplified into 7 channels. Each of the 7 Er-Yb doped fiber amplifiers delivers up to 3 W at 1543 nm. One of the fiber amplifiers is not phase controlled and is free to fluctuate in phase while in-between the MO and 6 of the amplifiers, we use fast electro-optic phase modulators with a 150-MHz bandwidth to control the phase of each channel and simultaneously tag each of these 6 channels at a specific frequency around 20 MHz with a low-depth phase modulation.

In this self-referenced LOCSET configuration, the 6 phase-controlled channels follow the phase fluctuations of the seventh channel that's not modulated (see Fig. 6).

Figure 6. On the left side, schematics of the experimental setup for frequency-tagging CBC (also known as LOCSET). On the right side, interference pattern in the far-field, captured on a 1.5-µm camera located at the same distance as the direct CBC detector.

The 7-channel CBC setup delivers a nice interference pattern in the far-field. We confirmed that in direct CBC, i.e. using a fast photodiode to capture part of the central lobe of interference in the far-field, it was easy to lock the phases of the 7 laser channels and stabilize the position of the interference pattern. In LOCSET, frequency tagging of the i th channel at a specific frequency νi results in an interference signal that generates an error signal after demodulation at the same frequency given by equation:

where φi is the i th amplifier chain output phase in the photodetector plane, and φu the unmodulated reference beam phase in the photodetector plane. Ei and Eu are the respective electric field amplitudes at the output of the i th amplifier and of the unmodulated amplifier. βi and βu are the respective modulation depths of the i th amplifier channel and of the unmodulated channel. Driving the 6 error signals simultaneously to zero in real time results in maintaining equal phases for the 7 channels. Any sudden phase shift is immediately compensated for, thanks to the very fast electro-optic phase modulators and fast detection.

CBC efficiency measured as the fraction of the total power located in the central lobe of the interference pattern was higher than 53 % (see Fig. 6). It's very close to the experimental maximum that can be achieved with Gaussian laser beams (theoretical maximum is 63 %), but a little lower due to the opto-mechanical constraints limiting the effective fillfactor in the near-field.

Residual phase error has been measured and is lower than λ/40, confirming again the high efficiency of the CBC process and the high speed of the feedback loop. The opto-mechanical head where the outputs of the 7 fiber amplifiers are collimated and sent through the atmosphere is completely adjustable (see Fig. 7). Piezo-electric motors are used to set the transverse position of the output fiber tip with respect to the collimating lens, in order to align the 7 collimated beams in the same direction. The focus of the collimating lenses is manually tuned beforehand. Figure 3 presents pictures of the first part of the testbed dedicated to laser emission.

For Target-In-the-Loop Coherent Beam Combining (TIL-CBC), a second stage of the testbed is dedicated to long range focusing of the laser beams and to collection and detection of the backscattered signal from the remote target. We chose to use a bi-static configuration (see Fig. 8) where the long range focusing and the reception apertures are separated, to limit the risk of narcissus effect, and decrease the optical complexity of the detection line with respects to a monostatic configuration.

) ( ) ( ) ( ) ( )       - + - ∝  - = 1 1 0 1 _ sin sin N j i j j j i u u i i i error J E E J E t i φ φ β φ φ β

Experimental demonstration of TIL-CBC

After some first demonstrations of CBC and TIL-CBC in the laboratory and then outdoor at shorter range (less than 100 m), the testbed that has been designed to be mobile was transferred to our laser site where we are able to shoot lasers safely up to 1-km range (see Fig. 9). The site is equipped with proper apparatus to monitor the weather and atmospheric conditions, and also to measure turbulence strength through the average value of the Cn² over 1 km.

The experiments took place during summer in the south west of France. As the laser beams propagated horizontally at 1.5 m from the ground, the level of turbulence was quite high and the average value of Cn² over 1 km was often close to or even much higher than 10 -13 m -2/3 . Despite these detrimental turbulence conditions, when the Cn² value was lower than 10 -13 m -2/3 , we were able to perform efficient TIL-CBC up to 1 km.

A transparent wedge plate was placed before the target, to reflect a small fraction of the combined beams power to a black screen located at the same distance as the target. Proceeding like this, we could record video sequences of the interference pattern displayed on black screen and observed with an InGaAs Raptor camera, without interfering with the TIL-CBC process in the target plane. Fig. 9 presents instantaneous images of the interference pattern with and without closing the CBC feedback loop. Even if turbulence induced perturbations clearly appear to decrease the efficiency of the CBC locking process, TIL-CBC still operates successfully and results in an increased power density deposited on the remote target at 1 km.

CONCLUSION

We developed an experimental setup for coherent combining of continuous wave OPOs with active phase control of their pump waves. To prevent fluctuations of the intracavity pump power, parasitic Fabry-Perot effects on the pump wave have to be mitigated Temperature tuning of NL crystal is enough to bring both signal wavelengths close enough for fine tuning through cavity length adjustment. However, this experimental test of OPO coherent combining was unsuccessful due to detrimental signal cavity modehops that prevented from smooth enough tuning of the signal wavelength.

Figure 1 .

 1 Figure 1. Schematics of the experimental set-up for demonstrating coherent combining of two 3.4-µm idler beams generated through DFG in PPLN crystals. Coherent combining is achieved by active phase-control of one of the pump waves.

Figure 2 .

 2 Figure 2. Time evolution of the interference signals of the idler beams when the phase control loop is open and then closed.

Figure 3 .

 3 Figure 3. Optimal continuous-wave OPO linear cavity. The output coupler is mounted on a piezo-electric translation stage to be able to finely tune the position of the spectrum emitted.

Figure 4 .

 4 Figure 4. Idler power vs. pump power for the continuous-wave OPO.. After duplicating the cw OPO cavity, we began testing CBC of OPOs. The experimental setup is arranged as in Fig. 5.

Figure 5 .

 5 Figure 5. OPO CBC experimental setup. For the sake of making the figure easier to read, ring cavities are drawn for the OPOs but we used the optimized linear cavity for both OPOs in the experiment.

Figure 7 .

 7 Figure 7. Pictures of the experimental setup parts.

(

Figure 8 .

 8 Figure 8. View of the complete testbed with both emission and reception lines.

Figure 9 .

 9 Figure 9. Instantaneous images of the interference pattern generated at 310 m when TIL-CBC feedback loop is active and inactive.

Fig. 10

 10 Fig.10presents averaged images from the same video sequence. The size of the interference lobes is not different in the instantaneous images and in the averaged image, demonstrating that the interference pattern was locked on a fixed position in the target plane.

Fig. 11

 11 Fig. 11 presents the averaged image obtained when target distance was 1 km.

Figure 10 .

 10 Figure 10. Averaged images of the interference pattern generated at 310 m when TIL-CBC feedback loop is active and inactive.

Figure 11 .

 11 Figure 11. Averaged images of the interference pattern generated at 1 km when TIL-CBC feedback loop is active and inactive.

--& / % / # * 0 2.53 4--& / % / # 6 & = 6 % -6 + " #

We also developed a mobile testbed for coherent beam combining of 7 fiber lasers emitting at 1.5 µm. This testbed operates through frequency-tagging phase difference monitoring. It can concentrate more than 53 % of the overall power in the central of the interference pattern generated in the far-field, after overlap of the laser beams. The testbed has been designed to use the backscattered signal coming from a remote target as a reference for phase difference monitoring and phase locking. It was operated in a target-in-the-loop configuration, first at short distance in the laboratory, and then at longer distance outdoors, propagating the beams through real, weak to strong atmospheric turbulence. The longer range outdoor experimental campaign demonstrated that TIL-CBC efficiency is preserved up to more than 310 m. Perturbations induced by strong atmospheric turbulence are significantly affecting TIL-CBC, but TIL-CBC operates successfully at 1-km range anyway, albeit with lower efficiency. Future work is dedicated to modifying the testebed to improve its performances, especially when dealing with strong atmospheric turbulence conditions.

ACKNOWLEGMENTS

This work was partially funded by French Delegation Generale pour l'Armement.


Laurent Chupin

Nicolae Cîndea

Geoffrey Lacour
email: geoffrey.lacour@uca.fr

Variational inequality solutions and finite stopping time for a class of shear-thinning flows

Keywords: Mathematical Subject Classification (2020): 35K55, 76D03, 35Q35, 76A05 non Newtonian, generalized Newtonian, shear thinning, variational inequalities

Introduction

The aim of this paper is to study the existence and some properties of solutions of the system

       ∂ t u + u • ∇u + ∇p -∆u -div (F (|D(u)|) D(u)) = f in (0, T ) × Ω div(u) = 0 in (0, T ) × Ω u = 0 on [0, T ) × ∂Ω u = u 0 on {0} × Ω (1.1)
in the form of nonlinear parabolic variational inequalities, where Ω is an open bounded subset of R N , for N ∈ {2, 3} with a regular enough boundary ∂Ω. Such nonlinear systems describe the flow of so-called generalized Newtonian fluids and give rise to several relevant models. Let us give some examples. First, if F (t) = C, the system (1.1) is nothing else than the Navier-Stokes equations for a viscous incompressible fluid. Also, by choosing

F (t) = (1 + t 2 )
p-2 2 , system (1.1) describes a Carreau flow. Another relevant case is obtained by choosing, for p ∈ (1, 2) by F (t) = t p-2 for t > 0 which leads to an Ostwald-De Waele (power-law) flow. In the particular case of Bingham arising for p = 1 which describes a viscoplastic behavior, we get F (0) ∈ [0, 1], where τ * = 1 > 0 is the so-called plasticity threshold, which is scaled here. In the latter case, the function is multivalued at the origin (note that a physical consequence of this phenomenon is the nonexistence of a reference viscosity for threshold fluids, see for example [START_REF] Becker | Simple non-newtonian fluid flows[END_REF]). It is now established that this problem can be circumvented by considering the function outside the origin by a regularization process and by giving a meaning to its limit, in sense of subdifferential. This approach has been successfully carried out in the case of a two-dimensional Bingham flow for example (see for example [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF]). In the present paper, we focus on the mathematical analysis of shear-thinning flows: a flow is said to be shear-thinning when its viscosity decreases as a function of the stresses applied to it, which is to say, in the flows we consider, that the function decreases as the shear rate increases. We mainly refer to [START_REF] Coussot | Rhéophysique: la matière dans tous ses états[END_REF][START_REF] Bird | Dynamics of polymeric liquids[END_REF][START_REF] Galdi | Hemodynamical flows[END_REF] for the physical motivations of such models. Throughout the present article, we will consider simple fluid flows, that is we will make the assumption that the shear rate is the second invariant of the strain-rate tensor, and moreover it is a scalar quantity given by |D(u)|.

Since the 1960s, the study of such systems has been the subject of numerous articles (see, for instance, [START_REF] Frehse | Non-homogeneous generalized Newtonian fluids[END_REF] and references therein). On one hand, the existence of weak solutions of (1.1) is a difficult problem which has been studied in many particular cases. When the shear tensor S has an p-structure (see [START_REF] Diening | Existence of weak solutions for unsteady motions of generalized Newtonian fluids[END_REF] for a definition), the existence of weak solutions has been established under certain hypotheses for p > 2N N +2 (see [START_REF] Eberlein | Existence of weak solutions for unsteady motions of Herschel-Bulkley fluids[END_REF], and also [START_REF] Málek | Global analysis of the flows of fluids with pressuredependent viscosities[END_REF]). The study of the existence of solutions under other assumptions has been done in [START_REF] Frehse | Non-homogeneous generalized Newtonian fluids[END_REF] in the case p > 2N N +2 , in [START_REF] Málek | On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2[END_REF] for the case N = 3 and p ≥ 9 4 , and in [START_REF] Wolf | Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity[END_REF] for p > 2N +2 N +2 , and in [START_REF] Berselli | Existence of strong solutions for incompressible fluids with shear dependent viscosities[END_REF] for the three-dimensional periodic case in space with p > 7 5 . One can avoid such hypotheses on p by using dissipative solutions. The existence of dissipative solutions has been proved in [START_REF] Abbatiello | On a class of generalized solutions to equations describing incompressible viscous fluids[END_REF] in the three-dimensional setting. Nevertheless, variational inequality solutions are particularly interesting in view of numerical simulations perspective (see for example [START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF]Chapter 4] or [START_REF] Saramito | Complex fluids[END_REF]) as for controllability (see for example [START_REF] Friedman | Optimal control for variational inequalities[END_REF] or [START_REF] Ito | Optimal control of parabolic variational inequalities[END_REF])

We recall that in the three-dimensional setting, it is known that we can prove the existence of a unique weak solution to the incompressible Navier-Stokes equations, as long as the norm of the initial velocity field is small enough and the force term is sufficiently regular. This theorem has been generalized for a large class of non-singular stress tensors in [START_REF] Amann | Stability of the rest state of a viscous incompressible fluid[END_REF], as long as the initial velocity field u 0 and the force term f are regular enough. The study of the regularity of solutions in a more general framework is a difficult problem, the case of the flow of an incompressible Newtonian fluid governed by the Navier-Stokes equations remaining open in the three dimensional case. However, the existence of regular solutions, sometimes giving rise to the uniqueness of solutions or even to the existence of strong solutions, has been established in the case of shear tensors having an (p, µ)-structure in [START_REF] Diening | Strong solutions for generalized Newtonian fluids[END_REF] and in [START_REF] Berselli | Existence of strong solutions for incompressible fluids with shear dependent viscosities[END_REF] in a three-dimensional periodic in space case. In the steady case, somes results have been obtained as in [START_REF] Frehse | On analysis of steady flows of fluids with sheardependent viscosity based on the Lipschitz truncation method[END_REF] (existence for p > 2N N +2 ), in [START_REF] Zhou | Regularity of weak solutions to a class of nonlinear problem with non-standard growth conditions[END_REF] (regularity), in [START_REF] Berselli | Global regularity properties of steady shear thinning flows[END_REF] (regularity), or in [START_REF] Crispo | On the existence, uniqueness and C 1,γ (Ω) ∩ W 2,2 (Ω) regularity for a class of shear-thinning fluids[END_REF] (existence and regularity).

Finally, a property of some non-Newtonian shear-thinning flows is the existence of a finite stopping time, that is, roughly speaking, a time from which the fluid is at the rest.This property has been established, for example, in the case of a two-dimensional Bingham flow in [START_REF] Jesús | Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support[END_REF], and in the case of some electrorheological fluids in [START_REF] Abbatiello | Existence of regular time-periodic solutions to shear-thinning fluids[END_REF].

In this paper, we firstly establish the existence of weak solutions by a parabolic variational inequality (see Theorem 3.1 and Definition 2.1) as used in [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF] for shear tensors τ of the form τ (D(u)) = F (|D(u)|)D(u) + D(u), by setting conditions directly on the viscosity coefficient F , restricting ourselves to the case of shearthinning flows. More precisely, we will make the following assumptions: (C1) F : (0, +∞) → (0, +∞);

(C2) F ∈ W 1,∞ loc ((0, +∞)); (C3) t → tF (t) is non-decreasing on (0, +∞); (C4) there exist p ∈ [1, 2], t 0 > 0 and K > 0 such that for every t ≥ t 0 , F (t) ≤ Kt p-2 .
Some examples of functions verifying the above assumptions are given in Appendix A. We emphasize in particular that this takes into account many physical models, such as the Carreau, Bingham, Herschel-Bulkley, Cross, or power law flows.

Remark 1. Assumption (C3) is equivalent to the fact that for all ε ≥ 0, the function t → tF √ ε + t 2 is non-decreasing. Indeed, we can write:

∀t ∈ (0, +∞), tF ε + t 2 = t √ ε + t 2 ε + t 2 F ε + t 2 .
Hence, t → tF √ ε + t 2 is the product of two non-negative and non-decreasing functions, so it is a nondecreasing function. The opposite implication being obvious by setting ε = 0.

Then, in Section 5 we establish the existence of a finite stopping time for solutions of (1.1) in the case of a function F verifying (C1)-(C4) and such that

F (t) ≥ Ct p-2 . (1.2)
This shows in particular the existence of a finite time from which the fluid is at rest, when the flow is comparable to that of a power-law or threshold fluid, thus for a shear-thinning flow. One of the main objectives of this paper, in addition to completing some existing results, is to provide simple hypotheses to verify the existence of solutions for shear-thinning flows. Emphasis is also placed on considering solutions whose regularization is that generally used in numerical simulations, in order to make sense of the finite stopping time observed numerically and experimentally. Let us conclude by observing that many thixotropic flows (such as blood) are represented by usual shear-thinning models, depending on the circumstances of the flow studied (see [START_REF] Robertson | Rheological models for blood[END_REF]).

We will note in a generic way the constants by the letter C throughout this article, and will omit their dependence on the parameters in the notations.

Weak characterization of solutions by a parabolic variational inequality

In this section we introduce a weak formulation of system (1.1) using a parabolic variational inequality (see Definition 2.1). Firstly, we point out that in the system (1.1), we do not consider any frictional force on ∂Ω. Recalling that

H 1 0 (Ω) is the closure of C ∞ 0 (Ω) into H 1 (Ω)
, it is thus natural to assume that the initial velocity field u 0 is of null trace on ∂Ω, that is u 0 belongs to H 1 0,σ (Ω), the space of functions v ∈ H 1 0 (Ω) such that div(v) = 0, where H 1 0 (Ω) is endowed with the norm u → ∇u L 2 . We denote H -1 σ (Ω) its dual and •, • is the duality product between H -1 σ (Ω) and H 1 0,σ (Ω). Following the ideas employed for showing the existence of solution to Bingham equations in [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF][START_REF] Kinderlehrer | An introduction to variational inequalities and their applications[END_REF], we define a functional j making appear the viscous non-linear term in (1.1) in its derivative. We fix for the moment 0 ≤ ε ≤ δ and we define a function G ǫ : (0, +∞) → (0, +∞) and a functional

j ε : H 1 0,σ (Ω) → R by G ε (t) = ˆt 0 sF ( ε + s 2 ) ds for every t ∈ (0, +∞) (2.1)
and

j ε (v) = ˆΩ G ε (|D(v)|) dx, (v ∈ H 1 0,σ (Ω)), (2.2)
respectively. We also denote j = j 0 and G = G 0 . One can check that G ε is a convex functional for ε small enough. Indeed, G ′ ε (t) = tF ( ε + t 2 ), for every t ∈ (0, +∞), and applying the hypothesis (C3) the convexity of G follows immediately.

Lemma 2.1. For every ε > 0 the functional j ε defined by (2.2) is convex and verifies

j ′ ε (v), w -1,1 = ˆΩ F ε + |D(v)| 2 (D(v) : D(w)) dx (v, w ∈ H 1 0,σ (Ω)). (2.3)
Proof. The convexity of j ε is immediately obtained from the hypothesis (C3) and (2.2). For every t ∈ R we have

d dt (G ε (|D(v + tw)|)) = G ′ ε (|D(v + tw)|) d dt (|D(v + tw)|) = F ε + |D(v + tw)| 2 |D(v + tw)| D(v + tw) : D(w) |D(v + tw)| = F ε + |D(v + tw)| 2 D(v + tw) : D(w).
Hence

j ′ ε (v + tw), w = d dt j ε (v + tw) = ˆΩ d dt (G ε (|D(v + tw)|)) dx = ˆΩ F ε + |D(v + tw)| 2 D(v + tw) : D(w) dx.
Letting t going to 0 we obtain (2.3).

Remark 2. We point out that j ′ is well defined. Firstly, by our assumptions (C2) and (C3), we can deduce that for all β ∈ 0, 1 2 , there exists δ 0 such that: 1+β) for every t ∈ (0, δ 0 ).

F (t) ≤ t -(
Indeed, assume that this last inequality does not hold, then for every δ 0 > 0, there exists t 0 ∈ (0, δ 0 ) such that:

F (t 0 ) > t -(1+β) 0 .
We can consider without loss of generality that δ 0 < min 1, F

-1

β , which implies, using our assumption (C3):

δ -β 0 < t -β 0 < t 0 F (t 0 ) ≤ F (1)
. This contradiction shows the result. We recall Korn's L 2 equality for divergence free vector fields:

ˆΩ|D(ϕ)| 2 dx = 1 2 ϕ 2 H 1 0 , (ϕ ∈ H 1 0,σ (Ω)).
Using these last results and applying Cauchy Schwarz's and Hölder's inequalities, we get:

| j ′ (u), ϕ -1,1 | = ˆΩ F (|D(u)|)D(u) : D(ϕ) dx ≤ 1 √ 2 ˆΩ F (|D(u)|) 2 |D(u)| 2 dx 1 2 ϕ H 1 0 = 1 √ 2 ˆ{|D(u)|≤δ 0 } F (|D(u)|) 2 |D(u)| 2 dx + ˆ{|D(u)|>δ 0 } F (|D(u)|) 2 |D(u)| 2 dx 1 2 ϕ H 1 0 ≤ 1 √ 2 ˆ{|D(u)|≤δ 0 } |D(u)| -2β dx + ˆ{|D(u)|>δ 0 } F (|D(u)|) 2 |D(u)| 2 dx 1 2 ϕ H 1 0 = 1 √ 2 1 1 -2β ˆ{|D(u)|≤δ 0 } ˆ|D(u)| 0 s 1-2β ds dx + ˆ{|D(u)|>δ 0 } F (|D(u)|) 2 |D(u)| 2 dx 1 2 ϕ H 1 0 .
And so j ′ is well-defined.

Definition 2.1 (Weak solution of (1.1)). We say that a function u ∈ L 2 (0, T ),

H 1 0,σ (Ω) ∩ C w ((0, T ), L 2 σ (Ω)) such that u ′ ∈ L 4 N (0, T ), H -1
σ (Ω) is a weak solution of (1.1) if and only if u verifies u |t=0 = u 0 ∈ H 1 0,σ (Ω), and for all ϕ ∈ C ∞ ((0, T ) × Ω):

ˆT 0 u ′ (t), ϕ(t) dt + 1 2 u 0 2 L 2 (Ω) -u(T ) 2 L 2 (Ω) + ˆT 0 ˆΩ D(u(t)) : D(ϕ(t) -u(t)) dx - ˆT 0 ˆΩ (u(t) • ∇u(t)) • ϕ(t) dx dt + ˆT 0 ˆΩ G (|D(ϕ(t))|) -G (|D(u(t))|) dx dt ≥ ˆT 0 f (t), ϕ(t) -u(t) dt. (2.4)
Let's quickly motivate this definition. First, we point out that since u belongs to C w ((0, T ), L 2 σ (Ω)), Definition 2.1 makes sense. Then, if we consider that the Lebesgue measure of the set

{(t, x) ∈ (0, T ) × Ω | |D(u)(t, x)| ≤ δ}
is equal to zero for a small δ > 0, we have, from an argument similar to the one in Lemma 2.1 that:

ˆT 0 j ′ (u), ϕ dt = ˆT 0 ˆΩ F (|D(u)|) (D(u) : D(ϕ)) dx dt.
Now, if we replace ϕ by u + sϕ, with s > 0, in the variational inequality (2.4), we obtain after dividing by s:

ˆT 0 ˆΩ D(u) : D(ϕ) dx dt + ˆT 0 ˆΩ G (|D(u + sϕ)|) -G (|D(u)|) s dx dt ≥ ˆT 0 ˆΩ f -u ′ , ϕ dt - ˆT 0 ˆΩ (u • ∇u) • ϕ dx dt.
Since j admits a Fréchet-derivative, it also admits a Gâteaux-derivative and both are the same. Hence, taking the limit as s → 0:

ˆT 0 ˆΩ D(u) : D(ϕ) dx dt + ˆT 0 ˆΩ F (|D(u)|) (D(u) : D(ϕ)) dx dt ≥ ˆT 0 ˆΩ f -u ′ , ϕ dt - ˆT 0 ˆΩ (u • ∇u) • ϕ dx dt.
Repeating once again the previous reasoning but writing u -sϕ instead of u + sϕ, we get the following equality:

ˆT 0 ˆΩ D(u) : D(ϕ) dx dt + ˆT 0 ˆΩ F (|D(u)|) (D(u) : D(ϕ)) dx dt = ˆT 0 ˆΩ f -u ′ , ϕ dt - ˆT 0 ˆΩ (u • ∇u) • ϕ dx dt.
Therefore, assuming that u is regular enough, we obtain

- 1 2 ˆT 0 ˆΩ ∆u • ϕ dx dt - ˆT 0 ˆΩ div (F (|D(u)|) D(u)) ϕ dx dt = ˆT 0 ˆΩ f -u ′ -u • ∇u • ϕ dx dt.
Furthermore De Rham's theorem for a domain with Lipschitz boundary states that there exists a pressure term p such that f = ∇p into some well chosen Sobolev space (see [17, section 2] for details). Considering such a function and also the two previous observations, we can write:

ˆT 0 ˆΩ u ′ + u.∇u - 1 2 ∆u + ∇p -div (F (|D(u)|) D(u)) -f ϕ dx dt = 0, (ϕ ∈ C ∞ ((0, T ) × Ω)) ,
which is almost everywhere equivalent to the equation (1.1) up to the multiplicative dynamic viscosity constant 1 2 . We have omitted this constant in Definition 2.1 for convenience, and note that it is enough to add the constant 2 in front of the term ´T 0 ´Ω D(u) : D(u -ϕ) dx dt in order to find exactly (1.1).

Finding a solution to the parabolic variational inequality thus amounts to giving meaning to the integral of the nonlinear viscosity coefficient term inherent in the problem, which can be a singular integral in the case of a power-law or a Bingham fluid.

Main results

We present in this section the main results of this article. The study of the existence of solutions by variational inequality has been developed following the classical Stampacchia's theorem and was further developed in [START_REF] Kinderlehrer | An introduction to variational inequalities and their applications[END_REF]. Then, this method was successfully applied for some nonlinear parabolic problems, as the two dimensional Bingham equations in [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF], or some power law systems in [START_REF] Málek | Weak and measure-valued solutions to evolutionary PDEs[END_REF]. Following the same approach, we get the following existence theorem. Theorem 3.1. Assume that the function F satisfies the hypotheses (C1)-(C4) and that Ω ⊂ R N , N ∈ {2, 3}, is a bounded domain with a Lipschitz boundary, T > 0 and consider an initial datum u 0 ∈ H 1 0,σ (Ω) and a force term f ∈ L 2 ((0, T ), H -1 σ (Ω)). Then, there exists a weak solution u of (1.1) having the following regularity

u ∈ C w (0, T ), L 2 σ (Ω) ∩ L 2 (0, T ), H 1 0,σ (Ω) and u ′ ∈ L 4 N ((0, T ), H -1 σ (Ω)
). This theorem thus ensures the existence of suitable solutions in the two-dimensional and three-dimensional cases. It follows from classical arguments that the solutions are Hölder continuous in time, for a well-chosen Hölder coefficient. Moreover, we show in the appendix, in Corollary ?? of Proposition B.1 that in some interesting cases as for power-law flows, that the solutions satisfy an energy equality.

Unlike the Navier-Stokes case, the nonlinear term in the Bingham equations allows us to obtain the rest of the fluid in finite time in the two-dimensional case. This has been demonstrated in [START_REF] Jesús | Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support[END_REF], using the following approach: it is assumed that the force term will compensate the initial kinetic energy of the fluid, which amounts to establishing a relation between the norm u 0 L 2 and an integral of f (t) L 2 . This argument is based on the use of the following Nirenberg-Strauss inequality:

∃γ > 0, ∀u ∈ H 1 0 (Ω), u L 2 ≤ γ ˆΩ|D(u)| dx.
We note that such an inequality cannot be true in dimension greater than two, because it would contradict the optimality of Sobolev embbeddings. We therefore propose to slightly adapt this approach to show the existence of a stopping finite time in both the two and the three-dimensional cases. Firstly, let us formalize the definition.

Definition 3.1 (Finite stopping time). Let u be a weak solution in the sense of Definition 2.1 of the system (1.1). We say that T 0 ∈ (0, T ) is a finite stopping time for u if:

u(T 0 ) L 2 (Ω) = 0.
In order to prove the existence of a finite stopping time for the solution u provided by Theorem 3.1, we do not make any assumption on the initial velocity field, but we assume that after a certain time the fluid is no longer subjected to any external force. More exactly we make some more assumption on F as stated by the following theorem.

Theorem 3.2 (Existence of a finite stopping time). Assume that the hypotheses of Theorem 3.1 are verified, that T > 0 is choosen large enough, and let p ∈ [1, 2). Moreover, we assume that there exists two positive constants κ and T 1 < T such that F (t) ≥ κt p-2 for every t ∈ (0, +∞) and f = 0 almost everywhere on (T 1 , T ).

(3.1)
Then, there exists a finite stopping time T 0 ∈ (0, T ) for u in the sense of Definition 3.1.

4 Proof of Theorem 3.1

In this section, we establish the proof of Theorem 3.1 in the bi-dimensional and three-dimensional settings.

In order to prove this result, we begin by establishing an energy estimate for solutions obtained by the Galerkin method in order to obtain uniform bounds with respect to the parameters. We note here that we will have two parameters: a first parameter due to Galerkin's approximation, and a second one due to the regularization proper to the viscosity coefficient F .

Existence of a Galerkin weak solution

We apply here the usual Galerkin method using the Stokes operator in homogeneous Dirichlet setting, and we use its eigenfunctions (w i ) i∈N as an orthogonal basis of H 1 0,σ (Ω) and orthonormal basis of L 2 σ (Ω) (see [START_REF] Evans | Partial differential equations[END_REF] for details about this property, and [34, Section 2.3] for details concerning the Stokes operator).

For every positive integer m, we denote by P m the projection of L 2 σ (Ω) onto Span ((w i ) 1≤i≤m ). We would like to formally define our Galerkin system as follows.

       ∂ t u m + P m (u m • ∇u m ) + ∇P m (p) -∆u m -P m (div (F (|D(u m )|) D(u m ))) = P m f div(u m ) = 0 on (0, T ) × Ω u m = 0 on [0, T ) × ∂Ω u m = P m (u 0 ) on {0} × Ω. (4.1)
In order to avoid the issue posed by the nonlinear term in domains for which the fluid is not deformed we consider the following regularized Galerkin system:

         ∂ t u m,ε + P m (u m,ε • ∇u m,ε )+ ∇P m (p) -∆u m,ε -P m div F ε + |D(u m,ε )| 2 D(u m,ε ) = P m f div(u m,ε ) = 0 on (0, T ) × Ω u m,ε = 0 on [0, T ) × ∂Ω u m,ε = P m (u 0 ) on {0} × Ω, (4.2
) with 0 < ε < 1. Applying a Galerkin method, we can see that, writing u m,ε (t) = m i=1 d i m (t)w i , we obtain the ordinary differential system for all 1 ≤ i ≤ m:

d i m ′ (t) = f, w i - ˆΩ 1 2 w i 2 H 1 0 d i m (t) dx -ˆΩ D(u 0 ) : D(w i ) dx - ˆΩ 1 2 w i 2 H 1 0 F   ε + m j=1 1 2 w j 2 H 1 0 (d j m (t)) 2 + 2(D(w j ) : D(u 0 ))d j m (t) + 1 2 u 0 2 H 1 0   d i m (t) dx -ˆΩ F   ε + m j=1 1 2 w j 2 H 1 0 (d j m (t)) 2 + 2(D(w j ) : D(u 0 ))d j m (t) + 1 2 u 0 2 H 1 0   (D(u 0 ) : D(w i )) dx - m j=1 ˆΩ w j • ∇w i d i m (t)d j m (t) dx, (4.3)
completed with initial condition d i m (0) = (u 0 , w i ) H 1 0 . This system is described by a locally Lipschitz continuous function with respect to d m . Indeed, applying the hypothesis (C2), the function ψ : R m → R defined by

ψ(x) = F   ε 2 + m j=1 1 2 w j 2 H 1 0 x 2 j + 2(D(w j ) : D(u 0 ))x j + 1 2 u 0 2 H 1 0   ∀x ∈ R m
is locally Lipschitz. The Picard-Lindelöf theorem shows the existence of a solution for system (4.2).

Energy estimate and consequences

We recall that the solution u m,ε of (4.2) belongs to Span ((w i ) 1≤i≤m ), for (w i ) i∈N the basis of H 1 0,σ (Ω) which are the eigenfunctions of the Stokes operator in the homogeneous Dirichlet setting.

In order to clarify our presentation, we specify that we consider the following notion of solution.

Definition 4.1 (Solution of (4.2)). We say that u m,ε ∈ L 2 ((0, T ),

H 1 0,σ (Ω)), u ′ m,ε ∈ L 2 ((0, T ), H -1 (Ω)
) is a weak solution of (4.2) if for every ϕ ∈ C ∞ ((0, T ) × Ω) and for a.e. t ∈ (0, T ) it satisfies

u ′ m,ε , ϕ + ˆΩ D(u m,ε ) : D(ϕ) dx + j ′ ε (u m,ε ), ϕ -ˆΩ(u m,ε • ∇u m,ε ) • ϕ dx = f, ϕ . (4.4)
We also say that (4.4) is the formulation in space of the solution of (4.2) when the time is fixed. We point out that this definition makes sense since we are studying smooth finite dimensional Galerkin solutions. Then, in order to obtain weak limits into the Galerkin formulation, we establish some estimates.

Proposition 4.1. Assume that u m,ε is a solution of (4.2) in the sense of Definition 4.1. Then, there exists a positive constant C depending on p, Ω, N , T , u 0 L 2 (Ω) and f L 2 ((0,T ),H -1 (Ω)) such that the following estimates hold:

1. u m,ε 2 L ∞ ((0,T ),L 2 σ ) + 1 2 u m,ε 2
L 2 ((0,T ),H 1 0,σ ) ≤ C f 2 L 2 ((0,T ),H -1 ) + u 0 2 L 2 ; 2. j ′ ε (u m,ε ) L 4 N ((0,T ),H -1 ) ≤ C 1 + f L 2 ((0,T ),H -1 ) + u 0 L 2 p-1 ; 3. u ′ m,ε L 4 N ((0,T ),H -1 ) ≤ C f 2 L 2 ((0,T ),H -1 ) + u 0 2 L 2 + C f 2 L 2 ((0,T ),H -1 ) + u 0 2 L 2 2 +C 1 + f L 2 ((0,T ),H -1 ) + u 0 L 2 p-1 .
Before the proof of Proposition 4.1, we state some useful results. We start by recalling a well known Gagliardo-Nirenberg inequality (for the proof, see, for instance, [START_REF] Nirenberg | On elliptic partial differential equations[END_REF] or [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]).

Theorem 4.1 (Gagliardo-Nirenberg inequality on bounded Lipschitz domain). Assume that Ω is a bounded domain in R N with Lipschitz boundary. Moreover, assume that there exists a couple (q, r)

∈ [1, +∞], θ ∈ [0, 1] and (l, k) ∈ N 2 such that:    1 p = k N + 1 r -l N θ + 1-θ q k l ≤ θ ≤ 1.
Then, there exists C := C(k, l, N, r, q, θ, Ω) > 0 such that the following inequality holds:

∇ k u L p (Ω) ≤ C u θ W l,r (Ω) u 1-θ L q (Ω) .
The following result formalizes some other properties. Lemma 4.1. Let X be a Banach space, and γ ≥ 1 2 . Then, the following inequality holds:

∀(u, v) ∈ X 2 , u + v γ X ≤ 2 (γ-1 2 ) u γ X + v γ X .
Proof. Using the convexity of t → t 2(2-p) and triangle's inequality of the norm, we get:

u + v 2γ X = 2 2γ u + v 2 2γ X ≤ 2 2γ-1 u 2γ X + v 2γ X .
Applying now the well-known inequality:

∀(a, b) ∈ [0, +∞) 2 , √ a + b ≤ √ a + √ b, we get the result.
Proof of Proposition 4.1.

1. Setting ϕ = u m,ε in the weak formulation, we get:

1 2 d dt u m,ε 2
L 2 + ˆΩ|D(u m,ε )| 2 dx + j ′ ε (u m,ε ), u m,ε ≥0 -ˆΩ(u m,ε • ∇u m,ε ) • u m,ε dx =0 = f, u m,ε .
Using the well-known Korn's L 2 equality for divergence free vectors fields, we get

d dt u m,ε (t) 2 L 2 + u m,ε (t) 2 H 1 0 ≤ 2 f (t), u m,ε (t) .
Moreover, we have:

2 f (t), u m,ε (t) ≤ 2 f (t) 2 H -1 + 1 2 u m,ε (t) 2 H 1 0 .
Then, using the above inequality and integrating on (0, t) we get

u m,ε (t) 2 L 2 + 1 2 ˆt 0 u m,ε 2 H 1 0 dt ≤ 2 ˆt 0 f 2 H -1 dt + u 0 2 L 2 . (4.5)
Indeed, we recall that (P m (u 0 ), w i ) L 2 = (u 0 , P m w i ) L 2 = (u 0 , w i ) L 2 , and the conclusion follows. From now on, we will omit to detail this last part which is usual.

2. We have, using Cauchy-Schwarz's inequality and Korn's equality in the divergence free L 2 setting:

j ′ ε (u m,ε ), ϕ = ˆΩ F ε + |D(u m,ε )| 2 D(u m,ε ) : D(ϕ) dx ≤ 1 √ 2 ˆΩ F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx 1 2 ϕ H 1 0 . (4.6)
From hypothesis (C4), setting A = Ω ∩ {|D(u m,ε )| ≤ t 0 } and B its complement in Ω, we obtain

ˆΩ F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx = ˆA F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx + ˆB F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx.
Let's estimate these two integrals independently. By assumption (C3), we have that the application

t → t 2 F √ ε + t 2 2
is non-decreasing, and we obtain directly:

ˆA F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx ≤ F ε + t 0 2 2 t 0 2 |A| ≤ F ε + t 0 2 2 t 0 2 |Ω| ≤ F 1 + t 0 2 2 1 + t 0 2 |Ω| ≤ C.
Then we have, using again (C4):

ˆB F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx ≤ K ˆB |D(u m,ε )| 2 (ε + |D(u m,ε )| 2 ) 2-p dx ≤ K ˆB|D(u m,ε )| 2(p-1) dx ≤ K ˆB|∇u m,ε | 2(p-1) dx ≤ C u m,ε 2(p-1) H 1 0 ,
where we used Jensen's inequality in the concave setting with t → t p-1 in the last line. So, we obtain:

ˆΩ F ε + |D(u m,ε )| 2 2 |D(u m,ε )| 2 dx 1 2 ≤ C + C u m,ε 2(p-1) H 1 0 1 2 . (4.7)
Thus, combining the inequality (4.6)-(4.7) and using Lemma 4.1 with γ = 2 N , we get:

j ′ ε (u m,ε ) 4 N H -1 ≤ C + C u m,ε 4(p-1) N H 1 0 .
Therefore, integrating in time over (0, T ):

j ′ ε (u m,ε ) 4 N L 4 N ((0,T ),H -1 ) ≤ C + C u m,ε 4(p-1) N L 4(p-1) N ((0,T ),H 1 0 )
.

Then, since 0 < 4(p-1) N ≤ 2, we get, using the embbedding L 2 ֒→ L

4(p-1) N

and Lemma 4.1 with X := H 1 0 , q = 4(p-1)

N and p = 2 on u m,ε L 4(p-1) N ((0,T ),H 1 0 )
:

j ′ ε (u m,ε ) 4 N L 4 N ((0,T ),H -1 ) ≤ C + C u m,ε 4(p-1) N L 2 ((0,T ),H 1 0 ) .
Using the first point of the proposition for t = T , and since 4(p-1) N ≥ 0, we get:

j ′ ε (u m,ε ) 4 N L 4 N ((0,T ),H -1 ) ≤ C + C( f L 2 ((0,T ),H -1 ) + u 0 L 2 ) 4(p-1) N .
Then, using the exponent N 4 on both sides and applying once again Lemma 4.1 with γ = N 4 on the right-hand side in the ineuality above leads us to:

j ′ ε (u m,ε ) L 4 N ((0,T ),H -1 ) ≤ C + C( f L 2 ((0,T ),H -1 ) + u 0 L 2 ) p-1 .
This is the wished result.

3. From the weak formulation (4.4) we get

u ′ m,ε , ϕ = -ˆΩ D(u m,ε ) : D(ϕ) dx -j ′ ε (u m,ε ), ϕ + ˆΩ(u m,ε • ∇u m,ε ) • ϕ dx + f, ϕ . (4.8)
Let us point out that

ˆΩ D(u m,ε ) : D(ϕ) dx = 1 2 ˆΩ ∇u m,ε • ∇ϕ dx ≤ 1 2 u m,ε H 1 0 ϕ H 1 0 . (4.9)
Also, setting p = 4, k = 0, l = 1, r = q = 2, and s = 2 into Theorem 4.1, we get the existence of a positive constant C which only depends on N and Ω such that:

u L 4 ≤ C ∇u N 4 L 2 u 4-N 4 L 2
and, renoting C, we get:

u 2 L 4 ≤ C ∇u N 2 L 2 u 4-N 2 L 2 . ( 4
ˆΩ(u m,ε .∇u m,ε ).ϕ dx ≤ 1 2 ˆΩ|u m,ε | 2 .|∇ϕ| dx ≤ u m,ε 2
L 4 ∇ϕ L 2 ≤ C u m,ε 4-N 2 L 2 u m,ε N 2 H 1 0 ϕ H 1 0 . (4.11)
So, putting (4.9)-(4.11) and the second estimate of the Proposition 4.1 in (4.8), we obtain

u ′ m,ε , ϕ ≤ 1 2 u m,ε H 1 0 ϕ H 1 0 + j ′ ε (u m,ε ) H -1 ϕ H 1 0 + C u m,ε 4-N 2 L 2 u m,ε N 2 H 1 0 ϕ H 1 0 + f H -1 ϕ H 1 0
, and, therefore,

u ′ m,ε (t) H -1 ≤ 1 2 u m,ε H 1 0 + j ′ ε (u m,ε ) H -1 + C u m,ε 4-N 2 L 2 u m,ε N 2 H 1 0 + f H -1 .
Now, using the following convexity inequality

∀k ∈ N, ∀(x i ) 1≤i≤k ∈ (0, +∞) k , ∃C > 0, k i=1 x i 4 N ≤ C k i=1 x 4 N i
we get:

u ′ m,ε (t) 4 N H -1 ≤ C u m,ε 4 N H 1 0 + j ′ ε (u m,ε ) 4 N H -1 + u m,ε 8-2N N L 2 u m,ε 2 H 1 0 + f 4 N H -1 .
Since N ∈ {2, 3}, we have 4 N ≤ 2. Hence, integrating in time over (0, T ) and using the embedding

L 2 (Ω) ֒→ L 4 N (Ω): u ′ m,ε 4 N L 4 N ((0,T ),H -1 ) ≤ C u m,ε 4 N L 2 ((0,T ),H 1 0 ) + j ′ ε (u m,ε ) 4 N L 4 N ((0,T ),H -1 ) + C u m,ε 8-2N N L ∞ ((0,T ),L 2 ) u m,ε 4 N L 2 ((0,T ),H 1 0 ) + C f 4 N L 2 ((0,T ),H -1 ) .
Using the previously given convexity inequality and the first and second points of the proposition we obtain the desired result.

Weak convergence

We are now interested in the weak convergence with respect to the estimates proven in Section 4.2. Here, we prove such convergences by passing to the limit with respect to the parameter ε in a first time, then by passing to the limit with respect to the Galerkin parameter m.

Before proving Theorem 3.1, we establish several useful lemmas.

Lemma 4.2. Consider that ϕ ∈ L 2 ((0, T ), H 1 0 (Ω)), then there exists a constant C(ε, ϕ) > 0 which goes to zero as ε does, such that the following inequality holds:

j ε (ϕ) + C(ε, ϕ) ≥ j(ϕ), (4.12)
where j ε and j are defined by (2.2).

Proof. Recalling that the assumption (C3) states that t → tF (t) is increasing, we get:

j(ϕ) := ˆΩ ˆ|D(ϕ)| 0 sF (s) ds dx ≤ ˆΩ ˆ√ε 0 sF (s) ds dx + ˆΩ ˆ√ε+|D(ϕ)| √ ε sF (s) ds dx ≤ ε √ εF (ε)|Ω| + ˆΩ ˆ√2|D(ϕ)| √ ε+|D(ϕ)| 2 0 sF ( ε + s 2 ) ds dx ≤ ε √ εF (ε)|Ω| + ˆΩ ˆ2 1 2 ε 1 4 |D(ϕ)| 1 2 +|D(ϕ)| |D(ϕ)| sF ( ε + s 2 ) ds dx :=C(ε,ϕ) +j ε (ϕ),
which is the wished result.

Lemma 4.3.

Consider Ω an open bounded subset of R N with Lipschitz boundary, and a sequence

(w n ) n∈N such that w n ⇀ n→+∞ w in L 2 ((0, T ), H 1 0,σ (Ω)).
Then, for almost all (t, x) ∈ (0, T ) × Ω, the following inequality holds:

|D(w n )(t, x)| ≥ |D(w)(t, x)|.
Proof. Firstly, let us recall that since w n ⇀ w in L 2 ((0, T ), H 1 0 (Ω)) then, for all Lebesgue points t 0 ∈ (0, T ) and x 0 ∈ Ω, for all δ > and R > 0 small enough, we have w n ⇀ w in L 2 ((t 0 -δ, t 0 + δ), H 1 (B(x 0 , R)). Indeed, we have for all test function ϕ :

ˆT 0 ˆΩ ∇w n • ∇ϕ dt dx -→ n→+∞ ˆT 0 ˆΩ ∇w • ∇ϕ dt dx.
Hence, we can take ϕ, which belongs to C ∞ 0 ((t 0 -δ, t 0 + δ) × B(x 0 , R)) (up to arguing by density thereafter), satisfying:

∇ϕ = ∇ψ on (t 0 -δ, t 0 + δ) × B(x 0 , R) 0 on (0, T ) × Ω\(t 0 -δ, t 0 + δ) × B(x 0 , R)
and so this leads to:

ˆt0 +δ t 0 -δ ˆB(x 0 ,R) ∇w n • ∇ψ dt dx -→ n→+∞ ˆt0 +δ t 0 -δ ˆB(x 0 ,R) ∇w • ∇ψ dt dx.
That is w n ⇀ w in L 2 ((t 0 -δ, t 0 + δ), H 1 (B(x 0 , R))). Now, applying Korn's L 2 equality and using that fact we get for all Lebesgue point t 0 of (0, T ) and x 0 ∈ Ω that:

ˆt0 +δ t 0 -δ ˆB(x 0 ,R) |D(w n )| 2 dx dt ≥ ˆt0 +δ t 0 -δ ˆB(x 0 ,R) |D(w)| 2 dx dt.
Dividing each side by 2δ|B(x 0 , R)|, we get:

t 0 +δ t 0 -δ B(x 0 ,R) |D(w n )| 2 dx dt ≥ t 0 +δ t 0 -δ B(x 0 ,R) |D(w)| 2 dx dt
then letting (δ, R) → (0, 0) leads to the result, applying Lebesgue's differentiation theorem.

The following lemma gives the convergence of u m,ε when ε goes to zero.

v m ∈ L 2 (0, T ), H 1 0,σ (Ω) ∩L ∞ (0, T ), L 2 σ (Ω) with v ′ m ∈ L 4 N
(0, T ), H -1 σ (Ω) such that, up to subsequences:

1. u ′ m,ε ⇀ v ′ m in L 4 N (0, T ), H -1 σ (Ω) ; 2. u m,ε ⇀ v m in L 2 (0, T ), H 1 0,σ (Ω) ; 3. u m,ε → v m in L 2 ((0, T ), L 2 σ (Ω)); 4. u m,ε * ⇀ v m in L ∞ (0, T ), L 2 σ (Ω) . Moreover, v m satisfies, for all ψ ∈ C ∞ ((0, T ) × Ω): 1 2 v m (T ) 2 L 2 - 1 2 u 0 2 L 2 - ˆT 0 v ′ m , ψ dt + ˆT 0 ˆΩ D(v m ) : D(v m -ψ) dx dt + ˆT 0 j(v m ) -j(ψ) dt - ˆT 0 ˆΩ(v m • ∇v m ) • ψ dx dt ≤ ˆT 0 f, v m -ψ dt. (4.13)
Proof. The first and second points follow from the reflexivity of L 4 N (0, T ), H -1 σ (Ω) and L 2 ((0, T ), H 1 0,σ (Ω)) respectively, the third one from Aubin-Lions' Lemma, and the last one by Banach-Alaoglu-Bourbaki's theorem.

Then, since u m,ε is a solution of (4.2), it satisfies (4.4). Testing against ϕ = u m,ε -ψ in (4.4) for a test function ψ, we have:

u ′ m,ε , u m,ε -ψ + ˆΩ D(u m,ε ) : D(u m,ε -ψ) dx + j ′ ε (u m,ε ), u m,ε -ψ -ˆΩ(u m,ε • ∇u m,ε ) • ψ dx = f, u m,ε -ψ . (4.14)
Applying Lemma 2.1 leads to the well-known convexity inequality:

j ε (u m,ε ) -j ε (ψ) ≤ j ′ ε (u m,ε ), u m,ε -ψ . (4.15)
Using now Lemma 4.2 for u m,ε in (4.15), we get:

j(u m,ε ) -C(ε, u m,ε ) -j ε (ψ) ≤ j ′ ε (u m,ε
), u m,ε -ψ and then, by (C3) and Lemma 4.3 applied to u m,ε for the convergence toward v m , we get:

j(v m ) -C(ε, u m,ε ) -j ε (ψ) ≤ j ′ ε (u m,ε ), u m,ε -ψ .
Then, we can write (see [START_REF] Evans | Partial differential equations[END_REF] part 5.9. for details):

∀ϕ ∈ H 1 0,σ (Ω), ˆΩ u m,ε (T )ϕ dx = u m,ε (T ), ϕ = ˆT 0 u ′ m,ε (t), ϕ dt + u 0 , ϕ . (4.16)
Now, we also have, using Proposition 4.1:

ˆT 0 u ′ m,ε (t), ϕ dt + u 0 , ϕ ≤ u ′ m,ε L 4 N ((0,T ),H -1 ) ˆT 0 ϕ 4 4-N H 1 0 dt 4-N 4 + C u 0 L 2 ϕ H 1 0 ≤ C T 4-N 4 + u 0 L 2 ϕ H 1 0 .
In the above inequality we considered ϕ as a function in L ∞ ((0, T ), H 1 0 (Ω)), so it belongs to L 4 4-N ((0, T ), H 1 0 (Ω)) and its left-hand side defines a linear form over L 4 N ((0, T ), H -1 (Ω)). Also, the weak convergence leads to:

ˆT 0 u ′ m,ε (t), ϕ dt -→ ε→0 ˆT 0 v ′ m (t), ϕ dt. (4.17)
Finally, (4.16) and (4.17) imply, up to apply a dominated convergence theorem, to:

u m,ε (T ) ⇀ ε→0 v m (T ) in L 2 (Ω). (4.18)
Then, (4.18) implies:

lim ε→0 1 2 u m,ε (T ) 2 L 2 -P m (u 0 ) 2 L 2 ≥ 1 2 v m (T ) 2 L 2 -P m (u 0 ) 2 L 2 (4.19)
Also, from usual estimates (see [START_REF] Robinson | The three-dimensional Navier-Stokes equations. Number 157 in Cambridge studies in advanced mathematics[END_REF]Chapter 4]), since u m,ε → ε v m in L 2 ((0, T ), H 1 0,σ (Ω)), we have:

ˆT 0 ˆΩ|D(u m,ε )| 2 dx dt -→ ε→0 ˆT 0 ˆΩ|D(v m )| 2 dx dt (4.20)
and

ˆT 0 ˆΩ(u m,ε • ∇u m,ε ) • ψ dx dt -→ ε→0 ˆT 0 ˆΩ(v m • ∇v m ) • ψ dx dt. (4.21)
Integrating in time (4.14), and passing to the limit over ε, combining with (4.20), (4.21), and (4.19) leads to (4.13).

Arguing in the same way, we obtain the following result.

Lemma 4.5. Under the assumptions of Proposition 4.1, there exists u ∈ L 2 (0, T ),

H 1 0,σ (Ω) ∩L ∞ (0, T ), L 2 σ (Ω) with u ′ ∈ L 4 N (0, T ), H -1
σ (Ω) such that the function v m given by Lemma 4.4 verifies.

1. v ′ m ⇀ u ′ in L 4 N (0, T ), H -1 σ (Ω) ; 2. v m → u in L 2 (0, T ), L 2 σ (Ω) ; 3. v m ⇀ u in L 2 ((0, T ), H 1 0,σ (Ω)); 4. v m * ⇀ u in L ∞ (0, T ), L 2 σ (Ω)
. Moreover, we point out that u ∈ C w ((0, T ), L 2 σ (Ω)) from the above estimates (see [10, Proposition V.1.7. p.363] for details).

Proof of Theorem 3.1. We point out that the coefficients of v m given by Lemma 4.4 satisfy an ODE as (4.3) with ε = 0, then v m is still smooth in space and time. Moreover, we can take up again the method previously used, that is we can write :

∀ϕ ∈ H 1 0,σ (Ω), ˆΩ v m (T )ϕ dx = v m (T ), ϕ = ˆT 0 v ′ m (t), ϕ dt + P m (u 0 ), ϕ . (4.22)
Using Proposition 4.1 then leads to:

ˆT 0 v ′ m (t), ϕ dt + u 0 , ϕ ≤ v ′ m L 4 N ((0,T ),H -1 ) ˆT 0 ϕ 4 4-N H 1 0 dt 4-N 4 + C u 0 L 2 ϕ H 1 0 ≤ C T 4-N 4 + u 0 L 2 ϕ H 1 0 . (4.23)
Then, the weak convergence leads to: Then, (4.18) implies:

ˆT 0 v ′ m (t), ϕ dt -→ ε→0 ˆT 0 v ′ m (t), ϕ dt. ( 4
lim m→+∞ 1 2 v m (T ) 2 L 2 -P m (u 0 ) 2 L 2 ≥ 1 2 u(T ) 2 L 2 -u 0 2 L 2 (4.26)
Using once again usual estimates for Navier-Stokes equation, since v m -→ m→+∞ u in L 2 ((0, T ), H 1 0,σ (Ω)), we have:

ˆT 0 ˆΩ|D(v m )| 2 dx dt -→ m→+∞ ˆT 0 ˆΩ|D(u)| 2 dx dt (4.27)
and

ˆT 0 ˆΩ(v m • ∇v m ) • ψ dx dt -→ m→+∞ ˆT 0 ˆΩ(u • ∇u) • ψ dx dt. (4.28)
Applying lemma 4.3 with our assumption (C3) and passing to the limit over m, we get:

lim m→+∞ ˆT 0 j(v m ) dt ≥ j(u). ( 4
1 2 u(T ) 2 L 2 (Ω) -u 0 2 L 2 (Ω) - ˆT 0 u ′ , ψ dt + ˆT 0 ˆΩ D(u) : D(u -ψ) dx dt + ˆT 0 j(u) -j(ψ) dt - ˆT 0 ˆΩ(u • ∇u) • ψ dx dt ≤ ˆT 0 f, u -ψ dt (4.30)
which is the desired result, that is u is a weak solution of (1.1).

Existence of a finite stopping time for shear-thinning flows

In this part, we assume that hypotheses of the Theorem 3.2 are fulfilled. We are interested to show the existence of a finite stopping time of weak solutions of (1.1) for a viscosity coefficient F which behaves at least as a power-law model, following classical methods for proving such an extinction profile. In fact, one can observe that the nonlinearity proper to Ostald-De Waele or Bingham flows in some special cases implies the existence of such a finite stopping time, as it has already been proved for the two-dimensional Bingham equation under some assumptions in [START_REF] Jesús | Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support[END_REF]. Moreover, the study of such a profile has been proved in the case of the parabolic p-Laplacian, see [15, section VII.2] for a bounded initial datum or [START_REF] Barbu | Controllability and stabilization of parabolic equations[END_REF]Theorem 4.6] for the case p = 1 and with initial datum belonging to L 2 (Ω). In this section, we will moreover assume for convenience that the force term belongs to L 2 ((0, T ), L 2 (Ω)) or, if necessary, we will identify the duality bracket •, • with the L 2 inner product. Note that this assumption is not necessary, the results remain valid for f ∈ L 2 ((0, T ), H -1 σ (Ω)).

Before proving the Theorem 3.2, we need to prove the following useful lemma.

Lemma 5.1. Assume that u ∈ L 6 (Ω). Then, for all r ∈ (0, 3), the following inequality holds:

u r L 2 (Ω) ≤ 3 -r 3 u 4r 3-r L 3 2 (Ω) + r 3 u 2 L 6 (Ω) . (5.1)
Proof of Lemma 5.1. First, we write for s ∈ (0, 2):

u 2 L 2 (Ω) := ˆΩ|u| s |u| 2-s dx. (5.2)
Now, passing to the power r 2 and using Hölder's inequality in (5.2) leads to:

u r L 2 (Ω) ≤ ˆΩ|u| sp dx r 2p ˆΩ|u| (2-s)q dx r 2q . (5.3)
Finally, we apply Young's inequality into (5.3) to obtain:

u r L 2 (Ω) ≤ 1 a ˆΩ|u| sp dx ar p + 1 b ˆΩ|u| (2-s)q dx br q . ( 5.4)
Fixing successively p = 3 2s , s = 4 3 , and b = 3 r leads to (5.1), that is the result is proved. We point out that it is necessary to have r ∈ (0, 3) in order to satisfy the necessary conditions in the inequalities used above:

       s ∈ 0, 3 2 , r > 0, 1 p + 1 q = 1, 1 a + 1 b = 1.
Moreover, we recall the Nirenberg-Strauss inequality: (Ω) the following inequality holds:

u L N N-1 (Ω) ≤ C D(u) L 1 (Ω) .
(5.5)

We are now able to prove Theorem 3.2. We point out that the proof being well-known in the two-dimensional case (see [START_REF] Jesús | Qualitative properties and approximation of solutions of Bingham flows: on the stabilization for large time and the geometry of the support[END_REF]) and can be in that last case a direct application of the Korn's inequality and Sobolev's embbedding theorem. For this reason, we only give a proof in the three-dimensional setting.

Proof of Theorem 3.2. Let u m,ǫ be the solution of (4.2). Choosing ϕ = u m,ε in (4.4) we get:

u ′ m,ε , u m,ε + ˆΩ|D(u m,ε )| 2 dx + j ′ ε (u m,ε ), u m,ε -ˆΩ (u m,ε ∇u m,ε ) u m,ε dx =0 = f, u m,ε . (5.6)
Combining (2.2) and (3.1), we obtain

j ′ ε (u m,ε ), u m,ε ≥ ˆΩ|D(u m,ε )| 2 ε + |D(u m,ε )| 2 p-2 2 dx (5.7)
Now, observing that:

|D(u m,ε )| 2 = ε + |D(u m,ε )| 2 -ε and ε + |D(u m,ε )| 2 p-2 2 ≤ ε p-2 2 ,
and since:

|D(u m,ε )| p ≤ (ε + |D(u m,ε )| 2 ) p 2 ,
we get from (5.6) and (5.7), we get:

1 2 d dt u m,ε (t) 2 L 2 (Ω) + ˆΩ|D(u m,ε )| 2 dx + C D(u m,ε ) p L p (Ω) ≤ f, u m,ε + |Ω|ε p 2 .
(5.8)

Then, using successively the embbedding L p (Ω) ֒→ L 1 (Ω), assumption (3.1) and the Theorem 5.2, we get from (5.8), for t ∈ (T 1 , t):

1 2 d dt u m,ε (t) 2 L 2 (Ω) + ˆΩ|D(u m,ε )| 2 dx + C u m,ε p L 3 2 (Ω) ≤ |Ω|ε p 2 .
(5.9)

Now, from the embbedding u ∈ H 1 0 (Ω)/ D(u) L 2 (Ω) < +∞ ֒→ L 6 (Ω) which can be obtained using Korn's L 2 equality and Sobolev embbedding H 1 0 (Ω) ֒→ L 6 (Ω) we get from (5.9):

1 2 d dt u m,ε (t) 2 L 2 (Ω) + C u m,ε 2
L 6 (Ω) + u m,ε p L 3 2 (Ω)
(5.10)

Conclusions and problems remaining open

In this paper, we have been able to establish the existence of variational inequality solutions for a large class of generalized Newtonian flows in both two-and three-dimensional settings, including threshold fluid flows. This result completes that of [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF] and should be related to that obtained in [START_REF] Abbatiello | On a class of generalized solutions to equations describing incompressible viscous fluids[END_REF] in the context of dissipative solutions. The issue of global regularity in the three-dimensional setting remains open, the question being still widely open for the Leray-Hopf (and Leray) solutions of the incompressible Navier-Stokes equations.

An important question would be to know if it is possible to control this extinction profile, the question of the control of solutions of quasilinear parabolic equations of p-Laplacian type remaining currently open.

A Some examples of functions F

In this section, we give some examples of functions F satisfying the conditions (C1)-(C4), most of which correspond to models of non-Newtonian coherent flows in the physical sense. This is the case for quasi-Newtonian fluids such as blood, threshold fluids such as mayonnaise, or more generally in the case of polymeric liquids.

1. Firstly, in order to describe power-law fluids (also known as Ostwald-De Waele or Norton-Hoff flows), we can consider functions (F p ) 1<p<2 given by: (0, +∞) → (0, +∞) F p :

t -→ t p-2 .

2. Considering functions (F µ,p ) µ>0,p∈ [START_REF] Abbatiello | On a class of generalized solutions to equations describing incompressible viscous fluids[END_REF][START_REF] Abbatiello | Existence of regular time-periodic solutions to shear-thinning fluids[END_REF] of the form (0, +∞) → (0, +∞) F µ,p : t -→ (µ + t 2 )

p-2 2 leads to Carreau flows.

3. Cross fluids are obtained by choosing function (F γ,α ) γ>0,p∈ [START_REF] Abbatiello | On a class of generalized solutions to equations describing incompressible viscous fluids[END_REF][START_REF] Abbatiello | Existence of regular time-periodic solutions to shear-thinning fluids[END_REF] given by: (0, +∞) → (0, +∞) F γ,p : t -→ γ + t p-2 .

4. Another possible choice is to take functions (F p,β,γ ) given (0, +∞) → (0, +∞) F p,β,γ :

t -→    t p-2 log(1 + t) -β
If t ∈ (0, γ] log(1 + γ) -β t p-2 If t ∈ (γ, +∞) for 1 < p < 2 and some β, γ > 0 with γ small enough.

B An energy equality

The purpose of this appendix is to establish, for some F and in the two-dimensional case, a energy equality (non-constructive) for the weak solutions of (1.1), which are continuous in time from Aubin-Lions's lemma.

More exactly, we have the following result.

Note that it is sometimes hard to establsih such equalities in the non-Newtonian setting, we quote as example the work of [START_REF] Beirão | On the energy equality for solutions to Newtonian and non-Newtonian fluids[END_REF] on the subject.

Proposition B.1. Assume that N = 2 and that u is a weak solution to (1.1) satisfying assumptions of Theorem 3.1. Moreover, assume that there exists θ > 0 such that for all t ∈ (0, T ), we have:

G(t) = θt 2 F (θt). (B.1)
Then we have for almost all t ∈ (0, T ) that there exists η(t) ∈ (0, 1] such that:

1 2 u(t) 2 L 2 + 1 2 u 2
L 2 ((0,T ),H 1 0 ) + ˆt 0 1 η(s) j ′ (η(s)u), u ds = 1 2 u 0 2 L 2 + ˆt 0 f, u ds.

Proof. Testing against χ (0,t) (ϕ -u m,ε ) in the weak formulation (4.4), for a well-chosen t and passing to the weak limit over m and ǫ, by Lemma 4.5, since v m (t) L 2 (Ω) ≥ u(t) L 2 (Ω) , we have for almost all time t ∈ (0, T ): Using now ϕ = (1 + δ)v m in Lemma 4.5 (for every δ > 0) as a test function, then dividing each side of the obtained inequality by δ > 0 and passing to the limit with respect to the parameter δ → 0 we obtain, after passing to the limit over m recalling that G admits a Gâteaux-derivative, that: Remark 3. Clearly, the case of threshold flows is contained into the previous proposition, since we can write, for t > 0: ˆt 0 s p-1 ds = 1 p t p = θt 2 (θt) p-2 with θ = p -1 p-1 if 1 < p < 2, θ = 1 if p = 1.

ˆt 0 u ′ , ϕ ds + 1 2 u 0 2 L 2 (Ω) -u(t

. 10 )

 10 Then, we have: Study of generalized Newtonian fluid flows

Lemma 4 . 4 .

 44 With the hypotheses of Proposition 4.1 there exists

. 24 )

 24 Finally, (4.22) and (4.24) imply: v m (T ) ⇀ ε→0 u(T ) in L 2 (Ω). (4.25)

Lemma 5 . 2 ([ 36 ,

 5236 Theorem 1]). Let Ω be an open bounded subset of R N with Lipschitz boundary, then there exists a constant C > 0 which depends of N and Ω such that for all u ∈ W

) 2 L 2 2 L 2 ( 2 L 2 (

 222222 (Ω) + ˆt 0 ˆΩ D(u) : D(ϕ -u) dx ds -ˆt 0 ˆΩ(u • ∇u) • ϕ dx ds + ˆt 0 j(ϕ) -j(u) ds ≥ ˆt 0 f, ϕ -u ds. (B.2)Testing against ϕ = 0 in (B.2), we get:Ω) -u(t) 2 L 2 (Ω) + ˆt 0 ˆΩ|D(u)| 2 dx ds + ˆt 0 ˆΩ G(|D(u)|) dx ds ≤ ˆt 0 f, u ds. (B.3)Now, we write:∀r ∈ (0, +∞), G(r) = ˆr 0 sF (s) ds = r 2 ˆ1 0 yF (ry) dy.We point out that the result still holds for r = 0. Combining the above equality and (B.1), we deduce that:ˆΩ G(|D(u)|) dx = ˆΩ θ|D(u)| 2 F (θ|D(u)|) dx = 1 θ j ′ (θu), θu . So, (B.3) leads to: Ω) -u(t) 2 L 2 (Ω) + ˆt 0 ˆΩ|D(u)| 2 dxds + 1 θ ˆt 0 j ′ (θu), θu ds ≤ ˆt 0 f, u ds. (B.4)

1 2 u 0 2 L 2 ( 2 u 0 2 L 2

 22222 Ω) -u(t) 2 L 2 (Ω) + ˆt 0 ˆΩ|D(u)| 2 dx ds + ˆt 0 j ′ (u), u ds ≥ ˆt 0 f, u ds. (B.5)Finally, using once again the assumption (C2), we obtain from the inequalities (B.4) and (B.5), that there exists η ∈ [θ, 1] such that:1 (Ω) -u(t) 2 L 2 (Ω) + ˆt 0 ˆΩ|D(u)| 2 dx ds + ˆt 0 1 η j ′ (ηu), ηu ds -ˆt 0 f, u ds = 0,which is the wished result.

Now, we can apply Lemma 5.1 with r = 3p p+4 to get:

(5.11)

Then, (5.11) combined with (5.10) leads to, for t ∈ (T 1 , T ):

(5.12)

Assume that for all t ∈ (T 1 , T ), we get that u m,ε L 2 (Ω) ≥ 1 µ |Ω|ε p 2 , for some µ > 0 small enough. Then dividing by u m,ε (t) L 2 (Ω) the both sides of (5.12), we obtain for almost all t ∈ (T 1 , T ):

which is equivalent to:

Up to take µ < C, integrating over (T 1 , t), we have for almost all t ∈ (T 1 , T ):

which leads to u m,ε (t)

8-p p+4

L 2 (Ω) < 0 for t large enough, up to take T large enough. This is a contradiction, so, there exists a time

, and so letting ε → 0 leads to v m (t) L 2 (Ω) = 0 for all t ∈ [T 0 , T ). Indeed, assume that t s is the time for which we have u m,ε (t s ) L 2 (Ω) ≤ 1 µ |Ω|ε p 2 . Then one can verify that t → u m,ε (t) 2 L 2 (Ω) is non-increasing on [max(t s , T 1 ), T ), and we get the result setting T 0 = max(t s , T 1 ).

So we get that there exists t ∈ [T 0 , T ) such that u( t) L 2 (Ω) = 0. If it was not the case, we would have:

This would be a contradiction with the uniform bound of (v m ) m∈N in L 2 ((T 0 , T ), L 2 σ (Ω)) and the convergence v m ⇀ u in L 2 ((T 0 , T ), L 2 σ (Ω)). Also, there exists t in [T 0 , T ) such that u( t) L 2 (Ω) = 0, and, arguing as we have already done for (v m ) m∈N , we have that t → u(t) 2 L 2 (Ω) is non-increasing on [ t, T ). Finally, since u ∈ C w ((0, T ), L 2 σ (Ω)), we get: ∀t ∈ [ t, T ), u(t) L 2 (Ω) = 0, which is the desired result and concludes the proof.