Sandrine Mathy
email: sandrine.mathy@upmf-grenoble.fr

Millennium goals and Climate-Change negotiations: for a climate and development convergence mechanism

Keywords: International climate negotiations, Millennium Development Goals, basic needs, bottom-up approach, climate and development convergence, output-based aid

In 2015 two major international events will coincide: the United Nations Climate Change Conference in Paris and negotiations on a new set of Millennium Development Goals for 2030, including measures to combat poverty. Until now these two dossiers have mainly been treated separately, but the stakes for development and the measures required to avoid unbridled climate change are inextricably linked, particularly in developing countries. In the build-up to the 2015 Paris Climate Conference, this article presents a proposal which seeks to reconcile development strategies with policies on climate change and adaptation, by promoting various forms of synergy between these two fields, in such a way as to make satisfying basic needs an absolute priority and to clear up the dispute between industrialized and developing countries regarding the latter's commitment to combating climate change. Our Climate and Development Convergence Mechanism advocates an approach based on voluntary, sector-based, flexible participation enabled by output-based aid and indicators of the satisfaction of basic needs.

POLICY RELEVANCE

The Climate and Development Convergence Mechanism could, in a single move, overcome several stumbling blocks within ongoing climate negotiations on climate change between developped and developing countries. First it could respond to the concerns of developing countries which do not accept constraints on their development on the grounds of combating climate change. On the contrary the mechanism represents a form of incentive to achieve convergence between development priorities, particularly poverty alleviation and satisfaction of several basic needs and the equally necessary efforts to cut greenhouse gas emissions, while developing in climate-change resilient ways. Second, if developing countries accepted the mechanism, industrialized countries could no longer maintain their current stance, demanding a formal commitment on emissions reduction. The potential of the mechanism relies on the credibility of financial support and technology-transfer mechanisms.

Introduction

Two key debates are set to occupy the international agenda in 2015: on the one hand the United Nations Climate Change Conference in Paris; on the other countries are due to set a new set of Millennium Development Goals for 2030, including combating poverty. So far these two processes have remained largely separate. However the stakes for development and the responses required to limit climate change are inextricably linked, particularly in developing countries.

It is extremely likely that the impacts of climate change due to an average temperature increase exceeding 2°C would decisively compromise economic development and efforts to effectively reduce poverty in developing countries. However, according to estimates published by the International Panel on Climate Change, without further commitments on emissions abatement, the planet could see temperatures rise by between 3.7°C and 4.8°C (and these are only global averages). The IPCC is also afraid that warming on this scale will exacerbate the plight of already vulnerable populations, such as slum dwellers, victims of malnutrition and those who lack access to improved sanitation facilities. Regions and whole countries, struggling to cope with rising population density or degraded ecosystems, could be destabilized.

However, since the Rio Earth Summit in 1992 and at subsequent annual international climate conferences, many developing countries have repeatedly set climate-change mitigation in opposition to development goals. They maintain that cutting greenhouse gas (GHG) emissions would place a further brake on growth. On these grounds they have, in particular, refused any commitment on emissions abatement, giving absolute priority to economic growth, industrial diversification and higher income in order to reduce poverty.

The scenarios in the latest IPCC report tell us that carbon neutrality must be achieved during the second half of the century in order to preserve the global emissions trajectories leading to the 2°C target; this requires cooperation by all parties.

It has to be admitted that another form of 'realpolitik' has prevailed. The industrialized nations are demanding a significant commitment on the part of developing and, particularly, emerging countries, which the latter have rejected on the grounds of development and the principle of common but differentiated responsibility, according to which it is up to industrialized nations to decarbonize their economies first. For a long time these positions have held back international negotiations on the climate.

In the build-up to the 2015 Climate Change Conference, this article presents a proposal designed to reconcile development strategies and climate policy, by targeting and supporting the deployment of effective synergies between these two concerns. In so doing it seeks to clear up the dispute between industrialized and developing countries.

In the first part we shall return to the contextual factors which have discouraged commitment by developing countries. In the second part we present the Climate and Development Convergence Mechanism (CDCM) for targeting low-carbon and climate-change resilient development goals. The final part shows how this proposal could overcome some of the obstacles holding up negotiations.

I. Context

Ending the monolithic view of developing countries

The principle of common but differentiated responsibility has, in the past, left a deep mark on international negotiations on climate change. It was first framed in Principle 7 of the Rio Declaration, as part of establishing 'a new and equitable global partnership'. It responded to the demands and views of developing countries which did not want to see environmental protection become an additional constraint on their development. This principle was operationalized in the Rio Convention and the Kyoto Protocol, through the distinction made between Annex I parties to the Convention, comprising industrialized countries, and non-Annex I parties, comprising developing countries. The Protocol (Article 3) only allocated binding targets for cutting GHG emissions to the first group. Furthermore, Annex II parties (a sub-group of Annex I), were required to provide financial support to non-Annex I parties to adapt to climate-change impacts and transfer technology (Article 11).

Until the 1990s this clear-cut distinction was justified by the substantial differences between developing and industrialized countries, in particular in terms of per capita income, historical responsibility for climate change, and per capita GHG emissions. But in the past 20 years this global setup has profoundly changed, leading to much more uneven conditions among non-Annex I parties, whereas for some characteristics the same conditions have prevailed in both developing and industrialized countries.

Firstly, the gap in terms of average per capita income between the least developed countries and emerging countries has widened (Table 1). The latter group has enjoyed powerful growth, closing the gap for GDP per capita based on purchasing power parity between China and the United States, for example, from 29 to 6.2. For India the gain was only from 25.5 to 13.3; for Brazil and South Africa it was largely steady; but for the least developed countries it deteriorated.

Source: Enerdata

The economic advances underpinning this trend over the past 20 years also raise the question of how much financial and technological support should be afforded to non-Annex I parties. The trend can no longer be treated as a single, uniform process. In just a few years China has become the world leader for wind turbines, with four Chinese manufacturers in the global top 10. China also manufactures almost half the world's solar panels. Obviously there is no question of adopting an exaggeratedly 'Sinocentric' outlook. [START_REF] Glachant | Promoting the International Transfer of Low-Carbon Technologies: Evidence and Policy Challenges[END_REF] show that to analyse technology transfer to the South and the framing of suitable policies we need to distinguish three groups of developing countries:

-China, Mexico and South Africa, and to a lesser extent Brazil, already appear to be well integrated in global trade in technology;

-India is some way behind, accounting for 4.9% of global GDP, whereas incoming technology transfer amounts to 1.5%;

-There is little technology-transfer to the least developed countries.

These observations cast doubt on the relevance of maintaining a strict distinction between Annex I and non-Annex I parties. At the very least they suggest the need for more flexible ways of applying the distinction, making allowance for the particularities of each country in the framing of a climate agreement.

The present 'firewall' between Annex I and non-Annex I parties fuels keen tension between states during international climate-change negotiations. In recent years two groups of negotiations between developing countries have been launched: the BASIC group comprising major emerging nations (Brazil, South Africa, India and China), which, it should be noted, played a very important part in the build-up to and during the Copenhagen Conference itself; and the group of Like-Minded Developing Countries (LMDC), a disparate assembly, given that India, China, Thailand and even Mali are among its members alongside several oil-exporting countries (such as Venezuela and Saudi Arabia). Due to the presence of countries which play a key role in both diplomacy and climate change (their action being required to keep global warming below 2°C), these groups exert real influence, which they use to advocate a series of demands specific to developing countries. However, even among developing countries there is disagreement on whether or not to maintain the distinction between Annex I and non-Annex I parties. Several of them are asking whether major emerging countries may legitimately remaining in the same category: the current classification exonerates non-Annex I parties from any quantified commitment to reduce emissions; furthermore, much as the least developed countries, it affords them the benefit of financial and technological support.

As we write a significant rift is opening up between the most fervent advocates of the 1992 status quo (the LDMC group) and Annex I parties (US, European Union, etc), the latter demanding that emerging countries commit themselves to reduce GHG emissions and campaigning for developing countries' entitlement to financial and technological support to be differentiated. For the least developed countries and the Alliance of Small Island States, the absolute priority is to reach an agreement which preserves a significant chance of keeping global warming below the 2°C ceiling, to ensure that their survival is not wholly compromised. They obviously realise that this goal will require leading emerging countries swiftly to adopt an emissions-reduction trajectory. They are also convinced that internal differentiation among non-Annex I parties could contribute to this process. The BASIC group have sometimes been accused of 'hiding behind the poor'1 to get out of any commitments under an international agreement.

Limiting global warming to 2°C requires an end to opposition between climate and development

The indisputable economic growth registered by most developing countries over the past 20 years must not be allowed to conceal the glaring inequality within these countries, in particular the chronic failure to satisfy certain basic needs of their population. The international community knows this, witness its 2000 commitment on the millennium development goals 2 . For example, the first of these goals is to halve, between 1990 and 2015, the proportion of people whose income is less than $1.25 a day, and also to halve the proportion of people who suffer from hunger. No one can deny that overall poverty has been reduced since the 1990s, but in 2010 there were still 1.2 billion people living on less than $1.25 a day 3 : 400 million in India, 400 million in sub-Saharan Africa and 230 million in China. Such poverty, which is more than just residual, goes hand-in-hand with large-scale failures to satisfy several basic needs: 840 million people in the world suffer from hunger; 2.5 billion lack access to improved sanitation facilities; 863 million urban residents live in slums. In sub-Saharan Africa, for example, only 34% of the population has access to electricity (58% in towns and only 8% in rural areas); in India the figure is 75%. To cope, people use hazardous, inefficient energy sources which are ultimately more costly. In Africa the poor spend $17 billion a year on oil lamps. Worldwide, 2.5 billion people use biomass to cook their food, with devastating effects on their health (IEA, 2011). So it is hardly surprising that developing countries should make satisfying such needs a priority, ahead of combating climate change.

On the other hand, we now know that even with global warming limited to 2°C, the many impacts of climate change will further endanger the most vulnerable communities, undermining recent gains against poverty. The World Bank's Turn Down the Heat report paints a very gloomy picture of a world 2 Eradicate extreme poverty and hunger; achieve universal primary education; promote gender equality and empower women; reduce child mortality; improve maternal health; combat HIV/Aids, malaria and other diseases; ensure environmental stability; global partnership for development. 3 Definition of extreme poverty according to the World Bank.

4°C hotter, on average, with extreme events and seriously disturbed climate causing far-reaching devastation and unbearable human suffering. These conclusions are endorsed by volume II of the latest IPCC assessment report and various national assessments.

To avoid this outcome, the IPCC explains, the world's population must become carbon neutral in the second half of the 21st century. The Deep Decarbonization Pathways Project4 also emphasizes the severity of the GHG pathways, within the same timeframe, to meet the 2°C target. If so, how can the solution to supplying electricity to 300 million Indians be to build new coal-fired power plants? Given the service life of infrastructure and facilities, with a very real risk of massive investment being made very misguidedly, it is surely essential that all the resources devoted to combating poverty should include a dual carbon and climate constraint.

In other words, just as the battle against climate change can only be stepped up if it makes allowance for the imperatives of development and combating poverty, so development strategies must integrate a decarbonized approach and adaptation to climate-change impacts. This opens the way for a positive agenda for achieving powerful synergies between international climate negotiations and new millennium development goals.

II. For convergence between poverty reduction, climate-change resilience and low-carbon development

It The corner stone of this mechanism would be a reporting system comprising a series of indicators reflecting the degree of satisfaction of certain basic needs. For each of these basic needs, any country falling below a set threshold would de facto be entitled to financial and/or technological support, to increase fulfilment of the need. If reporting is inadequate, the country would lose its entitlement for funding or technology transfer for the relevant basic need. The Green Fund would be responsible for funding the CDCM.

Identifying areas of convergence

The CDCM aims to focus on possible forms of convergence between poverty reduction, through more widespread fulfilment of basic needs, and the future reduction of GHG emissions and/or greater resilience to climate-change impacts. This concerns several sectors (Table 4). In many sectors synergy exists between all three tasks. With regard to housing, building bioclimatic5 dwellings would reduce the number of people living under foul conditions, limit GHG emissions from these dwellings and enhance the resilience of their homes to extreme events. Similarly, improving the global efficiency of farming would reduce the number of under-nourished people, limit GHG emissions in this sector and improve its climate-change resilience.

In some sectors there is only synergy between two dimensions. Efforts to reduce poverty go hand-inhand with greater resilience to the impacts of climate change, through enhanced healthcare. The development of health centres and access to treatment would reduce overall vulnerability to the public-health impacts of climate change, while satisfying an extremely basic need.

Choosing the criteria and thresholds for allocating aid

The choice of indicators must accurately reflect the degree to which needs in target sectors are fulfilled. Such indicators could be appraised either as a value relative to the population, or as an absolute value (for example the rate of electrification or the number of people without access to electricity6 , or indeed the concentration of particles). Several criteria may also be needed to give a true picture of the situation. So, sticking to access to energy, a distinction may be drawn between physical access to electricity and financial access, which depends on the relation between the price of electricity and household purchasing power.

Choosing the right criteria for each type of basic need should of course be the subject of detailed talks between developed and developing countries. The criteria should form the basis for reporting by non-Annex I countries. The agreed thresholds for allocation of CDCM-specific aid should be consistent with the revised poverty-reduction targets enshrined in the sustainable development goals. So, remaining with the example of electricity, if the new Millennium Development Goals stipulate that all global citizens should have access to electricity by 2030, the CDCM should make formal provision for this goal.

Output-based aid

The CDCM is an output-based approach. Aid will be provided on an ex-post basis once the targets and projects have been validated. This type of aid has already been experimented, in particular by the World Bank (see box), but on a limited scale.

Within a strict project-oriented framework, an output-based aid (OBA) is defined as a performancebased subsidy, under which service delivery is contracted out to a third party, usually a private firm. Most of the subsidy7 is paid only after the services or outputs have been delivered and verified by an independent agent. This mechanism has proved particularly effective for achieving project targets (GPOBA, 2009). Project appraisal shows that 85% of them achieved or exceeded their goals, inside the budget initially allocated, compared to only 49% for conventional projects [START_REF] Mumssen | Output-Based Aid: Lessons Learned and Best Practices[END_REF].

As part of the CDCM, output-based aid would be deployed on a 'macro' scale 8 and the third party, in this case a developing country, would receive the subsidy once progress in the satisfaction of a basic need had been observed and measured. For example, a project in the electricity sector would receive aid in proportion to the number of additional beneficiaries gaining access to electricity generated using decarbonized resources, expressed as a net change for the country as a whole, to prevent a gain in one region being lost in another. The same principle could be applied to promote salubrious housing built to meet bioclimatic standards, or to improved sanitation networks. Lastly, an independent body would carry out appraisals at regular intervals, every two to five years, for example.

The particular strategy for deploying such projects in each country, and contributing to better living conditions there, would not be discussed as part of the CDCM process; that is a matter of national sovereignty. Only results would be observed. The mechanism could equally well apply to a subsidy for infrastructure investment, for the costs of usage (funding a tariff scheme for instance) or other forms of aid. In the same way countries are free to choose how projects should be funded: out of a country's own funds, through private domestic or foreign investment, or development aid. The revenue contributed by the mechanism and covered by an ex post guarantee could be used to leverage project funding. The return-on-experience from output-based aid shows that a subsidy of $1 leverages $2 in private-sector funding [START_REF] Hussain | Results Based Financing Mechanisms: Experience from Output Based Aid Partnership Forum Panel Sessions -Financing Transformations[END_REF]. Figure 1 details the workings of the climate and development convergence mechanism.

8 This approach is close to the one described by S. [START_REF] Hallegatte | A Macro Output-Based Approach to Climate Change Adaptation Funding in Developing Countries[END_REF] for funding adaptation.

Box 1: Experience of output-based approaches (GPOBA, 2009)

The World Bank launched its first output-based projects in 2002. At that point this type of aid concerned 32 projects worldwide, representing a total of $1.5 billion in funding. This approach currently covers about 200 projects (131 backed by the World Bank and 66 others) benefiting about 60 million people.

The World Bank-sponsored projects represent a total subsidy of $3.5 billion provided by the Bank (supplementing $2.8 billion from other forms of public aid). These projects should, for example, afford access to electricity to nearly 9 million people (thanks to $204 million in output-based aid) or indeed benefit 4 million people with regard to water and sanitation ($166 million in output-based aid).

Of these projects, 52% concern South America and 33% Africa. Some 58% of the projects focus on transport, 6% on energy, 24% on healthcare, 5% on water and sanitation, 5% on education and 2% on telecommunications. The level of aid provided by the CDCM will be decided sector by sector, for each type of basic need, in such a way as to establish an international benchmarking system. The first step is to determine the cost of satisfying basic needs in a specific sector: the per-capita cost of access to electricity, improved sanitation or waste disposal, among others. Return on experience is valuable in this respect. The Camdessus report [START_REF] Camdessus | Report of the World Panel on Financing Water Infrastructure[END_REF] estimated that providing 19 million people living in rural areas of Africa with access to drinking water and a sewage network would cost $1.4 billion a year from 2003 to 2010. If this estimate was selected for all the countries, then for each additional person gaining access to these services, a country would receive $74 wholly or in part. Similarly, according to the International Energy Agency (2011), "to provide universal modern energy access by 2030 annual average investment needs to average $48 billion per year, more than five times the level of 2009". The same report asserts that electrification alone would require an investment of between $550 and $740 per person, depending on the type of access (on-grid, mini-grid or off-grid). Other cost estimates are provided by projects under the Global Partnership on Output-Based Aid (see http://www.oba-data.org/ for details of these projects). Yet another source is return on experience from the Clean Development Mechanism, which also enables us to estimate the surplus cost entailed by carbon-free technology. Predictably costs vary a great deal depending on the local context. Adopting a more macro-economic perspective (at country level, for instance) would certainly make it easier to estimate average rates.

The share of this cost covered by the CDCM would depend on various factors:

1. The financial resources available in the Green Fund. 2. Whether the project concerns emissions abatement.

If so, the amount of aid should be sufficient, on top of other funding and aid, to enable investors or service providers to see a return on facilities and in particular to cover any surplus cost entailed by using carbon-free technology or compliance with low-carbon development guidelines. 3. Whether the project concerns adaptation.

If so, is climate change the main reason for the need to invest? In the case of sectors or projects for which climate change is one reason among others justifying the project, the level of aid allocated by the CDCM would only cover part of the funding needs, the rest coming from conventional funding sources. 4. The aid ratios for adaptation and emissions abatement will be aggregated.

Talks on setting these ratios are likely to be difficult. Whatever happens they should involve the developing countries and will certainly require all such ratios to be adjusted to allow for the specific context in each country. The expertise of the Technology Executive Committee in identifying obstacles to the transfer of specific technologies to each of these countries will be valuable, particularly when framing guidelines.

At the same time care must be taken to avoid investments being concentrated in a particular country or sector, as was sometimes the case with the Clean Development Mechanism (Baker et al., 2011). It may prove necessary to establish ceilings for each sector and major geographical area, so that investment is not monopolized by the most advanced countries or the most profitable sectors.

III. Will this mechanism overcome the obstacles to negotiation on climate change? 1. Organizing 'intended nationally determined contributions' in an equitable frame of reference

The concept of 'intended nationally determined contributions' was introduced at the Warsaw Conference to replace the terms 'commitment' and 'nationally appropriate mitigation actions' (NAMAs) previously used for Annex I and non-Annex I parties, respectively. This raises two sorts of problem. The first one relates to how ambitious we may expect such contributions to be, in particular for them to remain consistent with the target of limiting global warming to 2°C once aggregated. The second one relates to assessment of voluntary contributions with regard to principles of equity. The Climate Action Network proposed to operationalize equity in a context of this sort in which commitments were not determined at domestic level, by establishing a frame of reference bringing together a series of indicators reflecting the main principles of equity. The CDCM is compatible with this framework, because the rate of satisfaction of basic needs count as an equity criterion to be taken into account when delimiting the margin of manoeuvre of each country regarding real reductions in GHG emissions, and lastly when assessing its needs in terms of adaptation.

Furthermore the CDCM creates a genuine incentive for countries to file conscientious reports, the prime condition for aid entitlement.

Respond to development stakes through country-specific strategies

The CDCM aims to make reducing poverty a top priority for climate policy. There can be no doubt that joint deployment along these lines would help overcome the traditional opposition between climate policy and development strategy. To achieve this, each country should, ideally, frame its own low-carbon, climate-change resilient development plan. This document would present in detail the sector-based approaches which could be adopted to satisfy the basic needs of a broader public. For example, such approaches might seek to deploy domestic policies tailored to attract private investors for infrastructure funding, or indeed tariff reforms guaranteeing real access to a range of services for the whole community.

A bottom-up approach with a sector-based pick-list

An additional advantage of the CDCM is its sector-based approach to the stakes for climate change and development. It thus responds to equity concerns with a bottom-up approach. Top-down strategies, which were preferred for many years, have failed to define a single framework applicable to all states in order to operationalize equity principles, in particular with respect to sharing the burden of reducing GHG emissions between different countries. So we shall certainly have give up the idea of a grand, all-embracing global structure.

The CDCM itself is strictly sector-based. By using a pick-list, it enables a country to commit itself to one or other field of action. As eligibility for financial support and technology transfer is also sectorbased, it is quite conceivable that a country may be eligible for some sectors, not for others.

An incentive for all developing countries to take part

The CDCM could attract widespread support on the part of developing countries, due to the flexible, incentive-based form of partnership it promotes. In this respect it resembles the model proposed as part of the Montreal Protocol for addressing equity issues. [START_REF] Wemaëre | Equity in the Ozone Layer Regime -Montreal Protocol on Substances that Deplete the Ozone Layer[END_REF] shows that large-scale support by countries, in particular developing countries, was facilitated by two principles: flexibility regarding the deadline set for achieving the objectives for countries with consumption of the substances targeted by the protocol below a certain level; and a support mechanism which represented both an incentive for action and a means of applying sanctions (withdrawal of support in the event of failure to meet targets or submit reports). With the CDCM there are no sanctions, it being difficult to impose binding targets within a framework of voluntary contributions.

The CDCM is close, in terms of its sector-based, voluntary approach, to the sector-based cleandevelopment mechanism [START_REF] Samaniego | A Sector-Based Clean Development Mechanism[END_REF], to 'no-loose targets' [START_REF] Schmidt | Sector-Based Approach to the Post-2012 Climate Change Policy Architecture[END_REF] or indeed sustainable development policies and measures (SD-Pams) [START_REF] Winkler | Sustainable Development Policies and Measures: Starting From Development to Tackle Climate Change[END_REF], which focus on, respectively, sector-based emissions mitigation programmes, sector-based targets for reducing energy intensity, or setting up policies in line with the development priorities of countries, while cutting emissions. However, in all three cases, support through emission-reduction credits is based on checking that emissions have been reduced by a certain amount compared to a hypothetical baseline scenario, which creates an incentive to fictitiously inflate the volume of emissions taken into account by the baseline scenario. Abuses of this sort were observed with the clean-development mechanism. In the present case aid is assessed ex-post on the basis of observed results. Lastly the CDCM overcomes the restrictive project-based dimension specific to CDMs to become a full-scale development strategy and programme.

IV. Conclusion

If a mechanism of this sort was debated at the international climate conference, it could, in a single move, overcome several stumbling blocks.

It would respond to the concerns of developing countries which will never accept constraints on their development on the grounds of combating climate change. On the contrary the mechanism represents a form of incentive to achieve convergence between development priorities and the equally necessary efforts to cut greenhouse gas emissions, while developing in climate-change resilient ways. However, it is of prime importance that financial support and technology-transfer mechanisms should be credible. As an essential condition for agreement by developing countries, this will probably require innovative funding sources, reaching beyond existing public development aid and the as yet nascent Green Fund.

At the same time, if developing countries accepted the mechanism, it would no longer be tenable for certain industrialized countries to maintain their current stance, demanding a formal commitment on emissions reduction, a further source of friction which has so far prevented a global climate agreement.

Public attention is already focused on Paris and the COP21 meeting to be held there at the end of 2015. Thinking back to the media excitement in the build-up to COP15, at Copenhagen in 2009, the stakes for the coming climate talks are clearly high, particularly for COP21, which is attracting the attention of many stakeholders, some of which exert considerable influence on public opinion. For many in the North and South the link between unpredictable climate conditions and public vulnerability is long established. But public opinion is also aware that achieving the Millennium Development Goals has failed to lift a substantial share of the world's population out of poverty; it is also increasingly doubtful about the difficult process of framing a new global climate agreement. COP21 has no plans to solve all the world's problems in just a few days, but expectations are so high that it is fair to say that the image of the international community would benefit from a diplomatic outcome which demonstrates that developed and developing countries share a common destiny, enshrined in efforts to bring global warming under control while satisfying the basic needs of which the poorest people are currently deprived.

  is consequently essential that the agreement reached at the Paris Climate Conference in 2015 should introduce a mechanism for directly targeting satisfaction of basic needs through decarbonized forms of development which are also climate-change resilient. The idea we shall develop below is based on a flexible architecture acting as an incentive for developing countries to contribute to combating climate change, backed by funding and technology transfer. In the following we shall refer to it as the Climate and Development Convergence Mechanism (CDCM).

Figure 1 :

 1 Figure 1: Operating diagram for the climate and development convergence mechanism. Continuous lines indicate financial streams, dotted lines information streams

Table 1 : Variation in per capita income between 1990 and 2010 in several developing countries based on purchasing power parity and in relation to US per capita income.

 1 Matching economic growth the balance of GHG emissions is also contrasting for all the non-Annex I parties. In 2010 China, India and Brazil ranked first, third and fifth, respectively, among the planet's largest sources of CO 2 emissions. But these figures are national aggregates and require closer analysis (Table2). The per capita emissions in India and Brazil are much lower than in China or South Africa, where the figures are comparable to some Annex I parties. Over and above the specific energy mixes of each country, these figures reflect the different levels of development reached by these countries.

		GDP per capita, PPP $	GDP per capita/US
		(thousands)
		1990	2010	1990	2010
	United States	23.0	46.6	-	-
	European Union	14.5	31.6	1.6	1.5
	India	0.9	3.5	25.5	13.3
	China	0.8	7.6	29.0	6.2
	South Africa	5.8	10.5	4.0	4.4
	Brazil	5.2	11.2	4.4	4.2
	Senegal	1.1	1.9	21.6	24.2
	Benin	0.8	1.6	27.7	29.6
	Burundi	0.5	0.6	46.3	80.4
	Cameroon	1.5	2.3	15.3	20.6
	Source: Enerdata

Table 2 : CO 2 emissions, in absolute value and per capita, for various developing countries, including several emerging countries, and for the US and France

 2

		Emissions (MtCO2)	Per capita emissions (tCO2)	US per capita emissions in relation to country
		1990	2010	1990	2010	1990	2010
	United States	4804	5432	19.2	17.6
	European Union	4028	3620	8.4	7.1	2.3	2.5
	India	547	1696	0.6	1.4	29.7	12.3
	China	2331	7134	2.1	5.3	9.4	3.3
	South Africa	246	367	7.0	7.4	2.8	2.4
	Brazil	186	375	1.2	1.9	15.5	9.1
	Benin	0.2	4.4	0.1	0.5	370.1	35.1
	Burundi	0.2	0.2	0.04	0.03	481.1	702.3
	Cameroon	2.6	5.0	0.2	0.3	89.9	69.4

Table 3 : Rate of satisfaction of certain basic needs in some developing countries

 3

		Electrification rate (%)	Proportion of the population using sanitation facilities (%)	Population below $1.25 (ppp) per day (%)	Exposure to PM10 (mg/m3)	Slum population as percentage of urban
	China	99.8	65	2.8	82	31
	India	75.3	36	7.5	100	32
	Brazil	99.3	81	3.6	36	28
	South Africa	84.7	74	2.3	40	29
	Mali	72.9	22	16.4	55	66
	Indonesia	80	59	2.7	47	23

*Urban-population weighted PM10 levels in residential areas of cities with more than 100.000 residents. The estimates represent the average annual exposure level of the average urban resident to outdoor particulate matter.

Source: http://mdgs.un.org/unsd/mdg/Data.aspx

Table 4 : Sector-based convergence between satisfying basic needs, combating climate change and enhancing climate-change resilience

 4

		Example of action	Poverty-reduction	GHG mitigation goal	Adaptation goal
			goal
	Housing	Promote bioclimatic	Reduce number of	Limit GHG emissions	Reduce exposure to
		building techniques	people living in	from new housing	extreme events
			slums
	Food	Improve agricultural	Reduce number of	Limit GHG emissions	Reduce vulnerability of
		efficiency from farm to	people suffering	from agriculture	agriculture to climate
		fork	from malnutrition		change
	Electricity	Develop decarbonized	Reduce number of	Limit GHG emissions	Contribute to resilience
		electricity generation	people without	from electricity	of electricity sector
		and energy efficiency	electricity	generation
	Waste	Waste-gas capture	Reduce pollution.	Limit GHG emissions	Limit health impacts of
		facilities	increase rate of	from waste	badly managed waste in
			waste processing		the context of climate
					change
	Sanitation	Develop efficient or	Reduce number of	Limit GHG emissions	Limit exposure to
		improved sewerage	people without	from effluents	flooding and/or
		networks	sanitation		epidemics (extreme
					events)
	Transport	Public transport	Broaden access to	Limit GHG emissions	Contribute to resilience
		networks. soft mobility	mobility and	from vehicles	of transport
		modes	improve public		infrastructures
			health in cities		(extreme events)
	Healthcare	Increase rate of	Improve public	-	Limit exposure to
		healthcare coverage	health and broaden		epidemics and extreme
			access to treatment		events

This expression refers to a publication by Greenpeace India, Hiding Behind the Poor -A report by Greenpeace on Climate injustice in

which prompted a controversy because it suggested that 'rich' Indians were hiding behind the hundreds of millions of poor to justify their refusal to make a commitment to climate policies.

http://unsdsn.org/what-we-do/deep-decarbonization-pathways/

Bioclimatic can refer to traditional construction techniques.

Defining thresholds as an absolute value or as a share relative to the population makes a difference. Despite having a 99.8% rate of electrification, 3 million Chinese lack physical access to electricity, which is more than other countries with a lower population and a much lower rate of electrification.

These subsidies may be one-off to subsidize investment, transitory or continuous to subsidize, use through a tariff suited to the poorest members of the community, for example.


Y Mammeri
email: youcef.mammeri@math.univ-lille1.fr

Unique continuation property for the KP-BBM-II equation

Keywords: KP equation, unique continuation property, compact support MS Codes. 35B60, 35Q53

We prove, by using a method introduced by Constantin [6], that if the solution of the Cauchy problem associated with the KP-BBM-II equation, has a compact support for all times, then this solution vanishes identically. The only restriction is that the support in the y-direction has to be small.

Introduction

The small amplitude long waves in shallow water moving mainly in the x-direction are modelled in dimension 2 by the Kadomtsev-Petviashvili equations [START_REF] Kadomtsev | Model equation for long waves in nonlinear dispersive systems[END_REF] u t + u x + αuu x + sβu xxx + γ∂ -1 x u yy = 0, called KP-I if s = -1 and KP-II if s = 1, according to whether the surface tension is or is not neglected, and where α denotes the quotient between the waves amplitude and the depth of the water, β the square of the quotient between the depth and the wavelength, γ the square of the quotient between the wavelength in the two directions of the plan and ∂ -1

x denotes the anti-derivative, defined by the Fourier symbol ∂ -1

x u(k) := û(k) ik . We recall that 0 ≤ α, β, γ ≤ 1. Since this equation is obtained correcting, at the second order, the transport equation u t + u x = 0, we find the KP-BBM equations [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]

u t + u x + αuu x -sβu xxt + γ∂ -1 x u yy = 0, called KP-BBM-I if s = -1 and KP-BBM-II if s = 1. The KP-BBM-I equation is not well posed in L 2 (R 2
) and so we study here only the KP-BBM-II equation. Indeed, the KP-BBM-I equation can be rewritten

(1 + β∂ 2 x )u t = -u x -αuu x -γ∂ -1
x u yy , and the operator (1 + β∂ 2

x ) is not invertible in L 2 (R 2 ). The goal of this work is to show that if the solution of the Cauchy problem associated with the KP-BBM-II equation has a compact support on a time interval, then this solution vanishes identically. In dimension 1, Saut and Scheurer [START_REF] Saut | Unique continuation for some evolution equations[END_REF] proved the unique continuation property for a general class of dispersive equation, in particular for the Korteweg-de Vries equation, by using the Carleman's estimates 1 [START_REF] Carleman | Sur les systèmes linéaires aux dérivées partielles du premier ordre à deux variables[END_REF]. In 1997, Bourgain introduced another method [START_REF] Bourgain | On the compactness of the support of solutions of dispersive equations[END_REF] based on entire function estimates. Thanks to the Paley-Wiener theorem, the entire function here is given by an analytic continuation of the Fourier transform in space of the Duhamel formula. In this way, he proved the unique continuation property for the KdV equation and Panthee [START_REF] Panthee | A note on the unique continuation property for Zakharov-Kuznetsov equation[END_REF][START_REF] Panthee | Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation[END_REF] expanded this property to the KP-II equation. But it is not yet clear if the proof can be expanded to the KP-BBM-II equation. Indeed, to use Bourgain's method, we need to prove that for all R > 0, there exists k ∈ R, with |k| > R, such that

|L ′ (k)| ≥ |f (k)|, with lim |k|→∞ |f (k)| = + ∞. (0.1)
where L is the symbol of the linear evolution. In the context of the KdV equation, we choose f (k) = k 2 [START_REF] Bourgain | On the compactness of the support of solutions of dispersive equations[END_REF]. We notice for the BBM equation

u t + u x + αuu x -βu xxt = 0,
the symbol of the linear evolution is given by 2 tends to zero at infinity and the condition (0.1) is not satisfied.

L(k) = k 1 + βk 2 , thus L ′ (k) = 1 -βk 2 (1 + βk 2 )
In dimension 2, this condition becomes for all R > 0, there exists

(k, l) ∈ R 2 , with |k| + |l| > R, such that |∂ k L(k, l)| ≥ |f (k, l)|, with lim |k|+|l|→∞ |f (k, l)| = + ∞. (0.2)
Let us see how behave the KP equations

u t + u x + αuu x + s βu xxx + γ∂ -1 x u yy = 0. We have L(k, l) = k -sβk 3 + γl 2 /k, and ∂ k L(k, l) = 1 -3sβk 2 -γ l 2 k 2 . If s = 1, i.e.
for the KP-II equation, we choose f (k, l) = |k| + |l| and the condition (0.2) is satisfied [START_REF] Panthee | Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation[END_REF]. If s = -1, i.e. for the KP-I equation, there exists (k, l) large (l ∼ k 2 ) such that ∂ k L(k, l) vanishes, and the method introduced by Bourgain no longer applies. In a similar way, the symbol of the linear evolution is given for the KP-BBM-II equation by

L(k, l) = k + γl 2 /k 1 + βk 2 , thus ∂ k L(k, l) = (1 -βk 2 )k 2 -γl 2 (1 + 3βk 2 ) k 2 (1 + βk 2 ) 2 .
For k and l large, ∂ k L(k, l) can be very small if |l| ≪ |k|.

In order to study the unique continuation property for the KP-BBM-II equation, we have to envisage another way. We are then inspired by a simpler method introduced by Constantin [START_REF] Constantin | Finite propagation speed for the Camassa-Holm equation[END_REF] based on integration by parts. If we denote X s (R 2 ) the space of functions f in

H s (R 2 ) such that ∂ -1 x f belongs to H s (R 2 ), provided with the norm |f | s = ||f || 2 s + ||∂ -1 x f || 2 s 1/2 ,
our result reads as follows. Then u vanishes identically.

We recall that, for all positive real number s, H s (R 2 ) is the Sobolev space of square integrable functions, which the s first derivatives are also square integrable, equipped with the norm

||f || 2 s = R 2 (1 + k 2 + l 2 ) s | f (k, l)| 2 dk dl,
where the Fourier transform in space is defined by

f (k, l) = R 2
e -i(kx+ly) f (x, y) dx dy.

Summary of existence theory

We consider the Cauchy problem

u t + u x + αuu x -βu xxt + γ∂ -1 x u yy = 0 (1.1) u(x, y, 0) = f (x, y). (1.2)
This Cauchy problem is dealt in the articles of Bona, Liu and Tom [START_REF] Bona | The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations[END_REF], or Saut and Tzvetkov [START_REF] Saut | Global well-posedness for the KP-BBM equations[END_REF]. They proved that for all initial datum in the subspace of L 2 (R 2 ) provided with the norm (||u|| 2 + ||u x || 2 ) 1/2 , there exists an unique global in time solution.

Unique continuation property

We start proving that the anti-derivative ∂ -1

x can be written as an integral.

∂ -1 x u(x, y, t) = x -∞ u(x ′ , y, t) dx ′ .
Moreover u is integrable with mean zero.

Proof. Let u ∈ C [-T, T ]; X s (R 2 ) solution of the Cauchy problem (1.1)-(1.2). We denote v(x, y, t) := ∂ -1

x u(x, y, t) and w(x, y, t

) := x -∞ u(x ′ , y, t) dx ′ .
We deduce that, for all t ∈ [-T, T ], v(t) = ŵ(t) in S ′ (R 2 ), where S ′ (R 2 ) is the space of tempered distributions. Since the Fourier transform is bijective on the tempered distribution, we conclude by definition of the Sobolev spaces with respect to S ′ (R 2 ). Let us prove now that u is integrable with mean zero. Since u has compact support, and u ∈ C [-T, T ]; X s (R 2 ) , with s > 2, the Cauchy-Schwarz inequality implies that u is integrable over the x and y-directions. Moreover,

∂ -1 x u ∈ C [-T, T ]; H s (R 2 ) , thus, by property of Sobolev space, lim x→+∞ ∂ -1 x u(x, y, t) = 0 = +∞ -∞ u(x, y, t) dx. Lemma 2.2 Let s > 2 and u ∈ C [-T, T ]; X s (R 2 ) solution of the Cauchy problem (1.1)-(1.2). If for all t ∈ [-T, T ], u(t) has a compact support, then for all t ∈ [-T, T ] and g ∈ C ∞ (R), R 2 e x √ β g(y)(u -βu xx )(x, y, t) dx dy = 0. Proof. Let G ∈ C ∞ (R 2 ; R) and t ∈ [-T, T ].
Since u(t) has a compact support, an integration by parts gives

R 2 G(x, y)(u -βu xx )(x, y, t) dx dy = 0, if G(x, y) -βG xx (x, y) = 0.
We set then G(x, y) = e

x √ β g(y).

We can prove now the main result of this paper. Proof of the theorem 0.1. We have, thanks to the differentiation under the integral sign theorem of Lebesgue,

0 = d dt R 2 e x √ β g(y)(u -βu xx )(x, y, t) dx dy = R 2 e x √ β g(y)(u t -βu xxt )(x, y, t) dx dy,
and since u is the solution of the equation (1.1), we find

0 = R 2 e x √ β g(y)(u x + αuu x + γ∂ -1
x u yy )(x, y, t) dx dy.

According to the lemma 2.1, we apply an integration by parts in the x-direction to find

0 = R 2 e x √ β 1 √ β g(y) u + α 2 u 2 + e x √ β
βγ g(y) u yy (x, y, t) dx dy.

Two new integrations by parts for the third term in the y-direction give

0 = R 2 e x √ β 1 √ β g(y) + βγg ′′ (y) u(x, y, t) + α 2 √ β e x √
β g(y)u 2 (x, y, t) dx dy.

Our aim is to solve the ordinary differential equation g(y) + βγg ′′ (y) = 0 with g(y) > 0.

(2.1)

One solution is given by g(y) = cos y + y 0 √ βγ , with y 0 to be chosen. Finally, we find

R 2 e x √ β cos y + y 0 √ βγ u 2 (x, y, t) dx dy = 0.
We choose then y 0 such that x u yy = 0, and since from the modelling point of view it is permissible to change u x by -u t in the fourth term [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF], we find another KP-BBM type equations u t + u x + αuu x -βu xxt + sγ∂ -1

y + y 0 √ βγ ∈ - π 2 , π 2
x u yy = 0.

(2.2)

In particular, the equation (2.2) with s = -1 is here well-posed [START_REF] Saut | Global well-posedness for the KP-BBM equations[END_REF], and we can expand the result of unique continuation property as follows. Let us notice that there is no restriction on the support in the y-direction in this case.

Theorem 0. 1

 1 Let s > 2 and u ∈ C [-T, T ]; X s (R 2 ) the solution of the Cauchy problem associated with the KP-BBM-II equation. Let us suppose that there exists 0 < B < +∞ and C > D in R, with |D -C| √ βγ < π, such that for all t ∈ [-T, T ] supp u(t) ⊆ [-B, B] × [D, C].

Lemma 2. 1

 1 Let s > 2 and u ∈ C [-T, T ]; X s (R 2 ) solution of the Cauchy problem (1.1)-(1.2). If for all t ∈ [-T, T ], u(t) has a compact support, then for all (x, y) ∈ R 2 and t ∈ [-T, T ],

Theorem 2. 4

 4 Let s > 2 and u ∈ C [-T, T ]; X s (R 2 ) solution of the Cauchy problem associated with the equation (2.2) for s = -1. Let us suppose that there exists 0 < B < +∞ such that for all t ∈ [-T, T ] supp u(t) ⊆ [-B, B] × [-B, B]. Then u vanishes identically. Proof. The proof is similar to the KP-BBM-II one by taking the function G(x, y) = e x √ β + y √ βγ .

  If we consider the KP equations written under the formu t + u x + αuu x + βu xxx + sγ∂ -1

	Remark 2.3
		. However, for all t ∈ [-T, T ], the support of u(t) verifies
	that supp u(t) ⊆ [-B, B] × [D, C], with	|D -C| √ βγ	< π, and this implies that e	x √ β cos	y + y 0 √ βγ	> 0 on the
	support of u(t). In particular u vanishes identically.


Y Mammeri
email: youcef.mammeri@math.univ-lille1.fr

On the decay in time of solutions of some generalized regularized long waves equations

Keywords: BO, BBM, KP equations, decay in time MS Codes. 35B05, 35B40, 35Q53, 76B03, 76B55

We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation ut +ux +u ρ ux +H(uxt) = 0, where H is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation ut + ux + u ρ ux -uxxt + ∂ -1

x uyy = 0. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power ρ of the nonlinearity is higher than 3, the respective solution of these equations tends to zero when time rises with a decay rate of order close to 1 2 .

Introduction

The small amplitude long waves moving inside a nonhomogeneous fluid are modelled in dimension 1 by the Benjamin-Ono equation [START_REF] Benjamin | Internal waves of permanent form in fluids of great depth[END_REF][START_REF] Davis | Solitary internal waves in deep water[END_REF][START_REF] Ono | Algebraic solitary waves in stratified fluids[END_REF]

u t + u x + uu x -H(u xx ) = 0,
where H indicates the Hilbert transform in the direction x. Since this equation is obtained correcting, at the second order, the transport equation u t + u x = 0, the BO-BBM equation is found [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]

u t + u x + uu x + H(u xt ) = 0.
We consider here a generalization of this equation, namely the gBO-BBM equation

u t + u x + u ρ u x + H(u xt ) = 0, (1.1)
where ρ is a nonnegative integer. We are interested here in the decay in time for small amplitude solution of the equation (1.1). For 1 < p < 2 and (m 0 , m 1 ) ∈ R 2 , we denote X m0,m1,p (R) = H m0 (R) ∩ W m1,p (R) the space of functions f such that the norm ||f || X m 0 ,m 1 ,p := ||f || H m 0 + ||f || W m 1 ,p is finite. Our result reads as follows.

Theorem 1.1 Let ρ ≥ 3. For 0 < δ < 1 3 -8 9ρ , we set m = 1 2δ -1. We choose 1 < p < 2 and q > 2 such that 1 p + 1 q = 1 and 0 < 1 q < 1 2 -4 3ρ (1 -3δ) .

Then there exists ε > 0 sufficiently small so that for all f ∈ X m+1,4-2 p ,p (R) such that ||f || X m+1,4-2 p ,p ≤ ε, there exists an unique global in time solution u ∈ C R; X m+1,4-2 p ,p (R) of the gBO-BBM equation (1.1) with initial datum f . Moreover this solution verifies that there exists a constant C > 0, depending only on m, p and ε such that for all time t ∈ R, we have

||u(t)|| L q + ||u x (t)|| L q ≤ C(1 + |t|) (-1 2 + 3δ 2 )(1-2 q ) .
We remark that if δ approaches zero then the decay rate of the norm L q of u(t), for q ∈ [2, +∞[, approaches t -1 2 (1-2 q ) , but we need to impose more regularity on the initial datum. We notice that t -1 2 (1-2 q ) is the decay rate of the norm L q of the linear evolution. We are inspired by the method described by Albert [START_REF] Albert | Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation[END_REF] for generalized Benjamin-Bona-Mahony equation

u t + u x + u ρ u x -u xxt = 0.
In this paper, Albert shows that, for a power ρ strictly higher than 4, the solution of the preceding equation decreases in time with a rate of order 1 3 , what is the same order as the rate of the decay in time of the solution of the generalized Korteweg-de Vries equation [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal and on a new type of long stationnary waves[END_REF][START_REF] Strauss | Dispersion of low-energy waves for two conservative equations[END_REF] u t + u x + u ρ u x + u xxx = 0.

In our work, we prove that the decay rate in time of the solution of the gBO-BBM equation is of order close to 1 2 , what is equal to the decay rate in time of the solution of generalized Benjamin-Ono equation [START_REF] Hayashi | Large time asymptotics of solutions to the generalized Benjamin-Ono equation[END_REF].

In a second time, we study the small amplitude long waves in shallow water moving mainly in the direction x, which are modelled in dimension 2 by the Kadomtsev-Petviashvili equations [START_REF] Kadomtsev | Model equation for long waves in nonlinear dispersive systems[END_REF] u t + u x + uu x + σu xxx + ∂ -1

x u yy = 0, called KP-I if σ = -1 and KP-II if σ = 1, according to whether the surface tension is or isn't neglected. Since this equation is also obtained correcting, at the second order, the transport equation u t + u x = 0, the KP-BBM equations are found [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]

u t + u x + uu x -σu xxt + ∂ -1 x u yy = 0, called KP-BBM-I if σ = -1 and KP-BBM-II if σ = 1. The KP-BBM-I is not well posed in L 2 (R 2 )
, and so we study here only the KP-BBM-II equation.

We consider a generalization of this equation, namely the gKP-BBM-II equation

u t + u x + u ρ u x -u xxt + ∂ -1 x u yy = 0, (1.2)
where ρ is a nonnegative integer. We prove a similar result as the theorem 1.1 for the equation (1.2).

Theorem 1.2 Let ρ ≥ 3. For 0 < δ < 1 5 - 8 15ρ
, we set m = 1 2δ -2. We choose 1 < p < 2 and q > 2 such that

1 p + 1 q = 1 and 0 < 1 q < 1 2 - 4 3ρ(1 -5δ) .
Then there exists ε > 0 sufficiently small so that for all f ∈ X 2m+1,7-4 p ,p (R 2 ) such that ||f || X 2m+1,7-4 p ,p ≤ ε, there exists an unique global in time solution u ∈ C R; X 2m+1,7-4 p ,p (R 2 ) of the gKP-BBM-II equation with initial datum f . Moreover this solution verifies that there exists a constant C > 0, depending only on m, p and ε such that for all time t ∈ R, we have

||u(t)|| L q + ||u x (t)|| L q ≤ C(1 + |t|) (-1 2 + 5δ 2 )(1-2 q ) .
We recall that, for the generalized KP equations, Hayashi, Naumkin and Saut [START_REF] Hayashi | Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation[END_REF] proved that for ρ ≥ 3, the decay rate of the solution of these equations is of order 1.

Whether it is for the gBO-BBM or the gKP-BBM-II, the Strauss method [START_REF] Strauss | Dispersion of low-energy waves for two conservative equations[END_REF] will be used to prove the decay in time. For T > 0, it consists of choosing a norm N T , which depends on time a priori, such that for a solution of the gBO-BBM equation, all the derivatives act like a linear term and are included in a Sobolev norm of sufficiently high order and such that the power ρ are linked with the decay in time.

More precisely, the Duhamel formula is written for the gBO-BBM equation (1.1) and for t ≥ 0

Φu(t) = S t f - 1 ρ + 1 t 0 S t-τ D x 1 + |D x | u ρ+1 (τ ) dτ,
where (S t ) t≥0 is the semi-group of the gBO-BBM evolution and

D x 1 + |D x |
denotes the operator defined by the Fourier multiplicator σ(k

) := ik 1 + |k|
. The fractional Leibniz rule implies for m ≥ 0 and 1 < p < +∞, there exists a constant C > 0 such that for t ∈ [-T, T ],

||D m u ρ+1 || L p (t) ≤ C||D m u|| L p (t) ||u|| ρ ∞ (t)
. The first term of this inequality is a linear term and a Sobolev norm, and for the second one, we choose

θ > 0 such that ||u|| ∞ (t) ≤ CN T (u)(1 + t) -θ .
Then we show that for sufficiently small initial datum, this norm N T is bounded independently of time.

The principal difficulty here is that the operator

D x 1 + |D x | is not bounded in L 1 (R) but only in L p (R) for 1 < p < +∞.
In dimension 2, we have to be careful that the operator

D x 1 + D 2 x is not bounded in L 1 (R 2 ) but only in L p (R 2 ) for 1 < p < +∞.
We will use the following notations: for n = 1, 2 and 1 ≤ p < ∞, we denote L p (R n ) the space of p-power integrable functions equipped with the norm

||f || L p := R n |f (x)| p dx 1/p , we denote L ∞ (R n ) the functions space equipped with the norm ||f || ∞ = sup ess (f ) := inf {c ; |f (x)| ≤ c almost everywhere in R n } . Let T an operator from L p (R n ) to L q (R n ) with 1 p + 1 q = 1
, we define the operator norm by

|||T ||| L q := sup ||f || L p =1 ||T f || L q .
The Schwartz space is denoted by S(R n ) and for 1

≤ p ≤ ∞, W m,p (R n ) is the Sobolev space equipped with the norm ||f || W m,p := (1 -∆) m/2 f L p .
In particular, we will note H m (R n ) the Sobolev space W m,2 (R n ).

We organise the paper as follows. In the first section, we will give estimates for the linear Cauchy problem associated with the gBO-BBM equation (1.1), and in the second one, the existence and uniqueness of global solution with the decay in time will be done. Finally, the decay in time for the gKP-BBM-II equation (1.2) will be studied.

Estimates for the linear gBO-BBM equation

We consider the linear Cauchy problem

u t + u x + H(u xt ) = 0 ; u(x, 0) = f (x).
Let us suppose that the initial datum f belongs to the Schwartz space S(R). Then the Fourier transform in space implies ût (1 + |k|)

+ ikû = 0 ; û(k, 0) = f (k).
The solution u of this ordinary differential equation is given by, for all x ∈ R and t ∈ R * ,

u(x, t) = 1 2π +∞ -∞ e -ithα(k) f (k) dk, (2.1)
with h α (k) = k 1 + |k| -αk and α = x t .
We recall the Van der Corput lemma [START_REF] Stein | Oscillatory integrals in Fourier analysis[END_REF]. (2.2)

Proof. In [START_REF] Stein | Oscillatory integrals in Fourier analysis[END_REF], the proof is done for λ = 1. Here it is enough to set g

(k) = h(k) λ . We have then b a e -ith(k) dk = b a e -i(λt)g(k) dk, with k ∈ [a, b], |g (k)| ≥ 1.
Now, the Van der Corput lemma is applied with the function h α .

Lemma 2.2 Let 0 ≤ δ < 1 3
. For all α ∈ R and all time |t| ≥ 1, we have

|t| δ -1 -|t| δ +1 e -ithα(k) dk ≤ 10 √ 2|t| -1 2 + 3δ 2 .
Proof.

We only consider the contribution for k ≥ 0, the contribution for k ≤ 0 being dealt with similarly. For k ≥ 0, we have

h α (k) = k 1 + k -α k , h α (k) = 1 (1 + k) 2 -α , h α (k) = -2 (1 + k) 3 . For k ∈ [0, |t| δ -1], we have |h α (k)| = -2 (1 + k) 3 = 2 (1 + k) 3 ≥ 2|t| -3δ .
The lemma 2.1 is then applied to find

|t| δ -1 0 e -ithα(k) dk ≤ 10 (2|t| -3δ |t|) 1/2 ≤ 5 √ 2|t| -1 2 + 3δ 2 .
The oscillating integral (2.1) can be majorized.

Lemma 2.3 Let 0 < δ < 1 3 and m = 1 2δ -1. For all f ∈ H m (R), α ∈ R and all time |t| ≥ 1, we have +∞ |t| δ -1 e -ithα(k) f (k) dk ≤ 2 m 2m -1 ||f || H m |t| -1 2 + 3δ 2 ,
and

-|t| δ +1 -∞ e -ithα(k) f (k) dk ≤ 2 m 2m -1 ||f || H m |t| -1 2 + 3δ 2 .
Proof. We have

+∞ |t| δ -1 e -ithα(k) f (k) dk ≤ +∞ |t| δ -1 f (k) dk = +∞ |t| δ -1 (1 + k) m (1 + k) m f (k) dk,
and the Cauchy-Schwarz inequality gives

+∞ |t| δ -1 e -ithα(k) f (k) dk ≤ +∞ |t| δ -1 (1 + k) 2m | f (k)| 2 dk 1/2 +∞ |t| δ -1 dk (1 + k) 2m 1/2 .
In one hand, since for 0 < δ < 1 3 and m = 1 2δ -1, the order m of the Sobolev space is strictly higher than 1 2 and we have

+∞ |t| δ -1 dk (1 + k) 2m 1/2 = -1 (2m -1)(1 + k) 2m-1 +∞ |t| δ -1 1/2 = 1 √ 2m -1 |t| δ 2 (1-2m) = 1 √ 2m -1 |t| -1 2 + 3δ 2 .
In the other hand, since for k ∈ R, we have (

1 + k) 2 ≤ 2(1 + k 2 ), the first integral becomes +∞ |t| δ -1 (1 + k) 2m | f (k)| 2 dk 1/2 ≤ √ 2 m +∞ |t| δ -1 (1 + k 2 ) m | f (k)| 2 dk 1/2 ≤ √ 2 m ||f || H m .
Finally, we find

+∞ |t| δ -1 e -ithα(k) f (k) dk ≤ 2 m 2m -1 ||f || H m |t| -1 2 + 3δ 2 .
By abuse, when there is no ambiguity, we will write C and C m the different constants appearing in the following results.

Proposition 1 Let 0 < δ < 1 3 and m = 1 2δ -1.
There exists a constant C m > 0, depending only on m, such that for all

function f ∈ L 1 (R) ∩ H m (R), α ∈ R and all time t ∈ R, we have +∞ -∞ e -ithα(k) f (k) dk ≤ C m (||f || L 1 + ||f || H m )(1 + |t|) -1 2 + 3δ 2 . (2.3) Proof. Let 0 < δ < 1 3 and |t| ≥ 1, we write +∞ 0 e -ithα(k) f (k) dk = |t| δ -1 0 e -ithα(k) f (k) dk + +∞ |t| δ -1 e -ithα(k) f (k) dk.
For the first integral, the Fubini theorem implies

|t| δ -1 0 e -ithα(k) f (k) dk = |t| δ -1 0 e -ithα(k) +∞ -∞ e -ikx f (x ) dx dk = +∞ -∞ |t| δ -1 0 e ik(x-x ) e -it k 1+k dk f (x ) dx .
We deduce from it that

|t| δ -1 0 e -ithα(k) f (k) dk ≤ |t| δ -1 0 e -ithα(k) dk ∞ ||f || L 1 .
The lemmas 2.2 and 2.3 are applied to give

+∞ 0 e -ithα(k) f (k) dk ≤ max 10 √ 2 ; 2 m 2m -1 (||f || L 1 + ||f || H m ) |t| -1 2 + 3δ 2 . Since |t| ≥ 1, we have |t| ≥ 1 + |t| 2 , thus we obtain finally +∞ 0 e -ithα(k) f (k) dk ≤ C m (||f || L 1 + ||f || H m ) (1 + |t|) -1 2 + 3δ 2 .
(2.4)

The contribution for k ≤ 0 is dealt with similarly. Let |t| ≤ 1. We have directly

+∞ -∞ e -ithα(k) f (k) dk ≤ +∞ -∞ f (k) dk = +∞ -∞ (1 + k 2 ) m/2 (1 + k 2 ) m/2 f (k) dk,
and the Cauchy-Schwarz inequality gives

+∞ -∞ e -ithα(k) f (k) dk ≤ C m ||f || H m . Since |t| ≤ 1, we have 1 ≥ 1 + |t| 2 , thus we obtain finally +∞ -∞ e -ithα(k) f (k) dk ≤ 2 1 2 -3δ 2 C m ||f || H m (1 + |t|) -1 2 + 3δ 2 .
(2.5)

The inequalities (2.4) and (2.5) give the result.

We can deduce the following corollary.

Corollary 1 Let 0 < δ < 1 3 and m = 1 2δ -1.
There exists a constant C m > 0, depending only on m, such that for all

function f ∈ L p (R) ∩ H m (R), with 1 ≤ p ≤ 2, for all α ∈ R and all time t ∈ R, we have +∞ -∞ e -ithα(k) f (k) dk L q ≤ C m (||f || L p + ||f || H m )(1 + |t|) (-1 2 + 3δ 2 )(1-2 q ) , (2.6)
where

1 p + 1 q = 1.
Proof. Let 0 < δ < 1 3 and |t| ≥ 1, we define the operator

I (1) t from L 1 (R) to L ∞ (R) by, for f ∈ L 1 (R) and x ∈ R, I (1) t f (x) := |t| δ -1 0 e -ithα(k) f (k) dk.
The lemma 2.2 gives that there exists a constant C > 0 such that

||I (1) t f || ∞ ≤ C|t| -1 2 + 3δ 2 ||f || L 1 , thus |||I (1) t ||| ∞ ≤ C|t| -1 2 + 3δ 2 .
(2.7)

The operator I

(1) t is rewritten

I (1) t f (x) := ∞ -∞ e ikx e -it k 1+|k| f (k) 1 [0,|t| δ -1] dk, thus ||I (1)
t f || L 2 ≤ ||f || L 2 , and |||I (1)
t ||| L 2 ≤ 1. (2.8)
The Riesz-Thorin interpolation theorem then implies that, for 1 ≤ p ≤ 2, and

1 p + 1 q = 1, the operator I (1) t
is continuous from L p (R) to L q (R), and we have according to the inequalities (2.7) and (2.8)

|||I

(1)

t ||| L q ≤ C|t| (-1 2 + 3δ 2 )(1-2 q ) .
We conclude that for f ∈ L p (R),

||I

t f || L q ≤ C|t| (-1 2 + 3δ 2 )(1-2 q ) ||f || L p ≤ C(1 + |t|) (-1 2 + 3δ 2 )(1-2 q ) ||f || L p . Let m = 1 2δ (1)
-1, we define now the operator

I (2) t from H m (R) to L ∞ (R) by, for f ∈ H m (R) and x ∈ R, I (2) t f (x) := +∞ |t| δ -1 e -ithα(k) f (k) dk.
The lemma 2.3 implies that there exists a constant C m > 0, depending only on m, such that

||I (2) t f || ∞ ≤ C m |t| -1 2 + 3δ 2 ||f || H m .
(2.9)

Since the operator I

(2) t can be rewritten

I (2) t f (x) := +∞ -∞ e ikx e -it k 1+|k| f (k) 1 [|t| δ -1,+∞] dk, we have ||I (2) t f || L 2 ≤ ||f || L 2 , and since m > 1 2 , ||I (2)
t f || L 2 ≤ ||f || H m . (2.10)
The Hölder inequality then implies that, for 2 ≤ q ≤ ∞, according to the inequalities (2.9) and (2.10), we have ||I

(2)

t f || L q ≤ ||I (2) t f || 1-2 q ∞ ||I (2) t f || 2 q L 2 ≤ C m (1 + |t|) (-1 2 + 3δ 2 )(1-2 q )
||f || H m . The contribution for k ≤ 0 is dealt with similarly. For |t| ≤ 1, we define the operator

I (3) t from H m (R) to L ∞ (R) by, for f ∈ H m (R) and x ∈ R, I (3) t f (x) := +∞ -∞ e -ithα(k) f (k) dk.
As for the proposition 1, we have directly that there exists a constant C m > 0, depending only on m, such that ||I

(3) t f || ∞ ≤ C m ||f || H m (1 + |t|) (-1 2 + 3δ 2 ) . (2.11)
Since the operator I

(3) t can be rewritten

I (3) t f (x) := +∞ -∞ e ikx e -it k 1+|k| f (k) 1 [0,+∞[ dk, we have, because m > 1 2 , ||I (3)
t f || L 2 ≤ ||f || H m .
(2.12)

The Hölder inequality then implies that, for 2 ≤ q ≤ ∞, according to the inequalities (2.11) and (2.12), we have ||I

(3)

t f || L q ≤ ||I (3) t f || 1-2 q ∞ ||I (3) t f || 2 q L 2 ≤ C m ||f || H m (1 + |t|) (-1 2 + 3δ 2 )(1-2 q ) .

Preliminary results

We now quote some useful results for the proof of our principal result. We will need the Mikhlin-Hörmander theorem proved in [12,

∈ R n \ {0}, |∂ α k σ(k)| ≤ C α |k| -|α| .
Then, for all 1 < p < +∞, there exists a constant C p > 0, depending only on p, such that for all u ∈ L p (R n ), we have

F -1 (σû) L p ≤ C p ||u|| L p .
The fractional Leibniz rule will be also used [9, lemma X.4].

Theorem 3.2

For all m ≥ 0 and 1 < p < +∞, there exists a constant C m,p > 0, depending on m and p such that for all u and v in S(R n ), we have

||u v|| W m,p ≤ C m,p (||u|| W m,p ||v|| ∞ + ||u|| ∞ ||v|| W m,p ) .
We need the Sobolev inequality [5, chapter 1 -theorem 9.3].

Theorem 3.3 Let 1 ≤ p, q ≤ +∞ and 0 ≤ j < m, there exists a constant C > 0, depending on p, q, j and m, such that for all u ∈ S(R n ), we have

|| (-∆) j 2 u|| L r ≤ C|| (-∆) m 2 u|| a L p ||u|| 1-a L q , where 1 r = j n + a 1 p - m n + 1 -a q and j m ≤ a < 1.
Finally, we prove an integration's lemma.

Lemma 3.4

Let α > 1 and β > 0. Then there exists 0 < β 1 < 1 and C > 0 such that for all time t ≥ 0

t 0 dτ (1 + τ ) α (1 + t -τ ) β ≤ C (1 + t) β1 , with β 1 = β if 0 < β < 1.
Proof. If t = 0, the result is obvious. Let t > 0, we separate the time integral in two parts

t 0 dτ (1 + τ ) α (1 + t -τ ) β = t/2 0 dτ (1 + τ ) α (1 + t -τ ) β + t t/2 dτ (1 + τ ) α (1 + t -τ ) β .
For the first integral, we have

t/2 0 dτ (1 + τ ) α (1 + t -τ ) β ≤ 1 + t 2 -β t/2 0 dτ (1 + τ ) α ≤ 1 + t 2 -β (1 + τ ) -α+1 -α + 1 t/2 0 , since -α + 1 < 0, we find t/2 0 dτ (1 + τ ) α (1 + t -τ ) β ≤ C 1 + t 2 -β 1 -1 + t 2 -α+1 ≤ C 1 + t 2 -β
, and it is enough to remark 1 +

t 2 ≥ 1 2 (1 + t) to obtain t/2 0 dτ (1 + τ ) α (1 + t -τ ) β ≤ C (1 + t) β .
Let us suppose first of all that 0 < β < 1. For the second integral, we have

t t/2 dτ (1 + τ ) α (1 + t -τ ) β ≤ 1 + t 2 -α t t/2 dτ (1 + t -τ ) β = 1 + t 2 -α - (1 + t -τ ) -β+1 -β + 1 t t/2
.

Since -β + 1 > 0 and -α + 1 < 0, we have

t t/2 dτ (1 + τ ) α (1 + t -τ ) β ≤ C 1 + t 2 -α-β+1 ≤ C (1 + t) -β .
Let us suppose now β ≥ 1. Then for 0 < β 1 < 1, we have for all t ≥ 0

t 0 dτ (1 + τ ) α (1 + t -τ ) β ≤ t 0 dτ (1 + τ ) α (1 + t -τ ) β1 ,
and the first part of the proof gives the result.

Existence and Uniqueness of global solution of the gBO-BBM equation

Let us return to the nonlinear problem. We consider the Cauchy problem

u t + u x + H(u xt ) + u ρ u x = 0 (4.1) u(x, 0) = f (x). (4.2)
We show from now the principal result of this paper.

Proof. [Proof of the theorem 1.1.] The proof is carried out for positive time. By abuse, we will write C m and C m,p the different constants depending respectively on m, and on m and p.

To simplify the writings, we denote

D x 1 + |D x |
the operator defined by the Fourier multiplicator σ(k) := ik 1 + |k| , and we denote also 0

< θ = 1 2 - 3δ 2 1 - 2 q < 1 2 .
We remark that 3θρ 4 > 1.

(4.3)

Indeed we have 3θρ 4 > 1 ⇐⇒ 3 4 1 2 - 3δ 2 1 - 2 q ρ > 1 ⇐⇒ (1 -3δ) 1 - 2 q > 8 3ρ , since 1 -3δ > 0, we find 3θρ 4 > 1 ⇐⇒ 1 q < 1 2 - 4 3ρ(1 -3δ) ,
which inequality is the assumption on q.

The Duhamel formula implies that u is the solution of the gBO-BBM equation (4.1)-(4.2) if and only if u is the solution of the following equation, for t ≥ 0

u(t) = Φu(t) := S t f - 1 ρ + 1 t 0 S t-τ D x 1 + |D x | u ρ+1 (τ ) dτ, (4.4)
where

S t f := 1 2π +∞ -∞ e ikx-it k 1+|k| f (k) dk.
We remark first of all that in one hand, we have for u ∈ S(R)

D x 1 + |D x | u ρ+1 H m = +∞ -∞ (1 + k 2 ) m ik 1 + |k| 2 | u ρ+1 | 2 dk 1/2 ≤ +∞ -∞ (1 + k 2 ) m | u ρ+1 | 2 dk 1/2 = ||u ρ+1 || H m . (4.5)
In the other hand, we have

for k ≥ 0, |σ (k)| = 1 (1 + k) 2 and for k ≤ 0, |σ (k)| = 1 (1 -k) 2 .
The Mikhlin-Hörmander theorem 3.1 is applied to give, for 1 < p < 2, there exists a constant C p > 0 depending only on p such that

D x 1 + |D x | u ρ+1 L p ≤ C p ||u ρ+1 || L p . (4.6)
Let T > 0, we define the norm N T by, for u ∈ X m+1,4-2 p ,p (R),

N T (u) := sup 0≤τ ≤T (||u|| L q (τ ) + ||u x || L q (τ ))(1 + τ ) θ + ||u|| X m+1,4-2 p ,p (τ ) . (4.7)
We will proved separately some technical lemmas.

Lemma 4.1

There exists a constant C m,p > 0 such that for all u and v in X m+1,4-2 p ,p (R), we have

||Φu -Φv|| L q (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i (1 + t) -θ N T (u -v), (4.8)
and

||(Φu -Φv) x || L q (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i (1 + t) -θ N T (u -v). (4.9)
Proof. Let u and v two elements of X m+1,4-2 p ,p (R), the Duhamel formula gives

||Φu -Φv|| L q ≤ 1 ρ + 1 t 0 S t-τ D x 1 + |D x | (u ρ+1 -v ρ+1 ) L q (τ ) dτ.
The corollary 1 implies that there exists a constant C m > 0, depending only on m, such that

||Φu -Φv|| L q ≤ C m t 0 Dx 1+|Dx| (u ρ+1 -v ρ+1 ) L p + Dx 1+|Dx| (u ρ+1 -v ρ+1 ) H m (τ ) (1 + t -τ ) θ dτ,
and the inequalities (4.5) and (4.6) imply

||Φu -Φv|| L q ≤ C m,p t 0 ||u ρ+1 -v ρ+1 || L p + ||u ρ+1 -v ρ+1 || H m (τ ) (1 + t -τ ) θ dτ.
From now, when there is no ambiguity, we will use the following notation: for any positive A and B, the notation A B means that there exists a constant C > 0 such that A ≤ C B.

Since

u ρ+1 -v ρ+1 = (u -v) ρ i=0
u ρ-i v i , we have according to the fractional Leibniz theorem 3.2 with p = 2 and the Minkowski inequality

||u ρ+1 -v ρ+1 || H m (τ ) ||u -v|| H m ρ i=0 ||u ρ-i v i || ∞ (τ ) + ||u -v|| ∞ ρ i=0 ||u ρ-i v i || H m (τ ) = I (τ ) + II (τ ).
On the other hand, the Sobolev inequality 3.3 with

1 p + 1 q = 1, r = ∞, j = 0, a = 1 4 and m = 3 - 2 p gives ||u|| ∞ ≤ C p ||u|| 1/4 W 3-2 p ,p ||u|| 3/4 L q . (4.10)
According to the definition (4.7) of the norm N T , we can see that the decay in time will be linked to the norm L q and the power 3/4. We deduce from the inequality (4.10) that for I (τ ), we have

I (τ ) ||u -v|| H m ρ i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i (τ ) ||u -v|| H m ρ i=0 ||u|| L q (1 + τ ) θ (1 + τ ) θ 3(ρ-i) 4 ||u|| ρ-i 4 W 3-2 p ,p ||v|| L q (1 + τ ) θ (1 + τ ) θ 3i 4 ||v|| i 4 W 3-2 p ,p ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
For II (τ ), the theorem 3.2 is again applied and we have

II(τ ) ||u -v|| ∞ ||v|| ρ-1 ∞ ||v|| H m + ρ-1 i=0 ||u|| ρ-i-1 ∞ ||u|| H m ||v|| i ∞ + ||u|| ρ-1 ∞ ||u|| H m + ρ i=1 ||u|| ρ-i ∞ ||v|| i-1 ∞ ||v|| H m (τ ),
and the inequality (4.10) gives

II(τ ) ||u -v|| 3/4 L q ||u -v|| 1/4 W 3-2 p ,p × ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p ρ-1 ||v|| H m + ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-1 ||u|| H m + ρ-1 i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i-1 ||u|| H m ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i + ρ i=1 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i-1 ||v|| H m (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
In the same manner, we have

||u ρ+1 -v ρ+1 || L p (τ ) ≤ ||u -v|| L p ρ i=0 ||u|| ρ-i ∞ ||v|| i ∞ (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
Finally, we find

||Φu -Φv|| L q (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i N T (u -v) t 0 dτ (1 + τ ) 3θρ 4 (1 + t -τ ) θ . Since 3θρ 4
> 1 by the inequality (4.3) and 0 < θ < 1 2 , the lemma 3.4 can be applied to give

||Φu -Φv|| L q (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i (1 + t) -θ N T (u -v).
We have, for the space derivate,

(Φu -Φv) x (t) = t 0 S t-τ D x 1 + |D x | (u ρ u x -v ρ v x ) (τ ) dτ,
the corollary 1 and the inequalities (4.5) and (4.6) imply

||(Φu -Φv) x || L q (t) ≤ C m,p t 0 (||u ρ u x -v ρ v x || L p + ||u ρ u x -v ρ v x || H m ) (τ ) (1 + t -τ ) θ dτ. Since u ρ+1 -v ρ+1 = (u -v) ρ i=0 u ρ-i v i , the fractional Leibniz rule gives ||u ρ u x -v ρ v x || H m (τ ) = ||(u-v) ρ i=0 u ρ-i v i || H m+1 (τ ) ||u-v|| H m+1 ρ i=0 ||u ρ-i v i || ∞ (τ ) + ||u-v|| ∞ ρ i=0 ||u ρ-i v i || H m+1 (τ ) = I (τ ) + II (τ ).
We deduce from the inequality (4.10) that for I (τ ), we have

I (τ ) ||u -v|| H m+1 ρ i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
For the second norm, the theorem 3.2 is again applied and we have

II (τ ) ||u -v|| ∞ ||v|| ρ-1 ∞ ||v|| H m+1 + ρ-1 i=0 ||u|| ρ-i-1 ∞ ||u|| H m+1 ||v|| i ∞ + ||u|| ρ-1 ∞ ||u|| H m+1 + ρ i=1 ||u|| ρ-i ∞ ||v|| i-1 ∞ ||v|| H m+1 (τ ),
and the inequality (4.10) gives

II (τ ) ||u -v|| 3/4 L q ||u -v|| 1/4 W 3-2 p ,p ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p ρ-1 ||v|| H m+1 + ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-1 ||u|| H m+1 + ρ-1 i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i-1 ||u|| H m+1 ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i + ρ i=1 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i-1 ||v|| H m+1 (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
In the same manner,

||u ρ u x -v ρ v x || L p (τ ) ||(u -v) x || L p ρ i=0 ||u|| ρ-i ∞ ||v|| i ∞ (τ ) + ||u -v|| L p ρ-1 i=0 ||u|| ρ-i-1 ∞ ||u x || ∞ ||v|| i ∞ + ρ i=1 ||u|| ρ-i ∞ ||v|| i-1 ∞ ||v x || ∞ (τ ).
According to the inequality (4.10),

||u x || ∞ ≤ ||u x || 3/4 L q ||u x || 1/4 W 3-2 p ,p = ||u x || 3/4 L q ||u|| 1/4 W 4-2 p
,p , we deduce

||u ρ u x -v ρ v x || L p (τ ) ||u-v|| W 1,p ρ i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i (τ ) + ||u -v|| L p ρ-1 i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i-1 ||u x || 3/4 L q ||u|| 1/4 W 4-2 p ,p ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i + ρ i=1 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i-1 ||v x || 3/4 L q ||v|| 1/4 W 4-2 p ,p (τ ),
and since 4 -

2 p > 1, we have ||u ρ u x -v ρ v x || L p (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
Thus we find

||(Φu -Φv) x || L q (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i (1 + t) -θ N T (u -v).

Lemma 4.2

There exists a constant C m,p > 0 such that for all u and v in X m+1,4-2 p ,p (R), we have

||Φu -Φv|| X m+1,4-2 p ,p (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i N T (u -v). (4.11)
Proof. We now majorize the norm

||(Φu -Φv) x || X m+1,4-2 p ,p (t), ||Φu -Φv|| X m+1,4-2 p ,p (t)
≤ 1 ρ + 1 t 0 S t-τ D x 1 + |D x | (u ρ+1 -v ρ+1 ) X m+1 ,4-2 p ,p (τ ) dτ, the inequalities (4.5) and (4.6) imply ||Φu -Φv|| X m+1,4-2 p ,p (t)
≤ 1 ρ + 1 t 0 u ρ+1 -v ρ+1 X m+1,4-2 p ,p (τ ) dτ. Since X m+1,4-2 p ,p (R) = H m+1 (R) ∩ W 4-2 p ,p (R) and u ρ+1 -v ρ+1 = (u -v) ρ i=0 u ρ-i v i , we have ||u ρ+1 -v ρ+1 || X m+1,4-2 p ,p (τ ) = ||u ρ+1 -v ρ+1 || H m+1 (τ ) + ||u ρ+1 -v ρ+1 || W 4-2 p ,p (τ ),
and as previously,

||u ρ+1 -v ρ+1 || H m+1 (τ ) ρ i=0 N T (u) ρ-i N T (v) i N T (u -v)(1 + τ ) -3θρ 4 .
Since 4 -2 p ≥ 0, the fractional Leibniz theorem 3.2 gives

||u ρ+1 -v ρ+1 || W 4-2 p ,p (τ ) ||u -v|| W 4-2 p ,p ρ i=0 ||u ρ-i v i || ∞ (τ ) + ||u -v|| ∞ ρ i=0 ||u ρ-i v i || W 4-2 p ,p (τ )
= I (τ ) + II (τ ).

We deduce from the inequality (4.10)

I (τ ) ||u -v|| W 4-2 p ,p ρ i=0 ||u|| 3/4 L q ||u|| 1/4 W 3-2 p ,p ρ-i ||v|| 3/4 L q ||v|| 1/4 W 3-2 p ,p i (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
For II (τ ), the theorem 3.2 is again applied and we have

II (τ ) ||u -v|| ∞ ||v|| ρ-1 ∞ ||v|| W 4-2 p ,p + ρ-1 i=0 ||u|| ρ-i-1 ∞ ||u|| W 4-2 p ,p ||v|| i ∞ + ||u|| ρ-1 ∞ ||u|| W 4-2 p ,p + ρ i=1 ||u|| ρ-i ∞ ||v|| i-1 ∞ ||v|| W 4-2 p ,p (τ )
, and the inequality (4.10) gives

II (τ ) ρ i=0 N T (u) ρ-i N T (v) i (1 + τ ) -3θρ 4 N T (u -v).
Finally we find

||u ρ+1 -v ρ+1 || X m+1,4-2 p ,p (τ ) ρ i=0 N T (u) ρ-i N T (v) i N T (u -v)(1 + τ ) -3θρ 4 , thus ||Φu -Φv|| X m+1,4-2 p ,p (t) ρ i=0 N T (u) ρ-i N T (v) i N T (u -v) t 0 (1 + τ ) -3θρ 4 dτ,
and thanks to the inequality (4.3),

||Φu -Φv|| X m+1,4-2 p ,p (t) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i N T (u -v).

Lemma 4.3

There exists a constant C m,p > 0 such that for all u and v in X m+1,4-2 p ,p (R), we have

N T (Φu -Φv) ≤ C m,p ρ i=0 N T (u) ρ-i N T (v) i N T (u -v), (4.12)
and

N T (Φu) ≤ C m,p ||f || X m+1,4-2 p ,p + N T (u) ρ+1 . (4.13)
Proof. The inequalities (4.8), (4.9) and (4.11) give the inequality (4.12). For the second inequality, it is enough to take v = 0.

Let M > 0, we consider the closed ball

B T,M := u ∈ C([-T, T ]; X m+1,4-2 p ,p (R)); N T (u) ≤ M .
We would like to show that there exists an unique solution u of the equation (4.4) in this ball by using the fixed point theorem. First, there exists ε > 0 sufficiently small such that if ||f || X m+1,4-2 p ,p ≤ ε, even if we take C m,p M instead of M , it is enough to take M > 0 satisfying ε + M ρ+1 ≤ M so that the inequality (4.13) implies that the image of the closed ball B T,M by the map Φ is include in itself. Here, the crucial point is that ε is independent of T . Secondly, we prove that the map Φ is a contraction on this ball for M sufficiently small. Let u and v two elements of the closed ball B T,M . The inequality (4.12) gives

N T (Φu -Φv) ≤ C m,p M ρ N T (u -v),
and it is enough to take M > 0 sufficiently small so that the quantity C m,p M ρ < 1. Then, the fixed point theorem is applied and there exists an unique solution of the equation (4.4) in the closed ball B T,M . It remains to prove that this unique solution can be prolonged in time with all [0, +∞[. By uniqueness of the solution, the inequality (4.13) is written

N T (u) ≤ C m,p ||f || X m+1,4-2 p ,p + N T (u) ρ+1 . (4.14)
Since there exists ε > 0 sufficiently small such that ||f || X m+1,4-2 p ,p ≤ ε, we can find M > 0 such that

N 0 (u) < M C m,p ε + M ρ+1 ≤ M.
Then for all T > 0, we have N T (u) < M . Indeed, if not by continuity, there exists a time T > 0 such that

N T (u) = M > C m,p ε + M ρ+1 > C m,p ε + N T (u) ρ+1 ,
what contradicts the inequality (4.14). Finally, there exists a constant M > 0 such that for all T > 0, N T (u) < M . In particular, we have for all time t ≥ 0

||u(t)|| L q + ||u x (t)|| L q ≤ C m,p ||f || X m+1,4-2 p ,p + N T (u) ρ+1 (1 + t) -θ ≤ C m,p ε + M ρ+1 (1 + t) -θ . (4.15)
We reason in a similar manner for negative times.

Decay in time of the gKP-BBM-II equation

We consider the linear Cauchy problem

u t + u x -u xxt + ∂ -1 x u yy = 0 ; u(x, y, 0) = f (x, y).
Let us suppose that the initial datum f belongs to the Schwartz space S(R 2 ). Then the Fourier transform in space implies

ût (1 + k 2 ) + i k + l 2 k û = 0 ; û(k, l, 0) = f (k, l).
The solution u of this ordinary differential equation is given by, for all (x, y) ∈ R 2 and t ∈ R * ,

u(x, y, t) = 1 (2π) 2 R 2 e -ith α,β (k,l) f (k, l) dk dl, (5.1) with h α,β (k, l) = k + l 2 k 1 + k 2 -αk -βl, α =
x t and β = y t . We have to majorize this oscillating integral in function of time.

Lemma 5.1 Let 0 ≤ δ < 1 5
. For all (α, β) ∈ R 2 , and all time |t| ≥ 1, we have

|t| δ -|t| δ +∞ -∞ e -ith α,β (k,l) dl dk ≤ 3 √ π |t| -1 2 + 5δ 2 .
Proof. We adapt the Haysahi-Naumkin-Saut method [START_REF] Hayashi | Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation[END_REF] to the generalized KP-BBM-II equation. It consists of transforming the oscillating double integral (5.1) in a simple one. For t ∈ R * , we do the change

of variables l = l |k|(1 + k 2 ) |t| to obtain |t| δ -|t| δ +∞ -∞ e -ith α,β (k,l) dl dk = |t| δ -|t| δ +∞ -∞ e ikx+iyl |k|(1+k 2 ) |t| -it k 1+k 2 -il 2 sgn( k t ) |k|(1 + k 2 ) |t| dl dk = 1 |t| 1/2 |t| δ -|t| δ |k|(1 + k 2 ) e ikx-it k 1+k 2 +∞ -∞ e iyl |k|(1+k 2 ) |t| -il 2 sgn( k t )
dl dk.

However,

iyl |k|(1 + k 2 ) |t| -il 2 sgn( k t ) = -i sgn( k t ) l 2 -l y |k|(1 + k 2 ) |t| = -i sgn( k t ) l 2 - y 2 |k|(1 + k 2 ) |t| 2 + i y 2 k(1 + k 2 ) 4t .
A last change of variable gives .

Finally, we find

|t| δ -|t| δ +∞ -∞ e -ith α,β (k,l) dl dk = √ πe ±iπ/4 |t| 1/2 |t| δ -|t| δ |k|(1 + k 2 ) e ixk+-it k 1+k 2 + y 2 4t 2 (k+k 3 ) dk. (5.2)
We deduce then

|t| δ -|t| δ +∞ -∞ e -ith α,β (k,l) dl dk = √ π |t| -1/2 |t| δ -|t| δ |k|(1 + k 2 ) dk ≤ 3 √ π |t| -1 2 + 5δ 2 .
The oscillating integral (5.1) can be majorized.

+∞ |t| δ +∞ -∞ e -ith α,β (k,l) f (k, l) dl dk ≤ C m ||f || H 2m |t| -1 2 + 5δ 2 ,
and

-|t| δ -∞ +∞ -∞ e -ith α,β (k,l) f (k, l) dl dk ≤ C m ||f || H 2m |t| -1 2 + 5δ 2 .
Proof. We have

+∞ |t| δ +∞ -∞ e -ith α,β (k,l) f (k, l) dl dk ≤ +∞ |t| δ +∞ -∞ f (k, l) dl dk, the Fubini theorem implies +∞ |t| δ +∞ -∞ e -ith α,β (k,l) f (k, l) dl dk ≤ +∞ -∞ (1 + l 2 ) m/2 (1 + l 2 ) m/2 +∞ |t| δ (1 + k) m (1 + k) m f (k, l) dk dl,
and the Cauchy-Schwarz inequality, first in k and secondly in l, gives the result.

Proposition 2

Let 0 < δ < 1 5 and m = 1 2δ -2. There exists a constant C m > 0, depending only on m, such that for all

function f ∈ L 1 (R 2 ) ∩ H 2m (R 2 ), (α, β) ∈ R 2 and all time t ∈ R, we have R 2 e -ith α,β (k,l) f (k, l) dl dk ≤ C m (||f || L 1 + ||f || H 2m )(1 + |t|) -1 2 + 5δ 2 .
(5.3)

Proof.

It is enough to separate only the integral in direction k and to take again the proof of the proposition 1.

We can deduce in the same way as the corollary 1 the following result. e -ith α,β (k,l) f (k, l) dl dk

L q ≤ C m (||f || L p + ||f || H 2m )(1 + |t|) (-1 2 + 5δ 2 )(1-2 q ) , (5.4
)

where 1 p + 1 q = 1.
Let us return to the nonlinear problem. We consider the Cauchy problem

u t + u x + u ρ u x -u xxt + ∂ -1
x u yy = 0 (5.5) u(x, y, 0) = f (x, y).

(5.6)

The existence and uniqueness of global solution, given by the theorem 1.2, is proved.

Proof. [Proof of the theorem 1.2] To simplify the writings, we denote D x 1 + D 2 x the operator defined by the Fourier multiplicator σ(k, l) := ik 1 + k 2 , and we denote also

0 < θ = 1 2 - 5δ 2 1 - 2 q < 1 2 .
We remark that 3θρ 4 > 1.

(5.7)

The Duhamel formula implies that u is the solution of the gKP-BBM-II equation (5 We remark first of all that in one hand, we have for u ∈ S(R 2 )

D x 1 + D 2 x u ρ+1 H m = R 2 (1 + k 2 + l 2 ) m ik 1 + k 2 2 | u ρ+1 | 2 dk dl 1/2 ≤ R 2 (1 + k 2 + l 2 ) m | u ρ+1 | 2 dk dl 1/2 = ||u ρ+1 || H m .
(5.9)

In the other hand, the Mikhlin-Hörmander theorem 3.1 is applied with σ, defined above, to give, for 1 < p < 2, there exists a constant C p > 0 depending only on p such that (5.12)

According to the definition (5.11) of the norm N T , we can see that the decay in time will be linked to the norm L q and the power 3/4.

Remark 1 If we compare the generalized Korteweg-de Vries equation and his BBM version, we remark that the decay rate in time is of the same order. In dimension 2, Haysahi, Naumkin and Saut [START_REF] Hayashi | Asymptotics for large time of global solutions to the generalized Kadomtsev-Petviashvili equation[END_REF] found, for the generalized KP equation, a decay rate in time of order 1. However, nothing says that our result is optimal and we wonder if we can find a decay rate in time better than 1/2, and approach 1 for the BBM version. We plan to study this issue in a future work.

Lemma 2. 1

 1 For all a ≤ b, λ > 0, t = 0 and for all function h ∈ C ∞ ([a, b]) real valued satisfying for all k ∈ [a, b], |h (k)| ≥ λ, we have b a e -ith(k) dk ≤ 10 (λ|t|) 1/2 .

- 2 .

 2 There exists a constant C m > 0, depending only on m, such that for allfunction f ∈ L p (R 2 ) ∩ H 2m (R 2 ), with 1 ≤ p ≤ 2, for all (α, β) ∈ R 2 and all time t ∈ R, we have R 2

≤

  C p ||u ρ+1 || L p . (5.10)Let T > 0, we define the norm N T by, for u ∈ X 2m+1,7-4 p ,p (R 2 ),N T (u) := sup 0≤τ ≤T (||u|| L q (τ ) + ||u x || L q (τ ))(1 + τ ) θ + ||u|| X 2m+1,7-4 p,p (τ ) .(5.11)From now, the end of the proof is similar to that of the theorem 1.1 by remarking that the Sobolev inequality 3

  chapter 6 -proposition 4.4].

	Theorem 3.1
	Let σ : R n \ {0} → C satisfying for all multi-index of length 0 ≤ |α| < l, where l is the smallest integer with l > n 2 , and for all k

  There exists a constant C m > 0, depending only on m, such that for all f ∈ H 2m (R 2 ), (α, β) ∈ R 2 and all time |t| ≥ 1, we have

	Lemma 5.2 Let 0 < δ < 1 5	and m =	1 2δ	-2.

  .5)-(5.6) if and only if u is the solution of the following equation, for t ≥ 0

			u(t) = Φu(t) := S t f -	1 ρ + 1	0	t	S t-τ	D x 1 + D 2 x	u ρ+1 (τ ) dτ,	(5.8)
	where S t f :=	1 (2π) 2	R 2	e ikx+ily-it	k+l 2 /k 1+k 2	f (k, l) dl dk.