Sendrine Constant

Balwant Godara

Thierry Agagliate

Nihaalini Kumar

Amara Amara

Peer Networking & Capacity Building for Child Protection Professionals -Lessons from "ChildHub"

Keywords: Blended (online and offline) approach, Capacity building, Child protection, Information and communications technologies, Peer networking

Child protection systems across the global South suffer from common problems, one of the most critical among which is low number and skills of relevant professionals to deliver services. Additionally, child protection professionals are often demotivated, uncoordinated and isolated, with limited access to continuous training and support. Peer learning and capacity building networks help address these issue, and often leverage the spread and scope of information and communications technologies. We present one such network, ChildHub, initially developed and deployed in South-East Europe, a region whose child protection systems present features similar to those in Africa and Asia. The success of this platform, evinced by a continuously growing community and confirmed by an evaluation after three years of operation, provides motivation and lessons for contextualization to sub-Saharan Africa and south Asia. Thanks to its inherent modularity, ChildHub will easily be adapted to the contexts and needs of the two regions, thus building on the interest generated in Asia and Africa for such networks. The paper also presents the approach that will be taken to implement the platform for Africa and Asia.

1

Introduction -Context and Landscape

Definition of Child Protection Systems

A child protection system is defined as [START_REF]UNICEF/UNHCR/Save the Children/World Vision[END_REF]: "certain formal and informal structures, functions and capacities that have been assembled to prevent and respond to violence, abuse, neglect and exploitation of children. A child protection system is generally agreed to be comprised of the following components: human resources, finance, laws and policies, governance, monitoring and data collection as well as protection and response services and care management. It also includes different actors -children, families, communities, those working and subnational and national level and those working internationally. Most important are the relationships and interactions between and among these component and these actors within the system. It is the outcomes of these interactions that comprise the system".

Child protection systems in the Global South

"Capacities" are a core element of child protection; formal examples of these include adequate number, capacity and skills of relevant professionals [2]. The child protection systems in the Global South -including sub-Saharan Africa [START_REF]Strengthening Child Protection Systems in Sub-Saharan Africa[END_REF]- [START_REF]Human capacity within child welfare systems -the social work workforce in Africa[END_REF] and south Asia [START_REF]Review of Child Protection Systems in Four Countries in South Asia[END_REF]- [START_REF]South Asia in Action: Preventing and responding to child trafficking[END_REF] -suffer from the same problems as concerns these capacities. More specifically:

 Social services are rarely a priority in the political agenda  Despite progress in the elaboration of regulatory systems, many countries show a lack of vision for quality improvement, no real preoccupation for "value for money" and no investment in the continuous development of the sector.  The professional groups (especially social workers, caregivers and medical professionals) are often demotivated and uncoordinated, are understaffed and have limited to no access, to continuous training and peer support. Their capacity of effectively supporting children is significantly affected in the long term.  Child protection systems are not capable of offering a continuum of care, from prevention to alternative support and reintegration.

Peer learning and capacity building networks in child protection

eLearning and Peer Networking have proven useful in addressing the following issues:

 Reducing fragmentation and inaccessibility of knowledge on specific topics;  Reducing isolation felt by professionals, and increasing support and motivation;  Increasing opportunities for development, through skill upgradation and recognition;  Strengthening the link between practices and policies on the one hand, and knowledge and evidence on the other; and  Improving synergies and coordination within and between teams.

There are some successful learning networks in the child protection space, such as the CPC Learning Network [START_REF]CPC Learning Network homepage[END_REF], the RISE Learning Network [START_REF]RISE Learning network homepage[END_REF] and the Alliance for Child Protection in Humanitarian Action [START_REF]The Alliance for Child Protection in Humanitarian Action homepage[END_REF]. These initiatives have proven to be feasible, relevant and useful in bringing together professionals to discuss, design and feed back into policies and programmes. Terre des hommes (Tdh) has initiated such a network, the Child Protection Hub (ChildHub) in Southeast Europe. ChildHub's architecture and success forms the basis for the current proposal.

Structure of this paper & its Relevance to the Conference

We present a peer learning and communication platform, ChildHub, initially developed for child protection professionals in Central and South Eastern Europe. Currently used by professionals in 10 countries, this platform has also been the subject of a detailed external evaluation. The results are promising, and will form the basis of improvements of the platform in Europe. The experience and evaluation of ChildHub also offers valuable lessons for its contextualization to other regions, notably South Asia and sub-Saharan Africa. Consequently, the sector is confronted with significant staff turnover; limited, irrelevant and uncoordinated training; and low motivation and performance. This comes at a time when the need for efficient child protection services is critical -there is a high incidence of sexual violence, bullying, domestic violence, neglect of children, and discrimination and marginalization of ethnic minorities.

At the same time, the study identified some interesting opportunities in the Region. The similarities between social protection systems in the region would make exchange of practices possible and relevant. There is an increased momentum in the child protection sector with the adoption of the Sustainable Development Goals (SDGs), and at the same time, there is a solid base of existing knowledge, promising practices, experts and organisations. These latter are willing to share experiences given the opportunity, and this sharing could be facilitated by the widespread access to digital technologies.

2.2

Approach, objectives and expected outcomes

Beneficiaries

 Indirect -500,000 children at risk or victims; families who will benefit from better services and enhanced policies.

 Direct -child protection professionals -with a strong focus on social workers, but including other professions.

Following the above analysis, Tdh's Regional Office decided to initiate the Child Protection Hub for South East Europe, ChildHub in 2015 [START_REF]Child Protection Hub for[END_REF]- [START_REF]Final Evaluation of the Child Protection Hub for South East Europe Project[END_REF]. The founding vision, key objectives, expected outcomes and beneficiaries are given in Table 1.

ChildHub -Components, Implementation & Performance

The Four Components of ChildHub

ChildHub allows professionals to (i) Learn and explore; (ii) Connect and exchange. These two core components are supplemented by a support component (knowledge management) and a crosscutting component (advocacy). These are available in all the major regional languages. The Learning component aims at providing opportunities for professionals to access resources for quick learning, such as podcasts and videos, as well as webinars, resources for trainers and materials for independent learning. ChildHub has built a significant portfolio of courses. It takes a blended approach -complementing offline with online learning. The Networking component proposes joint -online and face-to-face -activities such as regional conferences, national round tables, forums for discussions/sharing of news. It contains a roster of Resource Persons, as well as game-like elements that invite actors to cooperate and synchronise their understanding of common or urgent child protection issues and relevant policy aspects

The Knowledge Management and Resources Component aims at making knowledge, information and data, available online, to all and in a user-friendly format. However, ChildHub is not only a collection of resources. It develops its own knowledge. The library contains a database of publications and easy-to-absorb information, extracts and summary documents, infographics, videos, toolboxes and child friendly materials. ChildHub can also be used to promote organisations' work regionally and initiate collaborations.

Under each of the three components, efforts are made to advocate for change based on evidence collected. The cross-cutting component aims to provide professionals with the tools, methodologies and data that would allow for improved advocacy at national or regional levels. For instance, in one country, several NGOs have submitted a position paper on the deinstitutionalization process, resulting in some of the network's demands being achieved. Several countries held roundtables with local stakeholders to increase collaboration and input. Resource persons in each country have highlighted areas for continued advocacy.

Scale of ChildHub

Table 2 presents the main parameters of scale of ChildHub. More details are available on the ChildHub website (https://childhub.org/en).

Evaluation of ChildHub

An independent evaluation was commissioned by Tdh and the Austrian Development Agency, and conducted by inFocus Enterprises Ltd, at the end of the third year of ChildHub. Its overall purpose was to assess the extent to which the project has reached its objectives, for accountability purposes towards donors, beneficiaries and stakeholders and; and draw the main lessons and generate recommendations for the next phase. The evaluation methods span summative (extent to which anticipated outcomes were produced), formative (identifying improvements that could be made to future) and process (internal dynamics of how ChildHub was implemented) elements. A mixed method approach was taken for data collection, with primary and secondary data sources of both qualitative (key informant interviews and focus group discussions) and quantitative (survey based) data. With the limitations of budget and time and the availability of respondents, the evaluation focused on the primary beneficiaries of activities: a sample of the resource persons and the members of ChildHub. The evaluation encompassed visits to two sites, in two different countries. The data collection instruments are accessible publicly: Online survey questionnaire; Interview guide for Resource People; Interview guide for Coordinators, Staff and project advisors; Focus group guide.

Findings from the Evaluation of ChildHub

The main finding in relation to the output targets is that they have all been significantly over-achieved compared to original intended goals (figure 2). There have also been a number of early-stage outcomes, and there is an increasingly good understanding of the range of skills and capabilities required by both formal and informal actors. However, the timeframe for mid-to long-term changes to occur, for example in relation to widespread practice change, as a result of ChildHub's training and networking efforts, is likely to be longer than expected, with success dependent upon the inter-section of a number of factors, some of which are out of the control of Child-Hub. Some factors can be put into place to improve the likelihood of mid to longer term changes occurring in the next phase (see recommendations below).

There have also been a number of unintended outcomes. Networking appears to have provided a much wider set of connections for Resource People than expected. ChildHub has started to change attitudes and behaviours towards consuming information, accessing training and collaborating online, helping ensure professionals who are unfamiliar with the online world, are able to make full use of ChildHub in the future.

Recommendations and plans for the Future and Contextualisation

The experience in the design, implementation and scale-up of ChildHub in Europe, as well as the findings of the formal evaluation, will help shape the future of the platform while also providing the motivation and the guidelines for its contextualization to other regions. In this section, we present the recommendations and future plans for the evolution and contextualization. ChildHub can be enhanced with the following elements:

 Stronger advocacy at national and regional levels. On the one hand, ChildHub has gathered evidence through research; on the other, it has a core group of influential people who can convey the messages and represent the cause. Also, Child-Hub can be a platform for regional action on issues such as Children on the Move, as it is building a regional community and network of professionals. The advocacy pathway, in turn, should be connected with research and face-to-face meetings.  Seek to interface and/or collaborate with other, complementary platforms. For instance, a concept is currently being developed for the Alliance for Child Protection in Humanitarian Action to share resources and animate a community of practitioners.  Partnerships can also facilitate the daunting task of collecting resources, by establishing automatic systems of pulling regional and national information into Child-Hub. This would greatly facilitate the collection of resources -especially from important reference sites like the Save the Children Resource Library.  Innovative activities and services using the tools and communication/learning facilities of ChildHub. Accessibility for all represents a key innovative aspect and this aspect could be further developed. Improving the workplaces of professionals, especially in rural, isolated areas, is also a key topic for innovative projects.  Coaching and mentoring module highly demanded by professionals. It requires a dedicated pool of experts as coaches or supervisors. Using distance communication, the coaches should offer support to practitioners, especially those working in difficult, isolated areas, or those working with beneficiaries with complex needs.  'Offline' components like the regional conferences and the country visits, have proven much more useful than initially expected. These should be strengthened, to adequately complement the intensified and diversified online activities.  Sustainability: Introduce a more structured approach, best guided through the production of a common Theory of Change, which can then be used as a basis for the community developing learning questions to be further explored and guide the documentation of good practices. Adopt a Developmental Evaluation approach [START_REF] Patton | Developmental Evaluation: Applying Complexity Concepts to Enhance Innovation and Use[END_REF], better adapted to initiatives with multiple stakeholders, high levels of innovation, fast paced decision-making, and areas of uncertainty than the traditional forms of evaluation which work well in situations where the progression from problem to solution can be laid out in a relatively clear sequence of steps. Finally, consider a range of potential 'end games' or routes to sustainability as early as possible.

Contextualization of ChildHub to Africa & South Asia

As we saw in section 1.2, child protection systems in South Asia and sub-Saharan Africa share the same shortcomings with those in South-Eastern Europe. Tdh currently works in four countries in South Asia -Bangladesh, India, Myanmar and Nepal -and is exploring the contextualization of ChildHub there, as a tool for Tdh delegations to connect and learn, and, more importantly still, as an inter-agency platform for child protection professionals in the region. Similarly, the idea of ChildHub would be as relevant for Africa, and could learn from the effort underway to contextualize it in Asia.

Learnings from Europe -scope for contextualization

The modularity of ChildHub means that each pillar of the platform (learning, networking, resource center) allows for the use of a variety of methods and tools. The choice can be made at regional level and adjustments are possible regularly. The functionalities of each component (for example the use of blogs, forums, chat, etc) can be expanded or limited, in line with the specificities of users, with the general use of collaborative tools in the region, with the infrastructure and applications that are chosen by the implementing consortium. Experience shows that there are some minimum, necessary conditions for contextualization of ChildHub: (i) a strong regional executive team which guarantees that the development is responsive to the local needs and interests; (ii) a pool of resource persons to boost the activity, and to ensure a qualitative macro picture of the policies and practices in the region; (iii) online and face-to-face activities from the very beginning, to increase trust, exchanges and effective collaboration; and (iv) a carefully designed monitoring system, including a regional baseline on the policies and practices, as well as the workforce in the child protection systems.

Relevance for Asia

The need to establish a peer network around and beyond Tdh in Asia stems from Tdh's long presence and experience in this zone:

 There is a need for child protection actors (inc. those within Tdh and its partners) to continuously evaluate and update their skills. In-job training using a virtual plat-form could be feasible and effective way, causing least amount of disruption in operations.  There are similar projects (such as the continuum of care for children and their mothers) and common approaches (such as moving from direct implementation to institutional strengthening). Peer learning on these could be extremely beneficial.  Cross-border trafficking is a real menace, across several borders in the region. Collaboration between professionals across countries is thus a real need of the hour; a peer network between the Asian countries could help address this need.  Effective communication between Tdh persons -and Tdh's partners -could take place over a virtual platform, to complement in-person meetings and other avenues of exchange.

The process of contextualization is 'Southern-driven' for several reasons. Firstly, the need for peer networking has been expressed by several actors in south Asia, within, around and beyond Tdh. The potential users and beneficiaries of ChildHub in Asia are key and historic actors in the sector, with a good understanding of the ground needs. Moreover, the technical expertise and content would come from local organisations and contexts. The starting point would thus be the experience of ChildHub in Europe, but adapted to the context and tailored to the needs in south Asia.

4.3

Outline & Avenues for Implementation of ChildHub in Asia 1. Make ChildHub Asia a platform for strengthening collaborations in the sector. This implies seeing ChildHub for its full potential right from the outset -as a multi-agency tool. 2. Strong links with other regions, to strengthen cross-region learning. Our landscape analysis on ICT4D in and for Asia showed that such cross-learning is potentially very useful and under-exploited as of now. 3. Extend the platform to other strategic priorities -going beyond Child Protection, create a corresponding section for professionals working on maternal and child health. This will be planned for the future phases, once ChildHub Asia is up and running, and has achieved critical mass for child protection professionals. 4. Diversifying the information available and the sources it is collected from; it is increasingly important to involve user-generated information to make ChildHub truly participative and complete the 'top-down' structure of today (resources come from experts and are consumed by practitioners) with a 'bottom-up' approach. 5. Complementary and fruitful partnerships with other networking platforms. 6. Emphasis on using ChildHub for advocacy at national and regional levels. 7. Leveraging ChildHub to monitor the situation of child protection services. 8. Stronger coaching and mentoring to accompany capacity building. 9. A more innovation-driven approach, including child-led social innovation. 10. Stronger 'offline' components to reinforce and complement online activities.

Implementation

It is important to keep in mind the ultimate ambition for ChildHub Asia, irrespective of the short-term funding. So, a phased approach to implementation would help align our ambitions and timelines to probable funding options. Moreover, a phased approach will allow us to ensure a progressive evolution, from direct transposition to child protection, towards an inclusion of all the programmatic areas.

 Phase I (typically, 1 year): Creation of ChildHub Asia for child protection, thus allowing the use of existing resources from ChildHub Europe (theoretical bases, basic course structure, concepts, legal frameworks, international regulations, …). This will culminate in establishment of the overall structure and start of first training and networking activities. Creation of content in one dominant regional language. Start of main networking activities like webinars and blog posts.  Phase II (typically, 2 years):

─ Scale-up in number and variety of members; ─ Scale-across to other areas, with sections on MNCH and juvenile justice; ─ Scale-in, with creation of content in multiple languages. The content available to members -both the learning and the networking -will be built in collaboration with national and regional partners, and validated by external partners. We will also ensure a certification mechanism from academic institutions for eLearning. The Front-end (the website) will be designed for PC and mobile devices. Moreover, we propose to have offline access to courses and evaluations, to enable members to work (build and test skills) even in absence of connectivity.

Resources required: Since we propose to start the implementation using the same architecture as ChildHub Europe, as well as some of the same contents, this will ensure a dynamic and cost-effective start. The three-year experience in Europe has been well documented, including a cost analysis that provides the different expense items and the costs incurred (inc. the unit cost, since the document details the scale of each activity and its costs). This will help estimate and seek sufficient funds.

Results expected

We expect ChildHub to produce the following results in South Asia and Africa:

 A more extensive and linked child protection network, between the main stakeholders working on child protection in Asia;  A sustainable regional collaboration, with effective engagement between country delegations to share knowledge and experiences;  A stronger workforce, thanks to access to knowledge and modern practices which could enhance the skills of the professionals by learning from their peers;  A stronger link between professionals, and higher satisfaction and motivation;  Ultimately, improved outcomes for children.

Conclusions

Platforms like ChildHub enhance the resilient elements of the system by supporting and valuing the actors of child protection, creating a culture of collaboration, and a strong movement that can analyse and influence the evolution of children's welfare through strong, targeted and evidence-based advocacy. Professionals who are more connected to each other, with access to knowledge and practices, are more skilled, motivated and supported to make better-informed decisions for children.

Fig. 1 .

 1 Fig. 1. Structure and components of ChildHub

Fig. 2 .

 2 Fig. 2. Performance of ChildHub in terms of its main output targets The following technical and process improvements are relevant in view of the findings from the ChildHub evaluation:  Design: Clearly define who specific ChildHub activities are targeted towards; Leverage social media; Target supervisors and senior management more directly within organisations; Continue to address and intensify translation efforts; Address IT skills issues amongst users; Establish learning goals for training activities and consider post-activity support; and Co-create new solutions with Resource Persons and partners to address the lack of adequate supervision of social workers.  Governance and management: Set up steering groups for each country; Better cultivate the resource persons with 'system' leadership skills; participatory approaches to strategic planning; Ensure that Tdh country offices and in-country partner/ member organizations align their existing plans/ activities. Finally, guided by a clear strategic framework, define functional working groups (e.g. policy advocacy, M&E) and strategy working groups (e.g. juvenile justice) to drive the mutual alignment of member efforts and the implementation of ChildHub's overall strategic plan.  Sustainability: Introduce a more structured approach, best guided through the production of a common Theory of Change, which can then be used as a basis for the

Fig. 3 .

 3 Fig. 3. ChildHub Asia (a similar model can be developed for Africa)

Table 1 .

 1 The foundations of the Child Protection Hub

	Vision	To unlock the potentials of child protection actors by delivering innovative services, content and tools that can help them achieve positive changes for children.
		By end-2017, professionals and key stakeholders have increased access to technical
	Purpose	resources and support networks allowing them to expand, improve, and advocate for
		qualitative services and policies for children in need of protection and their families.
		 Create opportunity for strong networks at regional and national levels
	Objectives	 Increase opportunity for upgrading knowledge and skills of professionals & oth-ers
		 Equip professionals and stakeholders to advocate for better policies and practices
	Expected out-comes	 Professionals develop skills to implement activities in more responsive & effec-tive

Table 1 .

 1 Current scale of ChildHub in Europe

	Component	Scale
		 3 eLearning modules, available in 5 languages; accessed by 1'000 profession-
	Learning	als  422 persons trained in face to face meetings by the 50 trained trainers
		 400 people attended 43 case discussions; 600 people attended 26 webinars
	Networking	 229 resource persons trained to promote continuous improvement in the field

We saw above that there is a strong need to strengthen skills of child protection professionals and to connect them to each other; ChildHub has succeeded in doing this at the Central European region. We therefore strongly promulgate the contextualization of ChildHub to other regions, including Africa and Asia, where the challenges of the sector are essentially the same. Contextualization, rather than 'reinventing the wheel', would also be a good way to learn from past experiences (thus increasing chances of success) and increase resource-effectiveness (by reutilizing the technical resources). ChildHub Europe's experience is well documented. An internal report presents the motivation, contents, management, costs and partnership structure. The external evaluation done in February 2018 contains a set of recommendations for both the future evolution of ChildHub in Europe and its contextualization to other regions. Together, these would provide the 'blueprint' for contextualization to Asia.

We propose an architecture for ChildHub Asia that is similar to the existing Child-Hub. We plan to learn from the experience in Europe ("Recommendations and plans for the Future and Contextualization of ChildHub" section above). ChildHub Asia would involve some additional features. We also propose to take into account the technical and process recommendations, notably for more effective governance structures and higher sustainability. Therefore, ChildHub Asia and Africa would have the following 10 additional features:


Josephine Evans
email: jahe2@cam.ac.uk

HYPOCOERCIVITY IN PHI-ENTROPY FOR THE LINEAR RELAXATION BOLTZMANN EQUATION ON THE TORUS

Keywords: Convergence to equilibrium, Hypocoercivity, Linear Boltzmann Equation, φ-entropy, Logarithmic Sobolev inequality, Beckner Inequality

 entropic hypocoercivity for a class of PDEs in a Hörmander sum of squares form. It was an open question to prove such a result for an operator which does not share this form. We prove a closed entropy-entropy production inequality à la Villani which implies exponentially fast convergence to equilibrium for the linear Boltzmann equation with a quantitative rate. The key new idea appearing in our proof is the use of a total derivative of the entropy of a projection of our solution to compensate for an error term which appears when using non-linear entropies. We also extend the proofs for hypocoercivity for the linear relaxation Boltzmann to the case of Φ-entropy functionals.

Introduction

In this paper we constructively prove convergence to equilibrium for the linear relaxation Boltzmann equation on the torus in relative entropy. We also look at other entropy functionals, the Φ-entropies specifically p-entropies. The equation is

∂ t f + v • ∇ x f = λ Π(f ) -λf, (1)
where f = f (t, x, v) : R + × T d × R d → R and λ is a positive constant. We always consider f to be a probability density so it is positive and of mass one, this is well known to be preserved by the equation. It is straightforward to show that this equation is well posed in L 1 . The operator Π is defined by Π(f ) =:

R d f (t, x, u)du M(v), M(v) := (2π) -d/2 exp - |v| 2 2 .
The equilibrium state of this equation is µ(x, v) = M(v). We give two separate notations here to emphasize when we consider it as a function of v alone or a function of x and v. We will always work in terms of h = f /µ which satisfies,

∂ t h + v • ∇ x h = λΠh -λh, (2)
here we define Π by Πh = R d h(t, x, u)M(u)du.

So the function Πh does not depend on v.

We want to study the convergence to equilibrium for solutions to equation [START_REF] Achleitner | On linear hypocoercive BGK models[END_REF] in relative entropy, H, and Fisher information, I, of f to µ. Studying the relative entropy has been an important way of showing convergence to equilibrium for kinetic equations since Boltzmann's H-theorem [START_REF] Boltzmann | Lectures on gas theory[END_REF]. Fisher information was introduced into kinetic theory by McKean to study convergence to equilibrium for a caricature of the Boltzmann equation [START_REF] Mckean | Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas[END_REF]. These quantities are defined in terms of h = f /µ, and are

H(h) = T d ×R d h log(h)dµ, I(h) = T d ×R d |∇h| 2 h dµ.
1.1. Previous work. Villani and Desvillettes demonstrated convergence to equilibrium in weighted H 1 for spatially inhomogeneous kinetic equations including the Boltzmann equation in [START_REF] Desvillettes | On the trend to global equilibrium in spatially inhomogeneous entropydissipating systems: the linear Fokker-Planck equation[END_REF][START_REF] Desvillettes | On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation[END_REF], their techniques were also applied to the linear Boltzmann equation in [START_REF] Cáceres | Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles[END_REF] where they show convergence faster than any power of t. After this the theory of hypocoercivity was developed and the equation is shown to converge to equilibrium in weighted L 2 , [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation[END_REF], by Hérau in order to demonstrate the applicability of the tools used in [START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF]. Convergence in weighted H 1 is also demonstrated in section 5.1 of [START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF], by Mouhot and Neumann as a consequence of a more general theorem. The techniques used in both these papers exploit commutator relations between the transport and collision part of the equation using the tools of hypocoercivity also see [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF][START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF][START_REF] Helffer | Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians[END_REF] for hypoellipticity based approaches; [28] for Villani's method based on these earlier works; [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF] for work directly in weighted L 2 spaces developing methods similar to [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation[END_REF] to extend the work to a wider classe of operators and [START_REF] Mischler | Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation[END_REF] which extends these results to a wider class of function spaces. The paper [START_REF] Achleitner | On linear hypocoercive BGK models[END_REF], shows convergence in weighted L 2 spaces with improved rates, and studies the convergence in relative entropy for models with discrete velocities. A linearized version of the non-linear equation in the multi-species case is studied in [START_REF] Achleitner | On multi-dimensional hypocoercive BGK models[END_REF] and a similar problem for the Elipsoidal BGK model is considered in [START_REF] Yun | Classical solutions for the ellipsoidal BGK model with fixed collision frequency[END_REF]. In these references, convergence is shown for h in either L 2 (µ) or H 1 (µ). These norms control relative entropy, H(µ). Therefore these results do imply exponential convergence of relative entropy. This fact is written explicitly in [START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF], Corollary 1.2, in this paper we give a different result with a close form estimate. This means we do not require the initial data, h 0 , to be in L 2 (µ), which would exclude f 0 having heavy tails.

The convergence demonstrated in all these papers is of the form

(3) E(f (t)|µ) ≤ Ce -γt E(f (0)|µ),
where E is a functional or norm and C and γ are explicit constants. If C = 1 the equation would be coercive in this norm. When C > 1, we use the terminology introduced in [28] and say that it is hypocoercive. We now show briefly why our equation is not coercive. Let A be some set of functions. First we not that if (3) holds with C = 1 for every initial data f (0) ∈ A then this is equivalent to a functional inequality. Lets define another functional by

D(f 0 |µ) = - d dt t=0 E(f (t)|µ).
Then if (3) holds with C = 1 for every f (0) ∈ A this is equivalent to

(4) D(f ) ≥ γE(f |µ),
for every f ∈ A. We can see that ( 4) implies ( 3) with C = 1 by using Grönwall's inequality.

Conversely if (3) holds with C = 1 and f (0) then differentiating this inequality at t = 0 will allow us to recover (4). We can check that this last inequality does not hold for the functionals we consider when f is in local equilibrium (i.e. of the form ρ(x)M(v)). More precisely we can check that D(ρM|µ) = 0.

1.2. Entropic hypocoercivity. Studying such equations in relative entropy was introduced by Villani in [28]. More recently entropic hypocoercivity and hypocoercivity in different Φ entropies have been studied for diffusion operators [START_REF] Baudoin | [END_REF][START_REF] Bodineau | Lyapunov functionals for boundary-driven nonlinear drift-diffusion equations[END_REF][START_REF] Monmarché | Generalized Γ calculus and application to interacting particles on a graph[END_REF][START_REF] Baudoin | Wasserstein contraction properties for hypoelliptic diffusions[END_REF]. Whilst most hypocoercivity theory has been done in L 2 (µ -1 ), H 1 (µ -1 ) there are several motivations to try and push the theory in the context of relative entropy.

• We can enlarge the space of initial data for which we can show exponentially fast convergence to equilibrium. If we show a result for f ∈ L 2 (µ -1 ) then we are constrained to work with initial data in L 2 (µ -1 ). This means that f 0 must decay very fast at infinity. However, if µ = exp(-|v| 2 /2 + U (x)) then we have

H µ (f ) = f log(f /µ)dxdv = f log(f )dxdv + f (|v| 2 /2 + U (x))dxdv.
Similarly, for Fisher information we have

I µ (f ) ≤ I(f ) + f |∇(|v| 2 /2 + U (x))| 2 dvdx.
So these quantities will be finite provided we have some moment bounds (depending on U (x)) and finite entropy and Fisher information. This is true for many distributions which decay only polynomially at infinity. • If we want to eventually study non-linear equations then it is often the case that strong spaces like L 2 (µ -1 ) will not be a natural space for the equation. For initial data which is neither small nor close to the Maxwellian there is no well posedness theory for the Boltzmann equation in Hilbert spaces weighted against the equilibrium This problem is solved in the context of the Boltzmann equation by combining linearised theory with enlarging the space of solutions [START_REF] Gualdani | Factorization of non-symmetric operators and exponential H-theorem[END_REF] and Desvillettes-Villani results to show when the solution will enter the linearised regime. • The relative entropy and relative Fisher information functionals behave well with respect to the dimension of the phase space that the equation is set in. More specifically, suppose that F N = f ⊗N then we have

H(F N ) = f ⊗N (z) i log(f (z i ))dz = i f (z i ) log(f (z i ))dz i = N H(f ).
We can also show that if Π 1 (F N ) is its first marginal, and the particles are indistinguishable then

H(Π 1 (F N )) ≤ 1 N H(F N ).
Therefore, if we know that for all N that

H(F N (t)) ≤ Ce -λt H(F N (0)),
then we have that

H(Π 1 (F N (t))) ≤ C N e -λt H(F N (0)).
Furthermore if F N (0) is a tensor product or similar we will have

H(Π 1 (F N (t))) ≤ Ce -λt
where C does not depend on N . Therefore the rates of convergence to equilibrium are uniform in N . On the other hand for L 2 the distance F N 2 behaves like Π 1 F N N 2 . So if we try the same computation we get that

Π 1 F N (t) 2 ≤ Ce -λt/N .
This effect becomes particularly important if one wishes to study particle systems and derive convergence results which are uniform in the number of particles. Entropic hypocoercivity has been used in [START_REF] Letizia | Nonequilibrium isothermal transformations in a temperature gradient from a microscopic dynamics[END_REF] to show convergence to the limit equation for oscillator chains.

1.3. Villani's method for operators in Hörmander form and the problem for the linear relaxation Boltzmann equation. The main purpose of this work is to demonstrate that entropic hypocoercivity can be proved for an equation which is not in 'A * A + B' form where A, B are first order differential operators. The key difference between the proofs given here and those of previous hypocoercivity results arises because we do not have a diffusion operator. In order to understand this it is useful to compare the linear relaxation Boltzmann equation with the kinetic Fokker-Planck equation on the torus.

(5)

∂ t f + v • ∇ x f = ∇ v • (∇ v f + vf ) .
Here we put x ∈ T d , v ∈ R d as with the linear relaxation Boltzmann equation. This equation also has the same equilibrium µ. Therefore we can write an equation on h = f /µ in the same way (6)

∂ t h + v • ∇ x h = (∇ v -v) • ∇ v h.
We can look at the dissipation of H(h) for both these equation. We have,

D kF P (h) = T d ×R d |∇ v h| 2 h dµ, (7)
D LRB (h) = (h -Πh) log(h)dµ. ( 8
)
Here D kF P is the dissipation of relative entropy for the kinetic Fokker-Planck equation, [START_REF] Baudoin | [END_REF], and D LRB is the dissipation of relative entropy for the linear relaxation Boltzmann equation, [START_REF] Achleitner | On multi-dimensional hypocoercive BGK models[END_REF]. We can see that both these quantities will vanish when h is a function only of x. This is the local equilibria mentioned above. Here, we also see a crucial difference. The regularizing effect of the Fokker-Planck operator means that D kF P is a Fisher Information type term in the sense that it is of order one in terms of derivatives, where as D LRB is an entropy type term in the sense that it is order zero in terms of derivatives.

The proofs in [START_REF] Hérau | Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential[END_REF]28,[START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF] use crucially the way that the free transport operator interacts with mixed x and v derivatives of the solution. In the context of relative entropy and Fisher information we can state this precisely. If h(t, x, v) is a solution to the free transport equation

∂ t h + v • ∇ x h = 0, then we have that d dt T d ×R d ∇ x h • ∇ v h h dµ = - T d ×R d |∇ x h| 2 h dµ.
In the case of the kinetic Fokker Planck equation Villani uses a functional which involves both the terms

T d ×R d
h log(h)dµ, and

T d ×R d ∇ x h • ∇ v h h dµ.
These will then produce the terms

T d ×R d |∇ x h| 2 h
dµ, and

T d ×R d |∇ v h| 2 h dµ,
in their dissipation. In fact, Villani uses a functional of the form

F (h) = H(h) + T d ×R d a|∇ x h| 2 + 2b∇ x • ∇ v h + c|∇ v h| 2 h dµ.
The other Fisher information type terms are required to make F a positive functional. There will be a lot of error terms in the dissipation as well as the useful terms appearing above. We can differentiate F (h) along the flow of the kinetic Fokker-Planck equation ( 6). Here we give the calculations briefly and refer to [28] for more detail. If we choose b 2 < ac one can verify that

d dt F (h) ≤ -(1 + 2c) T d ×R d |∇ v h| 2 h dµ -2b T d ×R d |∇ x h| 2 h dµ -(2c + 2b) T d ×R d ∇ x h • ∇ v h h dµ.
Now if a, b, c are sufficiently small we can split up the last term by the Cauchy-Schwartz inequality. We want to control this term by a large amount of the Fisher information terms with gradients in v and a small amount of Fisher information terms with gradients in x. For appropriately chosen constants a, b, c, this will give us that

d dt F (h) ≤ -b T d ×R d |∇ x h| 2 + |∇ v h| 2 h dµ. ( 9
)
This strategy for choosing a, b and c relies on the fact that the dissipation of entropy for the kinetic Fokker-Planck term is a Fisher information type term and can be used to control other Fisher information type terms appearing as errors. Now the goal is to compare the dissipation of F to F via a functional inequality. For this we need another tool, the logarithmic-Sobolev inequality.

Definition 1. A measure µ in the space of positive measures on a state space Ω satisfies a logarithmic Sobolev inequality if for all h we have that

Ω h(z) log(h(z)) -h(z) + 1 dµ ≤ C LS Ω |∇ z h(z)| 2 h(z) dµ.
We know that the equilibrium state for the Fokker-Planck equation on the torus satisfies a logarithmic Sobolev inequality, see for example [START_REF] Gross | Hypercontractivity and logarithmic sobolev inequalities for the clifford-dirichlet form[END_REF]. Therefore we can substitute 1 into (9) to get

d dt F (h) ≤ - b 2 T d ×R d |∇ x h| 2 + |∇ v h| 2 h dµ - b 2C LS T d ×R d h log(h)dµ (10) ≤ -CF (h). ( 11
)
We can then conclude by Grönwall's inequality that F is decreasing exponentially fast. This implies that both the relative entropy and relative Fisher information will decrease.

This proof relies on the fact that the dissipation of relative entropy for the kinetic Fokker-Planck is a Fisher Information type term. In the case of the linear relaxation Boltzmann equation this is no longer the case. It is still possible to generate

T d ×R d |∇ v h| 2 h dµ,
we get this from because for h a solution to (2) then

d dt T d ×R d |∇ v h| 2 h dµ = - T d ×R d ∇ x h • ∇ v h h dµ - T d ×R d |∇ v h| 2 h 1 + Πh h dµ.
However, because we also generate a term like

T d ×R d ∇ x h • ∇ v h h dµ.
This makes it impossible to close a Grönwall inequality purely on components of Fisher Information. We must expect this to be true since if we could close a Grönwall inequality on a functional without using a logarithmic Sobolev inequality or similar then exactly the same calculations would work for the equation where x is in the whole space with no confining potential. In this situation we would not see exponential convergence of the relative entropy to zero. Therefore our strategy is to find another entropy type term whose dissipation is a Fisher Information type term. We are motivated by the fact that Πh the macroscopic density will gain some regularity due to the averaging lemma, which says that the free transport will generate H 1/2 regularity for Πh, see for example [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF]. Although we do not prove a regularizing result on Πh and are results are not directly related we still find it useful to look at entropies involving Πh. In fact we will prove in section 2 that for h a solution to (2)

(12) d dt T d ×R d Πh log (Πh) dµ = T d ×R d Π (∇ v h) • Π (∇ x h) Πh dµ.
This is the key new idea used in our proof. The full strategy is explained in section 2.

A similar problem occurs when working in H 1 norms. This situation is studied in [START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF]. Here they work with g = f / √ M instead of working with h = f /M, this is only possible in the Hilbert space setting. In this situation we can write an equation on g

(13) ∂ t g + v • ∇ x g = T d ×R d g(t, x, u) M(u)du M(v) -g(t, x, v) := Πg -g.
Working in terms of g rather than h means sacrificing simplicity in bounding the dissipation of

∇ v h 2 L 2 (µ) relative to ∇ v g 2 L 2
for simplicity in controlling the mixed term. Precisely we have that for every δ > 0

d dt ∇ v g 2 2 ≤ -2 ∇ v g, ∇ x g + δ ∇ v g 2 2 + C(δ) g 2 2
and for every η > 0

d dt ∇ v g, ∇ x g ≤ -∇ x g 2 2 + η ∇ v g 2 2 + C(η) ∇ x ( Πg -g) 2 2 .
This means we can control mixed derivatives appearing in the dissipation of our functional up to producing a large amount of g 2 2 and this can be controlled by adding g 2 2 to the original functional. It is currently unclear whether it is possible to make a similar strategy work for relative entropy and Fisher information.

1.4. Results. In fact we prove do not work only in relative entropy. We instead study general Φ-entropies and Φ-Fisher informations defined respectively by

H Φ = R d ×T d Φ(h)dµ I Φ = R d ×T d Φ ′′ (h)|∇h| 2 dµ.
We work with Φ a positive function such that Φ(1) = 0, Φ ′′ (t) > 0 ∀t, 1/Φ ′′ (t) a concave function and Φ(t)Φ ′′ (t) > 2Φ ′ (t) 2 ∀t. Definition 2. We say that a measure µ satisfies a Φ-logarithmic Sobolev inequality if there exists a constant C > 0 such that for all h with f dµ = 1 we have

H Φ (h) ≤ CI Φ (h).
Remark. The conditions of Φ are satisfied when Φ is one of

Φ 1 (t) := t log(t) -t + 1 and Φ p (t) := 1 p -1 t p -1 -p(t -1) ,
where p ∈ [START_REF] Achleitner | On linear hypocoercive BGK models[END_REF][START_REF] Achleitner | On multi-dimensional hypocoercive BGK models[END_REF]. These quantities interpolate between the quadratic functional case p = 2 which is the L 2 norm, and the Boltzmann entropy case, p ∼ 1. They are used in [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Bolley | Phi-entropy inequalities and Fokker-Planck equations[END_REF] to study Fokker-Planck equations and convergence to equilibrium. Here we have inequalities due to Beckner in [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF] which play the same role as the logarithmic Sobolev inequality does in showing hypocoercivity in Boltzmann entropy. They are of the form

T d ×R d h p -h p(p -1) dµ ≤ C T d ×R d h p-2 |∇ x,v h| 2 dµ.
These can be shown by interpolating between Poincaré and logarithmic Sobolev inequality [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF].

Beckner Inequalities are often stated in the form

u 2 dµ - u p dµ 2/p ≤ (2 -p)C |∇u| 2 dµ.
It is straightforward to show that this equivalent to the form given above. (Write q = 2/p, h = u p and assume by homogeneity that h = 1.)

Theorem 1. Let Φ satisfy the conditions in lemma 1 and also let Φ be such that the uniform measure on the torus satisfies a Φ-Sobolev inequality, Φ(t)Φ ′′ (t) > 2Φ ′ (t) 2 ∀t and 1/Φ ′′ is a concave function. If f is a solution to (2) with initial data h 0 such that

R d ×T d Φ ′′ (h 0 )|∇ x,v h 0 | 2 dµ < ∞, f 0 ∈ W 1,1 (µ),
then there exist constants Λ and A depending on λ and the constant in the Φ-Sobolev inequality, such that

I Φ µ (h t ) + H Φ µ (Πh t ) ≤ A exp (-Λt) I Φ µ (h 0 ) + H Φ µ (Πh 0
) . This implies that if the equilibrium measure satisfies a Φ-Sobolev inequality then for some γ,

H(h t ) ≤ γ exp (-Λt) I(h 0 ).
We can take

Λ = min 1, C 4(1 + λ) min{2, λ/2} and A = 4 max{2(1 + 1/λ) 2 , (1 + λ)}.
Here C is the constant in the Φ-Sobolev inequality for the uniform measure on the torus.

1.5. Perspectives. This work raises two natural questions. The first is whether a similar strategy can be used to show convergence to equilibrium for the linear relaxation Boltzmann equation when x is in the whole space and the transport operator also involves a confining potential term. For the kinetic Fokker-Planck equation Villani shows convergence in H 1 and Boltzmann entropy in the first section of [28]. In [START_REF] Monmarché | Generalized Γ calculus and application to interacting particles on a graph[END_REF] Monmarché proves a general theorem which shows that hypocoercivity holds for the kinetic Fokker-Planck equation with confining potential in a class of Φ entropies which include the p-entropies. The situation is different for the linear relaxation Boltzmann equation. It is shown to be hypocoercive in L 2 in [START_REF] Hérau | Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation[END_REF][START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF]. In these works they use the inverse of an elliptic operator in order to create norms where you can compare the effect of the transport to the effect of the collisions. Emulating this strategy would be difficult in Φ-entropies. To show hypocoercivity for the linear relaxation Boltzmann equation with a confining potential in Φ-entropies would involve a very different strategy to our proofs in this equation. However, in the near to quadratic case it is possible to exploit additional cancellations happening in the operator to show convergence as is shown in [START_REF] Monmarché | A note on Fisher Information hypocoercive decay for the linear Boltzmann equation[END_REF] using calculations based on the original version of this paper.

The second natural question is whether this strategy could be extended to different collision operators which are also not regularising. For example the more complex scattering operators of the form

C(h)(x, v) = R d k(u, v)h(u) -k(v, u)h(v) M(u)du,
where

R d k(u, v) -k(v, u) M(u)du = 0.
Here our main goal would be the linear Boltzmann operator where

C(h)(x, v) = Q(hM, M)M(v) -1 .
Where here Q is the Boltzmann collision operator

Q(f, g) = R d S d-1 f (v ′ )g(v ′ * ) -f (v)g(v * ) dσdv * , v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ.
Exactly the same proof will work for a scattering operator which satisfies that for any positive definite constant matrix S we have

d dt C T d ×R d ∇ x,v h • S∇ x,v h h dµ ≤ T d ×R d ∇ x,v Πh • S∇ x,v Πh Πh dµ - T d ×R d ∇ x,v h • S∇ x,v h h dµ.
Here Π is still the projection onto the space of functions depending only on x. Unfortunately we are currently unable to do this for collision operators which are not straightforwardly comparable to the linear relaxation Boltzmann collision operator.

Acknowledgements. I would like to thank Clément Mouhot for pointing me towards this problem, suggesting I tried to emulate the techniques in [START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF] and suggesting I look at other Φ-entropies. I would also like to thank Pierre Monmarché for many useful comments on the style, notation and references in an earlier draft of this paper. I also had several useful discussions with Tom Holding in the very early stages of this paper about possible forms for the derivatives of the X part of the Fisher information.

Proofs for General Φ-entropy

Throughout the main parts of this chapter we work with an h which is bounded above and below by constants and has bounded derivatives of all orders. In this set of possible h, all the integration by parts and differentiating through the integral are justified. In the appendix we show that these properties are propagated by the equation and that we can extend the result to a wider set using a density argument.

First we prove our entropies are well behaved. Lets define the functional

J Φ µ (h) = R d ×T d Φ ′′ (h) a|∇ x h| 2 + 2b∇ x h • ∇ v h + c|∇ v h| 2 dµ.
Lemma 1. Let Φ satisfy ∀t > 0:

• Φ(t) ≥ 0 • Φ ′′ (t) ≥ 0 • Φ ′′ (t)Φ (4) (t) > 2Φ (3) (t) 2
Then if b 2 ≤ ac then J is a convex functional.

Proof. Since b 2 < ab we can write J as the sum of functionals like

J(h) = R d ×T d Φ ′′ (h)|α∇ x h + β∇ v h| 2 dµ.
Then the if the function φ(x, y) = Φ ′′ (y)|x| 2

the whole functional will be convex. This is because if φ is convex then

J th + (1 -t)g = R d ×T d φ t(α∇ x h + β∇ v h) + (1 -t)(α∇ x g + β∇ v g), th + (1 -t)g dµ ≤ R d ×T d tφ(α∇ x h + β∇ v h, h) + (1 -t)φ(α∇ x g + β∇ v g, g) dµ =t J(h) + (1 -t) J(g).
So we have reduced to showing that φ is convex. The function φ is the sum of functions φ = Φ ′′ (y)x 2 where now x is one dimensional. So we only need to show that these are convex. The Hessian of φ is 2Φ ′′ (y) 2xΦ (3) (y) 2xΦ (3) (y) x 2 Φ (4) (y) .

This has positive trace as both diagonal terms are positive by our assumptions. It also has determinant 2x 2 Φ ′′ (x)Φ (4) (x) -4x 2 Φ (3) (x) 2 which is again positive due to he assumptions we made on Φ therefore the Hessian is positive definite so φ is convex.

We now outline our strategy for the proof. Our goal is to get constructive rates of convergence to equilibrium by closing a Grönwall estimate on a functional that we construct. This functional is composed from the components of Fisher information and an entropy term. In order to explain the strategy compactly we introduce the components of Fisher information.

I X :=I X (h) = R d ×T d Φ ′′ (h)|∇ x h| 2 dµ, I V :=I V (h) = R d ×T d Φ ′′ (h)|∇ v h| 2 dµ, I M :=I M (h) = R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ.
We note here that I M does not have a sign. We also introduce a projected entropy which we use in our functional,

H Π (h) = T d Φ(Πh)dx.
We have another term which only appears in the intermediate steps of the proof,

I ΠX := R d ×T d Φ ′′ (Πh)|∇ x (Πh)| 2 dµ,
We prove later in this section that I X -I ΠX ≥ 0.

By differentiating along the flow of the equation we show something close to the inequalities

d dt I X ≤ -λ I X -I ΠX , (14)
d dt I M ≤ -I X -λI M , (15)
d dt I V ≤ -2I M -λI V . ( 16
)
Actually there are extra elements appearing which would cancel out when these terms are combined into the type of functional we look at so these inequalities are not quite true. In fact we prove a global inequality on a functional like J defined bellow but it is clearer to separate the elements here. We begin by constructing a functional of the form

J = aI X + 2bI M + cI V ,
with ac -b 2 > 0. This inequality means that J is equivalent to the Fisher information I.

We now give a strategy for choosing a, b, c. We need that b is non-zero since inequality 15 provides the negative I X which we want in the derivative. The most natural next step would be to use the Cauchy-Schwartz inequality to control I M by I X , and I V . However, we can check that the quantity of I M is too large for this to be possible. We need to utilise inequality [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF]. We do this by showing that

-I M ≤ η 2 I V + 1 2η I X -I ΠX - d dt H Π . (17)
This is the key new element in our proof.

By adding a quantity of H Π to the functional and using inequality [START_REF] Gross | Hypercontractivity and logarithmic sobolev inequalities for the clifford-dirichlet form[END_REF], we can now control I M by I V and I X -I ΠX . Since the inequality ( 14) doesn't produce bad terms we are free to add as much I X to the functional as we need. Therefore, by adding a large amount of H Π and I X to our functional we can cancel out the positive I X -I ΠX . Therefore we can make η small. This means the sum of the positive I V from controlling I M and the negative I V from inequality (16) will sum to a negative amount of I V . We recall that we also have some negative I X for inequality [START_REF] Eckmann | Spectral properties of hypoelliptic operators[END_REF]. So we have,

d dt (J + A 4 H Π ) ≤ -C(I X + I V ).
We then use the equivalence between J and I and the logarithmic Sobolev inequality to get d dt

(J + A 4 H Π ) ≤ -C(J + A 4 H Π ).
So we can close a Gronwall estimate and then use the equivalence between J and I again to translate this to an inequality on I.

In order to prove our theorem we would like to study how a functional like J behaves under the action of the collision part of the operator. We write L = λ(Π -I) and T = -v • ∇ x and write (d/dt) O to write the derivative along the flow of the operator L. We have that Lemma 2. We can differentiate J along the flow of L to get that

d dt L J Φ µ (h) ≤a R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ - R d ×T d Φ ′ (h)|∇ x h| 2 dµ -2b R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ -c R d ×T d Φ ′′ (h)|∇ v h| 2 dµ.
Proof. As J Φ µ is convex we can see by Taylor expanding that

J Φ µ (e Ls h(t)) =J Φ µ (h(t) + λs(Π -I)h(t) + o(s)) ≤(1 -λs)J Φ µ (h(t) + o(s)) + λsJ Φ µ (Πh(t)). Now we calculate that J Φ µ (Πh) = R d ×T d Φ ′′ (Πh) a|∇ x Πh| 2 + 2b∇ x Πh • ∇ v Πh + c|∇ v Πh| 2 dµ =a R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ.
This means that

J Φ µ (e sL h(t)) -J Φ µ (h(t)) ≤λsa R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ - R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -λsb R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ -λsc R d ×T d Φ ′′ (h)|∇ v h| 2 dµ + J Φ µ (h(t) + o(s)) -J Φ µ (h(t)).
Dividing by s and taking the limit as s → 0 gives the result.

We now need to look at how J behaves under the flow of T .

Lemma 3. We can differentiate J along the flow of T to get that

d dt T J Φ µ (h) = -2b R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -2c R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ.
Proof. This is just a simple application of the chain rule. We have

d dt T J Φ µ (h) = -a R d ×T d Φ ′′′ (h)(v • ∇ x h)|∇ x h| 2 dµ -2a R d ×T d Φ ′′ (h)∇ x (v • ∇ x h) • ∇ x hdµ -2b R d ×T d Φ ′′′ (h)(v • ∇ x h)∇ x h • ∇ v hdµ -2b R d ×T d Φ ′′ (h)∇ x (v • ∇ x h) • ∇ v hdµ -2b R d ×T d Φ ′′ (h)∇ x h • ∇ v (v • ∇ x h)dµ -c R d ×T d Φ ′′′ (h)(v • ∇ x h)|∇ v h| 2 dµ -2c R d ×T d Φ ′′ (h)∇ v (v • ∇ x h) • ∇ v hdµ = R d ×T d v • ∇ x Φ ′′ (h)(a|∇ x h| 2 + 2b∇ x h • ∇ v h + c|∇ v h| 2 dµ -2b R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -2c R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ = -2b R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -2c R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ.
Now we need to show our helpful lemma relating projected entropy to the mixed term. This result relates the quantities involving only Πh to quantities coming from the full Fisher information. For this we define the local average speed U (x), of a solution to (1) by

U (x) := R d vh(v, x)M(v)dv = R d vf (v, x)dv.
Lemma 4. Suppose that the uniform measure on the torus satisfies a Φ-Sobolov inequality. Then for any h we have that

I ΠX (h) = R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ ≤ R d ×T d Φ ′′ (h)|∇ x h| 2 dµ.
This implies that for all h there exists a constant C such that

H Π (h) = T d Φ(Πh)dx ≤ C T d ×R d Φ ′′ (h)|∇ x h| 2 dµ. Finally, if h is a solution to (2) then d dt H Π (h(t)) = - T d Φ ′ (Πh)∇ x • U (x)dx.
Proof. We can see that the first inequality will follow from

Φ ′′ (Πh)|∇ x Πh| 2 ≤ Π Φ ′′ (h)|∇ x h| 2 .
Since Π is integrating against a probability measure we would like to use Jensen's inequality.

Instead of looking at h we consider h = (∇ x h, h) we have already shown the function φ(x, y) = Φ ′′ (y)|x| 2 is convex so from Jensen's inequality we have

φ(Πh) ≤ Π(φ(h)).
This implies our desired result since Π commutes with ∇ x . (Here Π acts component wise on vectors). Now since we have a Φ-Sobolev inequality for the uniform measure on the torus we have

T d Φ(Πh)dx ≤ C T d Φ ′′ (Πh)|∇ x Πh| 2 dx.
We can then conclude this part by the first inequality.

For the last part,

∂ t Πh = - R d v∇ x hM(v)dv + λΠ(Πh) -λΠh = -∇ x • U (x).
This implies that

∂ t H Π = T d Φ ′ (Πh)∂ t Πhdx = - T d Φ ′ (Πh)∇ x • U (x)dx.
We now need a lemma which will help us control the mixed derivative.

Lemma 5. If 1/Φ ′′ (t) is a concave function then for any positive η we have

- R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ ≤ η 2 R d ×T d Φ ′′ (h)|∇ v h| 2 dµ + 1 2η R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ - R d ×T d Φ ′′ (h)|∇ x h| 2 dµ - d dt R d ×T d Φ(Πh)dµ.
Proof. We need to rewrite the mixed term

- R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ = - R d ×T d ∇ x Φ ′ (h) • ∇ v hdµ = - R d ×T d (∇ x Φ ′ (h) -∇ x Φ ′ (Πh)) • ∇ v hdµ - R d ×T d ∇ x Φ ′ (Πh) • ∇ v hdµ ≤ η 2 R d ×T d Φ ′′ (h)|∇ v h| 2 dµ + 1 2η R d ×T d |∇ x Φ ′ (h) -∇ x Φ ′ (Πh)| 2 Φ ′′ (h) dµ - R d ×T d Φ ′ (Πh)∇ x • U (x)dµ.
We get the equality for the last term since

-∇ v hM(v)dv = -vhM(v)dv = U (x).
Then we can use the last part of lemma 4. Now we observe that

R d ×T d |∇ x Φ ′ (h) -∇ x Φ ′ (Πh)| 2 Φ ′′ (h) dµ = R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -2 R d ×T d Φ ′′ (Πh)∇ x h • ∇ x Πhdµ + R d ×T d Φ ′′ (Πh) 2 Φ ′′ (h) |∇ x Πh| 2 dµ.
Now we see in the second term the only part which depends on v is the ∇ x h so we can replace it by ∇ x Πh. The last term is positive and the only term which depends on v is 1/Φ ′′ (h) since we have that 1/Φ ′′ (h) is a concave function we have

Π 1 Φ ′′ (h) ≤ 1 Φ ′′ (Πh)
.

Therefore we have that

R d ×T d |∇ x Φ ′ (h) -∇ x Φ ′ (Πh)| 2 Φ ′′ (h) dµ ≤ R d ×T d Φ ′′ (h)|∇ x h| 2 dµ - R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ.
This completes the proof of our lemma.

Now we can prove the main theorem

Proof of Theorem 1. Using lemmas 2 and 3 we get that

d dt J Φ µ (h) ≤ -2b R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -cλ R d ×T d Φ ′′ (h)|∇ v h| 2 dµ -2(bλ + c) R d ×T d Φ ′′ (h)∇ x h • ∇ v hdµ + aλ R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ - R d ×T d Φ ′′ (h)|∇ x h| 2 dµ .
We now use lemma 5 to bound the mixed term.

d dt J Φ µ (h) ≤ -2b R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -(cλ -η(bλ + c)) R d ×T d Φ ′′ (h)|∇ v h| 2 dµ -aλ - 1 η (bλ + c) R d ×T d Φ ′′ (Πh)|∇ x Πh| 2 dµ - R d ×T d Φ ′′ (h)|∇ x h| 2 dµ -2(λb + c) d dt R d ×T d Φ(Πh)dµ. Now lets choose a = 2(1 + 1/λ) 2 , b = 1, c = 1, η = λ/(2(λ + 1)). This gives d dt J Φ µ (h) ≤ -2 R d ×T d Φ ′′ (h)|∇ x h| 2 dµ - λ 2 R d ×T d Φ ′′ (h)|∇ v h| 2 dµ -2(λ + 1) d dt R d ×T d Φ(Πh)dµ.
Lemma 6. The equation preserves bounded derivatives of all orders.

Proof. We rewrite the equation for h in a mild formulation as follows

e λt h(t, x, v) = h(0, x -vt, v) + λ t 0 e λs h(s, x -v(t -s), u)M(u)duds.
This leads to the following inequality

e λt D α x h(t) ∞ ≤ D α x h(0) + λ t 0 e λs D α x h(s) ∞ ds.
Therefore by Gronwall's inequality we have that

D α x h(t) ∞ ≤ D α x h(0) ∞ .
We also from this mild formulation that any mixed derivative can be written in terms of x derivative and derivatives of the initial data. Therefore, the derivatives will remain in L ∞ for all time.

Lemma 7. The equation preserves positivity and constants are a steady state of the equation therefore being bounded above and below is preserved.

Proof. We can show that ∂ t e λt h(t, x + vt, v) = λe λt h(t, x + vt, u)M(u)du.

Therefore if e λt h(t, x + vt, v) is positive for all x and v then so is its derivative. Therefore it will remain positive for all time.

It is easy to check that constants are a steady state so if h(0)-c is positive then since positivity is preserved so is h(t) -c and similarly if C -h(0) is positive then so is C -h(t).

Lemma 8. Suppose that we have h(0) is in W 1,1 (µ) with bounded Fisher information, and also suppose we have a sequence h n (0) which is bounded above and below, has bounded derivatives up to second order and converges to h(0) in L 1 (µ) with H Φ (h n (t)) ≤ Ae -Λt I Φ (h n (0)), for every n then we have H(h(t)) ≤ Ae -Λt I(h(0)).

Proof. Convergence in L 1 implies that h n tends to h a.e. along a subsequence. Also, suppose that h 1 and h 2 are two solutions to the equation then sup s≤t h 1 (s) -h 2 (s) L 1 (µ) ≤ e -λt h 1 (0) -h 2 (0) L 1 (µ) + sup s≤t h 1 (s) -h 2 (s) L 1 (µ) (1 -e -λt ).

Therefore, sup

s≤t h 1 (s) -h 2 (s) L 1 (µ) ≤ h 1 (0) -h 2 (0) L 1 (µ) , hence h n (t) tends to h(t) in L 1 therefore h n (t) also converges to h(t) almost everywhere along a subsequence.

Then since Φ(h n (t)) ≥ 0 by Fatou's lemma we have Φ(h(t))dµ ≤ lim inf n Φ(h n (t))dµ.

Therefore, if we have h a solution to the equation with initial data h(0) as defined above we have that H Φ (h(t)) ≤ lim inf n Ae -Λt I Φ (h n (0)).

So to prove our theorem holds in this larger set it remains to show that we can find a sequence h n (0) converging to h(0) in L 1 (µ) where for every n h n (0) is positive, integrates to 1 against µ, is bounded bellow and has derivatives bounded of all orders which also satisfies lim inf n I Φ (h n (0)) ≤ I Φ (h(0)).

To do this we make a very standard molifier argument. Let χ be a smooth function on R + with χ(x) = 1 for x < 1 and χ(x) = 0 for x > 2 and Φ ′′ (χ(x))|χ ′ (x)| 2 integrable. Then define χ R (x, v) = χ( v /R). Also let φ be a molifier integrating to one and compactly supported in B(0, 1) then set φ ǫ (x, v) = ǫ -2d φ((x, v)/ǫ). Take some h in W 1,1 (µ) with finite Φ-Fisher information. Let h R = hχ R , then set h ǫ,R = φ ǫ ⋆ h R and then h η,ǫ,R = (h ǫ,R + η)/( h ǫ,R 1 + η). So h η,ǫ,R is bounded below and has derivatives bounded of all orders and fairly clearly converges to h in L 1 (µ).

So first we try and get rid of η since ∇h η,ǫ,R = ∇h ǫ,R /( h ǫ,R 1 + η) we get that Φ ′′ (h η,ǫ,R )|∇h η,ǫ,R | 2 increases to Φ ′′ (h ǫ,R )|∇h ǫ,R | 2 . Therefore, by monotone convergence, lim η→0 I Φ (h η,ǫ,R ) = I Φ (h ǫ,R ). Now we work on ǫ, we have that ∇h ǫ,R = φ ǫ ⋆ ∇h R . We can now make a similar argument based on Jensen's inequality and the fact that Φ ′′ (y)|x| 2 is convex to get that

Φ ′′ (h ǫ,R )|∇h ǫ,R | 2 ≤ φ ǫ ⋆ Φ ′′ (h R )|∇h R | 2 .
Since, the mollification of and L 1 function converges in L 1 to that function we get that lim ǫ→0 I Φ (h ǫ,R ) ≤ I Φ (h R ). Now we work on R, we note that

Φ ′′ (h R )|∇h R | 2 = Φ ′′ (hχ R ) χ 2 R |∇h| 2 + χ R h∇h • ∇χ R + h 2 |∇χ R | 2
Since, h, ∇h, Φ ′′ (h)|∇h| 2 are all in L 1 (µ) we can see that lim R→∞ I Φ (h R ) = I Φ (h).

The author was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.

This gives that d dt (J Φ µ (h) + 2(λ + 1)H Φ µ (Πh)) ≤ -

Since we have that 2

Therefore if we set

we have that

) . Now we use that for all h,

This means that

We show for h, being bounded above and bellow and having bounded derivatives of all orders is propagated by the equation (this is similar to what is shown in the appendix of [START_REF] Cáceres | Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles[END_REF]). In this set we can do all the calculations given in the main part of the paper. We then show for h ∈ W 1,1 (µ) with finite Fisher information then we can make a density argument to show that the result still holds in this case.