Thomas Le Gallic
email: thomas.le_gallic@mines-paristech.fr

Edi Assoumou
email: edi.assoumou@mines-paristech.fr

Nadia Maïzi
email: nadia.maizi@mines-paristech.fr

Investigating long-term lifestyles changes in France: a statistical and modelling approach

Keywords: lifestyles, consumption, energy consumption, foresight, transition

, they are both a broad and complex object of thought and a key determinant of the sustainability of our societies. Conducted in a qualitative way, the analysis of future lifestyles offers great freedom to imagine dramatic changes of societies and to explore paradigm shifts. These changes could result from the widespread of various existing behaviours considered today as weak signals or from the emergence of social movements of great magnitude. Performing a quantitative analysis is less straightforward and could be conceptually questioned. Yet for the specific problem of sustainable lifestyles a quantitative approach can help ground the scenarios not only on technical solutions but also on elements of lifestyles change. What is missing is a quantitative method to define the possible long term impact of lifestyle change on mobility, housing and consumption patterns. In this paper we introduce a statistical model that we developed especially to address long-term lifestyles changes and their consequences on the consumption of goods and services and on the energy services demand in France. We make use of national mobility housing and consumption surveys to identify significant patterns. We propose a model of their diffusion in the long term according while also taking into account the demographic changes. Our contribution is organised as follows: we first describe the statistical model and the surveys; then an application to future societal trends is proposed in a prospective approach. A set of lifestyles anticipated for France in 2050 that explore various changes are considered. Finally we aim to discuss the contributions and limitations of the proposed quantitative model and how it can fit into futures thinking.

Introduction

Since the Rio Earth Summit in 1992 there has been an increased recognition of the role of societal changes in achieving a sustainable society. A typical quantitative technical and economic analysis will stress factors such as efficiency, carbon content of energy flows and costs or GDP; whereas a societal and lifestyle perspective will put the emphasis on the potential or barriers to a wider diffusion of sustainable practices in a society, and seek a stronger engagement of consumers or citizens to reduce the footprint of our lifestyles while avoid adverse rebound effects. Just as lifestyles relate to our ways of "doing", "having", "using" and "displaying", our behaviour and all of the related products, objects and infrastructures [START_REF] Røpke | The role of consumption in global warming -an ecological economic perspective[END_REF], they are the logical background of our ordinary energy uses and yet remain a broad and complex object of thought. As such understanding they role as a levy towards more sustainable society is essential. Furthermore, when looking forward by 2050 and beyond we are not irreversibly locked in an unchanging set of social patterns or lifestyle: the future decades could either be a movement towards a more energy intensive techno-society or a society with different degrees of sobriety. Conducted in a qualitative way, the analysis of future lifestyles thus offers great freedom to imagine dramatic changes of societies and to explore paradigm shifts. These changes could result from the widespread of various existing behaviours considered today as marginal or emerge from original social movements of great magnitude.

Yet a scientific and systematic treatment of lifestyle remains an emerging research field that encompasses several disciplines and multiple concepts. Starting from the abundance of the literature on consumption, [START_REF] Jackson | Live Better by Consuming Less?: Is There a "Double Dividend[END_REF] challenges the idea that consuming more always increases the welfare against the alternative understanding of consumption as a "social pathology". Beyond a criticism of "utilitarism" this work proposes grounds for a strong rehabilitation of a more complex understanding of the formation of needs than what is embedded in conventional consumers' preferences. The SEI (Kate [START_REF] Scott | Literature Review on Sustainable Lifestyles and Recommendations for Further Research[END_REF] report adopts a more practical and tool box approach and proposes a transversal description of some key themes and methodologies used to investigate either sustainable alternative lifestyles or, with a more restrictive focus, sustainable consumption. The UNEP's survey also provides evidence of differences and similarities in the declared priorities of young adult's across the world.

Though the sample might not be fully representative, this report gives multiple qualitative views on the aspirations associated to the notion of sustainable lifestyles (UNEP, 2011). Similarly the SPREAD project offers a comparable exercise centered on Europe [START_REF] Kuittinen | iFuture -The Diversity of Sustainable Lifestyles[END_REF] where participants to several workshops imagined a sustainable lifestyle in Europe in 2050 with the target of 8000 kg as material footprint per year and per person. The challenge of supporting sustainable consumption practices is addressed in [START_REF] Jaeger-Erben | Sustainable consumption through social innovation: a typology of innovations for sustainable consumption practices[END_REF] through a review and typology of several practical examples and, in [START_REF] Shao | Influencing factors to facilitate sustainable consumption: from the experts' viewpoints[END_REF] with an attention on the availability of information on products attributes to consumers for a better informed purchase decision. In UK, the sustainable lifestyle approach of DEFRA [START_REF] Eppel | A review of Defra's approach to building an evidence base for influencing sustainable behaviour[END_REF] provides a strong case of a government agency's effort to understand and promote sustainable behaviours and lifestyles. The proposed segmentation in population groups reveals the differences in willingness to engage in more sustainable behaviours and shows for instance that the two extreme groups of "Positive Greens" and "Honestly Disengaged" have similar size. In France (Le Gallic et al., 2014) the iterative and participative process described in [START_REF] Emelianoff | Modes de vie et empreinte carbone[END_REF] provides a comprehensive description of five possible future lifestyles defined within the PROMOV project. This overview of the growing and diverse literature on sustainable lifestyles or sustainable behaviours illustrate that the challenge has been a better understanding of the process of behavior itself, its psychological (values, meaning) or societal (social pressure, collective practices) dimensions, and possible governance schemes. This is done through both conceptual frameworks and practical case studies. Quantitative scenarios of future lifestyle have received less attention mainly because a shared understanding of the boundaries and meanings of lifestyle does not exist.

In this paper we propose a numerical model to project alternative social practices (mobility, housing and equipment levels) on the long term in France. It is intended as a dialog tool to bring assumptions formulated as lifestyle change in a form useable for techno-economic analysis and thus bridge the two approaches. A model cannot of course replace the cognitive, human and imaginative value of workshops such as the co-construction workshops cited above. Its interest should be seen as a complementary tool and one that can lightly explore alternative variants. We adopt a macroscopic perspective on social practices and quantify them using existing large scale statistical surveys. Compared to the literature on sustainable lifestyles our focus is put on the impact of changing lifestyles (more sustainable or not) on the demand for energy services. Longer term lifestyle transformation have explicitly been proposed in [START_REF] Mont | Sustainable lifestyles 2050: stakeholder visions, emerging practices and future research[END_REF] for Europe based on visions drawn from expert groups and a specified sustainability target. Using the information contained in current surveys, our core assumption is that the distribution of practices among the population captures the heterogeneity of practices. The proposed quantitative scenarios are of course exploratory and should not be considered as an attempt to "predict" future lifestyles. Section 2 presents the methodology while section 3 provides some illustrative results on different dimensions.

Methodology

Surveys as a substrate for a quantitative approach of social practices

Our aim is to investigate the consequences of future lifestyles changes on the energy services demand in France, i.e. on the housing demand, on the demand for mobility and on the consumption of goods and services. In order to achieve this goal, we need a quantitative approach of lifestyles. This constitutes a delicate task given that they are related to both qualitative and quantitative dimensions and to both conceptual and practical dimensions. Lifestyles are multifaceted and are marked by our relationships to time, to space, to others, and to ourselves (Héraut, 2013). Nevertheless, they have a visible resultant in the individual and collective practices in a broad sense. It is through these practices that we intend to somewhat "capture" lifestyles traits and apprehend them through data from French national surveys. Most of the surveys are conducted periodically every five to ten years providing a valuable material for foresight.

Five national surveys were used in order to cover several dimensions of social practices: the population census, the Housing Survey, the National Transport and Travel Survey, the time-use survey and the household budget survey. A short description of their goal, their scope, the existing editions and the size of the samples is provided in Table 1. We used the latest available edition of each of these surveys (except for the population census) to establish a baseline on current practices. To know and follow the evolution of the population of France, in its geographic diversity and its evolution 1966-1967 1974-1975 1986-1987 1998-1999 2009-2010 1955, 1961, 1963, 1967, 1970, 1973, 1978, 1984, 1988, 1992, 1996, 2002 et 2006, 2013 1966-1967 1973-1974 1981-1982 1993-1994 2007-2008 1979, 1984, 1989, 1995, 2001, 2006, 2011 Since 1801

Editions

Every 5 to 9 years from 1801to 1962Harmonized data : 1968, 1975, 1982, 1990, 1999[START_REF]Enquête logement[END_REF] A key step of our approach is to select a set of variables to build a representation of the people and their lifestyles. The Table 2 reports the types of variables that we used to address the different dimension of lifestyles and the questions addressed by each set of variable.

Table 2. Types of variables used to build a quantitative representation of "people and their lifestyles" and their availability in each survey. Legend: "+++" means "comprehensive data is available"; "++" means "partial data"; "+" means "a few information"; an empty cell means "no information available". By analyzing the surveys, we accede to the information which is contained in the correlation between variables. For example people aged 75 and older make an average of two times fewer trips per weekday than the rest of the population (see Figure 1). This reflects the process of biological and social ageing, i.e. it is more and more common for people beyond 75 years to experience physical discomfort to move/get around on one hand and to see their social networks decrease on the other hand. A different lifestyle characterized by an enhanced social participation or physical capability of this group with time, coupled with the change in population structure could then influence the total trip demand. Another example is given by the correlation between the size of household and the floor space per person in a dwelling (see Figure 2). This correlation reflects the process of space sharing which has a consequence on space needs. To give a concrete application of this process: if several types of rooms could be considered by most of people as a "need" in a dwelling (e.g. kitchen, bathroom, living-room), one of these type of room per household/dwelling is most of time considered as enough.

Types of variables

Surveys

Principle of the modelling approach

A survey basically consists of a matrix in which each row represents an individual and each column represents a variable. Some of the variables might describe individual or household attributes (e.g. age, gender), while others describe practices (e.g. number of trip per day per purpose, time spent at home per day). In its generic form a cell of coordinates contains the value or the modality of the variable for individual . Classification methods then allow for the identification of groups of individual according to the modalities of the variables used as basis for the segmentation. Finally we use the correlations between variables of the surveys as descriptor of social practices. These correlations are thus interpreted as the statistical manifestation of a combination of biological, psychological, cultural, juridical, geographical, demographical or economic processes. They are formalized in correlation matrices whose form is illustrated in Figure 3. Lisboa, Lisbon, Portugal, 13-15 July 2016 In this example, one categorical variablecalled and which have modalitiesis analyzed in relation to two explanatory variablescalled and and which have respectively and modalities. Each row represents an unique combination of modalities of and , in a group and each column represents one modality of the variable . The cells contain a value that is the frequency of a modality for one couple of values . If is a categorical variable, the cells contain the percentage for each modality; if is numerical, the cells contain a statistical measure (e.g. mean, percentage of cases between two values). The distribution consistency is checked (Equation ( 1)) for any couple ⟦ ⟧ ⟦ ⟧.

The statistical approach we propose uses dynamic population and practice matrixes as markers of lifestyles. Its practical starting point is a basic matrix containing a future population pyramid (which is extracted from demographic projections), i.e. only two variables are described: age and gender (an indication of the region could be added). The other variables are added one by one thanks to the correlation matrices. The Figure 4 illustrates the principle of adding a variable to the population matrix. In this case, adding a variable to this matrix increases the number of rows by a factor ( is the number of modalities of the added variable, ). The population matrix (a) has two variables (x and y). Analyzing the correlations between z (as response variable) and (x,y) (as explanatory variables) provides the correlation matrix (b). These two matrices are combined to create the population matrix (c) which disaggregates each row of (a) in k rows (for the k modalities of z) using the frequencies of the matrix (b). This elementary operation is included in a sequential process starting from the basic population matrix (population pyramid) and leading to a more comprehensive population matrix including several variables describing the future population and its lifestyles. Among the available methods to report the correlation between variables, we used decision trees. Firstly this method allows dealing easily with a mix of numerical and categorical covariates.

Secondly the implemented process provides a great flexibility, especially in the selection of variables that are taken into account. Decision trees use numerical categorization to build descriptors or predictors from a data source and are commonly used in data mining. In our case, they are used to group individuals with relatively homogeneous practices or situations with regard to a given variable. The practices or situations of each group are then characterized by standard statistical analyses (e.g. mean, distribution). This method -providing clarity and flexibility -also allows us to control the size of the formed groups to ensure their statistical significance. We use the RPART routines (as "Recursive PARTitioning") available in R to build the decision trees [START_REF] Breiman | Classification and regression trees[END_REF] and [START_REF] Therneau | An introduction to recursive partitioning using the RPART routines[END_REF].

The Figure 5 is showing an example of classification tree and its associated correlation matrix. In this example, the response variable is the space floor per person. The explanatory variables are the age of the household reference person (5-years classes), the household size (6 classes: 1

… … … … … … … … … group size … … … … group size … … … … … … (a) (b) (c) ∀ 𝑖 𝑗 ⟦1 𝑚⟧ ⟦1 𝑛⟧ ∑ 𝜌 𝑧 = 𝑧 = (1)
person, 2 people, 3 people, 4 people, 5 people, 6 or more), the location of the dwelling (4 classes: rural area, urban area of less than 100 000 inhabitants, urban area of more than 100 000 inhabitants, urban area of Paris) and the building type (2 class: apartment or flat). In this example, we have limited the recursive partitioning to 8 parts. The most explanatory variable is the household size. The sample is split into 2 groups: single-person-household and the others. Each group is then further split in subgroups using the most relevant criteria. Figure 5 also reports the mean and standard deviation of floor space per individual of each group.

Representing prospective changes in lifestyles indicators

We turn now to the critical question of the dynamics of our lifestyle indicators. Over several decades different social mechanisms can indeed lead to changes over time in the pattern of aggregated practices in a society (here at national level). Firstly we can mention demographic changes and the reconfiguration of the population structure. If it can be argued that there are as many original lifestyles as there are individuals, we can also identify some age based groups with relatively homogenous practices. Changes in the relative sizes of these different groupsall things equal otherwisewould also change the aggregate collection of lifestyles. Population pyramid and revenue distribution are some examples of such structural elements that will evolve in the future and potentially modify the dependent practices. Secondly we can mention some more progressive social changes such as changing in perception of societal priorities (awareness for the environment during the last decades), or new social trends such as remote activity (teleworking, teleshopping, teleconferencing) or the development of a leisure economy. Thirdly some changes are essentially disruptive and can be perceived as radical changes with fast rates of adoption. They can be triggered by unexpected economic crisis reducing brutally the wealth to share in a society, or can originate from a massive spread of new forms of social practices enabled by wider access to new social networking capabilities (car ownership, internet). In our approach they are treated as scenarios. The main principle is to convert narrative assumptions of change into quantitative practice changes in frequency in one or some of the correlation matrices. The Figure 6 depicts this simulation process where some changes are introduced through the replacement of reference matrices by modified matrices. Figure 6. Illustration of the process of changing matrices. In this generic case, the correlation matrix of is changed.

Results

Using the methodology presented above we tested the influence of lifestyle changes on housing demand, mobility needs and demand for selected goods in France. Our scope is 2050 and includes people living in Metropolitan France and belonging to a household1, representing about 92% of the whole population living in the French Republic. The housing demand is expressed in terms of number and type of dwellings, and the surface area which are the key determinants of the energy demand both at the household and country level [START_REF] Maresca | La transition énergétique à l'épreuve du mode de vie[END_REF]. The short distance passenger mobility is assessed through the number of trips made per year, the purposes of these trips, and the corresponding total distance. This structures the social network and the relation to space. The demand for selected goods concerns private cars, household appliances, digital equipment, computer equipment and audiovisual products.

Scenarios description

We highlight here four types of characteristic factors to concretely illustrate the proposed statistical model. The first one is the evolution in the household composition resulting from different cohabitation practices and more generally from changes in the relationship to others and to oneself. The second one is the changes in the location of the dwelling resulting from the societal appeal to metropolitan, urban or rural lifestyles. The different options include rural areas, urban areas of less than 100 000 inhabitants, urban areas of more than 100 000 inhabitants, and the urban area of Paris. The third one deals with the structural determinants of time use: sharing between work, leisure, sociability and other activities. The fourth one addresses the relationship to the space and sharing between real and virtual activities and also the part of each activity which is carried out at home or outside. These key dimensions are combined in three lifestyle scenarios exhibiting contrasting hypotheses of change (Table 3). They are inspired from a set of five lifestyles anticipated for France in 2050 and defined by [START_REF] Emelianoff | Modes de vie et empreinte carbone[END_REF]:

 The first scenario is a business as usual scenario. The household size continues to decrease as a result of the ageing population. The preference for houses continues to feed the urban sprawl. The main structure of time use is similar to the current one, i.e. especially work or study structure the organization of the week for most people.

 The second scenario called "individual and virtual society" consider a more individualistic and performance oriented society. The will of performance and personal development drives most people to live alone, without constraints of others. To access to the most sought leisure and services, they live in large cities in small apartments. More and more activities are carried out virtually: work, social relations, shopping… Individuals pay great attention to their home where they spend a lot of time.

 The third scenario called "social link society" is radically different. Social ties are placed in the heart of the society and its organization. It results in widespread forms of group housing. The relationship to time also changes and productive work gradually loses its structural role towards social tasks and civic activities.

Beside the three scenarios, the current situation is depicted as a reference. Demographic changes in volume and structure for the three scenarios is based on the central projection of the National Institute of Statistics and Economic Studies for 2050 ("Insee projection"). In this scenario the population size is increasing by 16% relatively to 2010 one and reaches 70.3 billion in our scope. In addition, the population ageing changes the age structure of the population between the current situation and the three scenarios.

Housing demand

The household is the driver for the housing demand in our modelling approach. Its composition (size, number of adults and children, age and situation of its members) will influence its aspiration in terms of location, type of dwelling and surface area. The current links between these variables are considered through their current correlations. The size and composition of the housing stock resulting from the simulation of the three scenarios varies significantly (see Figure 7). Firstly, the cohabitation practices have a great influence on the number of dwellings needed. The dynamics of the size of household is given in (Table 4). The preference for living alone in the "individual and virtual society" thus results in a need for about 42% more dwellings than in the "business as usual" scenario where the demographic effect conducted to an increase of 18% compared to the current situation. The will for more collective housing practices in the "social link society" involves that the size of the housing stock remains stable. However, changes in its composition would result in needs for housing construction or conversion. In this scenario, the main demand would concern the large apartments in urban area would be whereas the "individual and virtual society" scenario would strongly increase the demand for small apartments in big cities. In terms of surface, two effects influence the results (see Figure 8). On one hand, the share of space is accentuated in the "social link society" scenario contributing to a decrease of 8% of the mean surface area per person. Conversely, in the "individual and virtual society" scenario, the share of space is reduced contributing to the increase of 27% of the mean surface area per person. On the other hand, the preference for urban or metropolitan area in these two scenarios contributes to reduce the demand for surface area per person. This simulation thus reflects the highest pressure on space and on the housing market in dense areas. A small increase of the surface area per person is also visible in the business as usual scenario resulting from the single demographic effect (population ageing). The use of these three scenarios demonstrates the influence of lifestyles changes on the housing demand. These aggregated results may be useful for understanding their influence on the construction sector, on the urban dynamic or on the energy demand -which depends on the total surface area to be heated. More detailed results are available, for example an assessment of the housing demand per size of dwelling. But especially, depending on the goals in which the model is used, it would be interesting to explore more comprehensive scenarios, taking into account other type of lifestyle changes.

Demand for specific goods

We now consider the impact of our consumption of goods and services for cars, household appliances, digital equipment, computer equipment and audiovisual products. Whereas many other factors could be introduced for clarity we focus here on couple of them:

 The composition of households which reflects the effect of sharing goods at the household levelnote that some other hypotheses could be defined to simulate the effects of sharing at a district, apartment building or community level.

 The location of the dwelling which is a key determinant of the need for car to move around. The changes in age structure of the population could be mentioned as a third impact visible when the current situation and the business as usual scenario are compared.

The lifestyle assumption has a significant effect on the size of the car stock (see Figure 9). The increase of 24% of the car stock in business as usual scenario compared to the current situation show that the effect of changes in age structure of the population is not negligible (compared to the increase of 16% of the population size). Comparatively the "social link society" scenario combines the effects of car sharing at the household level and of a more urban society resulting in an increase of only 4% compared to 2010. The dynamics of location is given in (Table 5). Finally in the "individual and virtual society" scenario, the downsizing effect of location of household is largely compensated by the reduction of car sharing at household level due to the substantial increase of the number of households and a more individualistic approach. The other durable goods analyzed are also well anchored in our current society. While disruptive technological innovations could make them obsolete they have deeply integrated our practices and activities (leisure, productive work and domestic work). Figure 10 shows that for these goods and independently of the scenario, a significant reduction below today's levels is not at reach without targeting our practices. Here, the household composition and the associated sharing at household level are the main explanatory effect. The simulated scenarios do not outline a path towards a straightforward material sobriety but without efforts to better understand lifestyle related drivers (and their differences depending on the types of goods) we can miss 10% to 20% of reduction relatively to a "Business as usual" case. These differentiated outcomes illustrate the usefulness of a model to quantitatively challenge how far our initial narratives go. More complex scenarios, including changes in relationship to space (e.g. extent of the area where is organized the everyday life) or property (an increase of the sharing practices at community or building level and not only at household level) for instance could then be envisioned as extension to the present scenario set.

Needs for short distance mobility

Trips are not performed for themselves but to reach a desired location where an activity of interest is performed. In this section, we now discuss the social practices in terms of short distance mobility (and associated purposes) and as lifestyle markers for our scenarios. This relates to the dimension "ways of doing". Besides the household composition and the location of the dwelling the additional explanatory force highlighted here is the use of time (see also Table 3).

The "social link society" scenario tests such a structural change inspired by one of the five scenarios described by [START_REF] Emelianoff | Modes de vie et empreinte carbone[END_REF].In this society, time spent on productive work decreases by 40% for workers, representing three days a week. This has two consequences: the number of unemployed is reduced and most people spend two days in social tasks and civic activities. It also involves a reduction of the number of commuting for work, which are replaced by travels for the new practiced activities. Furthermore information and communications technologies change our relationship to space, leading to new arbitrages between real and virtual activities (work, leisure, learning, shopping). In "individual and virtual society" this share is largely moved towards more virtual activities. The withdrawal leads most people to prefer spending more time at home. The equivalent of 40% of fulltime workers work from home, pupils and students spend 40% percent at school, two-thirds of purchases are made online, an increasing part of the sociability and leisure activities are carried out from home…

The simulated impacts of these assumptions at a macroscopic level are illustrated in (Figure 11 and Figure 12). A first result is that compared to the 16% increase of population, the evolution of the number of travels (+3%) and of the total distance (+2%) are low, highlighting the effect of population ageing on the reduction of the intensity of practice in a "Business as usual" case. Here again altering this lower intensity of practice with further assumptions on good health and "socially reconnected aging" could be a potentially disruptive change. While this topic is beyond the scope of this paper, [START_REF] Moisescu | The Social Integration of Elders Through Free-time Activities[END_REF] and (Lumme-Sandt, 2011) illustrate this better ageing potentially disruptive evolution of old age lifestyle. The second main effect is that the virtualization of the society results in a drastic reduction of the short distance mobility (-41% by number of travels). In this scenario, our hypothesis does not consider any rebound effect linked to a resource given that the virtualization is the result of a will to stay at home. Here a parallel with the private car stock indicates that while owning a car remains a strong individualistic aspiration, the higher virtualization decouples this aspiration and a regular use. In this "private car for exceptional trips" world, the material and energy footprint of transportation could be less correlated. Thirdly and conversely, the release of time for workers in the "social link society" scenario causes some kind of rebound effect resulting in a higher number of travels per year (+7% compared to the current situation). However, these activities are closer to home, limiting the influence of this increasing on distance travelled. In addition the distance travelled is slightly reduced due to the more urban society.

Conclusive discussion

Lifestyles are a central yet complex dimension of the assessment of the different open options for our societies both in terms of desirability and sustainability. This work proposed a numerical model to project alternative social practices (mobility, housing and equipment levels) and to quantify some of their implications on the long term in France. The modelling approach is based on the use of official surveys as basic material and it aims at defining a consistent explanatory environment to question the broad idea of lifestyle change. It should again be stressed that its underlying paradigm is one of an exploratory tool and not a prediction.

From this rationale a statistical model was described and results for a set of contrasted future lifestyle scenarios were discussed. The simulations took into account the diversity of population, and demonstrated that the correlation between characteristics and practices can bring a renewed understanding to the macroscopic implications of alternative lifestyles change narratives. Such changes are not initially intended and formulated as either sustainable or unsustainable.

Nevertheless the rate of adoption of radical lifestyles or the rate of reconfiguration of our social practices (and meta-structures such as cities and institutions) might remain slow. The evaluation process and the large number of potential proposals might also appear counter-productive in terms of normative policy choices. On these two aspects a robust insight could be that despite the inherent difficulties ignoring the effect of lifestyle is also a first order approximation that can lead to half-blind estimates of perceived "needs": on one hand regarding the rationale of the Business as usual cases that are used as reference for mitigation, and on the other in the existence of opportunities for less costly mitigation options.

Figure 1 .

 1 Figure 1. Number of trips per age. Data source: National Transport and Travel Survey 2007-2008.

Figure 2 .

 2 Figure 2. Floor space per person depending on household size. Data source: Housing survey 2006.

Figure 3 .

 3 Figure 3. Form of a correlation matrix. The response variable is categorical and has modalities. and are the explanatory variables which have respectively and modalities.

Figure 4 :

 4 Figure 4 : Example of adding a variable to the statistical model.

Figure 5 .

 5 Figure 5. Regression tree related to the space floor per person and its associated matrix. Data source: Housing survey 2006 for France.

Figure 7 :

 7 Figure 7 : Size and composition of the housing stock for the current situation and the three scenarios. The lowest (and dark) part of the bars is for houses, the upper (and clear) part is for apartments.

Figure 8 :

 8 Figure 8 : Mean surface area per person and total housing area for the current situation and the three scenarios.

Figure 9 :

 9 Figure 9 : Size of the car stocks for current situation and the three scenarios.

Figure 11 :Figure 12 :

 1112 Figure 11: Total number of travels of the population per year in the current situation and the three scenarios.

Table 1 .

 1 Short description of the five used national surveys.

		Time use survey	Housing Survey	National	Household budget	Population census
	Name			Transport and	survey
				Travel Survey
		To collect	To collect	To improve the	To study the
		accurate	comprehensive	knowledge of	household
		information on	and accurate data	mobility of the	expenditure and
	Purpose	the use by individuals of	on the housing stock and the	households living in France	income at a microeconomic
		their time	conditions of		level
			occupation by
			household

Table 3 :

 3 Summary of the hypotheses distinguishing the three scenarios

		Business as usual	Individual and virtual society	Social link society
	Demography	INSEE projection Central hypotheses	INSEE projection Central hypotheses	INSEE projection Central hypotheses
	Household composition, relation to others	Current practices	More individual society Individual households are the norm.	More collective society Extended households and families are the norm
			Very strong attraction for
	Location of the dwelling	Current practices	metropolitan area to the detriment of rural areas in	Attraction for urban area and densification
			particular
				3 work days instead of 5, but
				2 days devoted to social
	Time use	Current practices	Current practices	tasks and civic activities
				More emphasis on sociability
				activities
	Location of activities, space relationship to	Current practices	Most of the time at home increases The share of virtual activities	Current practices

Table 4 :

 4 Dynamics of size of households for the three scenarios. B = business as usual; I = individual and virtual society; S = social link society.

			2010		Dynamic: average annual rate		2030			2050
	Scenario	B	I	S	B	I	S	B	I	S	B	I	S
	1 person		14%		0,00% 2,7% -0,4% 14%	27%	13%	14%	40%	12%
	2		28%		0,00% 0,3% -0,7% 28%	30%	25%	28%	31%	21%

Table 5 :

 5 Dynamics of location of the households for the three scenarios. B = business as usual; I = individual and virtual society; S = social link society. Expressed in percentage of households in each area. Consumption of four types of durable goods in the current situation and the three scenarios (using relative quantitative index)

		cars)	40 45
		(million of	25 30 35
		of the car stock	5 10 15 20
		Size	-	2010		2050	2050		2050
					Business as	Individual and	Social link
						usual	virtual society	society
			2010		Dynamic: average annual rate		2030			2050
	Scenario	B	I	S	B	I	S	B	I	S	B	I	S
	Rural area		23%		0% -2,3% -0,3% 23% 16% 22% 23%	9%	20%

22ndInternational Sustainable Development Research Society Conference (ISDRS 2016), School of Science and Technology, Universidade Nova deLisboa, Lisbon, Portugal, 13-15 July 2016

i.e. excluding for example people living in caserns or in retirement home.


Paola Francesca Antonietti
email: paola.antonietti@polimi.it

Benoît Merlet
email: merlet@cmap.polytechnique.fr

Morgan Pierre
email: morgan.pierre@math.univ-poitiers.fr

Marco Verani
email: marco.verani@polimi.it

Convergence to equilibrium for a second-order time semi-discretization of the Cahn-Hilliard equation

Keywords: Lojasiewicz-Simon inequality, gradient flow, Cahn-Hilliard, backward differentiation formula AMS Subject Classification: 65M10, 65P05, 37M

Introduction

In this paper, we consider a second-order time semi-discretization of the Cahn-Hilliard equation with an analytic nonlinearity, and we prove that any sequence generated by the scheme converges to a steady state as time goes to infinity, provided that the time-step is chosen small enough.

The Cahn-Hilliard equation [START_REF] Cahn | Free energy of a nonuniform system. I. Interfacial free energy[END_REF] reads

u t = ∆w w = -γ∆u + f (u) in Ω × (0, +∞), (1.1)
where Ω is a bounded subset of R d (1 ≤ d ≤ 3) with smooth boundary and γ > 0.

A typical choice for the nonlinearity is

f (s) = c(s 3 -s) (1.2)
with c > 0. More general conditions on f are given in Section 2, see (2.3)-(2.5). Equation (1.1) is completed with Neumann boundary conditions and an initial data. The Cahn-Hilliard equation was analyzed by many authors and used in different contexts (see, e.g., [START_REF] Cherfils | The Cahn-Hilliard equation with logarithmic potentials[END_REF][START_REF] Novick-Cohen | The Cahn-Hilliard equation[END_REF] and references therein). In particular, it is a H -1 gradient flow for the energy

E(u) = Ω γ 2 |∇u| 2 + F (u) dx,
where F is an antiderivative of f . Convergence of single trajectories to equilibrium for (1.1)-(1.2) has been proved in [START_REF] Rybka | Convergence of solutions to Cahn-Hilliard equation[END_REF]. The proof uses the gradient flow structure of the equation and a Lojasiewicz-Simon inequality [START_REF] Simon | Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems[END_REF].

In one space dimension, the set of steady states corresponding to (1.1)-(1.2) is finite [START_REF] Grinfeld | Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments[END_REF][START_REF] Kosugi | Stationary solutions to the onedimensional Cahn-Hilliard equation: proof by the complete elliptic integrals[END_REF]. In this case, the use of a Lojasiewicz-Simon inequality can be avoided [START_REF] Zheng | Asymptotic behaviour to the solution of the Cahn-Hilliard equation[END_REF] but otherwise, the situation is highly complicated; if d = 2 or 3, there may even be a continuum of stationary solutions (see, e.g., [START_REF] Wei | On the stationary Cahn-Hilliard equation: interior spike solutions[END_REF] and references therein). The Lojasiewicz-Simon inequality allows to prove convergence to an equilibrium without any knowledge on the set of steady states. This celebrated inequality is based on the analyticity of f (see [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF] for a recent overview). In contrast, for the related semilinear parabolic equation, convergence to equilibrium may fail for a nonlinearity of class C ∞ [START_REF] Poláčik | Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains[END_REF].

Using similar techniques, convergence to equilibrium for the non-autonomous Cahn-Hilliard equation was proved in [START_REF] Chill | Convergence to steady states in asymptotically autonomous semilinear evolution equations[END_REF], and the case of a logarithmic nonlinearity was considered in [START_REF] Abels | Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy[END_REF]. The Cahn-Hilliard equation endowed with dynamic or Wentzell boundary conditions was analyzed in [START_REF] Chill | Convergence to steady state of solutions of the Cahn-Hilliard and Caginalp equations with dynamic boundary conditions[END_REF][START_REF] Prüss | Maximal regularity and asymptotic behavior of solutions for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF][START_REF] Wu | Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition[END_REF][START_REF] Wu | Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF]. Coupled systems were also considered (see, e.g., [START_REF] Gal | Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions[END_REF][START_REF] Jiang | Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth[END_REF][START_REF] Prüss | Maximal L p -regularity and long-time behaviour of the non-isothermal Cahn-Hilliard equation with dynamic boundary conditions[END_REF]).

Since many space and/or time discretizations of the Cahn-Hilliard equation are available in the literature (see, e.g., [START_REF] Antonietti | A C 1 virtual element method for the Cahn-Hilliard equation with polygonal meshes[END_REF][START_REF] Elliott | The Cahn-Hilliard model for the kinetics of phase separation[END_REF][START_REF] Gomez | Isogeometric analysis of the Cahn-Hilliard phase-field model[END_REF][START_REF] Gomez | Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models[END_REF][START_REF] Goudenège | High order finite element calculations for the Cahn-Hilliard equation[END_REF][START_REF] Guo | An H 2 convergence of a secondorder convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation[END_REF][START_REF] Nabet | Convergence of a finite-volume scheme for the Cahn-Hilliard equation with dynamic boundary conditions[END_REF][START_REF] Shen | Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[END_REF][START_REF] Wu | Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models[END_REF]), it is natural to ask whether convergence to equilibrium also holds for these discretizations, by using similar techniques.

If we consider only a space semi-discretization of (1.1), and if this discretization can be shown to preserve the gradient flow structure, then convergence to equilibrium is a consequence of Lojasiewicz's classical convergence result [START_REF] Lojasiewicz | Une propriété topologique des sous-ensembles analytiques réels[END_REF] and its generalizations [START_REF] Bárta | Every ordinary differential equation with a strict Lyapunov function is a gradient system[END_REF][START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF]. Thanks to the finite dimension, the Lojasiewicz-Simon inequality reduces to the standard Lojasiewicz inequality. The latter is a direct consequence of analyticity of the discrete energy functional.

Thus, the situation regarding the space discretization is well understood, and we believe that the focus should be put on the time discretization, in the specific case where the time scheme preserves the gradient flow structure. In this regard, convergence to equilibrium for a fully discrete version of (1.1)-(1.2) was first proved in [START_REF] Merlet | Convergence to equilibrium for the backward Euler scheme and applications[END_REF]: the time scheme was the backward Euler scheme and the space discretization was a finite element method. Fully discretized versions of Cahn-Hilliard type equations were considered in [START_REF] Cherfils | A numerical analysis of the Cahn-Hilliard equation with non-permeable walls[END_REF][START_REF] Cherfils | A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions[END_REF][START_REF] Injrou | Error estimates for a finite element discretization of the Cahn-Hilliard-Gurtin equations[END_REF], where this nice feature of the backward Euler scheme was again demonstrated (see also [START_REF] Attouch | On the convergence of the proximal algorithm for nonsmooth functions involving analytic features[END_REF][START_REF] Guillén-González | Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows[END_REF]). In [START_REF] Alaa | Convergence to equilibrium for discretized gradientlike systems with analytic features[END_REF], convergence to equilibrium was proved for several fully discretized versions of the closely related Allen-Cahn equation; the time scheme was either first order or second order, conditionnally or unconditionnally stable, and the time-step could possibly be variable. In addition, general conditions ensuring convergence to equilibrium for a time discretization were given (see also [START_REF] Attouch | Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods[END_REF]).

Therefore, the fully discrete case is now also well understood. The last stage is to study the time semi-discrete case. This is all the more interesting since this approach is independent of a choice of a specific space discretization. Convergence to equilibrium was proved for the backward Euler time semi-discretization of the Allen-Cahn equation in [START_REF] Merlet | Convergence to equilibrium for the backward Euler scheme and applications[END_REF] (see also [START_REF] Bolte | Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity[END_REF]). A related damped wave equation was considered in [START_REF] Pierre | Convergence to equilibrium for a time semi-discrete damped wave equation[END_REF].

For schemes different from the backward Euler method, the situation is not so clear, and this is well illustrated by the second order case. Indeed, there exist several second-order time semi-discretizations of (1.1)-(1.2) which preserve the gradient flow structure (see, e.g., [START_REF] Shen | Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[END_REF][START_REF] Wu | Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models[END_REF] and references therein). Most of these schemes are onestep methods, which can be seen as variants of the Crank-Nicolson scheme, such as the classical secant scheme [START_REF] Du | Numerical analysis of a continuum model of phase transition[END_REF][START_REF] Elliott | The Cahn-Hilliard model for the kinetics of phase separation[END_REF] or the more recent scheme of Gomez and Hughes [START_REF] Gomez | Provably unconditionally stable, second-order time-accurate, mixed variational methods for phase-field models[END_REF], which is a Crank-Nicolson scheme with stabilization.

However, we have not been able to prove convergence to equilibrium for any of these second-order one-step schemes. One difficulty is that the gradient of E (cf. (3.2)) is treated in an implicit/explicit way, and another difficulty is that the discrete dynamical system associated with the scheme is defined on a space of infinite dimension. The first difficulty can be circumvented in finite dimension, as recently shown in [START_REF] Grasselli | Energy stable and convergent finite element schemes for the modified phase field crystal equation[END_REF], where convergence to equilibrium was proved for a fully discrete approximation of the modified phase-field crystal equation using the second-order time discretization of Gomez and Hughes. A related difficulty has been pointed out in [START_REF] Tone | On the long-time stability of the Crank-Nicolson scheme for the 2D Navier-Stokes equations[END_REF] where the stability of the Crank-Nicolson scheme for the Navier-Stokes equation was proved in a finite dimensional setting only.

In this paper, instead of a Crank-Nicolson type method, we use a standard twostep scheme with fixed time-step, namely the backward differentiation formula of order two. It is well-known [START_REF] Shen | Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[END_REF][START_REF] Stuart | Dynamical systems and numerical analysis[END_REF] that this scheme enjoys a Lyapunov stability, namely, if the time-step is small enough, a so-called pseudo-energy (cf. (2.17)) is nonincreasing at every time iteration. Thanks to the implicit treatment of the gradient of E (cf. (2.13)), the proof of convergence is similar to the case of the backward Euler scheme in [START_REF] Merlet | Convergence to equilibrium for the backward Euler scheme and applications[END_REF][START_REF] Pierre | Convergence to equilibrium for a time semi-discrete damped wave equation[END_REF]. Using the Lyapunov stability, we first prove Lasalle's in-variance principle by a compactness argument (Proposition 3.1). Convergence to a steady state is then obtained as a consequence of an appropriate Lojasiewicz-Simon inequality (Lemma 3.2), which is the most technical point. In order to derive the convergence rate in H 1 norm, we also take advantage of the fact that the scheme is more dissipative than the original equation (see Remark 2.4).

It would be interesting to extend our convergence result to first-order or secondorder schemes where the nonlinearity is treated explicitly. In order for such schemes to preserve the gradient structure, the standard approach is to truncate the cubic nonlinearity f (cf. (1.2)) at ±∞ so as to have a linear growth at most [START_REF] Shen | Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[END_REF]. However, it is not known if the energy associated with such a nonlinearity satisfies a Lojasiewicz-Simon inequality, in contrast with the finite-dimensional case where it can be proved for certain space discretizations [START_REF] Alaa | Convergence to equilibrium for discretized gradientlike systems with analytic features[END_REF].

It could also be of interest to investigate whether a similar convergence result holds for the p-step backward differentiation formula (BDF), with p ≥ 3. A favorable situation is the 3-step BDF method, which preserves the gradient flow structure, at least in finite dimension [START_REF] Stuart | Dynamical systems and numerical analysis[END_REF].

The paper is organized as follows. In Section 2, we introduce the scheme, we establish its well-posedness and we show that it is Lyapunov stable. In Section 3, we prove the convergence result.

2 The time semi-discrete scheme

Notation and assumptions

Let H = L 2 (Ω) be equipped with the L 2 (Ω) norm | • | 0 and the L 2 (Ω) scalar product (•, •). We denote V = H 1 (Ω) the standard Sobolev space based on the L 2 (Ω) space. We use the hilbertian semi-norm

| • | 1 = |∇ • | 0 in V , and the norm in V is v 2 1 = |v| 2 0 + |v| 2 1 .
We denote -∆ : V → V ′ the bounded operator associated with the inner product on V through

-∆u, v V ′ ,V = (∇u, ∇v), ∀u, v ∈ V,
where V ′ is the topological dual of V . As usual, we will denote W k,p (Ω) the Sobolev spaces based on the L p (Ω) space [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF].

For a function u ∈ L 2 (Ω), we denote

u = 1 |Ω| Ω u and u = u -u ,
where |Ω| is the Lebesgue measure of Ω. We also define

Ḣ = {u ∈ L 2 (Ω), u = 0}, V = V ∩ Ḣ.
We will use the continuous and dense injections

V ⊂ Ḣ = Ḣ′ ⊂ V ′ .
As a consequence of the Poincaré-Wirtinger inequality, the norms v 1 and

v → (|v| 2 1 + v 2 ) 1/2 (2.1)
are equivalent on V . The operator -∆ : V → V ′ , that is the restriction of -∆, is an isomorphism. The scalar product in V ′ is given by

( u, v) -1 = (∇(-∆) -1 u, ∇(-∆) -1 v) = u, (-∆) -1 v V ′ , V
and the norm is given by

| u| 2 -1 = ( u, u) -1 = u, (-∆) -1 u V ′ , V .
We recall the interpolation inequality

| u| 2 0 ≤ | u| -1 | u| 1 , ∀ u ∈ V . (2.2)
We assume that the nonlinearity f : R → R is analytic and if d ≥ 2, we assume in addition that there exist a constant C > 0 and a real number p ≥ 0 such that

|f ′ (s)| ≤ C(1 + |s| p ), ∀s ∈ R, (2.3) with p < 4 if d = 3. No growth assumption is needed if d = 1.
We also assume that

f ′ (s) ≥ -c f , ∀s ∈ R, (2.4)
for some (optimal) nonnegative constant c f , and that lim inf

|s|→+∞ f (s) s > 0. (2.5)
We define the energy functional

E(u) = γ 2 |u| 2 1 + (F (u), 1), (2.6)
where F (s) is a given antiderivative of f . The Sobolev injection V ⊂ L p+2 (Ω) and the growth assumption (2.3) ensure that E(u) < +∞ and f (u) ∈ V ′ , for all u ∈ V . In fact, by [START_REF] Kavian | Introduction à la théorie des points critiques et applications aux problèmes elliptiques[END_REF]Corollaire 17.8], the functional E is of class C 2 on V . For any u, v, w ∈ V , we have

dE(u), v V ′ ,V = Ω [γ∇u • ∇v + f (u)v]dx, (2.7)
d 2 E(u)v, w V ′ ,V = Ω [γ∇v • ∇w + f ′ (u)vw]dx, (2.8)
where

dE(u) ∈ V ′ is the first differential of E at u and d 2 E(u) ∈ L(V, V ′ ) is the differential of order two of E at u.
If u is a regular solution of (1.1), on computing we see that

d dt E(u(t)) = -|w| 2 1 = -|u t | 2 -1 t ≥ 0, (2.9)
so that E is a Lyapunov functional associated with (1.1).

Existence, uniqueness and Lyapunov stability

Let τ > 0 denote the time-step. The second-order backward differentiation scheme for (1.1) reads [START_REF] Shen | Numerical approximations of Allen-Cahn and Cahn-Hilliard equations[END_REF][START_REF] Stuart | Dynamical systems and numerical analysis[END_REF]:

let (u 0 , u 1 ) ∈ V × V and for n = 1, 2, . . . , let (u n+1 , w n+1 ) ∈ V × V solve    1 2τ (3u n+1 -4u n + u n-1 , ϕ) + (∇w n+1 , ∇ϕ) = 0 (w n+1 , ψ) = γ(∇u n+1 , ∇ψ) + (f (u n+1 ), ψ), (2.10)
for all (ϕ, ψ) ∈ V × V . For simplicity, we assume that

u 0 = u 1 , (2.11)
so that, by induction, any sequence (u n ) which complies with (2.10) satisfies u n = u 0 for all n (choose ϕ = 1/|Ω| in (2.10)). We note that w 0 and w 1 need not be defined.

For later purpose, we note that if

u n = u n-1 , then (2.10) is equivalent to              u n+1 = u n (-∆) -1 (3u n+1 -4u n + u n-1 ) 2τ + ẇn+1 = 0 ẇn+1 = -γ∆u n+1 + f (u n+1 ) -f (u n+1 ) w n+1 = f (u n+1 ) .
(2.12)

Eliminating w n+1 leads to

(-∆) -1 (3u n+1 -4u n + u n-1 ) 2τ -γ∆u n+1 + f (u n+1 ) -f (u n+1 ) = 0. (2.13)
Proposition 2.1 (Existence for all τ ). For all (u 0 , u 1 ) ∈ V ×V such that u 0 = u 1 , there exists at least one sequence (u n , w n ) n which complies with (2.10). Moreover, u n = u 0 for all n.

Proof. Existence can be obtained by minimizing an appropriate functional. By induction, assume that for some n ≥ 1, (

u n-1 , u n ) ∈ V × V is defined, with u n = u n-1 = u 0 .
Then, by (2.13), u n+1 can be obtained by solving

min {G n (v) : v ∈ V, v = u 0 } , (2.14)
where

G n (v) = 3 4τ | v| 2 -1 + 1 2τ (-4 un + un-1 , v) -1 + E(v).
By (2.5), there exist κ 1 > 0 and κ 2 ≥ 0 such that

F (s) ≥ κ 1 s 2 -κ 2 , ∀s ∈ R.
Thus, for all v ∈ V , (F (v), 1)

≥ κ 1 |v| 2 0 -κ 2 |Ω|,
and so

E(v) ≥ κ 3 v 2 1 -κ 2 |Ω|, (2.15)
with κ 3 = min{γ/2, κ 1 } > 0. Moreover, by the Cauchy-Schwarz inequality,

|(-4 un + un-1 , v) -1 | ≤ | v| -1 | -4 un + un-1 | -1 ≤ 3 2 | v| 2 -1 + C n ,
for some constant C n which depends on | un | -1 and | un-1 | -1 . Summing up, we have proved that

G n (v) ≥ κ 3 v 2 1 -κ 2 |Ω| - C n 2τ .
By considering a minimizing sequence (v k ) for problem (2.14), we obtain a minimizer, i.e. u n+1 . Then w n+1 can be recovered from u n+1 by (2.12).

Proposition 2.2 (Uniqueness). If 1/τ > c 2 f /(6γ), then for every (u 0 , u 1 ) ∈ V × V such that u 0 = u 1 , there exists at most one sequence (u n , w n ) n which complies with (2.10).

Proof. Assume that (u n+1 , w n+1 ) and (ũ n+1 , wn+1 ) are two solutions of (2.10), and denote δu = u n+1 -ũn+1 , δw = w n+1 -wn+1 . On subtracting, we obtain

3(δu, ϕ)/(2τ ) + (∇δw, ∇ϕ) = 0, (2.16)
(δw, ψ) = γ(∇δu, ∇ψ) + (f (u n+1 ) -f (ũ n+1 ), ψ),
for all (ϕ, ψ) ∈ V × V . Choosing ϕ = δw and ψ = δu, yields

-(2τ /3)|δw| 2 1 = γ|δu| 2 1 + (f (u n+1 ) -f (ũ n+1 ), δu).
Using the mean value inequality and (2.4) yields

(s -r)(f (s) -f (r)) = f ′ (ξ)(s -r) 2 ≥ -c f (s -r) 2 ,
for all r, s ∈ R, for some ξ ∈ R depending on r, s. Thus,

c f |δu| 2 0 ≥ γ|δu| 2 1 + (2τ /3)|δw| 2 1 .
Using now (2.16) with ϕ = δu, we obtain

c f |δu| 2 0 = -(2τ c f /3)(∇δw, ∇δu) ≤ γ|∇δu| 2 0 + τ 2 c 2 f 9γ |∇δw| 2 .
If τ c 2 f < 6γ, then δ ẇ = 0, and by (2.16), δu = 0 also. Uniqueness follows.

We define the following pseudo-energy

E(u, v) = E(u) + 1 4τ | v| 2 -1 , ∀(u, v) ∈ V × V ′ .
(2.17)

For a sequence (u n ) n , let also δu n = u n -u n-1 denote the backard difference. The following relation will prove useful,

3u n+1 -4u n + u n-1 = 2δu n+1 + (δu n+1 -δu n ). ( 2
E(u n+1 , δu n+1 ) + εγ 2 |u n+1 -u n | 2 1 + 1 τ - c 2 f 8γ(1 -ε) |u n+1 -u n | 2 -1 + 1 4τ |δu n+1 -δu n | 2 -1 ≤ E(u n , δu n ). ( 2

.19)

Proof. We take the L 2 scalar product of equation (2.13) by δu n+1 and we use (2.18). We obtain

1 τ |δu n+1 | 2 -1 + 1 2τ (δu n+1 -δu n , δu n+1 ) -1 + γ(∇u n+1 , ∇(u n+1 -u n )) = (f (u n+1 ), u n -u n+1 ).
By the Taylor-Lagrange formula, from (2.4), we deduce that

F (r) -F (s) ≥ f (s)(r -s) - c f 2 |r -s| 2 , ∀r, s ∈ R. Thus, (f (u n+1 ), u n -u n+1 ) ≤ (F (u n ), 1) -(F (u n+1 ), 1) + c f 2 |u n+1 -u n | 2 0 .
Next, we use the well-known identity

(a, a -b) m = 1 2 |a| 2 m - 1 2 |b| 2 m + 1 2 |a -b| 2 m ,
for m = -1 and m = 0. We find

1 τ |δu n+1 | 2 -1 + 1 4τ |δu n+1 | 2 -1 -|δu n | 2 -1 + |δu n+1 -δu n | 2 -1 + γ 2 |u n+1 | 2 1 -|u n | 2 1 + |u n+1 -u n | 2 1 ≤ (F (u n ), 1) -(F (u n+1 ), 1) + c f 2 |u n+1 -u n | 2 0 . (2.20)
The interpolation inequality (2.2) and Young's inequality yield

c f 2 |u n+1 -u n | 2 0 ≤ γ(1 -ε) 2 |u n+1 -u n | 2 1 + c 2 f 8γ(1 -ε) |u n+1 -u n | 2 -1 .
Plugging

Convergence to equilibrium

For a sequence (u n ) n in V , we define its omega-limit set by

ω((u n ) n ) := {u ⋆ ∈ V : ∃n k → ∞, u n k → u ⋆ (strongly) in V }.
Let M ∈ R be given and consider the following affine subspace of V ,

V M = {v ∈ V : v = M } = M + V . (3.1)
The set of critical points of

E (see (2.6)) in V M is S M = {u ⋆ ∈ V M : -γ∆u ⋆ + f (u ⋆ ) -f (u ⋆ ) = 0 in V ′ }.
Indeed, we already know that E ∈ C 2 (V M ; R). Observe that, for any u ∈ V M , v ∈ V , we have (see (2.7))

dE(u), v V ′ , V = Ω [γ∇u • ∇v + f (u)v]dx = Ω [γ∇u • ∇v + (f (u) -f (u) )v]dx = -γ∆u + f (u) -f (u) , v V ′ , V . (3.2) By definition, u ⋆ is a critical point of E in V M if dE(u ⋆ ) = 0 in V ′ .
The definition of S M follows.

Proposition 3.1. Assume that 1/τ > c 2 f /(8γ) and let (u n , w n ) n be a sequence which complies with (2.10)- (2.11). Then δu n → 0 in V and ω((u n ) n ) is a nonempty compact and connected subset of V which is included in S M with M = u 0 . Moreover, E is constant on ω((u n ) n ).

Proof. Using the assumption on τ , we may choose ε ∈ (0, 1) such that 1/τ = c 2 f /(8γ(1 -ε)). Then (2. [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]) reads

E(u n+1 , δu n+1 ) + εγ 2 |u n+1 -u n | 2 1 + 1 4τ |δu n+1 -δu n | 2 -1 ≤ E(u n , δu n ), (3.3)
for all n ≥ 1. In particular, (E(u n , δu n )) n is non increasing. Moreover, by (2.15),

E(u, v) ≥ κ 3 u 2 1 + 1 4τ | v| 2 -1 -κ 2 |Ω|, ∀(u, v) ∈ V × V ′ . (3.4) Since E(u 1 , δu 1 ) < +∞, we deduce from (3.4) that (u n , δu n ) is bounded in V × V ′
and that E(u n , δu n ) is bounded from below. Thus, E(u n , δu n ) converges to some E ⋆ in R. By induction, from (3.3)-(3.4) we also deduce that

∞ n=1 |u n+1 -u n | 2 1 ≤ 2 εγ (E(u 1 , δu 1 ) + κ 2 |Ω|) < +∞.
In particular, δu n → 0 in V . This implies that E(u n ) → E ⋆ , and so E is equal to

E ⋆ on ω((u n ) n ).
Next, we claim that the sequence (u n ) is precompact in V . Let us first assume d = 3. We deduce from the Sobolev imbedding [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] that (u n ) is bounded in L 6 (Ω). By the growth condition (2.3), there exists 2 ≥ q > 6/5 such that f (u n+1 ) L q (Ω) ≤ M 1 , where M 1 is independent of n. By elliptic regularity [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF], we deduce from (2.13) that (u n+1 ) n≥1 is bounded in W 2,q (Ω). Finally, from the Sobolev imbedding [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], W 2,q (Ω) is compactly imbedded in V , and the claim is proved.

In the case d = 1 or 2, we obtain directly from the Sobolev imbedding that f (u n+1 ) is bounded in L q (Ω), for any q < +∞, and we conclude similarly.

As a consequence, ω(( (2.13). Thanks to (2.11), the whole sequence (u n ) belongs to V M and u ⋆ as well, where M = u 0 . By (2.18), the term corresponding to the discrete time derivative tends to 0 in V , and we obtain that u ⋆ belongs to S M .

u n ) n ) is a nonempty compact subset of V . Since |u n+1 - u n | 1 → 0, ω((u n ) n ) is also connected. Let finally u ⋆ belong to ω((u n ) n ), with n k → ∞ such that u n k → u ⋆ in V . We let n k tend to ∞ in
If the critical points of E are isolated, i.e. S M is discrete, then Proposition 3.1 ensures that the sequence (u n ) n converges in V . However, as pointed out in the introduction, the structure of S M is generally not known, and there may even be a continuum of steady states. In such cases, the Lojasiewicz-Simon inequality which follows is needed to ensure convergence of the whole sequence (u n ). Lemma 3.2. Let u ⋆ ∈ S M . Then there exist constants θ ∈ (0, 1/2) and δ > 0 depending on u ⋆ such that, for any u ∈ V M satisfying |u -u ⋆ | 1 < δ, there holds

|E(u) -E(u ⋆ )| 1-θ ≤ | -γ∆u + f (u) -f (u) | -1 . (3.5)
Proof. We will apply the abstract result of Theorem 11.2.7 in [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF]. We introduce the auxiliary functional E M (v) = E(M + v) on V . We will also use the auxiliary functions f M (s) = f (M + s) and F M (s) = F (M + s).

It is obvious that

E M (v) = Ω γ 2 |∇v| 2 + F M (v) dx.
The function E M is of class C 2 on V and by (3.2), for any v ∈ V , we have

dE M (v) = -γ∆v + f M (v) -f M (v) in V ′ .
Similarly, by (2.8), for any v, ϕ ∈ V , we have

d 2 E M (v)ϕ = -γ∆ϕ + f ′ M (v)ϕ -f ′ M (v)ϕ in V ′ . (3.6) Let v ⋆ ∈ V be a critical point of E M , i.e. a solution of dE M (v ⋆ ) = 0 in V ′ . Us- ing (2.3) and elliptic regularity, we obtain that v ⋆ ∈ C 0 (Ω) ⊂ L ∞ (Ω). In particular, f ′ M (v) ∈ L ∞ (Ω). The operator A = d 2 E M (v ⋆ ) ∈ L( V , V ′ ) (cf. (3.6)) can be written A = -γ∆ + P 0 (f ′ M (v ⋆ )Id),
where -γ∆ : V → V ′ is an isomorphism, P 0 : H → Ḣ is the L 2 -projection operator, and f ′ M (v ⋆ )Id : V → H is a multiplication operator. Since V is compactly imbedded in Ḣ [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], f ′ M (v ⋆ )Id : V → H is compact, and P 0 (f ′ M (v ⋆ )Id) as well. Using [27, Theorem 2.2.5], we obtain that A is a semi-Fredholm operator.

Next, let N (A) denote the kernel of A, and Π : V → N (A) the L 2 projection. By [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF]Corollary 2.2.6], L := A + Π : V → V ′ is an isomorphism. We choose Z = Ḣ and denote W = L -1 (Z); W is a Banach space for the norm w W = |L(w)| 0 . We claim that W is continuously imbedded in W 2,2 (Ω). Indeed, by definition, w ∈ W if and only if w ∈ V and L(w) = g for some g ∈ Z, i.e.

w ∈ V and -γ∆w + f ′ M (v ⋆ )w -f ′ M (v ⋆ )w + Πw = g.
Thus, -∆w ∈ Ḣ. By elliptic regularity [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF], w ∈ W 2,2 (Ω). Moreover, by the triangle inequality,

γ| -∆w| 0 ≤ C f ′ M (v ⋆ ) L ∞ |w| 0 + |Πw| 0 + |L(w)| 0 ≤ C w W ,
where C is a constant independent of w. But, by elliptic regularity [START_REF] Agmon | Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions[END_REF], we also know that w W 2,2 ≤ C| -∆w| 0 for all w ∈ V . This proves the claim. The Nemytskii operator [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF]Example 2.3]). Using [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF]Proposition 2.3.4], we find that the functional v → Ω F M (v) is real analytic from L ∞ (Ω) into R. Thus, E M , which is the sum of a continuous quadratic functional and of a functional which is real analytic on W ⊂ W 2,2 (Ω) ⊂ L ∞ (Ω), is real analytic on W . We also obtain that dE M : W → Z is real analytic.

f M : v → f M (v) is analytic from L ∞ (Ω) into L ∞ (Ω) (see
We are therefore in position to apply the abstract Theorem 11.2.7 in [START_REF] Haraux | The convergence problem for dissipative autonomous systems[END_REF], which shows that there exist θ ∈ (0, 1/2) and δ > 0 such that for all v ∈ V ,

|v -v ⋆ | 1 < δ ⇒ |E M (v) -E M (v ⋆ )| 1-θ ≤ |dE M (v)| -1 .
(3.7)

Finally, we note that any u ⋆ ∈ S M can be written u ⋆ = M + v ⋆ , where v ⋆ is a critical point of E M ; by definition of V M , any u ∈ V M can be written u = M + v with v ∈ V . The expected Lojasiewicz-Simon inequality (3.5) is exactly (3.7) written in terms of u ⋆ , u, E and f . Theorem 3.3. Assume that 1/τ > c 2 f /(8γ) and let (u n , w n ) n be a sequence which complies with (2.10)- (2.11). Then the whole sequence converges to (u ∞ , w ∞ ) in V × V , with u ∞ ∈ S M , M = u 0 , and w ∞ constant. Moreover, the following convergence rate holds

u n -u ∞ 1 + w n -w ∞ 1 ≤ Cn -θ 1-θ , (3.8)
for all n ≥ 2, where C is a constant depending on u 0 1 , u 1 1 , f , γ, τ , and θ, while θ ∈ (0, 1/2) may depend on u ∞ .

Proof. Let M = u 0 . For every u ⋆ ∈ ω((u n ) n ), there exist θ ∈ (0, 1) and δ > 0 which may depend on u ⋆ such that the inequality (3.5) holds for every u

∈ B δ (u ⋆ ) = {u ∈ V M : |u -u ⋆ | < δ}. The union of balls {B δ (u ⋆ ) : u ⋆ ∈ ω((u n ) n )} forms an open covering of ω((u n ) n ) in V M .
Due to the compactness of ω((u n ) n ) in V , we can find a finite subcovering {B δ i (u i ⋆ )} m i=1 such that the constants δ i and θ i corresponding to u i ⋆ in Lemma 3.2 are indexed by i.

From the definition of ω((u n ) n ), we know that there exists a sufficiently large n 0 such that u n ∈ U = ∪ m i=1 B δ i (u i ⋆ ) for all n ≥ n 0 . Taking θ = min m i=1 {θ i }, we deduce from Lemma 3.2 and Proposition 3.1 that for all n ≥ n 0 ,

|E(u n ) -E ⋆ | 1-θ ≤ | -γ∆u n + f (u n ) -f (u n ) | -1 , (3.9)
where E ⋆ is the value of E on ω((u n ) n ). We may also assume (by taking a larger n 0 if necessary) that for all n ≥ n 0 , |δu n | -1 ≤ 1.

We denote Φ n = E(u n , δu n ) -E ⋆ , so that Φ n ≥ 0 and Φ n is nonincreasing. Let n ≥ n 0 . Using the inequality (a + b) 1-θ ≤ a 1-θ + b 1-θ , valid for all a, b ≥ 0, together with (3.9), we obtain

Φ 1-θ n+1 ≤ |E(u n+1 ) -E ⋆ | 1-θ + (4τ ) θ-1 |δu n+1 | 2(1-θ) -1 ≤ | -γ∆u n+1 + f (u n+1 ) -f (u n+1 ) | -1 + (4τ ) θ-1 |δu n+1 | -1 ≤ C (|u n+1 -u n | -1 + |δu n+1 -δu n | -1 ) , ≤ C |u n+1 -u n | 2 1 + |δu n+1 -δu n | 2 -1 1/2 (3.10) where C = C(τ, θ, (-∆) -1 L( V ′ , V ′ ) , (-∆) -1 L( V , V )
) (here and in the following, C denotes a generic positive constant independent of n). For the third inequality, we have used (2.13) and (2.18). Next, we choose ε ∈ (0, 1) such that 1/τ = c 2 f /(8γ(1ε)). Then (3.3) holds, and it can be written

Φ n -Φ n+1 ≥ C |u n+1 -u n | 2 1 + |δu n+1 -δu n | 2 -1 , (3.11)
with

C = C(τ, γ, ε) > 0. Assume first that Φ n+1 > Φ n /2. Then Φ θ n -Φ θ n+1 = θ Φn Φ n+1 x θ-1 dx ≥ θ Φ n -Φ n+1 Φ n ≥ 2 θ-1 θ Φ n -Φ n+1 Φ 1-θ n+1 .
Using (3.10) and (3.11), we find

Φ θ n -Φ θ n+1 ≥ C |u n+1 -u n | 2 1 + |δu n+1 -δu n | 2 -1 1/2 ,
where C = C(τ, θ, γ, ε,

(-∆) -1 L( V ′ , V ′ ) , (-∆) -1 L( V , V ) ). Now, if Φ n+1 ≤ Φ n /2, then Φ 1/2 n -Φ 1/2 n+1 ≥ (1 -1/ √ 2)Φ 1/2 n ≥ (1 -1/ √ 2)(Φ n -Φ n+1 ) 1/2
and using (3.11) again, we find

Φ 1/2 n -Φ 1/2 n+1 ≥ C |u n+1 -u n | 2 1 + |δu n+1 -δu n | 2 -1 1/2 .
Thus, in both cases, we have

|u n+1 -u n | 1 ≤ C(Φ θ n -Φ θ n+1 ) + C(Φ 1/2 n -Φ 1/2 n+1 ), (3.12)
for all n ≥ n 0 . Summing on n ≥ n 0 , we obtain Using the Cauchy criterion, we find that the whole sequence (u n ) converges to some u ∞ in V . By Proposition 3.1, u ∞ belongs to S M . Using the second equation in (2.12), we see that ẇn → 0. For the term w n , we write

Ω |f (u n ) -f (u ∞ )|dx = Ω 1 0 f ′ ((1 -s)u n + su ∞ )(u n -u ∞ )ds dx.
Using assumption (2.3), Hölder's inequality and Sobolev imbeddings, we find

Ω |f (u n ) -f (u ∞ )|dx ≤ C( u n 1 , u ∞ 1 ) u n -u ∞ 1 .
Since (u n ) is bounded in V , this yields, for all n ≥ 2,

| w n -w ∞ | = | f (u n ) -f (u ∞ )| ≤ |f (u n ) -f (u ∞ )| ≤ C u n -u ∞ 1 , (3.14)
where we have used the last equation in (2.12) and where w ∞ = f (u ∞ ) . This implies that w n → w ∞ in V (see (2.1)), and it concludes the proof of convergence.

For the convergence rate, we will first show that

0 ≤ Φ n ≤ Cn -1 1-2θ , (3.15)
for all n ≥ n 1 , for some n 1 > n 0 large enough. The exponent θ is the same as above. If Φ n 1 = 0 for some n 1 ≥ n 0 , then Φ n = 0 for all n ≥ n 1 , and estimate (3.15) is obvious. So we may assume that Φ n > 0 for all n. Let n ≥ n 0 and denote G(s) = 1 s 1-2θ . The sequence G(Φ n ) is nondecreasing and tends to +∞. If Φ n+1 > Φ n /2, then

G(Φ n+1 ) -G(Φ n ) = Φn Φ n+1 1 -2θ s 2-2θ ds ≥ (1 -2θ)2 2θ-2 Φ 2θ-2 n+1 [Φ n -Φ n+1 ] (3.11) ≥ CΦ 2θ-2 n+1 |u n+1 -u n | 2 1 + |δu n+1 -δu n | 2 -1 (3.10) ≥ C 1 ,
where C 1 is a positive constant independent of n.

If Φ n+1 ≤ Φ n /2 and Φ n ≤ 1, then

G(Φ n+1 ) -G(Φ n ) ≥ 2 1-2θ -1 Φ 1-2θ n ≥ 2 1-2θ -1.
Let n ′ 0 ≥ n 0 be large enough so that Φ n ′ 0 ≤ 1. Then, in both cases, for all n ≥ n ′ 0 , we have

G(Φ n+1 ) -G(Φ n ) ≥ C 2 ,

∞ n=n 0

 0 |u n+1 -u n | 1 ≤ CΦ θ n 0 + CΦ 1/2 n 0 < +∞. (3.13)

  Proposition 2.3 (Lyapunov stability). Let ε ∈ [0, 1). If (u n , w n ) n is a sequence which complies with (2.10)-(2.11), then for all n ≥ 1,

.18)

  this into (2.20) gives(2.19).

	Remark 2.4. If τ is small enough, then by choosing ε ∈ (0, 1), we see that the scheme (2.10) is more dissipative than the original equation (1.1), since the H 1
	norm |u n+1 -u n | 2 1 appears in (2.19); in contrast, only the H -1 norm |u t | 2 -1 appears in (2.9).

* Paola F. Antonietti has been partially supported by SIR Project No. RBSI14VT0S "PolyPDEs: Non-conforming polyhedral finite element methods for the approximation of partial differential equations" funded by MIUR.

where C 2 = min{C 1 , 2 1-2θ -1} > 0. By summation on n, we obtain

for all n ≥ n ′ 0 . Thus, by choosing n 1 > n ′ 0 large enough, we have

for all n ≥ n 1 , which yields (3.15). Now, by summing estimate (3.12) on n, we find

for all n ≥ n 1 . Using (3.15) yields

for all n ≥ n 1 . We may change the constant C in order for the estimate to hold for all n ≥ 2. From (3.16) and the second equation in (2.12), we obtain the convergence rate for ( ẇn ). The convergence rate for w n is a consequence of (3.16) and (3.14). This concludes the proof.

Remark 3.4. It is possible to show that a local minimizer of E in V M is stable uniformly with respect to τ . More precisely, let (u τ n ) n denote a sequence which complies with (2.13) and corresponding to a time-step τ . We assume τ ∈ (0, τ ⋆ ] where τ ⋆ > 0 is such that 1/τ ⋆ > c 2 f /(8γ). If u ⋆ ∈ V M is a local minimizer of E in V M , and if u τ 0 = u τ 1 is close enough to u ⋆ in V M , then the whole sequence (u τ n ) n remains close to u ⋆ , uniformly with respect to τ ∈ (0, τ ⋆ ]. The proof of this stability result is based on the Lojasiewicz-Simon inequality (it may be false for a C ∞ nonlinearity, see [START_REF] Absil | On the stable equilibrium points of gradient systems[END_REF]). It is proved in [START_REF] Alaa | Convergence to equilibrium for discretized gradientlike systems with analytic features[END_REF] for several fully discrete approximations of the Allen-Cahn equation. The case of the semi-discrete scheme (2.13) is more involved. Indeed, dissipative estimates (uniform in τ ) are needed to obtain precompactness of the set {u τ n : τ ∈ (0, τ ⋆ ], n ∈ N} in V M . Moreover, as τ → 0 + , the dissipation due to the scheme vanishes (cf. Remark 2.4). Thus, instead of the series

)), we have to deal with the series n |u τ n+1 -u τ n | -1 . We refer the interested reader to [START_REF] Huang | Gradient inequalities[END_REF][START_REF] Miranville | Local and asymptotic analysis of the flow generated by the Cahn-Hilliard-Gurtin equations[END_REF] for the proof of stability of a local minimizer in an infinite dimensional setting (for continuous dynamical systems).