Norbert Pachler

Caroline Daly

Yishay Mor

Harvey Mellar

Formative e-assessment: practitioner cases

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and literature review

This paper focuses on the cases which were developed as part of the JISC-funded project 'Scoping a vision of formative e-assessment' which aimed to identify current theories and practices relating to formative assessment where technologies play a key role. The project aimed to scope the 'domain' of formative e-assessment within post-16 contexts including Higher Education (HE), Further Education (FE) and Work-Based Learning (WBL), by developing cases of practice and identifying key formative processes within them which are affected by the use of technologies. A further aim, not explored in this paper, was then to derive patterns by analysing the cases, and capture the essential features of a range of formative e-assessment practices in patterns which would inform future software development. This paper presents two representative cases, and discusses the contribution they make to understanding the features of formative assessment where technologies have a range of roles in learning and teaching.

The literature presents a diverse set of perspectives on the nature and value of formative e-assessment. It is sometimes presented principally as a relatively easy way of introducing e-assessment within an institution, arguing that as 'low stakes' assessment it presents less of a threat to institutions. Formative assessment is often presented as being simply serial (or repeated) summative assessment, and it is even referred to as 'practice' assessment. However, many authors question the value of such an approach and argue for a view of formative assessment that places much more emphasis on the role of feedback in the learning process (e.g. [START_REF] Maughan | On-line formative assessment item banking and learning support[END_REF]. For others, formative e-assessment is seen as having the potential to support significant changes in the way in which learning occurs in HE. [START_REF] Whitelock | Computer assisted formative assessment: Supporting students to become more reflective learners[END_REF], for example, argues for formative e-assessment as a means of promoting self-reflection and students taking control of their own learning, a view which echoes the ideas of Nichol (2006), who proposes a series of design principles for using formative eassessment to support learner self regulation.

Writing in 1998 and looking to the future of e-assessment, Bennet (1998) describes three generations of e-assessment: First Generation tests using designs based closely on existing paper-based tests, Next-Generation tests which would use new formats including multimedia, constructed response, automatic item generation and automatic scoring, and Generation "R" (Reinvention) tests which would use complex simulations and intelligent tutors. Simulations such as the those now used in medical licensing examinations (see http://www.usmle.org/Orientation/2009/menu.html) enable detailed formative feedback on the execution of complex tasks, whilst intelligent tutoring systems (see for example http://www.carnegielearning.com/ and http://www.aleks.com/) use cognitive science and statistical techniques to generate detailed models of the student's understanding and use this to provide formative feedback and to modify the instructional sequence. A rather different approach to forming the future of formative assessment is seen in the call by [START_REF] Elliott | [END_REF] for the incorporation of perspectives based on Web 2.0 technologies and approaches, and the blurring of boundaries between formative and summative assessment.

In the Roadmap for e-assessment developed for JISC by Whitelock and Brashar (2006) the authors consulted widely with the UK e-assessment community considering the divergent views of the ways forward for e-assessment in HE. They concluded that the future roles of e-assessment in the medium term lie in the areas of on-demand testing, e-portfolios, student modelling, formative assessment supporting learner autonomy, and diagnostic assessment, and they stress the need for improved forms of feedback informed by pedagogical (or rather androgogical) principles.

Within this context JISC funded a range of projects (see http://www.jisc.ac.uk/assessment.html) as part of its e-Learning programme in order to examine various aspects of e-assessment. The project described here is one of several that dealt with formative e-assessment. The more advanced techniques belonging to Bennett's Generation "R", such as the use of simulations, were addressed in the Review of Advanced e-Assessment Techniques project [START_REF] Ripley | Review of Advanced e-Assessment Techniques (RAeAT) Final Report[END_REF]. The use of e-portfolios was addressed in a number of projects and the work of these is summarised in the JISC report 'Effective Practice with e-Portfolios' (JISC 2008). Therefore, positioning itself with respect to other ongoing work, the project reported in this present study concentrated on examples of formative e-assessment where the pedagogy was relatively well developed, but excluded both the use of eportfolios and the use of 'advanced' techniques.

The work of [START_REF] Black | Assessment and classroom learning[END_REF] is widely quoted in the literature in developing an argument for the effectiveness and value of formative e-assessment, though the argument is often made without regard for the nature of the feedback and interaction in teaching and learning contexts. Our own interest in formative e-assessment also starts from [START_REF] Black | Assessment and classroom learning[END_REF] work and in particular their recent theorisation of formative assessment [START_REF] Black | Developing the theory of formative assessment Assessment[END_REF], which proposes five key strategies: 1 engineering effective classroom discussion, questions, and learning tasks that elicit evidence of learning; 2 providing feedback that moves learners forward; 3 clarifying and sharing learning intentions and criteria for success; 4 activating students as owners of their own learning; and 5 activating students as instructional resources for one another.

Core to Black and Wiliam's developing theory of formative assessment is the concept of 'moments of contingency' [START_REF] Leahy | Classroom assessment: minute by minute, day by day[END_REF] which they argue to be 'hinge' points in the teaching and learning process where the flow of instruction will depend on how students respond and cannot be predetermined [START_REF] Black | Developing the theory of formative assessment Assessment[END_REF]. Moments of contingency contain within them the scope for learners' understanding to be 'otherwise'. Contingency is constituted by a multiplicity of factors which are present in the learning context, and which may be social, cognitive and psychological. A learner's response to a potential learning opportunity (provided for example by teacher questioning, stimulus material, automated scoring of performance or peer comment) is part of their unique engagement in the learning process and is autobiographical. Responsiveness (on the part of the learner, teacher and/or peers) is key to contingency, and is necessary to 'moments', which have formative effects on learning. The role of technologies in contributing to the potential for responsiveness within moments of contingency became a focus for analysing the cases, and identifying 'hinge' points in the learning process.

Our perspective on the use of technology to support formative assessmenthas also been strongly influenced by Lauruillard's Conversational Framework [START_REF] Laurillard | Rethinking university teaching. A conversational framework for the effective use of learning technologies[END_REF][START_REF] Laurillard | Pedagogical forms of mobile learning: framing research questions[END_REF]. We view learning as 'conversational', consisting of a series of iterative cycles of interaction between teacher/learner/peers in a variety of combinations which may make use of technologies to greater or lesser degrees.

Drawing on these perspectives, and for the purposes of developing cases of practice for this study, we defined formative e-assessment as the use of ICT to support the iterative process of gathering and analysing information about student learning by teachers as well as learners and of evaluating it in relation to prior achievement and attainment of intended, as well as unintended learning outcomes. We did so also with a view to allowing the teacher or student to adjust the learning trajectory by becoming activated in a variety of ways. Within this definition, technologies of eassessment are not seen as in themselves being inherently either summative or formative, but rather what is of interest is whether e-assessment is being used summatively or formatively.

Methodology

Formative assessment is continuously embedded in the teaching and learning process; its interaction with the multiple dimensions of this process create a highly complex domain. The project worked with a methodology that is capable of taking in the complexity and fluidity of this domain. We identified the potential of the patternbased methodology of the JISC-funded Planet project (see http://purl.org/planet/Outcomes/Methodology for details). The methodology centres on identifying ensembles of context, problem and solution in a domain of practice. These form the core of design patterns; a term coined by Christopher Alexander in the theory of architecture [START_REF] Alexander | A Pattern Language: Towns, Buildings, Construction[END_REF], and since then adopted widely in domains such as software engineering, interaction and interface design, and organizational change (cf. [START_REF] Mor | Design approaches in technology enhanced learning[END_REF] for a review).

Our initial review of the literature confirmed the domain of formative e-assessment as extremely complex (for details see http://patternlanguagenetwork.myxwiki.org/xwiki/bin/view/Groups.FormativeEAssess ment/Literature+Review). This complexity arises from the inherent nature of formative assessment. It is continuously embedded in the process of teaching and learning, and interacts with the multitude of factors, both social and cognitive, which constitute this process. This complexity is compounded by the "e" dimensions; technology introduces new opportunities along with new challenges and reshuffles the context of teacher-student interaction. A methodology was required which addressed the complex interaction of social, pedagogical and technical factors at play in a variety of learning contexts,.

In addition, a research approach was required which allowed us to capture the practical expertise of those engaged in formative assessment practices, together with opportunities for critical reflection on those practices. In exploring the practice dimensions, the project engaged participant informants in thinking through their professional practice in the field of formative e-assessment. We identified the designpattern based methodology, as developed by the JISC-funded Planet project [START_REF] Finlay | Planet: Pattern Language Network for Web 2.0 in Learning: Final Report[END_REF], as a suitable framework for user engagement, analysis, and modelling. We see this approach as supporting enquiry processes in relation to practice in the field of formative e-assessment, be they professional practice-or tool-design related. The process of case development discussed here was participatory and iterative. The project brought together education practitioners from a range of contexts to meet on 'Practical Enquiry Days' (PEDs), in order to collaboratively reflect on the aspects of their practice they considered to constitute formative e-assessment. Practitioners were prompted to recount their experiences of using formative eassessment as case stories, discuss them with their peers and document them on the project wiki using a template designed by the project team. The construction and discussion of these narratives was scaffolded by asking small groups of participants to consider the key features of their practice, concerning context, problem, task, results and lessons learned. These cases then constituted the data from which design patterns for formative e-assessment were developed. In contrast to other projects in the e-Learning programme, the cases were, therefore, not authored by members of the project team on the basis of data collected about practice through interviews, desk-based research etc. but by practitioners themselves.

The cases

Through participation in PEDs as well as (telephone) interviews, a number of cases were developed by participants and the project team and captured on the project wiki. Criteria were developed in order to identify the features of a range of cases from which processes could be identified and patterns relevant to formative eassessment were to be mainly derived. The following criteria were used:

The PEDs generated a large number of 'seed' cases describing practices in a variety of educational settings where elements of formative assessment were present. Within the scale of the project it was not possible to develop all of them into full cases from which patterns can be derived and five key examples were selected to develop detailed cases. There exists a variety of reasons for not developing some very interesting cases of practice. Some reasons were rooted in the challenges which are intrinsic to the collaborative participatory approach to case-building in the project. This needs to be considered in future development of the methodology, as practitioner engagement over time is an important feature in developing authentic cases from which patterns can be derived. Other reasons were rooted in the content focus of the practices which were captured. The reasons for excluding cases can be summarised:

• it was not practical for some practitioners to attend all the PEDs where the cases were developed over time, so practitioners could not take part in ongoing group review and refinement of the cases they owned; • formative assessment was not a major aspect of some of the practices described; • there were a number of cases based on similar pedagogical practices, and we tried to avoid duplication to present a broader range of contexts/ technologies/practices; • some cases were not sufficiently developed for inclusion because it was difficult to engage some practitioners in developing their cases on the wiki outside of the PEDs, due to pressures of time and workload; • some cases did not fall within the main focus of post-16 sectors; • some cases were already well-documented elsewhere.

We collected material to develop five cases, selecting these to ensure a range according to the criteria outlined in Table 1. Table 2 provides a summary overview of the cases. whilst this can be seen as a limitation, testing the effectiveness of each case would have gone far beyond the scope of this project. The cases are not intended as examples of 'good' practice, but as representative examples of current practice in the field of formative e-assessment in relation to which a 'vision' -the project team always preferred to think of it as a possible future -could be delineated with reference also to what is known from the literature.

Case 1: Audiofiles

Practice

Audiofiles have been piloted with twenty-five undergraduate university students in years 1, 2 and 3 in a school of sociology and social policy. They form part of ongoing research and development to improve strategies for tutors to give feedback to students on traditional written essays to help deal with a contemporary challengeincreasing class sizes and less time for staff to spend on feedback on written coursework, leading to inconsistency in feedback even where it is done well. The intervention sought to explore the value of replacing text-based feedback with audio feedback, and to find out 'Does the feedback change?' Tutors used both handheld dictaphones and audio software to record their feedback directly onto a PC. There were no rules about the length of the file. Feedback was recorded in a single audiofile at the conclusion to reading the whole piece of writing and lasted between ninety seconds and twenty-one minutes. Audiofiles were then returned to the students via the course Virtual Learning Environment (VLE). In the audiofiles, tutors tended to comment more freely than in their equivalent written feedback which conformed more closely to the content guidelines of a feedback template sheet. It took tutors longer to produce however, and it was considered that further development may help the staff with managing this. The student work was also being formally assessed, so the students were getting formative outcomes from a summative piece of work, and it was felt that the approach may be more effective in a purely formative context. Planned next steps in this institution were to: • explore audiofiles with other disciplines • explore the possibilities of inserting feedback at intervals into the body of the documents like 'comment' inserts on word files • enable tutors to easily edit the word files as they make them.

Analysis

A key feature of developing audiofiles to construct feedback is the dual contingency that is evident in the account. The technological intervention was designed to impact on both students and tutors, increasing the potential for responsiveness in both. For tutors, increased contingency can be interpreted in the changed nature of their feedback comments. The practice of audio-recording appears to develop heightened focus on the potential impact of what they say. This relates to [START_REF] Laurillard | Rethinking university teaching. A conversational framework for the effective use of learning technologies[END_REF] Conversational Framework, by which both teachers' and learners' conceptions go through a process of adaptation as a result of 'feedback on action'. The transmission of the audiofile itself takes place at the level of 'practice' within the framework. It is linked to conceptual adaptation by both teacher and learner by the reflective processes which are linked to engaging with feedback by both parties. In this case, the level of conceptual adaptation by the tutor was the most significant feature to emerge, and possibly greater levels of conceptual adaptation take place for the tutor than the learner in many instances. Comments were 'richer' and more emphatic. In comparing the audio transcripts with written feedback from the same tutors, the feedback tended to be richer, longer, personalised, more immediate and 'authentic'.

The process appeared to make tutors reflect more on their own feedback, leading to the desire to amend it. This is consistent with Black and Wiliam's view (1998) that pedagogical change in itself is a core aim of formative assessment, and that effects on learners may only become apparent over long feedback cycles.

In relation to the learners' responsiveness, the audiofiles case addresses somewhat Nicol and Macfarlane-Dick's ( 2006) criticisms of transmission-focused feedback, that 'feedback messages are invariably complex and difficult to decipher' in transmission contexts, because students find it hard to take appropriate actions to improve. The heightened tutor focus on providing emphasis and detail may counter these concerns, and certainly suggests that technology can enhance the capacity for transmission-focussed feedback to prompt 'moments of contingency' for learners as well as tutors. It is interesting to note the seemingly positive impact of oral feedback in this case given [START_REF] Shute | Focus on formative feedback[END_REF] classification of orally delivered feedback as a 'thing to avoid'. It might well be that in the audiofiles case the oral feedback is in the form of a disembodied voice, recorded digitally and 'delivered' by the computer that mitigates against the potential negative effects suggested by some of the literature.

Most importantly, the impact on the tutor is highlighted, and contingency is located firstly in the tutor's capacity to reflect more deeply on the nature of the feedback they will give, prompted by the technological intervention. The tutor's adaptation is the key to the second 'moment of contingency', which involves the learner's potentially enlarged range of responses to the enhanced feedback.

The full case can be found at: http://patternlanguagenetwork.myxwiki.org/xwiki/bin/view/Cases/Audiofiles.

Case 2: Como: mobiles + flickr = co-reflective practice

Practice

At the Royal Veterinary College, a group of students were engaged in practical work in a vet training hospital. As part of their training, the students were required to capture instances of practice on a mobile phone and the photos collected were automatically uploaded to flickr. The students worked in groups of four or five and each was provided with a mobile phone and given a short familiarisation session. In one scenario, during morning rounds students would be directed to monitor the progress of an animal being treated. Their task would be to document case progress over time. They took pictures throughout the day, uploaded them to a blog, tagged them with caseID and key features e.g. type of animal, the injury, condition. The students then used quiet moments to add details to the 'case' using blog postings.

During the evening rounds, the students presented their cases in group discussion sessions with their tutor, using the images, blog posts and a projector. The group reviewed the diagnosis and the actions which were taken, and reviewed these in the light of revisiting the images and postings which acted as catalysts for evaluation of practice. Co-reflection was enhanced because of the availability of images which bring the medical case into the seminar room. It affected the students' tutorial conversation, providing ongoing formative contributions to the case in the form of postings. The discussion moved from abstract "textbook theory" to what tutors called "case presentation": how the particular condition presents itself in a particular case, how to analyse symptoms in real-world conditions and how to assess treatment. These are key skills which are often neglected due to the inability to have a concrete presence of the case in the seminar room as a focus for reflective and analytical discussion. The process of using images to capture cases also provided feedback to tutors on the students' learning. Tutors reported that observing students' pictures gave them a window on their thinking: what they noticed, where their attention was and where they assigned importance. This was the basis for modifying tutor input and the focus of the tutorial discussion.

Analysis

A key to the role of technology in formative assessment in this case is the 'window on their thinking', which is opened up for the tutors in relation to learners' knowledge and understanding. [START_REF] Shute | Focus on formative feedback[END_REF], based on her survey of relevant literature on feedback, highlights the potential of multimedia to contribute to feedback practices, and recommends alternative modes of presentation. The project identified 'construction and representation' as one of six main affordances of digital technology in relation to formative e-assessment (see Pachler et al 2009, pp. 20-21). By this, the ability to represent ideas in a variety of ways and to make meaning with a variety of symbols and modes enhances opportunities for teachers and learners to engage in iterative exchanges, which support adaptation of thinking. The use of photographs in this case was core to the tutors' understanding of how the students were thinking in their work-based learning contexts. The process required the students to select, prioritise and synthesise in order to be able to articulate their cases using the images collected.

Again, dual contingency is at work here. Firstly, the learner is prompted to engage with the digital images they have made to decide how to use them to present a case. This does not require tutor intervention, but requires critical reflection at the level of the task via self-evaluation of the choices already made in taking the photographs. The affordances of the mobile technology mean that material can be re-purposed as a result of review, and speed and adaptability enable the process of case presentation to be enhanced. [START_REF] Laurillard | Rethinking university teaching. A conversational framework for the effective use of learning technologies[END_REF] presents such practices in her Conversational Framework as a feedback cycle in which the learner is able to develop conceptually by reflecting on their task and the actions they have undertaken to achieve a goal. There are potentials to engage with 'other learners' conceptions' (e.g. students could share their photographs and seek peer-advice on how to work with them) but self-evaluation is a valid iterative practice leading to enhanced understanding.

In a further stage, the tutor is able to respond to what the images reveal about a student's current state of understanding, and conduct the teaching session in the light of that knowledge. Further 'moments of contingency' are made possible here, by the tutor bringing their pedagogical and subject knowledge to inform their responses to the student presentations, based on what the images reveal about their thinking.

The full case can be found at: http://patternlanguagenetwork.myxwiki.org/xwiki/bin/view/Cases/CoMo.

Discussion

As part of our developing conceptualisation of formative e-assessment, we see key processes which have formative effects taking place within broader frameworks of learning. Such processes are based on the roles of key players (teachers, individual learners, peers) and a range of practical and discursive actions in which they participate.

As can be seen from the cases presented here, technology does not in itself bring about formative effects. 'Formative e-assessment', we argue, is better understood as multiple processes involving technologies to greater or lesser degrees, where evidence is generated about a learner's state of understanding relative to desirable goals, and where individuals are enabled to take actions which bring about changes in learners' skills, knowledge and understanding, or in teachers' pedagogical practice. What is key is not how we assess but what we do with the data we generate as a result of interventions which can be supported by technologies. The cases suggest that there are key technological attributes or 'resources' which appear to make a difference to the learners' potential for improvement, because of the way the technology contributed to creating moments of contingency. The technology does not in and of itself create these moments of contingency, however. These depend on the set of human responses, motivational factors and sociointeractive contexts which create opportunities for the choices learners make and actions taken in conjunction with feedback and interaction offered by electronic tools.

The tools do have particular shaping effects on the types of choices and actions which can emerge. The technologies we describe in the cases and patterns help to constitute the learning environment and contribute to shaping the contingent possibilities which are part of it. [START_REF] Black | Developing the theory of formative assessment Assessment[END_REF] outline 'key aspects' of formative assessment around three core learning scenarios (Table 3):

• where the learner is going • where the learner is • how to get there.

The cases, both the examples summarised in this paper and the complete collection from the project, include few examples of 'where the learner is going' with most of them providing examples of 'where the learner is' and 'how to get there'. There was little focus on understanding learning intentions as involving mutual review between teachers and learners. The study was small-scale and we can only make some cautious observations about this, but in these cases the majority of formative practice in post-16 formal education settings is geared towards achieving predetermined learning outcomes with relatively little time devoted to sharing and understanding learning goals. This is an area in need of further investigation.

Towards a vision of formative e-assessment

Two key points stand out from the theoretical work and case development within the project. Firstly, no technology-based assessment is in itself formative, but almost any technology can be used in a formative way -if the right conditions are set in place. This observation is in line with a socio-technical view of educational systems, which sees the technological dimensions (e.g. speed, storage capacity, processing, communication, construction and representation and mutability) as inseparable from the pedagogical conditions (e.g. verbal/electronic/ synchronous/asynchronous interaction between key players which brings about changes in concepts or skills). It is consistent with a view of learning as 'conversational' [START_REF] Laurillard | Rethinking university teaching. A conversational framework for the effective use of learning technologies[END_REF][START_REF] Laurillard | Pedagogical forms of mobile learning: framing research questions[END_REF], consisting of a series of iterative cycles of interaction between teacher/learner/peers in a variety of combinations which may make use of technologies to greater or lesser degrees. This range of combined technological and pedagogical resources impacts not only on how students act but also informs what teachers do to enhance learning.

From our analysis of the cases, 'moments of contingency' [START_REF] Leahy | Classroom assessment: minute by minute, day by day[END_REF] are identified in terms of how they act as 'hinge' points in the teaching and learning process. The technology itself does not create these moments; they are dependent on teachers' and learners' actions. But for technology to perform formatively, it needs to acknowledge and support these moments.

The second key point is that formative e-assessment is incredibly complex, since it requires the delicate orchestration of social, pedagogical and technological systems. Any framework of design and development attempting to address this domain needs to identify methodological tools which allow it to deal with such complexities. Such tools need to balance the need for a crisp directive for action with a rich representation of context, intentions and possible solutions. In our study, we found the Planet methodology -which develops cases, design patterns and future scenarios via collaborative structured groups conversations -to be a valuable asset. It enabled discussion between all stakeholders in an iterative, user-centric development process.

On the basis of our engagement with practice and the literature in the field of formative e-assessment a set of parameters emerged which, if they can be validated by further work, we consider to offer a potentially exciting way forward in analyzing and planning for formative e-assessment. They are:

• nature (extrinsic/intrinsic), frequency, role and functions (monitoring, diagnosing, instructionally tractable) of feedback; • affordance of moments of contingency;

• potential for learner self-regulation; • iteration; • scope for sharing outputs and ideas with peers;

• focus on where the learner is going; • length of cycle;

• potential for pedagogical modification;

• scope for closing the gap;

• contribution to future learning trajectories; • measurable attributes. We offer them here as possible focal points for discussion in the design of instructional episodes and/or the development of future e-assessment tools but wish to stress that they are neither systematic nor have they been empirically tested and verified as part of our project.

Conclusion

Formative e-assessment can be understood as a set of processes involving both technological and social resources by which individuals (both learners and teachers) are enabled to engage agentively with evidence of learning, in order to effect changes in understanding. Such engagement we see as crucial to 'moments of contingency'. Part of understanding formative assessment has to be the recognition that students may not improve despite engaging with technological and social resources; also, technology cannot guarantee moments of contingency. It is the learners and teachers as human actors who ultimately determine the formative effects of engaging with technologies, but technologies can shape the potential for this to happen. It is only when it is located in wider understandings of effective learning that the potentials of electronic tools to contribute to formative assessment can be understood and optimised. Our discussion of cases of formative e-assessment in this paper also clearly points to the conclusion that -on their own -cases are insufficient to move the sector forward. Whilst in themselves they may offer a useful stimulus for reflection on one's professional practice, in order to become an effective basis for the planning of intervention at a system level as well as for the design and development of software tools, a process of abstraction is necessary, which extracts the essential 'messages' contained in the cases, which lend themselves for generalisation and transfer across context. And, this distillation needs to be mapped tested against relevant literature and mapped against a domain map in order to ensure validity, reliability and comprehensiveness of coverage. The project discussed here affirms the view that patterns are a valuable extension of an inductive, case-based approach to improving pedagogical practice and better understanding the domain of formative eassessment. They can serve as meditational tools by providing a stimulus for practitioners to critically review and innovate their practice and by providing a basis for an understanding of key pedagogical issues for software developers. Patterns derived from cases can also form a useful set of criteria against which to assess the effectiveness of individual interventions -a process also outside the scope of the project discussed here.

For

  reasons of space, two of the five are presented below in summary form It is important to note that the cases are presented here mainly in a descriptive form -as developed by the practitioners -with only some reference to supporting evidence from the literature and with some analysis around the notion of contingency. This is done deliberately in order to demonstrate the nature of the data the project collected, on which the design patterns of the project, discussed in[START_REF] Pachler | Scoping a vision for formative e-assessment: a project report for JISC[END_REF] andDaly et al. (forthcoming) were derived. It is also important to stress that the project did not test the efficacy of the interventions described by practitioners in their cases;

For a comprehensive project report see[START_REF] Pachler | Scoping a vision for formative e-assessment: a project report for JISC[END_REF].

Acknowledgements

The project was funded by the Joint Information Systems Committee (JISC) and the Centre for Excellence in Work-Based Learning, Institute of Education, University of London. We wish to acknowledge the generous contribution of The Pattern Language Network (Planet) project, which was a collaboration between Leeds Metropolitan University, Coventry University, Glasgow Caledonian University, Kings College London and London Knowledge Lab. It was funded by JISC under the Users and Innovation Programme. For more information see http://patternlanguagenetwork.org. The Planet project provided us with the Participatory Methodology for Practical Design Patterns and the platform to support it. We would like to thank the Planet project members for their contributions: Janet Finlay, Isobel Falconer, Andrea Gorra, John Richard Gray, Jim Hensman, Jakki Sheridan-Ross, Yishay Mor and Steven Warburton. We would like to note our gratitude to the practitioners who developed the cases with us. A list of all the participants can be found in the project report, available at: http://patternlanguagenetwork.myxwiki.org/xwiki/bin/download/Groups.FormativeEAs sessment/Outcomes/formativee-assessmentfinal.pdf


Ibrahima Faye

F S T Emmanuel Frénod

Diaraf Seck

Long term behaviour of singularly perturbed parabolic degenerated equation

In this paper we consider models built in [3] for short-term, mean-term and long-term morphodynamics of dunes and megariples. We give an existence and uniqueness result for long term dynamics of dunes. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the mean-term and long-term models are homogenized.

Introduction and results

In Faye, Frénod and Seck [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF], based on works of Bagnold [START_REF] Bagnold | The movement of desert sand[END_REF], Gadd, Lavelle and Swift [START_REF] Gadd | Estimates of sand transport on the New York shelf using near-bottom current meter observations[END_REF], Idier [START_REF] Idier | Dunes et bancs de sables du plateau continental: observations in-situ et modélisation numérique[END_REF], Astruc and Hulcher [START_REF] Idier | Influence of bed roughness on dune and megaripple generation[END_REF], Meyer-Peter and Muller [START_REF] Meyer-Peter | Formulas for bed-load transport[END_REF] and Van Rijn [START_REF] Van Rijn | Handbook on sediment transport by current and waves[END_REF], we set out that a relevant model for short term dynamics of dunes, i.e. for their dynamics over several months, is

∂z ǫ ∂t - a ǫ ∇ • (1 -bǫm)g a (|u|)∇z ǫ = c ǫ ∇ • (1 -bǫm)g c (|u|) u |u| , (1.1)
where z ǫ = z ǫ (x, t), is the dimensionless seabed altitude at t and in x. For a given constant T, t ∈ [0, T ), stands for the dimensionless time and x = (x 1 , x 2 ) ∈ T 2 , T 2 being the two dimensional torus R 2 / Z 2 , is the dimensionless position variable. Operators ∇ and ∇• refer to gradient and divergence. Functions g a and g c are regular on R + and satisfy

               g a ≥ g c ≥ 0, g c (0) = g ′ c (0) = 0, ∃d ≥ 0, sup u∈R + |g a (u)| + sup u∈R + |g ′ a (u)| ≤ d, sup u∈R + |g c (u)| + sup u∈R + |g ′ c (u)| ≤ d, ∃U thr ≥ 0, ∃G thr > 0, such that u ≥ U thr =⇒ g a (u) ≥ G thr . (1.2)
Fields u and m are dimensionless water velocity and height. They are given by

u(t, x) = U(t, t ǫ , x), m(t, x) = M(t, t ǫ , x), (1.3)
where

                             U = U(t,
∀(t, θ, x) ∈ R + × R × T 2 , |U(t, θ, x)| ≤ U thr =⇒ ∂U ∂t = 0, ∂M ∂t = 0
, ∇M(t, θ, x) = 0 and ∇U(t, θ, x) = 0,

∃θ α < θ ω ∈ [0, 1] such that ∀ θ ∈ [θ α , θ ω ] =⇒ |U(t, θ, x)| ≥ U thr . (1.4)
A relevant model for mean term, i.e. when dune dynamics is observed over a few years, is

∂z ǫ ∂t - a ǫ ∇ • (1 -b √ ǫm)g a (|u|)∇z ǫ = c ǫ ∇ • (1 -b √ ǫm)g c (|u|) u |u| , (1.5)
with condition (1.2) on g a and g c and with u and m given by

u(t, x) = U(t, t √ ǫ , t ǫ , x), m(t, x) = M(t, t √ ǫ , t ǫ , x), (1.6)
For mathematical reasons, we assumed

U(t, τ, θ, x) = U(t, θ, x) + √ ǫ U 1 (t, τ, θ, x), (1.7)
where U = U(t, θ, x) and U 1 = U 1 (t, τ, θ, x) are regular. We also assumed that M = M(t, τ, θ, x) is regular and

                                           θ -→ (U, U 1 , M) is periodic of period 1, τ -→ (U 1 , M) is periodic
∀(t, τ, θ, x) ∈ R + × R × R × T 2 , | U(t, τ, θ, x)| ≤ U thr =⇒ ∂ U ∂t (t, τ, θ, x) = 0, ∂ U ∂τ (t, τ, θ, x) = 0, ∇ U (t, τ, θ, x) = 0, ∂M ∂t
(t, τ, θ, x) = 0, ∂M ∂τ (t, τ, θ, x) = 0 and ∇M(t, τ, θ, x) = 0,

∃θ α < θ ω ∈ [0, 1] such that ∀ θ ∈ [θ α , θ ω ] =⇒ | U(t, τ, θ, x)| ≥ U thr .
(1.8)

A relevant model for long-term dune dynamics is the following equation

∂z ǫ ∂t - a ǫ 2 ∇ • ((1 -bǫm)g a (|u|)∇z ǫ ) = c ǫ 2 ∇ • (1 -bǫm)g c (|u|) u |u| , (1.9)
where a, b and c are constants, where g a and g c satisfy assumption (1.2), and where z ǫ is defined on the same space as before. It is also relevant to assume

u(x, t) = U(t, t ǫ , x) = U 0 ( t ǫ ) + ǫU 1 ( t ǫ , x) + ǫ 2 U 2 (t, t ǫ , x), m(t, x) = M( t ǫ , x) + ǫ 2 M 2 (t, t ǫ , x), (1.10)
where

U 0 = U 0 (θ), U 1 = U 1 (θ, x), U 2 = U 2 (t, θ, x), M = M(θ, x) and M 2 = M 2 (t, θ, x) are regular and                                          θ -→ (U 0 , U 1 , U 2 , M, M 2 ) is periodic of period 1, |U 0 |, | ∂U 0 ∂θ |, |U 1 |, | ∂U 1 ∂θ |, |∇U 1 |, |U 2 |, | ∂U 2 ∂t |, | ∂U 2 ∂θ |, |∇U 2 |, |M|, | ∂M ∂θ |, |∇M|, |M 2 |, | ∂M 2 ∂t |, | ∂M 2 ∂θ |, |∇M 2 | are bounded by d, ∀(t, θ, x) ∈ R + × R × T 2 , |U 0 (θ) + ǫU 1 (θ, x) + ǫ 2 U 2 (t, θ, x)| ≤ U thr =⇒ ∂U 2 ∂t (t, θ, x) = 0, ∇U 1 (θ, x) = 0, ∇U 2 (t, θ, x) = 0, ∂M 2 ∂t (t, θ, x) = 0, ∇M(θ, x) = 0, ∇M 2 (t, θ, x) = 0, ∃θ α < θ ω ∈ [0, 1] such that ∀ θ ∈ R, θ ∈ [θ α , θ ω ] =⇒ |U 0 (θ) + U 1 (θ, x) + ǫ 2 U 2 (t, θ, x)| ≥ U thr .
(1.11)

Equations (1.1), (1.5) or (1.9) need to be provided with an initial condition

z ǫ |t=0 = z 0 , (1.12)
giving the shape of the seabed at the initial time.

In [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF], we then gave an existence and uniqueness result for short-term model (1.1) if hypotheses (1.2), (1.3) and (1.4) are satisfied and for the mean term one (1.5), if hypotheses (1.2), (1.6), (1.7) and (1.8) are satisfied. This result was built on a time-space periodic solution existence result for degenerated parabolic equation. Under the same assumptions, the asymptotic behaviour of z ǫ , as ǫ → 0, solution of short term model (1.1) is also given by homogenization methods. Futhermore if moreover U thr = 0, a corrector result was set out, which gives a rigorous version of asymptotic expansion of the sequence z ǫ :

z ǫ (t, x) = U (t, t ǫ , x) + ǫU 1 (t, t ǫ , x) + . . . , (1.13)
where U and U 1 are solutions to

∂U ∂θ -∇ • A∇U = ∇ • C, (1.14) ∂U 1 ∂θ -∇ • A∇U 1 = ∇ • C 1 + ∂U ∂t + ∇ • ( A 1 ∇U ), (1.15)
where A and C are given by

A = a g a (|U(t, θ, x)|) and C = c g c (|U(t, θ, x)|) U(t, θ, x) |U(t, θ, x)| , (1.16)
and A 1 and C 1 are given by

A 1 (t, θ, x) = -abM(t, θ, x) g a (|U(t, θ, x)|),
and

C 1 (t, θ, x) = -cbM(t, θ, x) g c (|U(t, θ, x)|) U(t, θ, x) |U (t, θ, x)| . (1.17)
In [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF], we did not state neither any existence result for long term model (1.9) nor any asymptotic behaviour result for mean term and long term models. Stating those results is the subject of the present paper. We will now state those main results. The first one concerns existence and uniqueness for the long-term model.

Theorem 1.1 For any T > 0, any a > 0, any real constants b and c and any ǫ > 0, under assumptions (1.2), ( 1) and (1.11), if

z 0 ∈ L 2 (T 2 ), (1.18)
there exists a unique function z ǫ ∈ L ∞ ([0, T ), L 2 (T 2 )), solution to equation (1.9) provided with initial condition (1.12). Moreover, for any t ∈ [0, T ], z ǫ satisfies

z ǫ L ∞ ([0,T ),L 2 (T 2 )) ≤ γ, (1.19)
for a constant γ not depending on ǫ and

d T 2 z ǫ (t, x) dx dt = 0. (1.20)
The proof of this theorem is done in section 2, except equality (1.20) which is directly gotten by integrating (1.9) with respect to x over T 2 .

We now give a result concerning the asymptotic behaviour as ǫ → 0 of the long term model. We notice that, since U and M + ǫ 2 M 2 do not depend on t and x when U ≤ U thr , we have the following property:

∀θ ∈ [0, 1], ∃(t, x) ∈ [0, T ) × T 2 such that U(t, θ, x) = 0 or M(θ, x) + ǫ 2 M 2 (t, θ, x) = 0 =⇒ ∀(t, x) ∈ [0, T ) × T 2 , U(t, θ, x) = 0 and M(θ, x) + ǫ 2 M 2 (t, θ, x) = 0 , (1.21) and {θ ∈ [0, 1], U(•, θ, •) = 0 and M(θ, •) + ǫ 2 M 2 (•, θ, •) = 0}
is an union of several intervals.

(1.22)

Moreover we denote

Θ = [0, T ) × {θ ∈ R, U(•, θ, •) = 0 and M(θ, •) + ǫ 2 M 2 (•, θ, •) = 0} × T 2 , (1.23) and Θ thr = {(t, θ, x) ∈ [0, T ) × R × T 2 , U(t, θ, x) < U thr }. (1.24)
Theorem 1.2 For any T > 0, under the same assumptions as in theorem 1.1, the sequence of solutions (z ǫ ) to equation (1.9) given by theorem 1.1 two-scale converges to a profile

U ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) which is the unique solution to -∇ • ( A∇U ) = ∇ • C on [0, T ) × R × T 2 \Θ, (1.25)
∂U ∂θ = 0 on Θ thr , (1.26)

and 1 0 T 2 U dθ dx = T 2 z 0 dx, (1.27)
where A and C are given by

A = a g a (|U(t, θ, x)|) and C = c g c (|U(t, θ, x)|) U(t, θ, x) |U(t, θ, x)| . (1.28)
Above and in the sequel, for all p ≥ 1 and q ≥ 1, we denote by

L p # (R, L q (T 2 )) = f : R -→ L q (T 2 ) mesurable and periodic of period 1 in θ such that θ → f (θ) L q (T 2 ) ∈ L p ([0, 1]) . Remark 1.1 Notice that [0, T ) × R × T 2 \Θ ∩ Θ thr is not empty. On this set 0 < U < U thr .
This contributes to make of (1.25),(1.26) a well posed problem. Now we turn to mean term model for which we set out asymptotic behaviours.

Theorem 1.3 Under assumptions (1.2), (1.6), (1.7) and (1.8), for any T, not depending on ǫ, the sequence (z ǫ ) of solutions to (1.5) built in [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF] provided with initial condition (1.12) two-scale converges to the profile

U ∈ L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 ))) solution to ∂U ∂θ -∇ • ( A∇U ) = ∇ • C, ( 1
U thr = 0, considering function z ǫ ∈ L ∞ ([0, T ), L 2 (T 2
)), solution to (1.5) with initial condition (1.12) and function

U ǫ ∈ L ∞ ([0, T ), L 2 (T 2 )) defined by U ǫ (t, x) = U (t, t √ ǫ , t ǫ , x), (1.31)
where U is the solution to (1.29), the following estimate is satisfied:

z ǫ -U ǫ √ ǫ L ∞ ([0,T ),L 2 (T 2 )) ≤ α, (1.32)
where α is a constant not depending on ǫ. Furthermore,

z ǫ -U ǫ √ ǫ two-scale converges to a profile U 1 2 ∈ L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 ))), (1.33)
which is the unique solution to

∂U 1 2 ∂θ -∇ • A∇U 1 2 = ∇ • C 1 + ∇ • A 1 ∇U - ∂U ∂τ (1.34)
where A and C are given by (1.30) and where A 1 and C 1 are given by

A 1 (t, τ, θ, x) = -abM(t, τ, θ, x) g a (|U(t, θ, τ, x)|),
and

C 1 (t, τ, θ, x) = -cbM(t, τ, θ, x) g c (|U(t, τ, θ, x)|) U(t, τ, θ, x) |U (t, τ, θ, x)| . (1.35)
2 Existence and estimates, proof of theorem 1.1

Setting:

A ǫ (t, x) = A ǫ (t, t ǫ , x), (2.1)
and

C ǫ (t, x) = C ǫ (t, t ǫ , x), (2.2)
where

A ǫ (t, θ, x) = a(1 -bǫM(t, θ, x)) g a (|U(t, θ, x)|), (2.3)
and

C ǫ (t, θ, x) = c(1 -bǫM(t, θ, x)) g c (|U(t, θ, x)|) U(t, θ, x) |U(t, θ, x)| , (2.4)
equation (1.9) with initial condition (1.12) can be set in the form

   ∂z ǫ ∂t - 1 ǫ 2 ∇ • (A ǫ ∇z ǫ ) = 1 ǫ 2 ∇ • C ǫ , z ǫ |t=0 = z 0 .
(2.5)

Because of hyptothesis (1.9) and under assumptions (1.2) and (1.11), A ǫ and C ǫ given by (2.3) and (2.4) satisfy the following hypotheses:

                       θ -→ ( A ǫ , C ǫ ) is periodic of period 1, x -→ ( A ǫ , C ǫ ) is defined on T 2 , | A ǫ | ≤ γ, | C ǫ | ≤ γ, ∂ A ǫ ∂t ≤ ǫ 2 γ, ∂ C ǫ ∂t ≤ ǫ 2 γ, ∂∇ A ǫ ∂t ≤ ǫ 2 γ, ∂ A ǫ ∂θ ≤ γ, ∂ C ǫ ∂θ ≤ γ, |∇ A ǫ | ≤ ǫγ, |∇ • C ǫ | ≤ ǫγ, ∂∇ • C ǫ ∂t ≤ ǫ 2 γ, (2.6)
             ∃ G thr , θ α < θ ω ∈ [0, 1] such that ∀ θ ∈ [θ α , θ ω ] =⇒ A ǫ (t, θ, x) ≥ G thr , A ǫ (t, θ, x) ≤ G thr =⇒        ∂ A ǫ ∂t (t, θ, x) = 0, ∇ A ǫ (t, θ, x) = 0, ∂ C ǫ ∂t (t, θ, x) = 0, ∇ • C ǫ (t, θ, x) = 0, (2.7)
and

       | C ǫ | ≤ γ| A ǫ |, | C ǫ | 2 ≤ γ| A ǫ |, |∇ A ǫ | ≤ ǫγ| A ǫ |, ∂ A ǫ ∂t ≤ ǫ 2 γ| A ǫ |, ∂(∇ A ǫ ) ∂t 2 ≤ ǫ 2 γ| A ǫ |, ∇ • C ǫ ≤ ǫγ| A ǫ |, ∂ C ǫ ∂t ≤ ǫ 2 γ| A ǫ |, ∂ C ǫ ∂t 2 ≤ ǫ 2 γ 2 | A ǫ |.
(2.8)

In this section we focus on existence and uniqueness of time-space periodic parabolic equations. From this, we then get existence of solution to equation (2.5). Existence of z ǫ over a time interval depending on ǫ, is a traightforward consequence of adaptations of results from LadyzensKaja, Solonnikov and Ural' Ceva [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF] or Lions [START_REF] Lions | Remarques sur les équations différentielles ordinaires[END_REF]. Our aim is to proove that z ǫ solution to (2.5) is bounded indepently of ǫ. We are going to introduce the following regularized equations. We recall that the method used is similar to the one used in [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF].

∂S ν ∂θ - 1 ǫ ∇ • Ãǫ (t, •, •) + ν ∇S ν = 1 ǫ ∇ • Cǫ (t, •, •), (2.9)
and

µS ν µ + ∂S ν µ ∂θ - 1 ǫ ∇ • Ãǫ (t, •, •) + ν ∇S ν µ = 1 ǫ ∇ • Cǫ (t, •, •), (2.10)
where µ and ν are positive parameters. We first prove existence of solutions S ν µ of (2.10) and we give estimates of S ν µ .

Theorem 2.1 Under assumptions (2.6), (2.7) and (2.8), for any µ > 0 and any ν > 0, there exists a unique

S ν µ = S ν µ (t, θ, x) ∈ C 0 ∩ L 2 (R × T 2
), periodic of period 1 with respect to θ, solution to (2.10) and regular with respect to the parameter t. Moreover, the following estimates are satisfied

sup θ∈R T 2 S ν µ (θ, x)dx = 0, (2.11) ∇S ν µ L 2 # (R,L 2 (T 2 )) ≤ γ ν , (2.12
)

∆S ν µ L 2 # (R,L 2 (T 2 )) ≤ √ 2 ǫγ ν γ 2 ν 2 + 1, (2.13) ∂S ν µ ∂θ L 2 # (R,L 2 (T 2 )) ≤ γ √ ǫν ( γ 2ν + 1), (2.14) ∇S ν µ L ∞ # (R,L 2 (T 2 )) ≤ γ 2 ν 2 + 2ǫγ 2 ν ( γ 2 ν 2 + 1), (2.15) S ν µ L ∞ # (R,L 2 (T 2 )) ≤ γ 2 ν 2 + 2ǫγ 2 ν ( γ 2 ν 2 + 1), (2.16) ∂S ν µ ∂t L ∞ # (R,L 2 (T 2 )) ≤ ǫ 3 γ ν (1 + γ ν ).
(2.17)

Proof . (of Theorem 2.1). The proof of this theorem is very similar to the one of Theorem 3.3 of Faye, Frénod and Seck [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF]. The big difference is the presence of 1 ǫ -factors in (2.10). Hence we only sketch the most similar arguments and focus on the management of those 1 ǫ -factors. Integrating equation (2.10) over T 2 gives µ

T 2 S ν µ dx + T 2 ∂S ν µ ∂θ dx - 1 ǫ T 2 ∇ • ( Ãǫ + ν)∇S ν µ dx = 1 ǫ T 2 ∇ • Cǫ dx, (2.18) then µ T 2 S ν µ dx + d( T 2 S ν µ dx) ∂θ = 0, which gives T 2 S ν µ (θ, x)dx = T 2 S ν µ ( θ, x)e -µ(θ-θ) dx.
Since S ν µ is periodic of period 1 with respect to θ, T 2 S ν µ (θ, x)dx is also periodic of period 1. Then (2.11) is true. Multiplying equation (2.10) by S ν µ , integrating over T 2 and from 0 to 1 with respect to θ gives

µ S ν µ 2 L 2 # (R,L 2 (T 2 )) + 1 2 1 0 d S ν µ 2 2 dθ dθ + 1 ǫ 1 0 T 2 ( Ãǫ + ν)|∇S ν µ | 2 dxdθ ≤ γ ǫ 1 0 T 2 |∇S ν µ |dxdθ.
Since Ãǫ + ν ≥ ν and taking into account that the above first term is positive and the second one equals zero, we have ν ǫ

1 0 T 2 |∇S ν µ | 2 dxdθ ≤ γ ǫ ∇S ν µ L 2 # (R,L 2 (T 2 )) , then ∇S ν µ 2 L 2 # (R,L 2 (T 2 )) ≤ γ ν ∇S ν µ L 2 # (R,L 2 (T 2
)) , which gives (2.12). Multiplying (2.10) by ∂S ν µ ∂θ , integrating over T 2 and integrating from 0 to 1 with respect to θ gives

∂S ν µ ∂θ 2 L 2 # (R,L 2 (T 2 )) = 1 2ǫ 1 0 T 2 ∂ Ãǫ ∂θ |∇S ν µ | 2 dxdθ + 1 ǫ 1 0 T 2 ∂ Cǫ ∂θ ∇S ν µ dxdθ (2.19) ≤ γ ǫ 1 2 ∇S ν µ 2 L 2 # (R,L 2 (T 2 )) + ∇S ν µ L 2 # (R,L 2 (T 2 )) , (2.20
|∇S ν µ | 2 dx + T 2 ∇S ν µ • ∇ ∂S ν µ ∂θ dx + 1 ǫ T 2 ∇ Ãǫ • ∇S ν µ ∆S ν µ dx+ 1 ǫ T 2 ( Ãǫ + ν)|∆S ν µ | 2 dx = - 1 ǫ T 2 ∇ • Cǫ ∆S ν µ dx, or µ ∇S ν µ 2 2 + 1 2 d ∇S ν µ 2 2 dθ + 1 ǫ T 2 ( Ãǫ + ν)|∆S ν µ | 2 dx = - 1 ǫ T 2 ∇ Ãǫ • ∇S ν µ ∆S ν µ dx - 1 ǫ T 2 ∇ • Cǫ ∆S ν µ dx.
Since for any real number U and V

|U V | ≤ Ãǫ + ν 4ǫ U 2 + ǫ Ãǫ + ν V 2 , (2.21) using this formula with U = ∆S ν µ , V = ∇ Ãǫ•∇S ν µ ǫ , we have 1 ǫ T 2 ∇ Ãǫ • ∇S ν µ ∆S ν µ dx ≤ T 2 Ãǫ + ν 4ǫ |∆S ν µ | 2 dx + T 2 1 ǫ( Ãǫ + ν) |∇ Ãǫ • ∇S ν µ | 2 dx.
Taking U = ∆S ν µ , V = ∇• Cǫ ǫ and using again (2.21) we obtain

1 ǫ T 2 ∇ • Cǫ ∆S ν µ ≤ T 2 Ãǫ + ν 4ǫ |∆S ν µ | 2 dx + T 2 1 ǫ( Ãǫ + ν) |∇ • Cǫ | 2 dx.
These two results give

µ ∇S ν µ 2 2 + 1 2 d ∇S ν µ 2 2 dθ + 1 ǫ T 2 ( Ãǫ + ν)|∆S ν µ | 2 dx ≤ T 2 Ãǫ + ν 2ǫ |∆S ν µ | 2 dx + 1 ǫ T 2 1 ǫ( Ãǫ + ν) |∇ Ãǫ • ∇S ν µ | 2 + |∇ • Cǫ | 2 dx, (2.22)
or, using (2.6),

µ ∇S ν µ 2 2 + 1 2 d ∇S ν µ 2 2 dθ + T 2 ( Ãǫ + ν) 2ǫ |∆S ν µ | 2 dx ≤ ǫ 2 γ 2 νǫ T 2 |∇S ν µ | 2 dx + 1 , (2.23)
and integrating from 0 to 1 with respect to θ, we have

µ ∇S ν µ L 2 # (R,L 2 (T 2 )) + 1 0 T 2 ( Ãǫ + ν) 2ǫ |∆S ν µ | 2 dxdθ ≤ ǫγ 2 ν 1 0 T 2 |∇S ν µ | 2 dxdθ + 1 .
From this last inequality, we deduce

ν 2ǫ ∆S ν µ 2 L 2 # (R,L 2 (T 2 )) ≤ ǫγ 2 ν γ 2 ν 2 + 1 , then ∆S ν µ 2 L 2 # (R,L 2 (T 2 )) ≤ 2ǫ 2 γ 2 ν 2 γ 2 ν 2 + 1 , which gives (2.13). As ∇S ν µ L 2 # (R,L 2 (T 2 )) is bounded by γ ν (see (2.
12)), we can deduce that there exists a

θ 0 ∈ [0, 1] such that ∇S ν µ (θ 0 , •) 2 ≤ γ ν . (2.24) From (2.23) we have d ∇S ν µ 2 2 dθ ≤ 2ǫγ 2 ν T 2 |∇S ν µ | 2 dx + 1 . (2.25) Integrating (2.25) from θ 0 to an other θ 1 ∈ [0, 1] gives ∇S ν µ (θ 1 , •) 2 2 -∇S ν µ (θ 0 , •) 2 2 ≤ 2ǫγ 2 ν θ1 θ0 T 2 |∇S ν µ | 2 dx + 1 dθ ≤ 2ǫγ 2 ν ∇S ν µ 2 L 2 # (R,L 2 (T 2 )) + 1 , (2.26)
giving the sought bound on

∇S ν µ (θ 1 , •) L ∞ # (R,L 2 (T 2
)) for any θ 1 or, in other words (2.15). Using Fourier expansion argument, because of (2.11), we have

S ν µ (θ, •) 2 2 ≤ ∇S ν µ (θ, •) 2 2 ≤ γ 2 ν 2 + 2 ǫγ 2 ν ( γ 2 ν 2 + 1), (2.27)
and then (2.16).

We have that

∂S ν µ ∂t is solution to µ ∂S ν µ ∂t + ∂ ∂S ν µ ∂t ∂θ - 1 ǫ ∇ • Ãǫ + ν ∇ ∂S ν µ ∂t = 1 ǫ ∇ • ∂ Cǫ ∂t + 1 ǫ ∇ • ∂ Ãǫ ∂t ∇S ν µ , (2.28)
from which we deduce

µ ∂S ν µ ∂t 2 2 + 1 2 d ∂S ν µ ∂t 2 2 dθ + 1 ǫ T 2 Ãǫ + ν ∇ ∂S ν µ ∂t 2 dx = - 1 ǫ T 2 Čǫ • ∇ ∂S ν µ ∂t dx, (2.29)
where

Čǫ = ∂ Cǫ ∂t + ∂ Ãǫ ∂t ∇S ν µ , ∇ • Čǫ = ∇ • ∂ Cǫ ∂t + ∂ Ãǫ ∂t ∇S ν µ .
(2.30) From (2.6), (2.12) and (2.13), we have

Čǫ 2 L 2 # (R,L 2 (T 2 )) ≤ ǫ 2 γ(1 + γ ν ), ∇ • Čǫ L 2 # (R,L 2 (T 2 )) ≤ ǫ 2 γ 1 + γ ν + ǫ √ ǫ γ ν γ ν + 1 . (2.31)
Integrating (2.29) from 0 to 1 with respect to the variable θ, we obtain

ν ǫ ∇ ∂S ν µ ∂t 2 L 2 # (R,L 2 (T 2 )) ≤ ǫ 2 γ(1 + γ ν ) ∇ ∂S ν µ ∂t L 2 # (R,L 2 (T 2 )) , then ∇ ∂S ν µ ∂t L 2 # (R,L 2 (T 2 )) ≤ ǫ 3 γ ν (1 + γ ν ).
Using the Fourier expansion of S ν µ , we have for a given θ 0

∇ ∂S ν µ ∂t (θ 0 , •) 2 ≤ ǫ 3 γ ν (1 + γ ν ).
Thus, as previously, we get

∇ ∂S ν µ ∂t L ∞ # (R,L 2 (T 2 )) ≤ ǫ 3 γ ν (1 + γ ν ), ∂S ν µ ∂t L ∞ # (R,L 2 (T 2 )) ≤ ǫ 3 γ ν (1 + γ ν ).
Since the estimates of theorem 2.1 do not depend on µ, making the process µ → 0 allows us to deduce the following theorem.

Theorem 2.2 Under assumptions (2.6),(2.7) and (2.8), for any ν > 0, there exists a unique S ν = S ν (t, θ, x) ∈ L 2 (R × T 2 ), periodic of period 1 with respect to θ solution to (2.9) and submitted to the constraint sup θ∈R T 2

S ν (θ, x)dx = 0.

(2.32)

Moreover, the following estimates are satisfied

∂S ν ∂θ L 2 # (R,L 2 (T 2 )) ≤ γ √ ǫν ( γ 2ν + 1), ∇S ν L ∞ # (R,L 2 (T 2 )) ≤ γ 2 ν 2 + 2ǫγ 2 ν ( γ 2 ν 2 + 1), (2.33) S ν L ∞ # (R,L 2 (T 2 )) ≤ γ 2 ν 2 + 2ǫγ 2 ν ( γ 2 ν 2 + 1), ∂S ν ∂t L ∞ # (R,L 2 (T 2 )) ≤ ǫ 3 γ ν (1 + γ ν ).
(2.34)

Proof . (of Theorem 2.2). As estimates of Theorem 2.1 do not depend on µ, to proof existence of S ν , it suffices to make µ tend to 0 in (2.10). Uniqueness is insured by (2.32), once noticed that, if S ν and S ν are two solutions of (2.9), with constraint (2.32), S ν -S ν is solution to

∂(S ν -S ν ) ∂θ - 1 ǫ ∇ • (( A ǫ + ν)∇(S ν -S ν )) = 0, (2.35)
from which we can deduce that

ν ∇(S ν -S ν ) 2 L 2 # (R,L 2 (T 2 )) = 0, (2.36)
and because of (2.32), and its consequence:

S ν -S ν L 2 # (R,L 2 (T 2 )) ≤ ∇(S ν -S ν ) L 2 # (R,L 2 (T 2 )) , (2.37)
that S ν = S ν .

(2.38)

Now we get estimates on S ν which do not depend on ν.

Theorem 2.3 Under the assumptions (2.6),(2.7) and (2.8), the solution S ν , of (2.9) given by theorem 2.2 satisfies the following properties

A ǫ |∇S ν | L 2 # (R,L 2 (T 2 )) ≤ γ, (2.39) θω θα T 2 |∇S ν | 2 dxdθ 1/2 ≤ γ G thr , (2.40) ∇S ν (θ 0 , •) 2 ≤ γ G thr , for a given θ 0 ∈ [θ α , θ ω ], (2.41)
S ν 2 L ∞ # (R,L 2 (T 2 )) ≤ γ Gthr + 2ǫγ 3 , (2.42)
∂S ν ∂t 2 L ∞ # (R,L 2 (T 2 )) ≤ ǫ γ + ǫγ 3 G thr + (γ 2 + ǫ 2 γ 4 ) . (2.43)
Proof . (of Theorem 2.3). Multiplying (2.9) by S ν and integrating over T 2 yields 1 2

d dθ T 2 |S ν | 2 dx + 1 ǫ T 2 ( A ǫ + ν)|∇S ν | 2 dx = - 1 ǫ T 2 C ǫ • ∇S ν dx. (2.44) Integrating (2.44) in θ over [0, 1] gives 1 ǫ 1 0 T 2 ( A ǫ + ν)|∇S ν | 2 dx ≤ γ ǫ 1 0 T 2 A ǫ |∇S ν |dx, (2.45)
then we obtain (2.39). Assuming (2.7), we have

G thr θω θα T 2 |∇S ν | 2 dxdθ 1/2 ≤ θω θα T 2 A ǫ |∇S ν | 2 dxdθ 1 2 ≤ A ǫ |∇S ν | L 2 # (R,L 2 (T 2 ))
.

(2.46) From (2.39) and this last inequality we get (2.40). Then, there exists a θ 0 ∈ [θ α , θ ω ] such that S ν satisfies (2.41).

Using the Fourier expansion of S ν and the relation (2.32) we get

S ν (θ 0 , •) 2 ≤ ∇S ν (θ 0 , •) 2 ≤ γ Gthr .
(2.47) Multiplying (2.9) by S ν , integrating over T 2 we obtain

1 2 d S ν (θ, •) 2 2 dθ + 1 ǫ T 2 ( A ǫ + ν)|∇S ν (θ, •)| 2 dx = 1 ǫ T 2 |∇ • C ǫ S ν (θ, •)|dx Applying formula (2.21) with V = |∇• Cǫ| ǫ and U = |S ν |, we get 1 2 d S ν (θ, •) 2 2 dθ + 1 ǫ T 2 ( A ǫ + ν)|∇S ν (θ, •)| 2 dx ≤ T 2 ( A ǫ + ν) 4ǫ |S ν (θ, •)| 2 + 1 ǫ( A ǫ + ν) |∇ • C ǫ | 2 dx, which gives 1 2 d S ν (θ, •) 2 2 dθ + 1 ǫ T 2 ( A ǫ + ν) |∇S ν (θ, •)| 2 - |S ν (θ, •)| 2 4 dx ≤ T 2 1 ǫ( A ǫ + ν) |∇ • C ǫ | 2 dx. (2.48)
Using Fourier expansion of S ν (θ, •), one can prove that the second term of the left hand side of (2.48) is positive, then we have

d S ν (θ, •) 2 2 dθ ≤ 2 T 2 1 ǫ( A ǫ + ν) |∇ • C ǫ | 2 dx. (2.49)
Using (2.6), (2.8) and integrating (2.49) from θ 0 to θ ∈ [0, 1], we obtain

S ν (θ, •) 2 2 ≤ S ν (θ 0 , •) 2 2 + 2ǫγ 3 , (2.50)
then inequality (2.42) is satisfied. Using inequality (2.39) and from hypothesis (2.8) we get

∂(∇ A ǫ ) ∂t ∇S ν L 2 # (R,L 2 (T 2 )) ≤ ǫ 2 γ A ǫ ∇S ν L 2 # (R,L 2 (T 2 )) ≤ ǫ 2 γ 2 .
(2.51) Multiplying (2.9) by -∆S ν and integrating in x ∈ T 2 we get 1 2

d dθ ∇S ν 2 2 + 1 ǫ T 2 A ǫ + ν |∆S ν | 2 dx + 1 ǫ T 2 ∇ A ǫ • ∇S ν ∆S ν dx = - 1 ǫ T 2 ∇ • C ǫ • ∆S ν dx. (2.52) Using (2.21) with U = |∆S ν | and V = ∇ Aǫ•∇S ν ǫ and with U = |∆S ν | and V = ∇• Cǫ ǫ , the equality (2.52) becomes 1 2 d dθ ∇S ν 2 2 + 1 2ǫ T 2 ( A ǫ + ν)|∆S ν | 2 dx ≤ 1 ǫ T 2 |∇ A ǫ | 2 A ǫ + ν |∇S ν | 2 + |∇ C ǫ | 2 A ǫ + ν dx, (2.53)
which, integrating from 0 to 1 yields

1 0 T 2 A ǫ |∆S ν | 2 dxdθ ≤ 2ǫγ 2 1 0 T 2 | A ǫ ||∇S ν | 2 dxdθ + γ ≤ 2ǫγ 2 (γ 2 + γ). (2.54) As ∂ A ǫ ∂t ≤ ǫ 2 γ| A ǫ |, (2.55)
we obtain

∂ A ǫ ∂t ∆S ν L 2 # (R,L 2 (T 2 )) ≤ ǫ √ 2ǫγ 2 1 + γ. (2.56)
Now we set out the equation to which ∂S ν ∂t is solution. We have

∂ ∂θ ∂S ν ∂t = ∂ ∂t ∂S ν ∂θ = 1 ǫ ∇ • ∂ A ǫ ∂t ∇S ν + ( A ǫ + ν)∇ ∂S ν ∂t + 1 ǫ ∇ • ∂ A ǫ ∂t , then ∂S ν ∂t is solution to ∂ ∂θ ∂S ν ∂t - 1 ǫ ∇ • ( A ǫ + ν)∇ ∂S ν ∂t = 1 ǫ ∇ • ∂ A ǫ ∂t ∇S ν + 1 ǫ ∇ • ∂ C ǫ ∂t . ( 2
+ 1 ǫ T 2 ( A ǫ + ν) ∇ ∂S ν ∂t 2 dx ≤ 1 ǫ T 2 ∂ A ǫ ∂t |∇S ν | ∇ ∂S ν ∂t dx + 1 ǫ T 2 ∂ C ǫ ∂t ∇ ∂S ν ∂t dx.
(2.58)

Using the fact that ∂ Cǫ ∂t 2 ≤ ǫ 2 γ 2 A ǫ , the second term of the right hand side of (2.58) satisfies

T 2 ∂ C ǫ ∂t ∇ ∂S ν ∂t dx ≤ ǫγ T 2 A ǫ ∇ ∂S ν ∂t dx ≤ ǫγ A ǫ ∇ ∂S ν ∂t 2 .
(2.59)

In the same way, using (2.8) we deduce the following estimate for the first term of the right hand side of (2.58)

T 2 ∂ A ǫ ∂t |∇S ν | ∇ ∂S ν ∂t dx ≤ ∂ A ǫ ∂t |∇S ν | 2 ∂ A ǫ ∂t ∇ ∂S ν ∂t 2 ≤ ǫ 2 γ 2 A ǫ |∇S ν | 2 A ǫ ∇ ∂S ν ∂t 2 . ( 2
( A ǫ + ν) ∇ ∂S ν ∂t 2 L 2 # (R,L 2 (T 2 )) ≤ ǫγ A ǫ ∇ ∂S ν ∂t L 2 # (R,L 2 (T 2 )) +ǫ 2 γ 3 A ǫ ∇ ∂S ν ∂t L 2 # (R,L 2 (T 2 ))
.

(2.61)

From this last inequality, we deduce

A ǫ ∇ ∂S ν ∂t L 2 # (R,L 2 (T 2 )) ≤ ǫ(γ + ǫγ 3 ), ( 2
∇ ∂S ν ∂t (θ 0 , •) 2 ≤ ǫ γ + ǫγ 3 G thr , (2.64)
and, since the mean value of ∂S ν ∂t (θ 0 , •) is zero,

∂S ν ∂t (θ 0 , •) 2 ≤ ǫ γ + ǫγ 3 G thr . (2.65)
To end the proof of the theorem it remains to show that

∂S ν ∂t is bounded independently of ν in L ∞ # (R, L 2 (T 2 )).
∂ ∂S ν ∂t 2 2 ∂θ + T 2 A ǫ + ν 2ǫ ∇ ∂S ν ∂t 2 dx ≤ T 2 ∂ C ǫ ∂t 2 ǫ( A ǫ + ν) dx + T 2 ∂ A ǫ ∂t 2 |∇S ν | 2 ǫ( A ǫ + ν) dx ≤ ǫγ 2 + ǫ 3 γ 2 T 2 A ǫ |∇S ν | 2 dx, (2.66)
where we used hypothesis (2.8) to get the last inequality. Integrating this last formula in θ over [θ 0 , σ] for any σ > θ 0 , we obtain, always remembering (2.39),

∂S ν ∂t (σ, •) 2 2 ≤ ∂S ν ∂t (θ 0 , •) 2 2
+ ǫ(γ 2 + ǫ 2 γ 4 ).

(2.67)

From inequality (2.67) we obtain directly the inequality of (2.43), using the periodicity of S ν .

Estimates (2.42) and (2.43) given in theorem 2.3 do not depend on ν. Making ν → 0, allows us to deduce that, up to a subsequence S ν -→ S ∈ L ∞ # (R, L 2 (T 2 )) weak - * . Concerning the limit S we have the following theorem.

Theorem 2.4 Under assumptions (2.6), (2.7), (2.8), there exists a unique function

S = S(t, θ, x) ∈ L ∞ # (R, L 2 (T 2
)), periodic of period 1 with respect to θ, solution to

∂S ∂θ - 1 ǫ ∇ • ( A ǫ (t, •, •)∇S) = 1 ǫ ∇ • C ǫ (t, •, •), (2.68)
and satisfying, for any

t, θ ∈ R + × R T 2 S(t, θ, x)dx = 0. (2.69) Moreover it satisfies: S 2 L ∞ # (R,L 2 (T 2 )) ≤ γ G thr + 2ǫγ 3 , (2.70) ∂S ∂t 2 L ∞ # (R,L 2 (T 2 )) ≤ ǫ γ + ǫγ 3 G thr + (γ 2 + ǫ 2 γ 4 ) . (2.71)
Proof . (of Theorem 2.4). Uniqueness of S is not gotten via the above evoked process ν -→ 0, but directly comes from (2.68). Assuming that there are two solutions S 1 and S 2 to (2.68), we easily deduce that

d S 1 -S 2 2 2 dθ + 1 ǫ T 2 A ǫ |∇(S 1 -S 2 )| 2 dx = 0, ( 2
A ǫ |∇(S 1 -S 2 )| 2 ≡ 0, (2.74) or, for any θ ∈ R, S 1 (θ + 1, •) -S 2 (θ + 1, •) 2 2 < S 1 (θ, •) -S 2 (θ, •) 2 2 .
(2.75)

As (2.75) is not possible because of the periodicity of S 1 and S 2 , we deduce that (2.74) is true. Using this last information, we deduce, for instance

∇(S 1 -S 2 )(θ ω , •) ≡ 0, (2.76)
yielding, because of property (2.69),

(S 1 -S 2 )(θ ω , •) 2 2 ≤ ∇(S 1 -S 2 )(θ ω , •) 2 2 .
(2.77)

Injecting (2.74) in (2.72) yields d S 1 -S 2 2 2 dθ = 0, (2.78)
and then (S 1 -S 2 )(θ, •)

2 2 = 0, (2.79)
for any θ ≥ θ ω and consequently or any θ ∈ R. This ends the proof of theorem 2.4.

With this theorem on hand we can get the following result concerning z ǫ solution of equation (2.5).

Theorem 2.5 Under properties (2.6), (2.7), (2.8), for any T, not depending on ǫ, equation (2.5), with coefficients given by (2.1) coupled with (2.3) and (2.2) coupled with (2.4) has a unique solution

z ǫ ∈ L ∞ ([0, T ]; L 2 (T 2 )
). This solution satisfies:

z ǫ L ∞ ([0,T ],L 2 (T 2 )) ≤ γ (2.80)
where γ is a constant which do not depend on ǫ.

Proof (of Theorem 1.1). Theorem 1.1 is a direct consequence of theorem 2.5.

Proof . (of Theorem 2.5). To prove uniqueness, we consider z ǫ 1 and z ǫ 2 two solutions of (2.5). Their difference is then solution to

   ∂(z ǫ 1 -z ǫ 2 ) ∂t - 1 ǫ 2 ∇ • A ǫ ∇(z ǫ 1 -z ǫ 2 ) = 0, (z ǫ 1 -z ǫ 2 ) |t=0 = 0, (2.81)
and multiplying the first equation of (2.81) by (z ǫ 1 -z ǫ 2 ) and integrating with respect to x gives

d z ǫ 1 -z ǫ 2 2 2 dt ≤ 0, (2.82) yielding z ǫ 1 -z ǫ 2 2 = 0, for any t, (2.83)
and giving uniqueness.

Existence of z ǫ is a straightforward of adaptations of results of Ladyzenskaja, Sollonnikov and Ural' Ceva [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF] or Lions [START_REF] Lions | Remarques sur les équations différentielles ordinaires[END_REF] on a time interval of length ǫ. Then, let us consider the function

Z ǫ = Z ǫ (t, x) = S(t, t ǫ , x)
where S is solution to (2.69). We obtain

∂Z ǫ ∂t = ∂S ∂t (t, t ǫ , x) + 1 ǫ ∂S ∂θ (t, t ǫ , x) (2.84)
Using equation (2.68) we deduce that Z ǫ is solution to

∂Z ǫ ∂t - 1 ǫ 2 ∇ • Ãǫ ∇Z ǫ = 1 ǫ 2 ∇ • Cǫ + ∂S ∂t (2.85)
then we deduce that

   ∂(z ǫ -Z ǫ ) ∂t - 1 ǫ 2 ∇ • Ãǫ ∇(z ǫ -Z ǫ ) = ∂S ∂t (z ǫ -Z ǫ ) |t=0 = z 0 -S(0, 0, x).
(2.86) Multiplying (2.86) by z ǫ -Z ǫ and integrating over T 2 , we have

d z ǫ -Z ǫ 2 2 dt + 1 ǫ 2 T 2 Ãǫ |∇(z ǫ -Z ǫ | 2 dx = T 2 ∂S ∂t (z ǫ -Z ǫ )dx (2.87) which gives d z ǫ -Z ǫ 2 2 dt ≤ ǫ γ + ǫγ 3 G thr + (γ 2 + ǫ 2 γ 4 ) z ǫ -Z ǫ 2 .
(2.88)

Then we have

z ǫ (t, •) -Z ǫ (t, •) 2 2 ≤ z 0 -S(0, 0, x) 2 ǫ γ + ǫγ 3 G thr + (γ 2 + ǫ 2 γ 4 ) T. (2.89) As S L ∞ # (R,L 2 (T 2 )) ≤ γ √ G thr
when ǫ → 0, then (2.80) is true.

3 Homogenization for long term dynamics of dunes, proof of theorem 1.2

We consider equation (2.5) where A ǫ and C ǫ are defined by formulas (2.1) coupled with (2.3) and (2.2) coupled with (2.4). Our aim consists in deducing the equations satisfied by the limit of z ǫ solution to (2.5) as ǫ -→ 0.

It is obvious that

A ǫ (t, x) two-scale converges to A(t, θ, x) ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) and C ǫ (t, x) two-scale converges to C(t, θ, x), (3.1)
with

A(t, θ, x) = a g a (|U(t, θ, x)|) and C(t, θ, x) = c g c (|U(t, θ, x)|) U(t, θ, x) |U(t, θ, x)| . (3.2)
Assumptions (1.23) and (1.24) have the following equivalence here:

Θ = [0, T ) × {θ ∈ R : A(•, θ, •) = 0} × T 2 , (3.3)
and

Θ thr = {(t, θ, x) ∈ [0, T ) × R × T 2 such that A(t, θ, x) < G thr }. (3.4)
Moreover, we notice that because of ( 1) and ( 1)

A(t, θ, x) = 0 if and only if (t, θ, x) ∈ Θ. (3.5)
We have the following theorem.

∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) solution to -∇ • ( A∇U ) = ∇ • C on [0, T ) × R × T 2 \Θ, (3.6
)

∂U ∂θ = 0 on Θ thr , (3.7
)

1 0 T 2 U dθ dx = T 2 z 0 dx, (3.8)
where A and C are given by (3.2); Θ and Θ thr are given by (3.3) and (3.4).

Proof . (of Theorem 1.2). Theorem 1.2 is a direct consequence of theorem 3.1 Proof . (of Theorem 3.1). Multiplying (2.5) by ψ ǫ (t, x) = ψ(t, t ǫ , x) regular with compact support in [0, T ) × T 2 and 1-periodic in θ, we obtain

- T 2 z 0 (x)ψ(0, 0, x)dx - T 2 T 0 ∂ψ ǫ ∂t z ǫ dt dx+ 1 ǫ 2 T 2 T 0 A ǫ ∇z ǫ ∇ψ ǫ dt dx = 1 ǫ 2 T 2 T 0 ∇ • C ǫ ψ ǫ dx. ( 3
we obtain

T 2 T 0 ∂ψ ∂t ǫ + 1 ǫ ∂ψ ∂θ z ǫ dt dx + 1 ǫ 2 T 2 T 0 z ǫ ∇ • A ǫ ∇ψ ǫ dt dx + 1 ǫ 2 T 2 T 0 ∇ • C ǫ ψ ǫ dt dx = - T 2 z 0 (x)ψ(0, 0, x)dx (3.12)
Multiplying by ǫ 2 and using the two-scale convergence due to Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF], Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], Frénod, Raviart and Sonnendrucker [START_REF] Frénod | Asymptotic expansion of the Vlasov equation in a large external magnetic field[END_REF], as z ǫ is bounded in L ∞ (0, T, L 2 (T 2 )), there exists a profile U (t, θ, x), periodic of period 1 with respect to θ, such that for all ψ(t, θ, x), regular with compact support with respect to (t, x) and periodic of period 1 with respect to θ, we have

- T 2 T 0 1 0 U ∇ • Ã∇ψ dθ dt dx = T 2 T 0 1 0 ∇ • C ψ dθ dt dx, (3.13)
then

-∇ • Ã∇U = ∇ • C, (3.14)
with à and C given by (3.2).

Since A and C vanish on Θ, we deduce (3.6) from (3.14). Moreover, because of (2.7), in points where A(t, θ, x) < G thr , ∇ • C = 0 and A does not depend on t and x. Hence U depends only on θ. In other words, U (t, θ, x) = U (θ) on Θ thr .

(3.15)

Taking now test functions ψ not depending on x in (3), the two last terms of the left hand side of (3.12) vanish. Then passing to the limit, we obtain the weak formulation of

∂ T 2 U (t, θ, x)dx ∂θ = 0 (3.16)
which yields because of (3.15) ∂U ∂θ = 0 on Θ thr .

(3.17)

Finally, taking test function ψ depending only on t we obtain

1 0 T 2 U (t, θ, x)dθ dx = T 2 z 0 (x)dx, (3.18)
ending the proof of the theorem.

4 Homogenization and corrector result for mean-term dynamics of dunes, proof of theorem 1.3 and 1.4

Making the same as in the begining of section 2, setting:

A ǫ (t, x) = A ǫ (t, t √ ǫ , t ǫ , x), (4.1)
and

C ǫ (t, x) = C ǫ (t, t √ ǫ , t ǫ , x), (4.2)
where

A ǫ (t, τ, θ, x) = a(1 -b √ ǫM(t, τ, θ, x)) g a (|U(t, τ, θ, x)|), (4.3)
and

C ǫ (t, τ, θ, x) = c(1 -b √ ǫM(t, τ, θ, x)) g c (|U(t, τ, θ, x)|) U(t, τ, θ, x) |U(t, τ, θ, x)| , (4.4)
equation (1.5) with initial condition (1.12) can be set in the form

   ∂z ǫ ∂t - 1 ǫ ∇ • (A ǫ ∇z ǫ ) = 1 ǫ ∇ • C ǫ , z ǫ |t=0 = z 0 . (4.5)
Under assumptions (1.2) and (1.8), A ǫ and C ǫ given by (4.3) and (4.4) satisfy the following hypotheses:

                             τ -→ ( A ǫ , C ǫ ) is periodic of period 1, θ -→ ( A ǫ , C ǫ ) is periodic of period 1, x -→ ( A ǫ , C ǫ ) defined on T 2 , | A ǫ | ≤ γ, | C ǫ | ≤ γ, ∂ A ǫ ∂t ≤ γ, ∂ C ǫ ∂t ≤ γ, ∂∇ A ǫ ∂t ≤ γ. ∂ A ǫ ∂θ ≤ γ, ∂ C ǫ ∂θ ≤ γ, |∇ A ǫ | ≤ γ, |∇ • C ǫ | ≤ γ, ∂∇ • C ǫ ∂t ≤ γ, (4.6)
             ∃ G thr , θ α < θ ω ∈ [0, 1] such that ∀ θ ∈ [θ α , θ ω ] =⇒ A ǫ (t, τ, θ, x) ≥ G thr , A ǫ (t, τ, θ, x) ≤ G thr =⇒        ∂ A ǫ ∂t (t, τ, θ, x) = ∂ A ǫ ∂τ (t, τ, θ, x) = 0, ∇ A ǫ (t, τ, θ, x) = 0, ∂ C ǫ ∂t (t, τ, θ, x) = ∂ C ǫ ∂τ (t, τ, θ, x) = 0, ∇ • C ǫ (t, τ, θ, x) = 0, (4.7)
             | C ǫ | ≤ γ| A ǫ |, | C ǫ | 2 ≤ γ| A ǫ |, |∇ A ǫ | ≤ γ| A ǫ |, ∂ A ǫ ∂t ≤ γ| A ǫ |, ∂(∇ A ǫ ) ∂t 2 ≤ γ| A ǫ |, ∇ • C ǫ ≤ γ| A ǫ |, ∂ C ǫ ∂t ≤ γ| A ǫ |, ∂ C ǫ ∂t 2 ≤ γ 2 | A ǫ | ∂ A ǫ ∂τ 2 ≤ ǫγ| A ǫ |, ∂∇ A ǫ ∂τ 2 ≤ ǫγ| A ǫ |. (4.8)
For (4.5), if hypotheses (4.6), (4.7) and (4.8) are satisfied, an existence and uniqueness result is given in [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF].

Homogenization

Let us consider equation (4.5) with A ǫ and C ǫ given by (4.1) and (4.2); θ -→ A, C is periodic of period 1, τ -→ A, C is periodic of period 1, (4.9)

A ǫ (t, x) 3-scale converges to A(t, τ, θ, x) ∈ L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 ))) and C ǫ (t, x) 3-scale converges to C(t, τ, θ, x), ( 4
∈ L ∞ ([0, T ]×R, L ∞ # (R, L 2 (T 2 ))) solution to ∂U ∂θ -∇ • ( A∇U ) = ∇ • C, (4.12)
where A and C are given by (4.11).

Proof . (of Theorem 1.3 z 0 (x)ψ(0, 0, 0, x)dx -

T 2 T 0 ∂ψ ǫ ∂t z ǫ dt dx - 1 ǫ T 2 T 0 z ǫ ∇ • A ǫ ∇ψ ǫ dt dx = 1 ǫ T 2 T 0 ∇ • C ǫ ψ ǫ dt dx.
Replacing ∂ψ ǫ ∂t by the relation (4.13), we have

T 2 T 0 z ǫ ∂ψ ∂t ǫ + 1 √ ǫ ∂ψ ∂τ ǫ + 1 ǫ ∂ψ ∂θ ǫ dt dx + 1 ǫ T 2 T 0 z ǫ ∇ • A ǫ ∇ψ ǫ dt dx + 1 ǫ T 2 T 0 ∇ • C ǫ ψ ǫ dt dx = - T 2
z 0 (x)ψ(0, 0, 0, x)dx.

Multiplying by ǫ we have

T 2 T 0 z ǫ ǫ ∂ψ ∂t ǫ + √ ǫ ∂ψ ∂τ ǫ + ∂ψ ∂θ ǫ + ∇ • A ǫ ∇ψ ǫ dt dx+ T 2 T 0 ∇ • C ǫ ψ ǫ dt dx = -ǫ T 2
z 0 (x)ψ(0, 0, 0, x)dx.

The functions ∂ψ ∂t ǫ , ∂ψ ∂τ ǫ and ∂ψ ∂θ ǫ are periodic with respect to the two variables τ, θ. Here we use the 3-scales convergence see [START_REF] Lions | Reiterated homogenization of monotone operators[END_REF].

Taking the limit as ǫ → 0, using the 3-scales convergence, we have There is indeed existence and uniqueness of the equation (4.15) according to the application of the theorem 3.15 of [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF]; thus (4.15) is the homogenized equation. In (4.15), τ and t are only parameters. ))) defined by U ǫ (t, x) = U (t, t √ ǫ , t ǫ , x), where U is the solution to (4.12), the following estimate is satisfied:

A corrector result

z ǫ -U ǫ √ ǫ L ∞ ([0,T ),L 2 (T 2 )) ≤ α, (4.21)
where α is a constant not depending on ǫ. Furthermore

∂z ǫ ∂t - 1 ǫ ∇ • A ǫ ∇z ǫ = 1 ǫ ∇ • C ǫ + √ ǫ∇ • C 1 + ǫ∇ • C 2 + √ ǫ∇ • A ǫ 1 ∇z ǫ + ǫ∇ • A ǫ 2 ∇z ǫ . (4.24)
As U is solution to (4.12) and taking into account that

∂U ǫ ∂t = ∂U ∂t ǫ + 1 √ ǫ ∂U ∂τ ǫ + 1 ǫ ∂U ∂θ ǫ , (4.25)
we obtain the following equation

∂U ǫ ∂t - 1 ǫ ∇ • A ǫ ∇U ǫ = 1 ǫ ∇ • C ǫ + √ ǫ ∂U ∂τ ǫ + ǫ ∂U ∂t ǫ . (4.26)
Considering equation (4.24) and (4.26), z ǫ -U ǫ is solution to

∂( z ǫ -U ǫ √ ǫ ) ∂t - 1 ǫ ∇• A ǫ + √ ǫ A ǫ 1 +ǫ A ǫ 2 ∇( z ǫ -U ǫ √ ǫ ) = 1 ǫ ∇• C ǫ 1 + √ ǫ∇• C ǫ 2 +∇• A ǫ 1 ∇U ǫ + √ ǫ∇• A ǫ 2 ∇U ǫ - √ ǫ ∂U ∂t ǫ - ∂U ∂τ . ( 4

.27)

Using the fact that U solution to (4.12) belongs to L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 ))), U ǫ is solution to (4.26) and a results of Ladyzenskaja, Solonnikov and Ural'Ceva [START_REF] Ladyzenskaja | Linear and quasi-linear equations of parabolic type[END_REF], all the terms ∂U ∂τ , ∂U ∂t are bounded. The terms A ǫ 1 , A ǫ 2 , C ǫ 1 and C ǫ 2 are also bounded by hypotheses and then so are ∇• C ǫ 1 , ∇• C ǫ 2 and ∇ • A 1 ∇U ǫ , ∇ • A 2 ∇U ǫ . Using the same arguments as in the proof of Theorem 1.1 in [START_REF] Faye | Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment[END_REF] we obtain that z ǫ -U ǫ √ ǫ converges to a profile

U 1 2 ∈ L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 ))) solution to ∂U 1 2 ∂θ -∇ • A∇U 1 2 = ∇ • C 1 + ∇ • A 1 ∇U - ∂U ∂τ .
(4.28)

2 C

 2 • ∇ψ dτ dθ dt dx.Then, the limit U of z ǫ solution to (2.5) satisfies the following equation∂U ∂θ -∇ • Ã∇U = ∇ • C. (4.15)

Theorem 4 . 2

 42 Under assumptions (4.6), (4.7), (4.8),(4.1), (4.10) and (4.11), considering functionz ǫ ∈ L ∞ ([0, T ), L 2 (T 2 )), solution to (2.5) and function U ǫ ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2

z ǫ -U ǫ √ ǫ 3 - 2 =

 32 scale converges to a profile U 1 2 ∈ L ∞ ([0, T ] × R, L ∞ # (R, L 2 (T 2 )∇ • C 1 + ∇ • A 1 ∇U -∂U ∂τ . (4.23) Proof . (of Theorem1.4). Theorem 1.4 is a direct consequence of theorem 4.2. Proof . (of Theorem 4.2). Using the relations (4.18), (4.19) and (4.20), equation (4.5) becomes

  of period 1,

	|U|, |	∂U ∂t	|, |	∂U ∂θ	|, |∇U|, |U 1 |, | |M|, | ∂M ∂U 1 ∂t ∂t |, | |, | ∂M ∂U 1 ∂τ ∂θ |, | |, | ∂M ∂U 1 ∂θ ∂τ |, |∇M| are bounded by d, |, |∇U 1 |,

  For this we will estimate the right hand side of (2.58) by applying formula (2.21) with V = 1 ǫ | ∂ Cǫ ∂t | and U = |∇ ∂S ν ∂t | to treat the second term of the right hand side of (2.58) and with V = 1 ǫ | ∂ Aǫ ∂t ||∇S ν | and U = |∇ ∂S ν

∂t | to treat the first. It gives: 1 2

  regular with compact support on [0, T ) × T 2 and periodic of period 1 with respect to τ and θ.

								∂ψ ǫ ∂t	=	∂ψ ∂t	ǫ +	1 √ ǫ	∂ψ ∂τ	ǫ +	1 ǫ	∂ψ ∂θ	ǫ ,				(4.13)
	where
	∂ψ ∂t	ǫ (t, x) =	∂ψ ∂t	(t,	t √ ǫ	,	t ǫ	, x),		∂ψ ∂τ	ǫ =	∂ψ ∂τ	(t,	t √ ǫ	,	t ǫ	, x),	∂ψ ∂θ	ǫ =	∂ψ ∂θ	(t,	t √ ǫ	,	t ǫ	, x).
																									(4.14)
	Multiplying (4.5) by ψ ǫ (t, t √ ǫ , t

). Theorem 1.3 is a direct consequence of theorem 4.1. Proof . (of Theorem 4.1). Considering test functions ψ ǫ (t, x) = ψ(t, t √ ǫ , t ǫ , x) for all ψ(t, τ, θ, x) ǫ , x) and integrating on [0, T ) × T 2 , we get

-T 2

  Because of hypotheses (4.6), (4.7) and (4.8), A, A 1 , A 2 , A ǫ , A ǫ 1 , A ǫ 2 , C, C 1 , C 2 , C ǫ , C ǫ 1 and C ǫ

						√	ǫ C ǫ 1 (t, x) + ǫ C ǫ 2 (t, x)				(4.17)
	where	A ǫ (t, x) = A(t,	t √ ǫ	,	t ǫ	, x),	C ǫ (t, x) = C(t,	t √ ǫ	,	t ǫ	, x)	(4.18)
		A ǫ 1 (t, x) = A 1 (t,	t √ ǫ	,	t ǫ	, x),	C ǫ 1 (t, x) = C 1 (t,	t √ ǫ	,	t ǫ	, x)	(4.19)
		A ǫ 2 (t, x) = A 2 (t,	t √ ǫ	,	t ǫ	, x),	C ǫ 2 (t, x) = C 2 (t,	t √ ǫ	,	t ǫ	, x)	(4.20)

Considering equation (4.5) with coefficients (4.1) and (4.2) and hypothesis (4.10) leads to

A ǫ (t, x) = A ǫ (t, x) + √ ǫ A ǫ 1 (t, x) + ǫ A ǫ 2 (t, x),

(4.16)

C ǫ (t, x) = C ǫ (t, x) + 2 are regular and bounded coefficients.

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emmanuel.frenod@univ-ubs.fr

diaraf.seck@ucad.edu.sn