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Introduction -Higher education and its clients: lnstitutional responses to changes indemand and in environment

THIERRY CHEVAILLIER IREDU -CNRS, Université de Bourgogne Article publié dans Higher Education, 44, 2002, pp.303-308 DOI : https://doi.org/10.1023/A:1019802608400

Introduction

Over a few decades, all over the world, higher education institutions have been confronted with new demands in a number of areas. They have responded to them more or less rapidly, more or less willingly, according to national and local circumstances. These demands were expressions of deep changes in society and in the roles assigned by it to universities. In a world where knowledge was becoming the essential resource to foster and sustain economic development, they were increasingly expected not only to produce new knowledge but also to apply it to production by taking part into the innovation process. They were asked to train a growing number of highly skilled people, the knowledge workers, prepared to constantly update their skills to make the most of the outcomes of research. These aspirations were accompanied, in most developed countries, by a substantial financial effort from public budgets and private sources. By bringing the attention to bear on higher education institutions, they also made them accountable to society at large. As long as these new demands remained on the boundaries of institutions, they had little effect on their modes of operation. When they start competing with or even replace 'traditional' missions and activities, they create tensions inside institutions and force adjustments that can deeply affect them. What were these new demands? How were they formulated and channelled to universities? Sorne of these new demands came from students. New types of students entered universities or became a significant part of the student population. Mature student, part-time students, working students, students attending short programmes may currently make up a majority of 'learners' in universities and colleges. To know them, to understand their needs and motivations, to provide them with the kind of education they expect, in terms of content and modes of delivery, has become crucial for the survival of institutions. Either they pay for their education and training or they attract public money in countries relying on student-related funding schemes. They have a choice and therefore bring competition to bear on institutions. They act as clients on various markets or on various segments of an education market. In order to attract them and to keep them, universities may resort to marketing techniques developed by business. Mission statements are turned into business plans. New demands also corne from industry. Firms want flexible and adaptable knowledge workers. Researchers and research-trained workers are essential to rapidly evolving industrial sectors where a sizeable part of the research and development activities that lead to innovations and commercial successes. They also need new knowledge and are ready to buy it from university laboratories that produce relevant concepts and techniques. In a similar way, new demands may originate in various sectors of society when groups of interest express needs that can be met by production and application of new knowledge. Once recognised, such groups become stakeholders in higher education and start competing with others for a share of the attention of institutions. They also may offer financial resources to get satisfaction of their demands, thus creating other markets. Local authorities, professions, even government become purchasers of specific services that universities offer in competition with others: local development trough creation of employment, building of collective infrastructure, advising, consultation, and so on. In a few institutions, a vast amount of energy is devoted to identifying existing needs that are not covered and developing services or 'products' likely to meet such needs or to match what is offered by competitors. They are 'Entrepreneurial universities' [START_REF] Clark | Creating Entrepreneurial University Organizations: Pathways of Transformation[END_REF]. The majority of institutions, which takes a more cautious approach, cannot but feel pressures mounting around them. At the periphery, they recognise that they are on some markets and that they deal with clients, but the core of their activity is hardly affected. Differences in national traditions, in policies and in legal or institutional settings among higher education systems of developed countries make it likely that the strength of external pressures on universities and the type of response they devise will vary widely. Analysing these variations and comparing national or institutional evolutions is a fascinating task for researchers on higher education. The Consortium of Higher Education Research (CHER) held its annual conference in September 2001 in Dijon (France) on the following theme: "Higher Education and it clients: Institutional responses to changes in demand and in environment". This issue of Higher Education presents a selection of articles originating in papers presented at the conference. They offer detailed account of some of the changes and responses observed in a few countries as well as comparative studies conducted in a number of countries of the developed world. A first concem is the study of the changes in the student population and of the extent to which universities responded to them. A second has to do with the research function of universities as places where knowledge is produced and disseminated.

New students, changing expectations towards learning and teaching

Hans Schütze and Maria Slowey study 'non-traditional students', i.e. groups of students who "for a complex range of social, economic and cultural reasons were traditionally excluded from, or underrepresented in, higher education". To what extent are they different from 'traditional' students? A comparison of their situation across ten countries allows to identify factors that influence their participation and to realise how traditional structures and concepts of learning poorly address their needs. Alan Woodley and Jane Wilson concentrates on 'mature students', a heterogeneous category that is becoming a majority in some institutions in the UK. He distinguishes several groups with different needs and trajectories. For those who enter or re-enter higher education after a significant working experience, he observes how little flexibility universities have introduced in their teaching provision to cater for their specific needs. Brenda Little deals with another group of students, full time undergraduates working during term time in British universities. A consequence of changes in the student support system, student work, probably damaging to academic performance, provides a valuable learning experience which is increasingly being recognised and supported by institutions. Karen Jensen and Per Olaf Aamodt address the issue of student motivations. Faced with the diminishing attraction of nursing or social work studies, fields obviously useful to society but that do not have a long-standing university tradition, Norwegian institutions discuss their admission procedures and their entry requirements to take into account students' moral and professional motivations. As the widening access to higher education enables new groups of students to enter university, the issue of standards cannot be avoided. Jake Murdoch draws from a large European survey (extended to Japan) on students careers to raise questions on institutional selectivity: Have prestigious universities opened their doors to new students in the same way as others or have they remained selective? Does competition for students favour élite institutions? He concludes that student motivations and behaviour in choosing their university as well as admission policies of institutions are highly dependent on the national context and vary between subjects and types of institutions. In a study of the attitude of Norwegian universities towards continuing education, Ellen Brandt identifies different strategies to enter this competitive and fast growing market. Although market demands may contribute to create innovative courses and to introduce new perspectives that will benefit 'traditional' education, universities chose not to respond to all market demands and to base their offer of continuing education on research-oriented teaching and leaming. Reflecting on the much-publicised concept of lifelong learning, Ben Jongbloed assesses its implications for higher education providers. It requires them to move as much as they can to a student-centered provision of education and training. Mass individualisation will have far-reaching consequences for the curriculum, the concept of research, the interaction with students, and the relationships with other institutions in the education system. Boundaries between different types of providers, sectors (higher education, vocational education, continuing education) and disciplines will become irrelevant as students will cross them at different points of their life. This is in line with the new mode of knowledge production which requires a trans-disciplinary approach, characterised by interaction between users, producers and brokers of knowledge [START_REF] Gibbons | The New Production of Knowledge: The Dynamics of Science and Research in Contemporary Societies[END_REF] In a broader perspective, Margarita Jeliazkova and Don Westerheijden look at the evolution of the concept and methods of quality assurance in western European universities. After a few decades of unprecedented growth in student enrolment, concern with quality of provision appeared and institutions were subjected to an evaluation of their teaching programmes. They identify several stages in the development of quality assurance with notable shifts in aims and purposes. Starting with a need for public accountability, quality assurance ends up as a requirement of the clients of institutions that increasingly diversify their education supply under the pressure of competition.

University research in the knowledge society

Donald Fisher and Janet Atkinson-Grosjean refer to the Canadian universities industry liaison offices (ILOs) as the universities' brokers in the knowledge market. They describe how these offices were institutionalised when universities started commercialising their research. Some universities operate with an 'internal' model where the ILO is fully integrated into the university's structure. Other universities operate with an 'external' model where the ILO is set up outside the university either as a non-profit or a for-profit corporation. Conflict over ownership and disclosure is most likely in universities that adopt the 'internal' model and least likely in those who adopt the 'external' model. The gap between university culture and commercial culture is narrowing with the new generations of researchers especially in biological and health sciences. Kay Harman presents an evaluation of the Co-operative Research Centres (CRCs), a feature of Australian universities that has a strong impact on research training, particularly in developing in students more favourable attitudes to university-industry collaboration in R&D and more favourable orientations towards careers in industry. Most doctoral students linked to CRCs are full-time and supported by industry-based scholarships. She compares doctoral students working in CRCs or funded by them with other research students. A surveys shows that they fare better on a number of indicators and that their attitude towards working in industry or applied research organisations is significantly more positive. For Jürgen Enders, doctoral training nowadays stands at the heart of the traditional idea of the university as an institution not only combining but cross-fertilising education and science, teaching and research, in the Humboltian tradition. In many countries, efforts are made to organise more systematically the training of researchers and to increase its relevance to the world outside higher education. However, a research conducted in German universities challenges the view that decisions made by governments, intermediate bodies and universities to reorganise and streamline doctoral training will bring the degree of change that is being thought of among other things to broaden and strengthen the employability of doctoral candidates In a reflection on the concept of knowledge and its changing status in our society, Ivar Bleiklie and Haldor Byrkjeflot observe that higher education has come under pressure to expand the kinds and types of knowledge it provides and to diversify the criteria by which it is judged. This is the result of a struggle to define the true nature of knowledge among different actors, states and politicians, institutional leaders and students, researchers and intellectuals, consultants and business leaders. Until recently, public attention has focused on the formal education system, because it is so amenable to state steering, and the "other educational system" -adult education, further education, life-long learning, etc. -has received less attention. However, with the transition to a more utilityoriented knowledge concept, and with the various efforts to create a more knowledge-intensive private business sector, the focus is now moving towards this alternative system.


Conclusion

1 Introduction

Model description

Simulation is performed using the numerical model hereafter called Shaltop 2d resulting from a long term collaboration between Département de Mathématiques et Applications (DMA), École Normale Supérieure (ENS, Paris) and Institut de Physique du Globe de Paris (IPGP) in France [START_REF] Bouchut | A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows[END_REF][START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF][START_REF] Bouchut | Gravity driven shallow water models for arbitrary topography[END_REF][START_REF] Mangeney-Castelnau | On the use of Saint-Venant equations for simulating the spreading of a granular mass[END_REF], 2007). This model was developed after former studies performed by our group in the context of a collaboration between IPGP, ENS-DMA, and INRIA. A first model was developed based on the model developed for shallow river flows by Audusse et al. (2000) and [START_REF] Bristeau | Boundary conditions for the shallow water equations solved by kinetic schemes[END_REF]. Mangeney et al., 2003 extended this model based on a kinetic scheme to the classical Savage-Hutter model for granular flows over sloping topography with a Coulomb-type basal friction involving either a constant friction coefficient or the empirical flow rule proposed by [START_REF] Pouliquen | Scaling laws in granular flows down rough inclined planes[END_REF]. Finally, the projection of the gravity field on the reference frame linked to the topography was performed during the PhD of M. Pirulli together with the active/passive earth pressure coefficient [Pirulli et al., 2007] and several basal friction law have been tested on real events [START_REF] Pirulli | Numerical modelling of landslide runout, a continuum mechanics approach[END_REF]Mangeney, 2007). This model after called RASH 3D by Pirulli et al., (2007) is used in the Benchmark exercise by Pirulli and Scavia (report of the 2007 forum). The main advantage of this model is the unstructured finite element mesh best suited to deal with complex topography. However, the numerical method is only of order 1 and the friction, the projection of the gravity field and the introduction of the earth pressure coefficients are not compatible with the preservation of steady state and do not respect all the properties required by the numerical resolution of the equations (see [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF], for details about numerical methods to solve these problems). Furthermore this kinetic model (i.e. RASH 3D model) is based on the Savage-Hutter equations developed in a reference frame linked to the topography including only the curvature terms in the x and y direction. Note that the afore-mentioned problems are shared with most of the numerical models proposed in the literature.

Contrary to the classical approaches, the new model Shaltop 2d is based on the equations developed by [START_REF] Bouchut | A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows[END_REF] and [START_REF] Bouchut | Gravity driven shallow water models for arbitrary topography[END_REF] developed in a fixed cartesian reference frame with the thin layer approximation (TLA) imposed in the direction perpendicular to the topography (see figure 1). The rigorous asymptotic analysis makes it possible for the first time to account for the whole curvature tensor. Furthermore, the numerical method is of order 2 requiring less refined grid to reach the same precision. The numerical method is based on the work of François [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources[END_REF] and preserve the steady states as well as other requirements related to the resolution of hyperbolic equations.

The overall idea is to develop the equations in a fixed reference frame (x, y, z), for example horizontal/vertical, as opposed to the equations developed by Hutter and co-workers in a variable reference frame linked to the topography. However, the shallowness assumptions are still imposed in the local reference frame (X, Y, Z) linked to the topography (fig. 1). Indeed, to satisfy the hydrostatic assumption for shallow flow over inclined topography, it is the acceleration normal to the topography that must be neglected compared to the gradient of the pressure normal to the topography. The reference frame is shown in figure 1. The 2D horizontal coordinate vector is x = (x, y) 2 R 2 and the topography is described by the scalar function b(x, y) with a 3D unit upward normal vector

ñ ⌘ ( s, c) 2 R 2 ⇥ R, (1)
where,

s = r xy z p 1 + ||r xy z|| 2 , (2)
and

c = 1 p 1 + ||r xy z|| 2 , ( 3
)
where r xy is the gradient of the topography in both directions, and c = cos ✓ with ✓ the angle between ñ and the vertical. In the horizontal/vertical Cartesian coordinate and for a inclined plane, formulation the equations reduce to

@ t h + c • @ x (hu) + @ y (hv) = 0, (4)
@ t u + cu • @ x u + v • @ y u + c • @ x (ghc) = g sin ✓ + ffx , (5)
@ t v + cu • @ x v + v • @ y v + @ y (ghc) = ffy , (6)
where h is the depth-thickness of the flow, g is the vertical acceleration and u(u, v) is the depth-average velocity. The two-dimensional Saint-Venant system with topography and friction is written 8 > < > :

@ t h + @ x (hu) + @ y (hv) = 0, @ t (hu) + @ x (hu 2 + gh 2 /2) + @ y (huv) + hg@ x b = hf x , @
t (hv) + @ x (huv) + @ y (hv 2 + gh 2 /2) + hg@ y b = hf y , where b(x, y) represents the 2D topography.

Debris avalanche is treated here as a single-phase, dry granular flow with Coulomb-type behavior. The transition between a static state and flowing state is modeled by introducing a threshold allowing or not the material to flow. The friction forces f (x, y, t) = (f x , f y ) (see ( 5) and ( 6)) must satisfy

||f (x, y, t)||  gµ, (7) u(x, y, t) 6 = 0 ) f (x, y, t) = gµ u(x, y, t) ||u(x, y, t)|| . ( 8
)
The friction coefficient is either supposed to be constant (Coulomb friction law) or depending on the thickness h and the Froude number of the flow u/ p gh (hereafter called Pouliquen flow rule): (1) Coulomb friction law defined as follow

µ = tan , (9)
where is the constant friction angle.

(2) Pouliquen friction empirical rule [START_REF] Pouliquen | Scaling laws in granular flows down rough inclined planes[END_REF]:

µ(||u||, h) = tan 1 + (tan 2 tan 1 ) exp ✓ h d p gh ||u|| ◆ , (10)
where 1 and 2 are characteristic friction angles of the material, d is a lengthscale of the order of a grain diameter, and = 0.136 is a dimensionless parameter as it has been proposed by [START_REF] Pouliquen | Scaling laws in granular flows down rough inclined planes[END_REF]. Basically, the friction parameter is increasing for small thickness h and high velocity u. The Coulomb type friction law applied on the averaged media has been tested by comparing depth-averaged model Shaltop 2d with discrete element simulations [START_REF] Mangeney | Comparison Between Discrete and Continuum Modeling of Granular Spreading[END_REF] on the collapse of granular column of variable aspect ratio (a = H i /R i , where H i and R i are the initial thickness and radius of the granular column). The main result was that vertical acceleration has to be included in the model if aspect ratio a > 1 are deal with.

Shaltop 2d has been used successfully to simulate laboratory experiments of granular collapse [START_REF] Mangeney-Castelnau | On the use of Saint-Venant equations for simulating the spreading of a granular mass[END_REF], levees-channel formation [START_REF] Mangeney | Numerical modeling of self-channeling granular flows and of their levee/channel deposits[END_REF] as well as real 3D avalanches in a natural context as it has been recently shown for large Martian landslides [START_REF] Lucas | Mobility and topographic effects for large Valles Marineris landslides on Mars[END_REF].

Computing resources

We benefit of the computing facilities from the Metacentre de Données et de Calcul Parallèle of IPGP: Time of computation could rise up to several days on the IBM e325 depending of the grid space increment and the time of the simulation. Parallelization of the code (using MPI) has been carried out by P. Stoclet at IPGP.

1.2.

Softwares used

DEM (Digital Elevation Model) processing (e.g. gridding) were performed with Surfer (Goldensof tware). Visualization and rendering have been carry out under GMT [START_REF] Wessel | Free software helps map and display data[END_REF] and MatLab.

Cases studied

As previously mentioned, Shaltop 2d is appropriate for dry granular monophasic flow such as landslides, rock avalanches and dry sand flows. In this context, we focused our runs on the following cases eventhought water could be present in the real landslides:

Description of the scenario

This test makes it possible to calibrate the numerical modeling with analytical solution. The scenario involves movement of mass in a wide inclined channel, triggered by sudden removal of dam (fig. 2). From the analytical solution (AS) derived by Mangeney et al., (2000), profiles of the debris flow at any given time t are:

h(t) = 8 > < > : H ; x  x r 1 9g cos ✓ 2c 0 x t mt 2 2 ; x r < x < x l 0 ; x l  x with, m = g(cos ✓ tan sin ✓), (11)
c 0 = p gH cos ✓, (12)
x l = 2c 0 t 1 2 mt 2 , ( 13
) x r = c 0 t + 1 2 mt 2 , ( 14
)
where H is the initial height of debris mass, g the gravity, ✓ is the slope angle and is the angle of friction. We perform test using the following settings (fig. 3):

• Initial height of the mass: 10m

• Slope angle: 30 o

• Friction angle: 25 o

Results

Shaltop 2d is able to predict very well the analytical solution (fig. 3). et al., 2003). (Bottom) -Using the same parameters, we compare here Shaltop 2d with analytical solution.

On figure 4, we show that, for the same parameters, Shaltop-2d better fits the analytical solutions than RASH 3D model [START_REF] Mangeney-Castelnau | Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme[END_REF]. For = 4 o , ✓ = 5 o and x = 20, Shaltop 2d reproduces better the runout distance predicted by the analytical solution (fig. 4). The overestimation of this runout distance in the kinetic model is related to numerical diffusion. As a result, the calibration of the kinetic model in order to fit the right runout distance would lead to friction angle almost one degree higher than the real friction angle. In the simulation of real avalanches over complex topography, numerical diffusion is expected to generate larger runout distances (i.e. larger value of the fitted friction angle) using kinetic scheme or similar schemes of order 1 than that obtained with Shaltop 2d. A series of numerical test on real topography with increasing discretization is therefore necessary if the models (i.e. the fitted friction coefficients) are to be compared on real events.

3 Dry sand behavior within a 3D channel

Description of the simulation

Simulation of the experimental results of [START_REF] Iverson | Granular avalanches across irregular threedimensional terrain: 2. Experimental tests[END_REF] dealing with dry granular flows over irregular 3D channel (see figure 3) is performed here.

The basal topography has the following characteristics (fig. 5):

• Mean angle of sloping flume bed: 31.6 o

• Headgate aperture width: 12 cm

• X location of headgate: 8.858 cm.

• Location of urethane topographic insert: entire flume width (20 cm), and covering from x = 9.14 cm to x = 39.12 cm.

• The right triangular prism that spans the full width of the flume ( 20cm) is approximately h 0 = 4.35 cm thick (measured vertically) at x = 8.858 cm (headgate) and tapers to 0 cm thick upflume from the headgate at x ' 1.9 cm. We interpolate (using kriging algorithms) the DEM so as to get a space increment x = y = 0.24 cm. Results obtained using a constant friction coefficient all over the topography are obviously not accurate because substrates with different roughness were used in the tank and in the irregular topography. Numerical simulation usign = 29 all over the numerical domain is however shown for comparison (see fig. 7). The bottom part of the mass spreading is quiet well reproduce by the code, whereas the top part does not fit with [START_REF] Iverson | Granular avalanches across irregular threedimensional terrain: 2. Experimental tests[END_REF] observations. Mass staying in the tank is overestimated within the sand tank at the end of the simulation. As consequences, 15% of the bulk mass don't participate in the spreading.

As mentioned in [START_REF] Iverson | Granular avalanches across irregular threedimensional terrain: 2. Experimental tests[END_REF], two kind of materials are used in this 3D channel. The tank and the bottom parts are made of Formica with a bed friction angle f = 23.5 o and the channel part with urethane characterized by a bed friction angle u = 19.8 o . We thus adapt our code so as to take into account the variation of the bed friction angle due to the different floor (Formica and urethane). Results are obviously significantly improved as shown in figure 8. Mass profile is more alike experimental observations (top of fig. 6).

Results

The friction angle is calibrated to fit the runout distance of the landslide. Best agreement is obtained using = 18 o . The mean velocity increases with a factor two during the 10 first seconds. The flow stops at t ' 30 s (see table 1).

Results

Better results are obtained with = 26 o as it is shown on Figure 12. On the right side of Figure 12, we show the evolution of the velocity field at several times. The mean velocity increases rapidly after triggering and rise up to 70 m/s at its maximum. The flow stops at t ' 15 s ( (see table 2).

Finest analysis shows that between t = 8 s and t = 14 s, the back of the mass is still under movement whereas the front does not.

As consequences, the flow spread on each side (along the road) and not in the slope direction. We thus overestimate the width of the final deposits.

Results

Pirulli and Mangeney (2007) proposed a Coulomb-type friction angle = 14 o for this specific landslide. Using the same angle Shaltop 2d underestimates the observed runout as it is shown on figure 14. We get better results with = 12 o (fig. 15). Using a Pouliquen-type friction law, we get very similar results (fig. 16). Sensitivity of the numerical results to mesh size has to be performed before getting any conclusion from this comparison. Actually, the differences could only result from numerical dissipation (see discussion in the analytical solution section). Using Coulomb-type friction or Pouliquen-type friction law has consequences on velocity field. With the second law, the flow gets faster but slows down quickly compare to Coulomb-type friction law (see table 3).

Conclusion

Our simulations shows that despite the accordingly nonphysical empirical friction coefficient used in the numerical model, the area of the deposit is quite well reproduced in many natural cases. However, we showed that this coefficient ranges between 12 o and 28 o depending on the studied case. This dispersion of the friction coefficient questions the predictable power of this approach. The presence of a fluid phase, granulometry distribution, the presence of tree roots at the top of the released mass as this is the case for Shum Wan slide or Fei Tsui slide could explain the difference between the friction coefficient. Only calibration in the similar geological context would possibly make it possible to predict further events.

Using both friction laws (Coulomb and Pouliquen) may have inferences on velocity field. In terms of prediction, this is a key point, because of the pressure impact on the foundations.

  description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Computing resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Cases studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Dam-Break Scenario 2.1 Description of the scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Dry sand behavior within a 3D channel 3.1 Description of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Shum Wan landslide, HK (1995) 4.1 Description of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fei Tsui landslide, HK (1995) 5.1 Description of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Frank Slide, CA (1902) 6.1 Description of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Frames and variables used in the model (after Mangeney et al., 2007).

• 1 .

 1 Dam Break Scenario from analytical solution • 2.Laboratory Test of dry sand flow from USGS • 3.Shum Wan landslide, HK • 4.Fei Tsui Road landslide, HK • 5.FrankSlide, CA 2 Dam-Break Scenario

Figure 2 :

 2 Figure2: Initial conditions of dam-break scenario. H is the initial height of the debris mass, ✓ is the slope angle and is the angle of friction at channel bed.

Figure 3 :

 3 Figure 3: Evolution of the flow height for both analytical and numerical solutions respectively named AS and NS. Using a reasonable space increment x, Shaltop 2d is able to reproduce very well the analytical solution.

Figure 4 :

 4 Figure 4: (Top) -Comparison between RASH 3D model with analytical solution (figure from Mangeney et al., 2003). (Bottom) -Using the same parameters, we compare here Shaltop 2d with analytical solution.

Figure 5 :

 5 Figure 5: 3D irregular channel with the initial mass (in color).

Figure 6 :

 6 Figure 6: Evolution of the dry granular media across from Iverson et al., 2004.

Figure 7 :

 7 Figure 7: Evolution of the dry granular media across an irregular 3D channel using a single friction angle = 29 o all over the numerical domain (DEM courtesy[START_REF] Iverson | Granular avalanches across irregular threedimensional terrain: 2. Experimental tests[END_REF]

4

  Shum Wan landslide, HK (1995) 4.1 Description of the simulation Analysis of Fei Tsui landslide has been carried out using Coulomb-type friction law. The initial volume in our simulation is 26.10 3 m 3 . Using griding algorithms (e.g. kriging) we get a 2 m grid spacing. The grid is 113 x 85 points. We performe several analysis dealing with Coulomb-type friction angle 2 [16 o , 26 o].

Figure 9 :

 9 Figure 9: (Left) -DEM of Shum Wan landslide, HK. Slide path is indicated by black lines over the topographic map. (Right) -Slope map calculated from the DEM.

Figure 10 :

 10 Figure 10: (Left) -Evolution of the mass (Right) -Velocity field. Results from simulation using friction angle = 18 o .

5

  Fei Tsui landslide, HK (1995) 5.1 Description of the simulation Analysis of Fei Tsui landslide has been carried out using Coulomb-type friction and Pouliquen friction laws. The initial volume in our simulation is 14.10 3 m 3 with a 1m grid spacing. The grid is 97x117 points. A series of simulation is performed using friction angles 2 [22 o , 30 o].

Figure 11 :

 11 Figure 11: (Left) -DEM of Fei Tsui landslide, HK. Slide path is indicated by black lines over the topographic map. (Right) -Slope map from the DEM.

Figure 12 :

 12 Figure 12: Evolution of the flow height and velocity at several times using Coulomb type friction angle = 26 o . Isovalues of the thickness are represented every 1m. On velocity field, only 1/4 vectors are displayed for a better visualization.

Figure 13 :

 13 Figure 13: (Left) -DEM of Frank Slide, CA. Slide path is indicated by black lines over the topographic map. (Right) -Slope map calculated from the DEM.

Figure 14 :

 14 Figure 14: Evolution of the slide using friction angle = 14 o .

Figure 15 :

 15 Figure 15: Evolution of the slide using friction angle = 12 o .

Figure 16 :

 16 Figure 16: Evolution of the slide using a Pouliquen friction law where 1 = 12 o , 2 = 26 o and d = 1.5m.

Figure 17 :

 17 Figure 17: Velocity field at time = 10, 40 and 70 secs. (Left) -Using Coulomb-type friction (Right) -Using Pouliquen-type friction law.

  1 CPU Server:

	• Server Bi-Xeon 5160 3.00GHz
	-Mem: 3Gb
	-OS: Linux Fedora 8
	-Compiler: Intel Fortran 10.1
	• IBM e325 server bi-cpu AMD Opteron 256
	-Mem: 4Gb
	-OS: Linux RedHat Entreprise 3
	-Compiler: pgf95 -The Portland Group Inc. Fortran 90/95 compiler
	1.2.2 Parallel Computing Cluster:
	• 64 x IBM x3550 bi-cpu Intel Xeon Quad-Core E5420 (512 cores)
	-Mem: 512 Gb
	-OS: RedHat EL release 5.1
	-Compiler: Intel Fortran 10.1

Table 1 :

 1 ). 13 Velocity evolution and statistics for Shum Wan landslide

		Velocity Stats.
	Time (sec.) u (m.s 1 ) U max (m.s 1 )	/u
	4	2.39	11.18	2.71 1.14
	16	5.16	16.53	4.85 0.94
	28	2.56	13.35	4.17 1.62

Table 2 :

 2 Velocity evolution and statistics for Fei Tsui landslide

		Velocity Stats.
	Time (sec.) u (m.s 1 ) U max (m.s 1 )	/u
	2	4.5	29.05	3.40 0.76
	8	2.38	68.33	3.08 1.29
	14	1.34	47.39	2.35 1.75

Table 3 :

 3 Velocity Stats. (Coulomb-type Friction) Time (sec.) u (m.s 1 ) U max (m.s 1 ) Velocity evolution and statistics for both behaviors.

	/u


Éric Séré

A SURJECTION THEOREM FOR MAPS WITH SINGULAR PERTURBATION AND LOSS OF DERIVATIVES

. For singularly perturbed functionals Fε, we allow v to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrödinger Cauchy problem with concentrated initial data studied by Texier-Zumbrun [26], and we show that our result improves significantly on theirs.

Introduction

The basic idea of the inverse function theorem (henceforth IFT) is that, if a map F is differentiable at a point u 0 and the derivative DF (u 0 ) is invertible, then the map itself is invertible in some neighbourhood of u 0 . It has a long and distinguished history (see [START_REF] Krantz | The Implicit Function Theorem, History, and Applications[END_REF] for instance), going back to the inversion of power series in the seventeenth century, and has been extended since to maps between infinite-dimensional spaces. If the underlying space is Banach, and if one is only interested in the local surjectivity of F , that is, the existence, near u 0 , of a solution u to the equation F (u) = v for v close to F (u 0 ), one just needs to assume that F is of class C 1 and that DF (u 0 ) has a right-inverse L(u 0 ). The standard proof is based on the Picard scheme:

u n = u n-1 -L(u 0 )(F (u n-1 ) -v)
which converges geometrically to a solution of F (u) = v provided F (u 0 ) -v is small enough. In the C 2 case, the Newton algorithm:

u n = u n-1 -L(u n-1 )(F (u n-1 ) -v)
uses the right-invertibility of DF (u) for u close to u 0 , and provides local quadratic convergence.

In functional analysis, u will typically be a function. In many situations the IFT on Banach spaces will be enough, but in the study of Hamiltonian systems and Date: May 27, 2021.

PDEs, one encounters cases when the right-inverse L(u) of DF (u) loses derivatives, i.e. when L(u)F (w) has less derivatives than u and w. In such a case, the Picard and Newton schemes lose derivatives at each step. The first solutions to this problem are due, on the one hand, to Kolmogorov [START_REF] Kolmogorov | On the conservation of conditionally periodic motions by small perturbations of the Hamiltonian[END_REF] and Arnol'd [START_REF] Arnol'd | Generation of quasi-periodic motion from a family of periodic motions[END_REF], [START_REF] Arnol'd | Small denominators I. Mapping the circle onto itself[END_REF], [START_REF] Arnol'd | Small denominators and problems of stability of motion in classical and celestial mechanics[END_REF] who investigated perturbations of completely integrable Hamiltonian systems in the analytic class, and showed that invariant tori persist under small perturbations, and, on the other hand, to Nash [START_REF] Nash | The imbedding problem for Riemannian manifolds[END_REF], who showed that any smooth compact Riemannian manifold can be imbedded isometrically into an Euclidian space of sufficiently high dimension 1 .

In both cases, the fast convergence of Newton's scheme was used to overcome the loss of regularity. Since Nash was considering functions with finitely many derivatives, he had to introduce a sequence of smoothing operators S n , in order to regularize L(u n-1 )(F (u n-1 ) -v), and the new scheme was

u n = u n-1 -S n L(u n-1 )(F (u n-1 ) -v) .
An early presentation of Nash's method can be found in Schwartz' notes [START_REF] Schwartz | Nonlinear functional analysis[END_REF]. It was further improved by Moser [START_REF] Moser | A new technique for the construction of solutions to nonlinear differential equations[END_REF], who used it to extend the Kolmogorov-Arnol'd results to C k Hamiltonians. The Nash-Moser method has been the source of a considerable amount of work in many different situations, giving rise in each case to a so-called "hard" IFT. We will not attempt to review this line of work in the present paper. A survey up to 1982 will be found in [START_REF] Hamilton | The Inverse Function Theorem of Nash and Moser[END_REF]. In [START_REF] Hörmander | The Boundary Problems of Physical Geodesy[END_REF], Hörmander introduced a refined version of the Nash-Moser scheme which provides the best estimates to date on the regularity loss. We refer to [START_REF] Alinhac | Opérateurs Pseudo-différentiels et Théorème de Nash-Moser[END_REF] for a pedagogical account of this work, and to [START_REF] Baldi | A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs[END_REF] for recent improvements. We also gained much insight into the Nash-Moser scheme from the papers [START_REF] Berti | Cantor families of periodic solutions for completely resonant nonlinear wave equations[END_REF], [START_REF] Berti | Cantor families of periodic solutions for completely resonant nonlinear wave equations[END_REF], [START_REF] Berti | Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions[END_REF], [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF].

The question we want to address here is the following. The IFT implies that the range of F contains a neighborhood V of v 0 = F (u 0 ). What is the size of V?

In general, when one tries to apply directly the abstract Nash-Moser theorem, the estimates which can be derived from its proof are unreasonably small, many orders of magnitude away from what can be observed in numerical simulations or physical experiments. Moreover, precise estimates for the Nash-Moser method are difficult to compute, and most theoretical papers simply do not address the question.

So we shall address instead a "hard" singular perturbation problem with loss of derivatives. The same issue appears in such problems, as we shall explain in a moment, but it takes a simpler form: one tries to find a good estimate on the size of V as a power of the perturbation parameter ε. Such an asymptotic analysis has been carefully done in the paper of Texier and Zumbrun [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] which has been an important source of inspiration to us, and we will be able to compare our results with theirs. As noted by these authors, the use of Newton's scheme implies an intrinsic limit to the size of V.

Let us explain this in the "soft" case, without loss of derivatives. Suppose that for every 0 < ε ≤ 1 we have a C 2 map F ε between two Banach spaces X and Y , such that F ε (0) = 0, and, for all u ≤ R,

||| D u F ε (u) -1 ||| ≤ ε -1 M ||| D 2
uu F ε (u) ||| ≤ K 1 Nash's theorem on isometric embeddings was later re-proved by Gunther [START_REF] Günther | Isometric embeddings of Riemannian manifolds[END_REF], who found a different formulation of the problem and was able to use the classical IFT in Banach spaces.

Then the Newton-Kantorovich Theorem (see [START_REF] Ciarlet | Linear and nonlinear functional analysis with applications[END_REF], section 7.7 for a comprehensive discussion) tells us that the solution u ε of F ε (u) = v exists for v < ε 2 2KM 2 , and this is essentially the best result one can hope for using Newton's algorithm, as mentioned by Texier and Zumbrun in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], Remark 2.22. Note that the use of a Picard iteration would give a similar condition.

However, in this simple situation where no derivatives are lost, it is possible, using topological arguments instead of Newton's method, to find a solution u provided v ≤ εR/M : one order of magnitude in ε has been gained. The first result of this kind, when F is C 1 and dim X = dim Y < ∞ , is due to Wazewski [START_REF] Wazewski | Sur l'évaluation du domaine d'existence des fonctions implicites réelles ou complexes[END_REF] who used a continuation method. See also [START_REF] John | On quasi-isometric mappings. I[END_REF] and [START_REF] Sotomayor | Inversion of smooth mappings[END_REF] and the references in these papers, for more general results in this direction. In [START_REF] Ekeland | An inverse function theorem in Fréchet spaces[END_REF] (Theorem 2), using Ekeland's variational principle, Wazewski's result is proved in Banach spaces, assuming only that F is continuous and Gâteaux differentiable, the differential having a uniformly bounded right-inverse (in §2 below, we recall this result, as Theorem 5).

Our goal is to extend such a topological approach to "hard" problems with loss of derivatives, which up to now have been tackled by the Nash-Moser algorithm. A first attempt in this direction was made in [START_REF] Ekeland | An inverse function theorem in Fréchet spaces[END_REF] (Theorem 1), in the case when the estimates on the right-inverse do not depend on the base point, but it is very hard to find examples of such situations. The present paper fulfills the program in the general case, where estimates on the inverse depend on the base point.

In [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], Berti, Bolle and Procesi prove a new version of the Nash-Moser theorem by solving a sequence of Galerkin problems

Π ′ n F (u n ) = Π ′ n v, u n ∈ E n
, where Π n and Π ′ n are projectors and E n is the range of Π n . They find the solution of each projected equation thanks to a Picard iteration:

u n = lim k→∞ w k with w 0 = u n-1 and w k+1 = w k -L n (u n-1 )(F (w k ) -v) , where L n (u n-1 ) is a right inverse of D(Π ′ n F |E n )(u n-1
). So, in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] the regularized Newton step is not really absent: it is essentially the first step in each Picard iteration. As a consequence, the proof in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] involves quadratic estimates similar to the ones of more standard Nash-Moser schemes. Moreover, Berti, Bolle and Procesi assume the right-invertibility of

D(Π ′ n F |E n )(u n-1
). This assumption is perfectly suitable for the applications they consider (periodic solutions of a nonlinear wave equation), but in general it is not a consequence of the right-invertibility of DF (u n-1 ), and this restricts the generality of their method as compared with the standard Nash-Moser scheme.

As in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], we work with projectors and solve a sequence of Galerkin problems. But in contrast with [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], the Newton steps are completely absent in our new algorithm, they are replaced by the topological argument from [START_REF] Ekeland | An inverse function theorem in Fréchet spaces[END_REF] (Theorem 2), ensuring the solvability of each projected equation. Incidentally, this allows us to work with functionals F that are only continuous and Gâteaux-differentiable, while the standard Nash-Moser scheme requires twice-differentiable functionals. Our regularity assumption on v also seems to be optimal, and even weaker than in [START_REF] Hörmander | The Boundary Problems of Physical Geodesy[END_REF]. Moreover, our method works assuming either the right-invertibility of D(Π ′ n F |E n )(u) as in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], or the right-invertibility of DF (u) (in the second case, our proof is more complicated). But in our opinion, the main advantage of our approach is the following: there are no more quadratic terms in our estimates, as a consequence we can deal with larger v's, and this advantage is particularly obvious in the case of singular perturbations.

To illustrate this, we will give an abstract existence theorem with a precise estimate of the range of F for a singular perturbation problem: this is Theorem 3 below. Comparing our result with the abstract theorem of [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], one can see that we have weaker assumptions and a stronger conclusion. Then we will apply Theorem 3 to an example given in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], namely a Cauchy problem for a quasilinear Schrödinger system first studied by Métivier and Rauch [START_REF] Métivier | Dispersive stabilization[END_REF]. Texier and Zumbrun use their abstract Nash-Moser theorem to prove the existence of solutions of this system on a fixed time interval, for concentrated initial data. Our abstract theorem allows us to increase the order of magnitude of the oscillation in the initial data. After reading our paper, Baldi and Haus [START_REF] Baldi | Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems[END_REF] have been able to increase even more this order of magnitude, using their own version [START_REF] Baldi | A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs[END_REF] of the Newton scheme for Nash-Moser, combined with a clever modification of the norms considered in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] and an improved estimate on the second derivative of the functional. In contrast, our proof follows directly from our abstract theorem, taking exactly the same norms and estimates as in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], and without even considering the second derivative of the functional.

The paper is constructed as follows. In Section 2, we present the general framework: we are trying to solve the equation F ε (u) = v near F ε (0) = 0, when F ε maps a scale of Banach spaces of functions into another and admits a right-invertible Gâteaux differential with "tame estimates" involving losses of derivatives and negative powers of ε. After giving our precise assumptions, we state our main theorem. Section 3 is devoted to its proof. In Section 4, we apply it to the example taken from Texier and Zumbrun [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], and we compare our results with theirs.

Main assumptions and results.

2.1. Two tame scales of Banach spaces. Let (V s , • s ) 0≤s≤S be a scale of Banach spaces, namely:

0 ≤ s 1 ≤ s 2 ≤ S =⇒ [V s2 ⊂ V s1 and • s1 ≤ • s2 ] .
We shall assume that to each Λ ∈ [1, ∞) is associated a continuous linear projection Π(Λ) on V 0 , with range E(Λ) ⊂ V S . We shall also assume that the spaces E(Λ) form a nondecreasing family of sets indexed by [1, ∞), while the spaces Ker Π(Λ) form a nonincreasing family. In other words:

1 ≤ Λ ≤ Λ ′ =⇒ Π(Λ)Π(Λ ′ ) = Π(Λ ′ )Π(Λ) = Π(Λ) .
Finally, we assume that the projections Π(Λ) are "smoothing operators" satisfying the following estimates:

Polynomial growth and approximation: There are constants A 1 , A 2 ≥ 1 such that, for all numbers 0 ≤ s ≤ S, all Λ ∈ [1, ∞) and all u ∈ V s , we have:

∀t ∈ [0, S] , Π(Λ)u t ≤ A 1 Λ (t-s) + u s (2.1) ∀t ∈ [0, s] , (1 -Π(Λ))u t ≤ A 2 Λ -(s-t) u s (2.2)
When the above properties are met, we shall say that (V s , • s ) 0≤s≤S endowed with the family of projectors { Π(Λ) , Λ ∈ [1, ∞) } , is a tame Banach scale.

It is well-known (see e.g. [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]) that (2.1,2.2) imply:

Interpolation inequality: For 0 ≤ t 1 ≤ s ≤ t 2 ≤ S , (2.3) u s ≤ A 3 u t 2 -s t 2 -t 1 t1 u s-t 1 t 2 -t 1 t2
.

Let (W s , • ′ s ) 0≤s≤S be another tame scale of Banach spaces. We shall denote by Π ′ (Λ) the corresponding projections defined on W 0 with ranges E ′ (Λ) ⊂ W S , and by A ′ i (i = 1, 2, 3) the corresponding constants in (2.1), (2.2) and (2.3). Remark. In many practical situations, the projectors form a discrete family as, for instance, {Π(N ) , N ∈ N * }, or {Π(2 j ) , j ∈ N}. The first case occurs when Π(N ) acts on periodic functions by truncating their Fourier series, keeping only frequencies of size less or equal to N , as in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]. The second case occurs when truncating orthogonal wavelet expansions as in an earlier version of the present work [START_REF] Ekeland | An implicit function theorem for non-smooth maps between Fréchet spaces[END_REF]. Our choice of notation and assumptions covers these cases, taking Π(Λ) = Π(⌊Λ⌋) or Π(Λ) = Π(2 ⌊log 2 (Λ)⌋ ), where ⌊•⌋ denotes the integer part.

2.2. Main theorem. We state our result in the framework of singular perturbations, in the spirit of Texier and Zumbrun [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]. The norms • s , • ′ s on the tame scales (V s ), (W s ) may depend on the perturbation parameter ε ∈ (0, 1], as well as the projectors Π(Λ) , Π ′ (Λ) and their ranges E(Λ), E ′ (Λ) . But we impose that S and the constants A i , A ′ i appearing in estimates (2.1, 2.2, 2.3) be independent of ε. In order to avoid burdensome notations, the dependence of the norms, projectors and subspaces on ε will not be explicit in the sequel. Denote by B s the unit ball in V s :

B s = {u | u s ≤ 1}
In the sequel we fix nonnegative constants s 0 , m, ℓ, ℓ ′ and g, independent of ε. We will assume that S is large enough.

We first recall the definition of Gâteaux-differentiability, in a form adapted to our framework: Definition 1. We shall say that a function F : B s0+m → W s0 is Gâteauxdifferentiable (henceforth G-differentiable) if for every u ∈ B s0+m , there exists a linear map DF (u) : V s0+m → W s0 such that for every s ∈ [s 0 , S -m], if u ∈ B s0+m ∩ V s+m , then DF (u) maps continuously V s+m into W s , and

∀h ∈ V s+m , lim t→0 1 t [F (u + th) -F (u)] -DF (u) h ′ s = 0 .
Note that, even in finite dimension, a G-differentiable map need not be C 1 , or even continuous. However, if DF : V s+m → L(V s+m , W s ) is locally bounded, then F : V s+m → W s is locally Lipschitz, hence continuous. In the present paper, we will always be in such a situation.

We now consider a family of maps (F ε ) 0<ε≤1 with F ε : B s0+m → W s0 . We are ready to state our assumptions on this family: Definition 2.

• We shall say that the maps F ε : B s0+m → W s0 (0 < ε ≤ 1) form an S-tame differentiable family if they are G-differentiable with respect to u, and, for some positive constant a , for all ε ∈ (0, 1] and all s

∈ [s 0 , S -m] , if u ∈ B s0+m ∩ V s+m and h ∈ V s+m , then DF ε (u) h ∈ W s with the tame direct estimate (2.4) DF ε (u) h ′ s ≤ a h s+m + u s+m h s0+m . • Then we shall say that (DF ε ) 0<ε≤1 is tame right-invertible if there are b > 0 and g, ℓ, ℓ ′ ≥ 0 such that for all 0 < ε ≤ 1 and u ∈ B s0+max{m,ℓ} , there is a linear map L ε (u) : W s0+ℓ ′ → V s0 satisfying (2.5) ∀k ∈ W s0+ℓ ′ , DF ε (u) L ε (u) k = k and for all s 0 ≤ s ≤ S -max {ℓ, ℓ ′ }, if u ∈ B s0+max{m,ℓ} ∩ V s+ℓ and k ∈ W s+ℓ ′ , then L ε (u) k ∈ V s , with the tame inverse estimate (2.6) L ε (u) k s ≤ bε -g k ′ s+ℓ ′ + k ′ s0+ℓ ′ u s+ℓ .
• Alternatively, we shall say that (DF ε ) 0<ε≤1 is tame Galerkin right-invertible if there are Λ ≥ 1 , b > 0 and g, ℓ, ℓ ′ ≥ 0 such that for all Λ ≥ Λ , 0 < ε ≤ 1 and any u ∈ B s0+max{m,ℓ} ∩ E(Λ), there is a linear map L Λ,ε (u) :

E ′ (Λ) → E(Λ) satisfying (2.7) ∀k ∈ E ′ (Λ) , Π ′ (Λ)DF ε (u) L Λ,ε (u) k = k
and for all s 0 ≤ s ≤ S -max {ℓ, ℓ ′ }, we have the tame inverse estimate:

(2.8) ∀k ∈ E ′ (Λ) , L Λ,ε (u) k s ≤ bε -g k ′ s+ℓ ′ + k ′ s0+ℓ ′ u s+ℓ .
In this definition, the integers m, ℓ, ℓ ′ denote the loss of derivatives for DF ε and its right-inverse, and g > 0 denotes the strength of the singularity at ε = 0. The unperturbed case of a fixed map can be recovered by setting ε = 1.

We want to solve the equation F ε (u) = v. There are three things to look for. How regular is v ? How regular is u, or, equivalently, how small is the loss of derivatives between v and u ? How does the existence domain depend on ε ? The following result answers them in a near-optimal way. Theorem 3. Assume that the maps F ε (0 < ε ≤ 1) form an S-tame differentiable family between the tame scales (V s ) 0≤s≤S and (W s ) 0≤s≤S , with F ε (0) = 0 for all 0 < ε ≤ 1. Assume, in addition, that (DF ε ) 0<ε≤1 is either tame right-invertible or tame Galerkin right-invertible. Let s 0 , m, g, ℓ, ℓ ′ be the associated parameters.

Let

s 1 ≥ s 0 + max{m, ℓ}, δ > s 1 + ℓ ′ and g ′ > g.
Then, for S large enough, there is r > 0 such that, whenever 0 < ε ≤ 1 and v ′ δ ≤ rε g ′ , there exists some u ε ∈ B s1 satisfying:

F ε (u ε ) = v u ε s1 ≤ r -1 ε -g ′ v ′ δ
As we will see, the proof of Theorem 3 is much shorter under the assumptions that DF ε is Galerkin right-invertible. But in many applications, it is easier to check that DF ε is tame right-invertible than tame Galerkin right-invertible. See [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], however, where an assumption similar to (2.7, 2.8) is used.

All other "hard" surjection theorems that we know of require some additional conditions on the second derivative of F ε . Here we do not need such assumptions, in fact we only assume

F ε to be G-differentiable, not C 2 .
As for the three questions we raised, let us explain in which sense the answers are almost optimal in Theorem 3. For the tame estimates (2.4),(2.6) to hold, one needs u ∈ B s1 with s 1 ≥ s 0 + max{m, ℓ}. When solving the linearized equation

DF ε (u) h = k in V s1 by h = L ε (u) k, one needs k ∈ W s1+ℓ ′ ,
so it seems necessary to assume δ ≥ s 1 + ℓ ′ , and we find that the strict inequality is sufficient. Replacing s 1 with its minimal value, our condition on δ becomes

δ > s 0 + max{m, ℓ} + ℓ ′ .
We have not found this condition in the literature: in [START_REF] Hörmander | The Boundary Problems of Physical Geodesy[END_REF] for instance, a stronger assumption is made, namely δ > s 0 + max{2m + ℓ ′ , ℓ} + ℓ ′ .

For the dependence of v ′ δ on ε, the constraint g ′ > g also seems to be nearly optimal. Indeed, the solution u ε has to be in B s1 , but the right-inverse L ε of DF ε has a norm of order ε -g , so the condition v ′ δ ε g seems necessary. We find that the condition v ′ δ ε g ′ is sufficient. Our condition on S is of the form S ≥ S 0 where S 0 depends only on the parameters s 0 , m, g, ℓ, ℓ ′ and g ′ , s 1 , δ. Then r depends only on these parameters and the constants A i , A ′ i associated with the tame scales. In principle, all these constants could be made explicit, but we will not do it here. Let us just mention that one can take

S 0 = O 1 g ′ -g
as g ′ → g, all other parameters remaining fixed. This follows from the inequality σ < ζg/η in Lemma 1.

In the case of a tame right-invertible differential, we can restate our theorem in a form that allows direct comparison with [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]: Theorem 2.19 and Remarks 2.9, 2.14. For this purpose, we consider two tame Banach scales (V s , • s ) and (W s , • ′ s ) with associated projectors Π Λ , Π ′ Λ , we take γ > 0 and we introduce the norms

| • | s := ε γ • s and | • | ′ s := ε γ • ′ s . We then denote B s (ρ) = {u | |u| s ≤ ρ} and we consider functions F ε of the form F ε (u) = Φ ε (a ε + u) -Φ ε (a ε ) , where Φ ε is defined on B s0+m (2ε γ ) and a ε ∈ B S (ε γ ) is chosen such that v ε := -Φ ε (a ε ) is very small. A point u in B s0+m satisfies F ε (u) = v ε if and only if it solves the equation Φ ε (a ε + u) = 0 in B s0+m (ε γ ) .
We make the following assumptions on Φ ε :

For some γ > 0 and any 0 < ε ≤ 1, the map Φ ε : B s0+m (2ε γ ) → W s0 is G-differentiable with respect to u, and there are constants a, b and g > 0 such that:

• for all 0 < ε ≤ 1 and s 0 ≤ s ≤ S -m , if u ∈ B s0+m (2ε γ ) ∩ V s+m and h ∈ V s+m , then DΦ ε (u) h ∈ W s , with the tame direct estimate (2.9) |DΦ ε (u) h| ′ s ≤ a |h| s+m + ε -γ |u| s+m |h| s0+m • for all 0 < ε ≤ 1 and u ∈ B s0+max{m,ℓ} (2ε γ ) , there is L ε (u) : W s0+ℓ ′ →
V s0 linear, satisfying:

(2.10) ∀k ∈ W s0+ℓ ′ , DΦ ε (u) L ε (u) k = k and for all s 0 ≤ s ≤ S -max {ℓ, ℓ ′ } , if u ∈ B s0+max{m,ℓ} (2ε γ ) ∩ V s+ℓ and k ∈ W s+ℓ ′ , then L ε (u) k ∈ V s , with the tame inverse estimate (2.11) |L ε (u) k| s ≤ bε -g |k| ′ s+ℓ ′ + ε -γ |k| ′ s0+ℓ ′ |u| s+ℓ
Under these assumptions, the maps F ε : B s0+m → W s0 form an S-tame differentiable family for the "old" norms • s , • ′ s . So the following result holds, as a direct consequence of our main theorem:

Corollary 4. Consider two tame Banach scales (V s , |•| s ) 0≤s≤S and (W s , |•| ′ s ) 0≤s≤S , nonnegative constants s 0 , m, ℓ, ℓ ′ ,
g, γ, and two positive constants a, b. Take any g ′ > g, s 1 ≥ s 0 + max{m, ℓ} and δ > s 1 + ℓ ′ . For S large enough and r > 0 small, if a family of G-differentiable maps Φ ε : B s0+m (2ε γ ) → W s0 (0 < ε ≤ 1) satisfies (2.9,2.10,2.11), and, in addition, for some

a ε ∈ B S (ε γ ) , |Φ ε (a ε )| ′ δ ≤ rε γ+g ′ , then there exists some u ε ∈ B s1,ε (ε γ ) such that: Φ ε (a ε + u ε ) = 0 |u ε | s1 ≤ r -1 ε -g ′ |Φ ε (a ε )| ′ δ
In [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] (Theorem 2.19 and Remarks 2.9, 2.14), the assumptions are stronger, since they involve the second derivative of Φ ε . More importantly, we only need the norm of Φ ε (a ε ) to be controlled by ε γ+g ′ with g ′ > g, provided S ≥ S 0 with [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] (Assumption 2.15 and Remark 2.23), due to quadratic estimates, one needs g ′ > 2g with the faster growth

S 0 = O 1 g ′ -g , while in
S 0 = O 1 (g ′ -2g) 2 .

Proof of Theorem 3

The proof consists in constructing a sequence (u n ) n≥1 which converges to a solution u of F (u) = v. At each step, in order to find u n , we solve a nonlinear equation in a Banach space, using Theorem 2 in [START_REF] Ekeland | An inverse function theorem in Fréchet spaces[END_REF], which we restate below for the reader's convenience (the notation ||| L ||| stands for the operator norm of any linear continuous map L between two Banach spaces): Theorem 5. Let X and Y be Banach spaces. Let f : B X (0, R) → Y be continuous and Gâteaux-differentiable, with f (0) = 0. Assume that the derivative Df (u) has a right-inverse L (u), uniformly bounded on the ball B X (0, R):

∀(u, k) ∈ B X (0, R) × Y , Df (u) L (u) k = k sup { ||| L (u) ||| : u X < R} < M .
Then, for every v ∈ Y with v Y < RM -1 there is some u ∈ X satisfying:

f (u) = v and u X ≤ M v Y < R .
Note first that this is a local surjection theorem, not an inverse function theorem: with respect to the IFT, we lose uniqueness. On the other hand, the regularity requirement on f and the smallness condition on v are much weaker. As mentioned in the Introduction, for a C 1 functional in finite dimensions, this theorem has been proved a long time ago by Wazewski [START_REF] Wazewski | Sur l'évaluation du domaine d'existence des fonctions implicites réelles ou complexes[END_REF] by a continuation argument (we thank Sotomayor for drawing our attention to this result). For a comparison of the existence and uniqueness domains in the C 2 case with dim X = dim Y , see [START_REF] Hartman | Ordinary differential equations[END_REF], chapter II, exercise 2.3.

It turns out that the proof of Theorem 3 is much easier if one assumes that the family (DF ε ) is tame Galerkin right-invertible. But most applications require that (DF ε ) be tame right-invertible. Let us explain why the proof is longer in this case. In our algorithm, we will use two sequences of projectors

Π n := Π(Λ n ) and Π ′ n := Π ′ (M n ) with associated ranges E n = E(Λ n ) and E ′ n = E ′ (M n ), where Λ 0 ≈ ε -η for some small η > 0, Λ n = Λ α n 0 for some α > 1 close to 1, and M n = Λ ϑ n for some ϑ ≤ 1 such that ϑα > 1.
The algorithm consists in finding, by induction on n and using Theorem 5 at each step, a solution

u n ∈ E n of the problem Π ′ n F ε (u n ) = Π ′ n-1 v.
For this, we need Π ′ n DF ε (u) |E n to be invertible for u in a certain ball B n , with estimates on the right inverse for a certain norm • Nn .

When the family (DF ε ) is tame Galerkin right-invertible, we can take ϑ = 1 so that M n = Λ n , instead of assuming ϑ < 1. Then the right-invertibility of Π ′ n DF ε (u) |E n follows immediately from the definition. But when (DF ε ) is only tame right-invertible, it is crucial to take ϑ < 1. The intuitive idea is the following. One can think of DF ε (u) as very large right-invertible matrix. The topological argument we use requires Π ′ n DF ε (u) |E n to have a rightinverse for u in a suitable ball. If we take M n = Λ n , this is like asking that a square submatrix of a right-invertible matrix be invertible. In general this is not true. But a rectangular submatrix, with more columns than lines, will be right-invertible if the full matrix is and if there are enough columns in the submatrix. This is why we impose M n < Λ n when we do not assume the tame Galerkin right-invertibility.

In the sequel, we assume that the family (DF ε ) is tame right-invertible, so we take ϑ < 1, and we point out the specific places where the arguments would be easier assuming, instead, that (DF ε ) is tame Galerkin right-invertible.

The sequence u n depends on a number of parameters η, α, β, ϑ and σ satisfying various conditions: in the first subsection we prove that these conditions are compatible. In the next one, we construct an initial point u 1 depending on η, α and ϑ. In the third one we construct, by induction, the remaining points u n which also depend on β and σ. Finally we prove that the sequence (u n ) converges to a solution u of the problem, satisfying the desired estimates.

3.1.

Choosing the values of the parameters. We are given s 1 ≥ s 0 +max {m, ℓ} , δ > s 1 + ℓ ′ and g ′ > g. These are fixed throughout the proof.

We introduce positive parameters η, α, β, ϑ and σ satisfying the following conditions:

η < g ′ -g max {ϑℓ ′ , ℓ} (3.1) 1 α < ϑ < 1 (3.2) (1 -ϑ) (σ -δ) > ϑm + max {ℓ, ϑℓ ′ } + g η (3.3) σ > αβ + s 1 (3.4) (1 + α -ϑα) (σ -s 0 ) > αβ + α (m + ℓ) + ℓ ′ + g η (3.5) (1 -ϑ) (σ -s 0 ) > m + ϑℓ ′ + g αη (3.6) δ > s 0 + α ϑ (σ -s 0 -αβ + ℓ") (3.7) (α -1) β > (1 -ϑ) (σ -s 0 ) + ϑm + ℓ" + g η (3.8) ℓ" = max {(α -1) ℓ + ℓ ′ , αϑℓ ′ } (3.9)
Note that condition (3.3) implies that δ < σ . Note also that condition (3.7) may be rewritten as

β > 1 α (σ -δ) + 1 - ϑ α δ -s 0 α + ℓ" α
which implies the simpler inequality

(3.10) β > 1 α (σ -δ)
Inequality (3.10) will also be used in the proof.

If we assume tame Galerkin right-invertibility instead of tame right-invertibility, we can replace condition (3.3) by the weaker condition δ < σ, we do not need conditions (3.5), (3.6) any more, and we can take ϑ = 1 instead of ϑ < 1.

Lemma 1. The set of parameters (η, α, β, ϑ, σ) satisfying the above conditions is non-empty. More precisely, there are some α > 1 and ζ > 0 depending only on (s 0 , m, ℓ, ℓ ′ , s 1 , δ), such that, for ϑ = α -1/2 and for every 0 < η < 1, there exist (β, σ) with σ < ζg/η such that the constraints (3.3) to (3.9) are satisfied.

Proof. Since δ > s 1 + ℓ ′ , and ℓ" → ℓ ′ when both α and ϑ tend to 1, it is possible to choose ϑ and α = ϑ -2 close enough to 1 so that δ > s 0 + α ϑ (s 1 -s 0 + ℓ"). Take some τ with 0 < τ < ϑ α (δ -s 0 ) -s 1 + s 0 -ℓ", and set:

(3.11) β = σ α - s 1 + τ α
Then conditions (3.2), (3.4) and (3.7) are satisfied.

The remaining inequalities are constraints on β and σ. They can be rewritten as follows:

σ > δ + 1 1 -ϑ ϑm + max {ℓ, ϑℓ ′ } + g η (3.12) β < 1 α + 1 -ϑ σ -m -ℓ - ℓ ′ α - 1 α + 1 -ϑ s 0 - g αη (3.13) σ > s 0 + 1 1 -ϑ m + ϑℓ ′ + g αη (3.14) β > 1 -ϑ α -1 σ + 1 α -1 ϑm + ℓ" + g η -(1 -ϑ) s 0 (3.15)
These inequalities define half-planes in the (σ, β)-plane. Since αϑ > 1, the slopes in (3.11), (3.13) and (3.15) are ordered as follows:

0 < 1 -ϑ α -1 < 1 α < 1 α + 1 -ϑ < 1
As a consequence, for the chosen values of α, ϑ and τ , the domain defined by these three conditions in the (σ, β)-plane is an infinite half-line stretching to the North-East. The remaining two, (3.12) and (3.14), just tell us that σ should be large enough. So the set of solutions is of the form σ > σ, β = σ α -s1+τ α and σ is clearly a piecewise affine function of g/η. We may thus choose σ < ζg/η for some constant ζ .

Remark. As already mentioned, if we assume that (DF ε ) is tame Galerkin right-invertible, (3.3) can be replaced by the condition δ < σ, and (3.5) and (3.6) are not needed. The remaining conditions can be satisfied by taking ϑ = 1 and for a larger set of the other parameters. The corresponding variant of Lemma 1 has a simpler proof. We can choose α > 1 such that δ > s 0 + α (s 1 -s 0 + ℓ") and τ such that 0 < τ < 1 α (δ -s 0 ) -s 1 + s 0 -ℓ", and we may impose condition (3.11). Then conditions (3.12), (3.13) and (3.14) are no longer required, and the last conditions δ < σ and (3.15) are easily satisfied by taking σ large enough.

The values (η, α, β, ϑ, σ) are now fixed. For the remainder of the proof we introduce an important notation. By

x y

we mean that there is some constant C such that x ≤ Cy. This constant depends on A i , A ′ i , a, b, s 0 , m, ℓ, ℓ ′ , g, g ′ , s 1 , δ and our additional parameters (η, α, β, ϑ, σ), but NOT on ε, nor on the regularity index s ∈ [0, S] or the rank n in any of the sequences which will be introduced in the sequel. For instance, the tame inequalities become:

DF ε (u) h s u s+m h s0+m + h s+m L ε (u) k s ε -g u s+l k s0+l ′ + k s+l ′
In the iteration process, we will need the following result:

Lemma 2. If the maps F ε form an S-tame differentiable family and F ε (0) = 0, then, for u ∈ B s0+m ∩ V s+m and s 0 ≤ s ≤ S -m, we have:

F ε (u) s u s+m
Proof. Consider the function ϕ (t) = F ε (tu) s . Since F ε is G-differentiable, we have:

ϕ ′ (t) = DF ε (tu) u, F ε (tu) F ε (tu) s s
≤ a t u s0+m u s+m + u s+m and since ϕ (0) = 0, we get the result.

3.2. Initialization.

3.2.1.

Defining appropriate norms. This subsection uses condition (3.2) and the inequalities s 1 + ℓ ′ < δ < σ , which, as already noted, follows from (3.3).

We are given (η, α, ϑ, δ, σ). We fix a large constant K > 1, to be chosen later independently of 0 < ε ≤ 1.

We set Λ 0 = (Kε -η ) 1/α , Λ 1 := (Λ 0 ) α = Kε -η , M 0 := (Λ 0 ) ϑ = (Kε -η ) ϑ/α and M 1 := (Λ 1 ) ϑ = (Kε -η ) ϑ . We then have the inequalities M 0 < Λ 0 < M 1 < Λ 1 .
Let

E 1 := E(Λ 1 ) , Π 1 := Π(Λ 1 ) , E ′ 1 = E(M 1
) and Π ′ i := Π ′ (M i ) for i = 0, 1 . We choose the following norms on E 1 , E ′ 1 :

h N1 : = h δ + Λ -ϑ α (σ-δ) 1 h σ k ′ N1 := k ′ δ + Λ -ϑ α (σ-δ) 1 k ′ σ
Endowed with these norms, E 1 and E ′ 1 are Banach spaces. We shall use the notation ||| L ||| N 1 for the operator norm of any linear continuous map L from the Banach space E ′ 1 to a Banach space that can be either

E 1 or E ′ 1 . The map F ε induces a map f 1 : B s0+m ∩ E 1 → E ′ 1 defined by f 1 (u) := Π ′ 1 F ε (u) for u ∈ B s0+m ∩ E 1 .
Note that f 1 (0) = 0. We will use the local surjection theorem to show that the range of f 1 covers a neighbourhood of 0 in E ′ 1 . We begin by showing that Df 1 has a right inverse.

Note that, if we assume that DF is tame Galerkin right-invertible, we can take M 1 = Λ 1 ≥ Λ, and Df 1 is automatically right-invertible, with the tame estimate (2.8). So the next subsection is only necessary if we assume that DF is tame right-invertible.

3.2.2.

Df 1 (u) has a right inverse for u N1 ≤ 1 . This subsection uses condition (3.3). We recall it here for the reader's convenience:

(1 -ϑ) (σ -δ) > ϑm + max {ℓ, ϑℓ ′ } + g η Lemma 3
. For K large enough and for all u ∈ E 1 with u N1 ≤ 1:

||| Π ′ 1 DF ε (u) (1 -Π 1 ) L ε (u) ||| N 1 ≤ 1 2
Proof. From u N1 ≤ 1, it follows that u δ ≤ 1, and since δ > s 0 +max {ℓ, m}+ℓ ′ , the tame estimates hold at u. Take any k ∈ E ′ 1 and set h = (1 -Π 1 ) L ε (u) k.

We have h δ Λ δ-σ 1 L ε (u) k σ , and:

Π ′ 1 DF ε (u) h ′ δ-m h s0+m u δ + h δ h δ Π ′ 1 DF ε (u) h ′ δ M m 1 Π ′ 1 DF ε (u) h δ-m M m 1 h δ .
Hence:

Π ′ 1 DF ε (u) h ′ δ M m 1 Λ δ-σ 1 L ε (u) k σ . Writing Π ′ 1 DF ε (u) h ′ σ M σ-δ 1 Π ′ 1 DF ε (u) h δ we finally get: (3.16) Π ′ 1 DF ε (u) h ′ N1 M m 1 Λ δ-σ 1 1 + Λ -ϑ α (σ-δ) 1 M σ-δ 1 L ε (u) k σ .
We now have to estimate L ε (u) k σ . By the tame estimates, we have:

L ε (u) k σ ε -g k ′ σ+ℓ ′ + u σ+ℓ k ′ s0+ℓ ′ ε -g M ℓ ′ 1 k ′ σ + Λ ℓ 1 u σ k ′ δ Since u N1 ≤ 1, we have u σ ≤ Λ ϑ α (σ-δ) 1
. Substituting, we get:

L ε (u) k σ ε -g M ℓ ′ 1 k ′ σ + Λ ϑ α (σ-δ)+ℓ 1 k ′ δ ε -g Λ ϑ α (σ-δ) 1 M ℓ ′ 1 + Λ ℓ 1 k ′ N1 (3.17)
Putting (3.16) and (3.17) together, we get:

Π ′ 1 DF ε (u) h ′ N1 ε -g M m 1 Λ δ-σ 1 Λ ϑ α (σ-δ) 1 + M σ-δ 1 M ℓ ′ 1 + Λ ℓ 1 k ′ N1 Since α > 1, we have Λ ϑ α (σ-δ) 1 ≤ Λ ϑ(σ-δ) 1 = M σ-δ 1
, so that:

Π ′ 1 DF ε (u) h ′ N1 ε -g M m 1 Λ δ-σ 1 M σ-δ 1 M ℓ ′ 1 + Λ ℓ 1 k ′ N1 ε -g Λ ϑm-(1-ϑ)(σ-δ)+max{ℓ,ϑℓ ′ } 1 k ′ N1
Since Λ 1 = Kε -η , the inequality becomes:

Π ′ 1 DF ε (u) h ′ N1 K -C0 ε -g+ηC0 k ′ N1 with C 0 := (1 -ϑ) (σ -δ) -ϑm -max{ℓ, ϑℓ ′ }.
By condition (3.3), the exponent C 0 is larger than g/η, and the proof follows by choosing K large enough independently of 0 < ε ≤ 1.

Introduce the map L 1 (u) = Π 1 L ε (u) | E ′ 1 . Since DF ε (u) L ε (u) = 1, it follows from Lemma 3 that, for k ∈ E ′ 1 , u ∈ E 1 and u N1 ≤ 1 , we have: k -Df 1 (u) L 1 (u) k ′ N1 ≤ 1 2 k ′ N1
This implies that the Neumann series

i≥0 I E ′ 1 -Df 1 (u) L 1 (u) i converges in operator norm. Its sum is S 1 (u) = (Df 1 (u)L 1 (u)) -1
and it has operator norm at most 2.

Then T 1 (u) := L 1 (u)S 1 (u) is a right inverse of Df 1 (u) and ||| T 1 (u) ||| N 1 ≤ 2 ||| L 1 (u) ||| N 1 . By the tame estimates, if u ∈ E 1 , u N1 ≤ 1 and k ∈ E ′
1 , we have:

L 1 (u) k δ L ε (u) k δ ε -g k ′ δ+ℓ ′ + u δ+ℓ k ′ s0+ℓ ′ ε -g M ℓ ′ 1 + Λ ℓ 1 k ′ δ
Combining with (3.17), we find:

sup u N 1 ≤1 ||| T 1 (u) ||| N 1 ε -g M ℓ ′ 1 + Λ ℓ 1 = m 1 3.2.3. Local inversion of f 1 .
Applying Theorem 5, we find that if

Π ′ 0 v ′ N1 < 1/m 1 , then equation f 1 (u) = Π ′ 0 v has a solution u 1 ∈ E 1 with u 1 N1 ≤ 1 and u 1 N1 ≤ m 1 Π ′ 0 v ′ N1 . Note that Π ′ 0 v ′ σ M σ-δ 0 Π ′ 0 v ′ δ Λ ϑ α (σ-δ) 1 Π ′ 0 v ′ δ . It follows that Π ′ 0 v ′ N1 = Π ′ 0 v ′ δ + Λ -ϑ α (σ-δ) 1 Π ′ 0 v ′ σ Π ′ 0 v δ
Assume from now on:

(3.18) v ′ δ ε g M ℓ ′ 1 + Λ ℓ 1 -1 Then Π ′ 0 v ′ N1
m -1 1 , and Theorem 5 applies. The estimate on u 1 implies:

(3.19) u 1 δ ε -g M ℓ ′ 1 + Λ ℓ 1 v ′ δ ≤ 1
It also implies an estimate in higher norm:

(3.20)

u 1 σ ε -g Λ ϑ α (σ-δ) 1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ Λ ϑ α (σ-δ) 1
.

3.3. Induction.

3.3.1.

Finding uniform bounds. In addition to (α, ϑ, δ, ε, η) we are given β satisfying relations (3.4) and (3.10) . We recall them here for the reader's convenience. With

s 1 ≥ s 0 + max {m, ℓ} and δ > s 1 + ℓ ′ , σ > αβ + s 1 β > 1 α (σ -δ)
We also inherit Λ 1 = Kε -η and u 1 from the preceding section. Combining (3.10) and (3.20), we immediately obtain the estimate

(3.21) u 1 σ ε -g Λ β 1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ Λ β 1 .
Consider the sequences of integers M n and Λ n , n ≥ 1, defined by

Λ n := Λ α n-1 1 and M n := Λ ϑ n . Let Π n := Π(Λ n ) , Π ′ n := Π ′ (M n ) , E n := E(Λ n ) , E ′ n := E ′ (M n ) .
We will construct a sequence u n ∈ E n , n ≥ 1, starting from the initial point u 1 we found in the preceding section. For all n ≥ 2 the remaining points should satisfy the following conditions:

Π ′ n F ε (u n ) = Π ′ n-1 v (3.22) u n -u n-1 s0 ≤ ε -g Λ αβ-σ+s0 n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ (3.23) u n -u n-1 σ ≤ ε -g Λ αβ n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ (3.24)
We proceed by induction. Suppose we have found u 2 , ..., u n-1 satisfying these conditions. We want to construct u n . Lemma 4. Let us impose K ≥ 2. For all t with s 0 ≤ t < σ -αβ, and all i with 2 ≤ i ≤ n -1, we have:

n-1 i=2 u i -u i-1 t ≤ ε -g M ℓ ′ 1 + Λ ℓ 1 Σ (t) v ′ δ
where Σ (t) is finite and independent of n , ε.

Proof. By the interpolation formula,

u i -u i-1 t ≤ ε -g Λ αβ-σ+t i-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ for all 2 ≤ i ≤ n . Since Λ 1 = Kε -η ≥ 2, we have: n-1 i=2 u i -u i-1 t ≤ ε -g M ℓ ′ 1 + Λ ℓ 1 ∞ i=2 Λ αβ-σ+t i-1 v ′ δ ≤ ε -g M ℓ ′ 1 + Λ ℓ 1 ∞ j=0 2 α j(αβ-σ+t) v ′ δ
By (3.4) we can take t = s 1 , and we find a uniform bound for u n-1 in the s 1 -norm, namely:

u n-1 s1 ≤ u 1 δ + n-1 i=2 u i -u i-1 s1 ε -g M ℓ ′ 1 + Λ ℓ 1 (1 + Σ(s 1 )) v ′ δ ε -g M ℓ ′ 1 + Λ ℓ 1 v ′ δ
In particular, we will have

u n-1 s1 ≤ 1 if v ′ δ ε g M ℓ ′ 1 + Λ ℓ 1 -1
, so the tame estimates hold at u n-1 .

Similarly, if v ′ δ ε g M ℓ ′ 1 + Λ ℓ 1 -1
we find a uniform bound in the σ-norm. We have:

u n-1 σ ≤ u 1 σ + n-1 i=2 u i -u i-1 σ and n-1 i=2 u i -u i-1 σ ε -g M ℓ ′ 1 + Λ ℓ 1 n-1 i=1 Λ β i v ′ δ ε -g M ℓ ′ 1 + Λ ℓ 1 Λ β n-1 v ′ δ
so, combining this with (3.21), we get:

(3.25) u n-1 σ ε -g Λ β n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ Λ β n-1 .

3.3.2.

Setting up the induction step.

Suppose, as above, that v ′ δ ε g M ℓ ′ 1 + Λ ℓ 1 -1
and that u 2 , ..., u n-1 have been found. We have seen that u n-1 s1 ≤ 1, so that the tame estimates hold at u n-1 , and we also have u n-1 σ Λ β n-1 . We want to find u n satisfying (3.22), (3.23) and (3.24). Since

Π ′ n-1 F ε (u n ) = Π ′ n-2 v
, we rewrite the latter equation as follows:

(3.26) Π ′ n (F ε (u n ) -F ε (u n-1 )) + Π ′ n -Π ′ n-1 F ε (u n-1 ) = Π ′ n-1 -Π ′ n-2 v Define a map f n : E n → E ′ n with f n (0) = 0 by: f n (z) = Π ′ n (F ε (u n-1 + z) -F ε (u n-1
)) Equation (3.26) can be rewritten as follows:

f n (z) = ∆ n v + e n (3.27) ∆ n v = Π ′ n-1 1 -Π ′ n-2 v (3.28) e n = -Π ′ n 1 -Π ′ n-1 F ε (u n-1 ) (3.29)
We choose the following norms on E n and E ′ n :

x Nn = x s0 + Λ -σ+s0 n-1 x σ y ′ Nn = y ′ s0 + Λ -σ+s0 n-1 y ′ σ
Endowed with these norms, E n and E ′ n are Banach spaces. We shall use the notation ||| L ||| Nn for the operator norm of any linear continuous map L from the Banach space E ′ n to a Banach space that can be either E n or E ′ n .

Lemma 5. If 0 ≤ t ≤ σ -s 0 , then:

x s0+t Λ t n-1 x Nn y ′ s0+t Λ t n-1 y ′ Nn
Proof. Use the interpolation inequality.

We will solve the system (3.27), (3.28), (3.29) by applying the local surjection theorem to f n on the ball B Nn (0, r n ) ⊂ E n where:

(3.30) r n = ε -g Λ αβ-σ+s0 n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ
Note that if the solution z belongs to B Nn (0, r n ), then

z s0 ≤ ε -g Λ αβ-σ+s0 n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ and z σ ≤ ε -g Λ αβ n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ .
In other words, u n = u n-1 + z satisfies (3.23) and (3.24), so that the induction step is proved.

We begin by showing that Df n (z) has a right inverse.

Note that, if we assume that DF ε is tame Galerkin right-invertible, we can take M n = Λ n , and the result of the next subsection is obvious. This subsection is only useful if we assume that DF is tame right-invertible but not tame Galerkin right-invertible.

3.3.3.

Df n (z) has a right inverse for z Nn ≤ r n . In this subsection, we use conditions (3.5) and (3.6). We recall them for the reader's convenience:

(1 + α -ϑα) (σ -s 0 ) > αβ + α (m + ℓ) + ℓ ′ + g η (1 -ϑ) (σ -s 0 ) > m + ϑℓ ′ + g αη Take now any z ∈ B Nn (0, r n ).
Arguing as above, we find that if , then:

u n-1 + z s1 ≤ 1 (3.31) u n-1 + z σ Λ β n (3.32)
By (3.31) the tame estimates hold on z ∈ B Nn (0, r n ). Lemma 6. Take Λ 1 = Kε -η with K > 1 chosen large enough, independently of n and ε ∈ (0, 1]. Then, for all z ∈ B Nn (0, r n ):

||| Π ′ n DF ε (u n-1 + z) (1 -Π n ) L ε (u n-1 + z) ||| Nn ≤ 1 2 Proof. We proceed as in the proof of Lemma 3. For k ∈ E ′ n , we set h = (1 -Π n ) L ε (u n-1 + z) k .
We have: (3.32) and the tame estimates for L ε , we get:

h s0+m Λ -σ+s0+m n L ε (u n-1 + z) k σ By
L ε (u n-1 + z) k σ ε -g u n-1 + z σ+ℓ k ′ s0+ℓ ′ + k ′ σ+ℓ ′ ε -g Λ β+ℓ n Λ ℓ ′ n-1 + M l ′ n Λ σ-s0 n-1 k ′ Nn (3.33)
where we have used Lemma 5. Substituting in the preceding formula, we get:

h s0+m ε -g Λ β+ℓ-σ+s0+m n Λ l ′ n-1 + M ℓ ′ n Λ -(α-1)(σ-s0)+αm n-1 k ′ Nn
By the tame estimate (2.4), we have:

Π ′ n DF ε (u n-1 + z) h ′ s0 h s0+m
From this it follows that:

Π ′ n DF ε (u n-1 + z) h ′ σ M σ-s0 n h s0+m
Hence:

Π ′ n DF ε (u n-1 + z) h ′ Nn 1 + Λ -σ+s0 n-1 M σ-s0 n h s0+m
We have

Λ -σ+s0 n-1 M σ-s0 n Λ (αϑ-1)(σ-s0) n-1
. Since αϑ > 1, the dominant term in the parenthesis is the second one, and:

Π ′ n DF ε (u n-1 + z) h Nn Λ -σ+s0 n-1 M σ-s0 n h s0+m ε -g M σ-s0 n Λ α(β+ℓ-σ+s0+m)+ℓ ′ -σ+s0 n-1 + M ℓ ′ n Λ -α(σ-s0)+αm n-1 k ′ Nn
From (3.5) and (3.6), it follows that the right-hand side is a decreasing function of n. To check that it is less than 1/2 for all n ≥ 2, it is enough to check it for n = 2. Since Λ 1 = Kε -η , substituting in the right-hand side, we get:

Π ′ n DF ε (u n-1 + z) h Nn K -min{C1,C2} α n-2 ε min{C1,C2}-α 2-n g/η ηα n-2 k ′ Nn with C 1 = -α(β + ℓ + m) -ℓ ′ + (1 + α -αϑ) (σ -s 0 ) C 2 = α ((1 -ϑ)(σ -s 0 ) -ϑℓ ′ -m)
By (3.5) and (3.6), both exponents C 1 and C 2 are larger than g/η. As a consequence,

Π ′ n DF ε (u n-1 + z) h Nn ≤ 1 2 k ′ Nn for K chosen large enough, indepen- dently of n and 0 < ε ≤ 1. Define L n (z) = Π n L ε (u n-1 + z) | E ′ n . Arguing as in subsection 3.2.2, we find that the Neumann series i≥0 I E ′ n -Df n (u) L n (u) i converges in operator norm. Its sum is S n (u) = (Df n (u)L n (u)) -1
and it has operator norm at most 2. Then

T n (u) := L n (u)S n (u) is a right inverse of Df n (u) , with the estimate ||| T n (u) ||| Nn ≤ 2 ||| L n (u) ||| Nn .
We have already derived estimate (3.33) which immediately implies:

L n (z) k σ ε -g Λ σ-s0 n-1 Λ β+ℓ n Λ -σ+s0+ℓ ′ n-1 + M ℓ ′ n k ′ Nn
From the tame estimates and Lemma 5, we also have:

L n (z) k s0 ε -g k ′ s0+ℓ ′ ε -g Λ ℓ ′ n-1 k ′ Nn Since αϑ > 1, we have Λ ℓ ′ n-1 M ℓ ′ n .
So the two preceding estimates can be combined, and we get the final estimate for the right inverse in operator norm:

(3.34) ||| T n (z) ||| Nn ε -g Λ β+ℓ n Λ -σ+s0+ℓ ′ n-1 + M ℓ ′ n 3.3.4. Finding u n .
In this subsection, we use relations (3.4),(3.7), (3.8) and (3.9). We recall them for the reader's convenience:

σ > αβ + s 1 δ > s 0 + α ϑ (σ -s 0 -αβ + ℓ") (α -1) β > (1 -ϑ) (σ -s 0 ) + ϑm + ℓ" + g η ℓ" = max {(α -1) ℓ + ℓ ′ , αϑℓ ′ }
Let us go back to (3.27). By Theorem 5 to solve

Π ′ n f n (z) = ∆ n v + e n with z ∈ B Nn (0, r n ) it is enough that: (3.35) ||| T n (z) ||| Nn ∆ n v Nn + e n Nn ≤ r n
Here r n is given by (3.30). We can estimate ||| T n (z) ||| Nn using (3.34). We need to estimate ∆ n v Nn and e n Nn .

From (3.28) we have:

∆v ′ s0 M s0-δ n-2 v ′ δ ∆v ′ σ M σ-δ n-1 v ′ δ ∆v ′ Nn max M s0-δ n-2 , Λ -σ+s0 n-1 M σ-δ n-1 v ′ δ
An easy calculation yields:

σ -s 0 -ϑ(σ -δ) + ϑ α (s 0 -δ) = (1 -ϑ)(σ -δ) + (1 - ϑ α )(δ -s 0 )
Since s 0 < δ < σ and ϑ < 1 < α, the two terms on the right-hand side are positive, so

Λ -σ+s0 n-1 M σ-δ n-1 M s0-δ n-2 . It follows that: (3.36) ∆v ′ Nn M s0-δ n-2 v ′ δ
From (3.29), we derive:

e n ′ s0 M -σ+m+s0 n-1 e n ′ σ-m
By Lemma 2, F ε (u n-1 ) σ-m u n-1 σ . So, remembering (3.29) and (3.25), we get:

e n ′ s0 M -σ+m+s0 n-1 u n-1 σ ε -g M -σ+m+s0 n-1 Λ β n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ
Similarly,

e n ′ σ u n-1 σ+m Λ m n-1 u n-1 σ ε -g Λ β+m n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ
Finally, since M n-1 < Λ n-1 and σ > m + s 0 , we get:

(3.37) e n ′ Nn ε -g Λ β n-1 M -σ+m+s0 n-1 M ℓ ′ 1 + Λ ℓ 1 v ′ δ Substituting (3.
+ M ℓ ′ n M s0-δ n-2 + ε -g Λ β n-1 M -σ+m+s0 n-1 Λ αβ-σ+s0 n-1
We estimate both sides separately. Remembering that M n-i = (Λ n-1 ) α 1-i ϑ and

Λ n-1 = (Kε -η ) α n-2
, we find

Λ β+ℓ n Λ -σ+s0+ℓ ′ n-1 + M ℓ ′ n M s0-δ n-2 + ε -g Λ β n-1 M -σ+m+s0 n-1 ε -ηα n-2 max{C3,C4}+max{C5,C6}
and

Λ αβ-σ+s0 n-1 ε -ηα n-2 C7
with

C 3 :=α(β + ℓ) -σ + s 0 + ℓ ′ C 4 :=αϑℓ ′ C 5 :=ϑα -1 (s 0 -δ) C 6 :=g/η + β + ϑ(-σ + m + s 0 ) C 7 :=αβ -σ + s 0
By (3.4), we have σ -αβ > s 1 > s 0 + max {m, ℓ}. It follows that:

C 3 < (α -1) ℓ + ℓ ′ .
So, defining ℓ" = max {(α -1) ℓ + ℓ ′ , αϑℓ ′ } as in (3.9), we see that

ε g Λ ℓ 1 + M ℓ ′ 1 -1
on v , and we

have M 1 ε -ϑη , Λ 1 ε -η , hence Λ ℓ 1 + M ℓ ′ 1 ε -η max{ϑℓ ′ ,ℓ} . So the condition v ′ δ
ε g+η max{ϑℓ ′ ,ℓ} guarantees the existence of the sequence (u n ). But (3.1) may be rewritten in the form g + η max {ϑℓ ′ , ℓ} < g ′ , so the preceding condition is implied by the estimate v ′ δ ε g ′ , which is thus sufficient, as desired. Now we can translate the symbol into more explicit estimates. Choosing r > 0 small enough, our construction gives, for every v ∈ W δ with v

′ δ ≤ r ε g ′ a sequence u n , n ≥ 1, such that u n ∈ E n , u n s1 ≤ r -1 ε -g ′ v ′ δ ≤ 1 , and Π ′ n F ε (u n ) = Π ′ n-1 v .
It follows from Lemma 4 that for any t < σ -αβ , (u n ) is a Cauchy sequence for the • t . We recall that, by condition (3.4), s 1 < σ -αβ. So we can choose t 1 ∈ (s 1 , σ-αβ) . Then (u n ) converges to some

u ε in V t1 with u ε s1 ≤ r -1 ε -g ′ v ′ δ ≤ 1 . Since t 1 ≥ s 0 + m, the map F ε is continuous from the t 1 -norm to the (t 1 -m)- norm, so F ε (u n ) converges to F ε (u ε ) in W t1-m . Then F ε (u n ) is a bounded sequence in W t1-m ,
and t 1 -m > s 0 . So, using the approximation estimate (2.2), we find that (1

-Π ′ n )F ε (u n ) s0 → 0 , and finally Π ′ n F ε (u n ) -F ε (u ε ) s0 → 0 as n → ∞ .
On the right-hand side, using (2.2) again, we find that Π ′ n-1 v converges to v in W s0 , since δ > s 0 .

We conclude that F ε (u ε ) = v, as desired, and this ends the proof of Theorem 3.

4. An application of the singular perturbation theorem 4.1. The result. In this section, we consider a Cauchy problem for nonlinear Schrödinger systems arising in nonlinear optics, a question recently studied by Métivier-Rauch [START_REF] Métivier | Dispersive stabilization[END_REF] and Texier-Zumbrun [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]. Métivier-Rauch proved the existence of local in time solutions, with an existence time T converging to 0 when the H s norm of the initial datum goes to infinity. Texier-Zumbrun, thanks to their version of the Nash-Moser theorem adapted to singular perturbation problems, were able to find a uniform lower bound on T for certain highly concentrated initial data.

The H s norm of these initial data could go to infinity. By applying our "semiglobal" version of the Nash-Moser theorem, we are able to extend Texier-Zumbrun's result to even larger initial data. In the sequel we follow closely their exposition, but some parameters are named differently to avoid confusions with our other notations.

The problem takes the following form:

(4.1) ∂ t u + iA(∂ x )u = B(u, ∂ x )u, u(0, x) = ε κ (a ε (x), āε (x)) with u(t, x) = (ψ(t, x), ψ(t, x)) ∈ C 2n , (t, x) ∈ [0, T ] × R d , A(∂ x ) = diag(λ 1 , • • • , λ n , -λ 1 , • • • , -λ n )∆ x and B = B C C B
The coefficients b jj ′ , c jj ′ of the n × n matrices B, C are first-order operators with smooth coefficients:

b jj ′ = d k=1 b kjj ′ (u)∂ x k , c jj ′ = d k=1 c kjj ′ (u)∂ x k
, with b kjj ′ and c kjj ′ smooth complex-valued functions of u satisfying, for some integer p ≥ 2, some C > 0, all 0 ≤ |α| ≤ p and all u = (ψ, ψ) ∈ C 2n :

|∂ α b kjj ′ (u)| + |∂ α c kjj ′ (u)| ≤ C|u| p-|α| .
Moreover, we assume that the following "transparency" conditions hold: the functions b kjj are real-valued, the coefficients λ j are real and pairwise distinct, and for any j, j ′ such that λ j + λ j ′ = 0, c jj ′ = c j ′ j .

We consider initial data of the form ε κ (a ε (x), āε (x)) with a ε (x) = a 1 (x/ε) where 0 < ε ≤ 1, a 1 ∈ H S (R d ) for some S large enough and a 1 H S small enough.

Our goal is to prove that the Cauchy problem has a solution on [0, T ] × R d for all 0 < ε ≤ 1 , with T > 0 independent of ε. Texier-Zumbrun obtain existence and uniqueness of the solution, under some conditions on κ, which should be large enough. This corresponds to a smallness condition on the initial datum when ε approaches zero. Our local surjection theorem only provides existence, but our condition on κ is less restrictive, so our initial datum is allowed to be larger. Note that, once existence is proved, uniqueness is easily obtained for this Cauchy problem, indeed local-in-time uniqueness implies global-in-time uniqueness. Our result is the following: Theorem 6. Under the above assumptions and notations, let us impose the additional condition

(4.2) κ > d 2(p -1) . Let s 1 > d 2 + 4 . If 0 < ε ≤ 1, a 1 ∈ H S (R d )
for S large enough, and a 1 H S is small enough, then the Cauchy problem (4.1) has a unique solution in the functional space

C 1 [0, T ], H s1-2 (R d ) ∩ C 0 [0, T ], H s1 (R d ) .
Metivier-Rauch already provide existence for a fixed positive T when κ ≥ 1 . So we obtain something new in comparison with them when d

2 1 p-1 < 1 , that is, when p > 1 + d 2 .
Let us now compare our results with those of Texier-Zumbrun [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]. In order to do so, we consider the same particular values as in their Remark 4.7 and Examples 4.8, 4.9 pages 517-518. Let us illustrate this in 2 and 3 space dimensions.

In two space dimensions, d = 2 (Example 4.8 in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]): Our condition becomes 1 p-1 < κ. In their paper, Texier and Zumbrun need the stronger condition 9 2(p+1) < κ. In three space dimensions, d = 3 (Example 4.9 in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]): Our condition becomes Remark. After reading our paper, Baldi and Haus [START_REF] Baldi | Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems[END_REF] have been able to relax even further the condition on κ, based on their version [START_REF] Baldi | A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs[END_REF] of the classical Newton scheme in the spirit of Hörmander. A key point in their proof is a clever modification of the norms considered by Texier-Zumbrun, allowing better C 2 estimates on the functional. They also explain that their approach can be extended to other C 2 functionals consisting of a linear term perturbed by a nonlinear term of homogeneity at least p + 1. Our abstract theorem, however, seems more general since we do not need such a structure.

4.2.

Proof of Theorem 6. We have to show that our Corollary 4 applies. Our functional setting is the same as in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], with slightly different notations.

We introduce the norm

f H s ε (R d ) = (-ε 2 ∆ + 1) s/2 f L 2 (R d )
, and we take

V s = C 1 ([0, T ], H s-2 (R d )) ∩ C 0 ([0, T ], H s (R d )) , |u| s = sup 0≤t≤T ε 2 ∂ t u(t, •) H s-2 ε (R d ) + u(t, •) H s ε (R d )
and

W s = C 0 ([0, T ], H s (R d )) × H s+2 (R d ) , |(v 1 , v 2 )| ′ s = sup 0≤t≤T v 1 (t, •) H s ε (R d ) + v 2 H s+2 ε (R d )
Our projectors are

Π Λ u = F -1 x (1 |εξ|≤Λ F x u(t, ξ)) , Π ′ Λ (v 1 , v 2 ) = F -1 x (1 |εξ|≤Λ F x v 1 (t, ξ)), F -1 (1 |εξ|≤Λ F v 2 (ξ))
We take Φ ε (u) = ε 2 ∂ t u + iA(ε∂ x )u -εB(u, ε∂ x )u , u(0, •) -ε κ (a ε , āε ) and a ε (t, x) = ε κ (exp(-itA(∂ x ))a ε , exp(itA(∂ x ))ā ε ) . We have Φ ε (a ε ) = (-εB(a ε , ε∂ x )a ε , 0). A solution of the functional equation Φ ε (u) = 0 is a solution on [0, T ] × R d of the Cauchy problem 4.1.

Our Corollary 4 requires a direct estimate (2.9) on DΦ ε and an estimate (2.11) on the right-inverse L ε . Take s 0 > d/2 + 2, m = 2, γ = dp 2(p-1) and S large. Since κ > d 2(p-1) , we have an estimate of the form |a ε | S ε γ a 1 H S , so, taking a 1 H S small, we can ensure that a ε ∈ B S (ε γ ). Moreover the inequality κ > d 2(p-1) implies the condition

1 - dp 2 + pγ ≥ 0 .
So we see that the assumptions of Lemma 4.4 in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] are satisfied by the parameters γ 0 = γ 1 = γ (note that our exponent p is denoted ℓ in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]). The direct estimate (2.9) thus follows from Lemma 4.4 in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]. Note that Lemma 4.4 of [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF] also gives an estimate on the second derivative of Φ ε (•), but we do not need such an estimate.

Choosing, in addition, ℓ = 2, ℓ ′ = 0, g = 2, our inverse estimate (2.11) follows from from Lemma 4.5 in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF].

To summarize, the assumptions (2.9, 2.10, 2.11) of our Corollary 4 are satisfied for s 0 > d/2, m = 2, γ = p p-1 d 2 , g = 2, ℓ = 2, ℓ ′ = 0. Moreover, in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], Proof of Theorem 4.6, one finds an estimate which can be written in the form |Φ ε (a ε )| ′ s1-1 ≤ r ε 1+κ(p+1)+d/2 where r is small when a 1 H s 1 is small. So, using our Corollary 4, taking S large enough we can solve the equation Φ ε (u) = 0 in X s1 under the additional condition 1 + κ(p + 1) + d/2 > γ + g , which can be rewritten as follows:

Remark.

In the examples 4.8 and 4.9 of [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF], Texier and Zumbrun also study the case of oscillating initial data, i.e. a ε = a(x)e ix•ξ0/ε , and in the first submitted version of this paper we considered it as well. However, a referee pointed out to us that the corresponding statements were not fully justified in [START_REF] Texier | Nash-Moser iterations and singular perturbations[END_REF]. Indeed, in the proof of their Theorem 4.6, Texier and Zumbrun have to invert the linearized functional DΦ ε (u) for u in a neighborhood of the function a ε , denoted a f in their paper. For this purpose, it seems that they need the norm of their function a f to be controlled by ε γ . This condition appears in their Remark 2.14 and their Lemma 4.5, but not in the statement of their Theorem 4.6. This additional constraint does not affect their results for concentrating initial data in Examples 4.8, 4.9. But in the oscillating case, their statements seem overly optimistic. We did not want to investigate further that issue, this is why we only deal with the concentrating case. Note, however, that this difficulty with the oscillating case is overcome in the recent work [START_REF] Baldi | Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems[END_REF], thanks to improved norms and estimates.

Conclusion

The purpose of this paper has been to introduce a new algorithm into the "hard" inverse function theorem, where both DF (u) and its right inverse L (u) lose derivatives, in order to improve its range of validity. To highlight this improvement, we have considered singular perturbation problems with loss of derivatives. We have shown that, on the specific example of a Schrödinger-type system of PDEs arising from nonlinear optics, our method leads to substantial improvements of known results. We believe that our approach has the potential of improving the known estimates in many other "hard" inversion problems.

In the statement and proof of our abstract theorem, our main focus has been the existence of u solving F (u) = v in the case when S is large and the regularity of v is as small as possible. We haven't tried to give an explicit bound on S, but with some additional work, it can be done. In an earlier version [START_REF] Ekeland | An implicit function theorem for non-smooth maps between Fréchet spaces[END_REF] of this paper, the reader will find a study of the intermediate case of a tame Galerkin right-invertible differential DF , with precise estimates on the parameter S depending on the loss of regularity of the right-inverse, in the special case s 0 = m = 0 and ℓ = ℓ ′ .

max{C 3 , C 4 } 6 }3. 4 .

 3464 + max{C 5 , C 6 } ≤ max{ℓ" + C 5 , ℓ" + C So condition(3.38) is implied by the inequalities ℓ" + C 5 < C 7 and ℓ" + C 6 < C 7 , which are the same as conditions (3.7) and(3.8). So inequality (3.35) holds, and the induction holds by Theorem 5 End of proof. First of all, for the above construction to work, the only constraint on S is S > σ, and Lemma 1 gives us the estimate σ < ζg/η. The constant η is only constrained by condition (3.1), and we can choose, for instance, η = g ′ -g 2 max{ϑℓ ′ ,ℓ} . So we only need a condition on S of the form S ≥ S 0 with S 0 = O( 1 g ′ -g ) as g ′ → g , all the other parameters being fixed. Let us now check that the estimate v ′ δ ε g ′ is sufficient for the above construction. In(3.18) we made the assumption v ′ δ

3 2 (

 2 p-1) < κ. In their paper, Texier-Zumbrun need the stronger condition 4 p+1 < κ. In both cases, we improve over Métivier-Rauch when p ≥ 3, while Texier-Zumbrun need p ≥ 4.

Since d ≥ 2 ,

 2 this inequality is a consequence of our assumption κ ≥ d 2(p-1) . So our Corollary 4 implies the existence of a solution to the Cauchy problem (4.1). The uniqueness of this solution comes from the local-in-time uniqueness of solutions to the Cauchy problem. This proves Theorem 6 as a consequence of Corollary 4.

Acknowledgement. We are grateful to Massimiliano Berti, Philippe Bolle, Jacques Fejoz and Louis Nirenberg for their interest in our work and their encouragements. It is a pleasure to thank Pietro Baldi for stimulating discussions in Naples and Paris, for a careful reading of the present paper and for a number of suggestions. We also thank the referees, whose remarks have helped us to improve this manuscript.