Michiel Doorman Luhuan Huang
email: l.huang@uu.nl

Michiel Doorman
email: m.doorman@uu.nl

Wouter Van Joolingen
email: w.r.vanjoolingen@uu.nl

Chinese and Dutch mathematics teachers' beliefs about inquiry-based learning

Keywords: Mathematics education, inquiry-based learning, comparative study, teacher belief, lower-secondary education

Introduction

As an intentional student-centered pedagogy rooted in the Western teaching culture, inquiry-based learning (IBL) encourages students to take responsibilities in the learning process, to explore by themselves, and to construct knowledge through actively participating in cycles like questioning, hypothesizing, designing, investigating, analyzing and reflecting [START_REF] Swan | Designing and using professional development resources for inquiry-based learning[END_REF].

The understanding and implementing of IBL may be impacted by teaching cultures, while teaching culture of the East Asia is considered remarkably different from that of the West. These two teaching cultures have been identified and compared, which seemed to produce some stereotypes [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF]. From these stereotypes, the East Asia emphasizes learning content and related skills [START_REF] Correa | Connected and culturally embedded beliefs: Chinese and US teachers talk about how their students best learn mathematics[END_REF][START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF] and values in-depth knowledge [START_REF] Norton | Primary mathematics teacher education in Australia and China: What might we learn from each other?[END_REF], while the West emphasizes learning process [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF] and values practical knowledge [START_REF] Norton | Primary mathematics teacher education in Australia and China: What might we learn from each other?[END_REF]; the East Asia conducts instruction to the whole class of students, while the West adopts individualized learning and group work [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF]; the East Asia considers teachers as the center and conducts well-organized directive instruction to deliver knowledge to students [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF], while the West considers students as the center and encourages them to construct knowledge actively [START_REF] Liu | How culture matters in educational borrowing? Chinese teachers' dilemmas in a global era[END_REF]; the East Asia makes students learn by memorizing and practicing repetitively [START_REF] Liu | How culture matters in educational borrowing? Chinese teachers' dilemmas in a global era[END_REF][START_REF] Tan | Education policy borrowing and cultural scripts for teaching in China[END_REF], while the West encourages meaningful learning [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF]; the East Asia motivates students by external factors such as examinations, while the West motivates students by internal factors such as interests [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF].

These stereotypes about teaching cultures in the East Asia and the West also apply to the subject of mathematics. While the conceptualization of IBL in mathematics is less obvious [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF]. A large part of existing research focused on IBL in science, including research on science teachers' beliefs and practices of IBL [START_REF] Wallace | An investigation of experienced secondary science teachers' beliefs about inquiry: An examination of competing belief sets[END_REF], a few of these research also included mathematics teachers [START_REF] Marshall | K-12 science and mathematics teachers' beliefs about and use of inquiry in the classroom[END_REF][START_REF] Song | Linking teacher beliefs, practices and student inquiry-based learning in a CSCL environment: A tale of two teachers[END_REF]. More attention needs to be paid to IBL specifically in mathematics.

As for the concept of IBL itself, no consensus has been reached about the definition, there exist a variety of interpretations about the IBL way of teaching approach [START_REF] Furtak | Experimental and Quasi-Experimental Studies of Inquiry-Based Science Teaching: A Meta-Analysis[END_REF], especially about the amount of support provided to students, which makes IBL complicated to understand and implement for practitioners. Teachers may not have a complete understanding of IBL [START_REF] Chin | A survey of the practice of a large-scale implementation of inquiry-based mathematics teaching: from Taiwan's perspective[END_REF], and their beliefs about the detailed content of IBL tend to be diverse [START_REF] Hok | How do teachers' beliefs affect the implementation of inquiry-based learning in the PGS Curriculum? A case study of two primary schools in Hong Kong[END_REF]. In addition, IBL seems not yet a common practice embedded in daily teaching [START_REF] Dobber | Literature review: The role of the teacher in inquiry-based education[END_REF]. Teachers' beliefs may shape their decisions and practice of implementing IBL [START_REF] Saad | The relationship between teachers' knowledge and beliefs about science and inquiry and their classroom practices[END_REF][START_REF] Song | Linking teacher beliefs, practices and student inquiry-based learning in a CSCL environment: A tale of two teachers[END_REF][START_REF] Wallace | An investigation of experienced secondary science teachers' beliefs about inquiry: An examination of competing belief sets[END_REF]. While some studies also found a disconnect [START_REF] Engeln | Inquiry-based learning in mathematics and science: A comparative baseline study of teachers' beliefs and practices across 12 European countries[END_REF][START_REF] Ramnarain | Teacher beliefs and attitudes about inquiry-based learning in a rural school district in South Africa[END_REF] or a more complicated impact [START_REF] Hok | How do teachers' beliefs affect the implementation of inquiry-based learning in the PGS Curriculum? A case study of two primary schools in Hong Kong[END_REF] between the beliefs and practice in regard to IBL. A deep understanding of teachers' beliefs towards the "complicated" IBL makes sense and may provide potential for better understanding their practice.

In an investigation into the beliefs and practices of IBL in mathematics, we explored from a students' perspective, and took China and the Netherlands as representing countries of the two teaching cultures (the east Asia and the West). The results showed that Chinese students reported more experience and preference of IBL than Dutch students, which challenge the stereotypes about the two teaching cultures. In this study, a teachers' perspective is to be explored.

The aim of this study is to present and compare Chinese and Dutch mathematics teachers' beliefs about IBL. The research questions are: What kinds of beliefs do lower-secondary mathematics teachers in China and the Netherlands have about inquiry-based learning (IBL)? What are the main similarities and differences on this issue between the two countries?

Methods

Participants

We interviewed 30 teachers from 15 Chinese schools and 19 teachers from 13 Dutch schools, all of them were teaching lower-secondary mathematics. 28 of the Chinese teachers and 9 of the Dutch teachers are female. The average age was 38 for Chinese teachers and 42 for Dutch teachers. As for the average years of teaching, it was 15 for Chinese teachers and 11 for Dutch teachers.

In China, generally a permission from school leaders makes it convenient to enter a school and conduct interviews, while teachers in the Netherlands have more freedom to accept an interview. Therefore in China we mainly contacted school leaders first, also a few local administrations, and some mathematics teachers directly, while in the Netherlands we invited individual teachers and included all teachers with interests. In both countries, participants were contacted mainly through an interpersonal network. Because of the large areas in China, we only collected data at Beijing, where differences exist between urban and suburban schools, thus we ensured a balanced selection of eight urban schools and seven suburban schools to better represent the situation. In the Netherlands, schools at different areas are quite similar, we included different types of secondary schools1 .

Instruments

Data was collected through semi-structured interviews, which provided opportunities for teachers to express views and suit the research questions well. Without much presume about the definition or model of IBL, we left it open for teachers about their understanding of IBL.

We constructed an interview outline including general questions such as "what is your understanding of IBL" and two IBL example tasks as context to promote discussion. The tasks were chosen from materials of Primas project2 and had potential for IBL, while they were not defined as IBL tasks in the interview, participants were asked questions such as "can it be used in an IBL lesson/do both versions represent IBL". The interview outline and example tasks were originally in English and translated into Chinese. The outline and tasks were piloted with two Dutch teachers and two Chinese teachers to make sure the questions and guidelines were clear enough and led to information we expected to collect. The pilot interviews also helped to prepare for practical issues that may happen during interviews.

Data collection and analysis

Dutch participants were interviewed from April to June of 2017, and Chinese participants from October to November in the same year. Each participant was interviewed individually for around 40 minutes, the language was English for Dutch teachers and Chinese for Chinese teachers. The participant was asked if recording can be accepted. If not, the interview would not go on. Generally, questions from the interview outline were asked in sequence, and extensive questions related to the topic were also allowed. Similar introduction and guideline were given to each participant.

All the interview recordings were turned into transcripts. The original codes were constructed mainly based on the questions from the interview outline, "teachers' responsibility" and "students' responsibility" were derived from literature. All transcripts were imported into Nvivo 11 and divided into sets of fragments. Each fragment represents a single idea. In the process of coding -including individual coding and discussing differencessub codes with example quotations were developed to create a better understanding of the main codes. The coding scheme (shown in Table 1) was adjusted for several rounds to be better connected to the transcripts3 .  "Not to take too big steps, make sure that is the right question for the right age, make sure it's a statement that they really discover something, like this."

When about half of the interviews had been coded, we introduced an external researcher to the interview outline and the coding scheme, and asked her to randomly choose one Chinese interview and one Dutch interview to code. The Chinese interview was coded together, and next the Dutch interview was coded by herself, resulting in a 69% agreement.

After coding, we arranged fragments with similar views together and extracted representative statements from those groups of fragments. For each statement, we counted how many teachers expressed this view during interviews. Based on the number, we ranked all the statements within each main code and kept the four highest-ranking statements to provide an overview of beliefs that the two groups of teachers have about IBL. If there existed multiple statements at the same ranking, all of them were kept or quitted. We counted and ranked for Chinese teachers and Dutch teachers respectively, then we compared the results between two countries.

Results

As preliminary results of this study, Table 2 shows 30 Chinese teachers and 19 Dutch teachers' beliefs on three important aspects through the four highest ranking statements within each main code. For example, 12 of the 30 Chinese teachers talked about "students explore and find a way to the problem".

Chinese teachers' beliefs about IBL

As is shown in Table 2, Chinese teachers talked about students' exploring, solving, thinking and getting the conclusion in IBL, they also connected IBL with student discussion and collaboration. In addition, emphasis was put on teacher guidance during the process. As for the reasons to implement IBL, Chinese teachers mainly focused on the benefits of IBL on mathematical thinking, knowledge and motivation, while only a few teachers paid attention to general skills developed from IBL. As for difficulties that teachers encountered in IBL, Chinese teachers pointed out factors related to students including the lack of motivation and performance in IBL, they also talked about factors related to conditions including lack of time and lack of suitable tasks.

Dutch teachers' beliefs about IBL

As is shown in Table 2, Dutch teachers emphasized students' taking responsibility in IBL. They paid attention to students' exploring, solving, thinking and doing activities (such as drawing and calculating) in IBL, and they made it explicit that students explore by themselves without getting explanations from teachers about the theory or problem-solving procedures. They also noticed that the problem used in IBL usually provided enough space for students to explore from different approaches. When it comes to the reasons for IBL, Dutch teachers talked about the benefits of IBL on knowledge, motivation, general skills and mathematical thinking, they considered IBL as preparation for examinations as well. As for the difficulties in IBL, Dutch teachers pointed out factors related to conditions including lack of time and lack of suitable tasks, they also included factors related to teachers, namely they were required a lot from IBL, and they expressed dislike about the uncertain and uncontrollable feature of IBL.

Comparisons between Chinese and Dutch teachers' beliefs about IBL

As is shown by statements in bold in Table 2, Chinese and Dutch teachers shared some beliefs about IBL. Teachers in both countries paid attention to students' responsibility in IBL and mentioned students' exploring, solving and thinking. They both took the four benefits of IBL (on knowledge, mathematical thinking, motivations and general skills) as reasons to implement it. Moreover, they listed two common difficulties related to conditions, namely lack of time and lack of suitable tasks.

The differences between Chinese and Dutch teachers' beliefs about IBL are apparent in Table 2. As for the general views of IBL, Dutch teachers mainly emphasized students' responsibility in IBL while Chinese teachers also paid attention to teacher guidance (such as promoting students by questions) during the process. Chinese teachers talked about student discussion and collaboration and the attainment of results in IBL as well. The shared four benefits of IBL rank differently as reasons for IBL that Chinese teachers emphasized mathematics thinking most while Dutch teachers focused on knowledge most. Dutch teachers talked more about general skills, they also mentioned IBL as preparation for examinations. As for the difficulties in IBL, Chinese teachers listed factors related to students about their lack of motivation and performance in IBL, while Dutch teachers mentioned factors related to teachers that they considered IBL as demanding, uncertain and difficult to control.

Discussion

Our findings are based on the samples from convenient sampling. Chinese data was only collected at Beijing with more advantaged educational resources, most Dutch teachers had connections with universities or research institutes, participants may be more active in exploring new teaching approaches. In addition, this study was limited to reported beliefs and lacked observations of actual lessons. Finally, although we tried to ensure a shared understanding of IBL by providing example tasks during interviews, the data might be biased if participants having different interpretations of a term like "inquiry" that originated from science education [START_REF] Beumann | Sometimes it goes wrong!" -Teachers' beliefs concerning experiments in mathematics[END_REF] Despite the limitations of the study, some findings are in line with stereotypes about teaching cultures in the East Asia and the West. The attention of Chinese teachers on attainment of results, student performance, mastery and appliance of knowledge as well as teacher guidance in IBL match the stereotypes that the East Asia emphasizes learning content [START_REF] Correa | Connected and culturally embedded beliefs: Chinese and US teachers talk about how their students best learn mathematics[END_REF][START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF] and teachers' role in instruction. The attention of Dutch teachers on students' doing activities, general skills, students' taking responsibility before teacher explanation and student motivation match the stereotypes that the West emphasizes learning process [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF] and practical knowledge [START_REF] Norton | Primary mathematics teacher education in Australia and China: What might we learn from each other?[END_REF], encourages students' constructing knowledge [START_REF] Liu | How culture matters in educational borrowing? Chinese teachers' dilemmas in a global era[END_REF] and meaningful learning, and values students' internal motivations [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF].

However, some findings are not in line with the stereotypes. The attention of Chinese teachers on students' exploring, solving and thinking, on student discussion and collaboration as well as on student motivation do not match the stereotypes that the East Asia conducts teacher-centered directive instruction to deliver knowledge [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF], makes students learn by memorizing and practicing repetitively [START_REF] Liu | How culture matters in educational borrowing? Chinese teachers' dilemmas in a global era[END_REF][START_REF] Tan | Education policy borrowing and cultural scripts for teaching in China[END_REF], conducts instruction to the whole class with little group work [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF] and ignores students' internal motivations. The attention of Dutch teachers on knowledge and preparing students for examinations do not match the stereotypes that the West focuses on learning process more than learning content, and emphasizes students' internal motivations more than external motivations such as examinations [START_REF] Leung | In search of an East Asian identity in mathematics education[END_REF].

Follow-up study will analyze other topics from the interviews such as IBL activities and IBL outcomes, we will also include the reported IBL practice of Chinese and Dutch teachers to see to what extent the reported practice match their beliefs about IBL, whether the beliefs shape the practice in regard to IBL or there exists a more complicated relation between them as suggested by literature.

Table 1 : Coding scheme of the study

 1

	Main code	Sub code	Example
	• General views-Basic		 "It is...about doing something different than just making the
	understanding of IBL		exercises from the book"
	• Attitudes-Overall tendency		 "I think it is very useful for the basis, but I think you cannot
	towards IBL, Whether in favor		use it as the only way of teaching."
	of IBL
	• Prerequisite-Factors	Task	 "I think this one (Unstructured Version B) will (represent
	considered before		IBL), this one (Structured Version A) not, this is too
	implementing IBL in lessons		structured."
		Students	 "I think it is a very good group, I would maybe change this a
			bit, then I would give (Unstructured) Version B. In a lower
			grade like HAVO, then I would do this (Structured Version A),
			because otherwise they will not come any further."
		Teacher	 "You need to have a repertoire of dealing with different things,
			you have to build that repertoire also"
		Context	 "They have to be ready to think about something new, even if
			it's very small detail, you must make them ready to do it, and
			sometimes you do not succeed because the circumstances are
			not ideal."
	• Activity-Things going on	Students' responsibility	 "They need some time for themselves to try and succeed or try
	when implementing IBL in		and fail"
	lessons; They are expected or	Teachers' responsibility	 "I think here you should be more encouraging, and maybe
	planned to happen in the		come and ask how things are going, and if they need help, then
	teacher's lessons		ask questions."
	• Outcome-Results of	Cognitive & Positive	 "They learned to think, and to think deeper, and to persuade
	implementing IBL		the other person"
		Cognitive & Negative	 "And the risks, risk of if every student learns enough, it's not
			guaranteed"
		Motivational & Positive	 "I think they would like mathematics more"
		Motivational & Negative	 "I think some of the kids will get discouraged, less
			encouraged, demotivated from like this."
	• Reasons For-Factors for	Lead to positive results	 "I think it can really motivate students, it can really help them
	teachers to implement IBL		learning fast"
	• Difficulties-difficulties in	Related to conditions	 "I just don't find the time to do it"
	implementing IBL
		Related to students	 "That the kids get stuck"
		Related to teachers	 "They can go really deep, and then they come into an area that
			you really don't know the answer any more. That's scary."
	• Strategies-Tips for
	implementing IBL

Table 2 : High-ranking statements of Chinese and Dutch teachers on three aspects

 2

	Main code	CN statements	n	NL statements	n
	General	 Students explore and find a way to the	12	 Students explore and find a way to the problem	15
	Views	problem		by themselves
		 Students discuss and collaborate with peers	12	 Teachers do not provide explanations before	5
				students' exploration
		 Teachers guide the IBL process	8	 The problem can be solved in different ways	4
		 Students think during the process	8	 Students think during the process	4
		 Students come to the conclusion	8	 Students do activities to solve the problem	4
	Reasons For	 IBL develops mathematical thinking	17	 IBL leads to a better understanding, mastery	8
				and appliance of knowledge
		 IBL leads to a better understanding, mastery	14	 IBL gives rise to more interests and motivates	7
		and appliance of knowledge		students
		 IBL gives rise to more interests and	11	 IBL is
		motivates students

a way to develop general skills also necessary outside school and in future academic life and professional life 6  IBL is a way to develop general skills also necessary outside school and in future academic life and professional life

			4	 IBL develops mathematical thinking	4
				 To prepare for examinations	4
	Difficulties	 Some students do not think or participate	11	 Lack
		actively

of time to prepare and do IBL sufficiently 12  Lack of time to prepare and do IBL sufficiently

		10	 IBL asks for a lot from teachers to design and	10
			implement it well
	 Students may not perform well in IBL tasks	9	 Lack

of suitable IBL tasks at hand 8  Lack of suitable IBL tasks at hand

  CN" is the abbreviation of China, and "NL" of the Netherlands. "n" means the number of teachers who expressed this view during interviews. Content in bold were the shared statements for Chinese and Dutch teachers.

	7	 Teachers don't want the unpredictable results and	6
		insecurity that IBL brings
	Note: "

In the Netherlands, students choose after primary school (grade 6) for one of three types of secondary education: pre-vocational secondary education (VMBO), senior general secondary education (HAVO) or pre university education (VWO) (source: https://www.government.nl/topics/secondary-education).

The Primas project: Promoting inquiry-based learning (IBL) in mathematics and science education across Europe.

For Chinese data, we analyzed and coded the original transcripts, then we translated statements and important quotations from Chinese into English.

Acknowledgment

This work was supported by scholarship from the China Scholarship Council (CSC) [grant number 201606040155]. The authors thank Prof. Dongjiao Zhang, Dr. Xianglan Zhang as well as colleagues and friends that helped to contact teachers, and participants involved in this study for their contribution.


Lvqiao Liu
email: lvqiaoliu@whu.edu.cn

Jin Tan
email: jin.tan@u-pec.fr

GLOBAL WELL-POSEDNESS FOR THE HALL-MAGNETOHYDRODYNAMICS SYSTEM IN LARGER CRITICAL BESOV SPACES

Keywords: 2010 Mathematics Subject Classification. 35Q35, 76D03, 86A10 Hall-MHD, Well-posedness, Critical space, Decay estimates

. Meanwhile, we analyze the long-time behavior of the solutions and get some decay estimates. Finally, a stability theorem for global solutions is established.

Introduction

This paper focuses on the following three dimensional incompressible resistive and viscous Hall-magnetohydrodynamics system (Hall-MHD) in R 3 :

∂ t u + u • ∇u + ∇π = b • ∇b + µ∆u, (1.1)
div u = div b = 0, (1.2)
∂ t b -∇ × ((u -ε∇ × b) × b) = ν∆b, (1.3)
with the initial data:

(u(0, x), b(0, x)) = (u 0 (x), b 0 (x)), x ∈ R 3 .

(1.4)

Where π = (P + |b| 2 2 ), u, b and P stand for the velocity field, the magnetic field and the scalar pressure, respectively. The parameters µ and ν denote the fluid viscosity and the magnetic resistivity respectively, while the dimensionless number ε measures the magnitude of the Hall effect compared to the typical length scale of the fluid.

Hall-MHD is much different from the classical MHD equation, due to the appearance of the so-called Hall-term : ε∇ × ((∇ × b) × b). It does play an important role in magnetic reconnection, as observed in e.g. plasmas, star formation, solar flares, neutron stars or geo-dynamo. For more explanation on the physical background of Hall-MHD system, one can refer to [START_REF] Balbus | Linear analysis of the hall effect in protostellar disks[END_REF][START_REF] Braiding | Star formation and the hall effect[END_REF][START_REF] Huba | Hall magnetohydrodynamics of neutral layers[END_REF][START_REF] Somov | Magnetic reconnection in solar flares[END_REF]. Meanwhile, it looks that the mathematical analysis of the Hall-MHD system is more complicated than that for the MHD system, since Hall-term makes Hall-MHD a quasi-linear PDEs.

Considering its physical significance and mathematical applications, Hall-MHD system has been considered by many researchers. The authors in [START_REF] Acheritogaray | Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system[END_REF] had derived the Hall-MHD equations from a two-fluid Euler-Maxwell system for electrons and ions by some scaling limit arguments, which also provided a kinetic formulation for the Hall-MHD. Then, in [START_REF] Chae | Well-posedness for Hall-magnetohydrodynamics[END_REF], Chae, Degond and Liu showed the global existence of weak solutions as well as the local well-posedness of classical solutions with initial data in sobolev spaces H s with s > 5/2. Weak solutions have been further investigated by Dumas and Sueur in [START_REF] Dumas | On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magneto-hydrodynamic equations[END_REF]. Moreover, Blow-up criteria for smooth solutions and the small data global existence of smooth solutions are obtained in [START_REF] Chae | On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics[END_REF][START_REF] Wan | On global existence, energy decay and blow-up criteria for the Hall-MHD system[END_REF]. Later, in [START_REF] Weng | On analyticity and temporal decay rates of solutions to the viscous resistive Hall-MHD system[END_REF][START_REF]Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations[END_REF], Weng studied the long-time behaviour and obtained optimal space-time decay rates of strong solutions. More recently, [START_REF] Benvenutti | Existence and stability of global large strong solutions for the Hall-MHD system[END_REF][START_REF] Wan | Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita-Kato type initial data[END_REF] established the well-posedness of strong solutions with improved regularity conditions for initial data in sobolev or Besov spaces, and smooth data with arbitrarily large L ∞ norms giving rise to global unique solutions have been exhibited in [START_REF] Li | A class large solution of the 3D Hall-magnetohydrodynamic equations[END_REF]. Very recently, Danchin and the second author in [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF][START_REF]The Global Solvability Of The Hall-magnetohydrodynamics System In Critical Sobolev Spaces[END_REF] establish well-posedness in critical spaces based on a new observation of the Hall-MHD system.

Our first goal here is to prove the global well-posedness of Hall-MHD system with initial data in larger critical spaces compared with [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF]. Let us first recall the classical MHD system (corresponding to ε = 0):

(M HD)      ∂ t u + u • ∇u + ∇π = b • ∇b + µ∆u, div u = div b = 0, ∂ t b -∇ × (u × b) = ν∆b,
it is invariant for all λ > 0 by the rescaling (u(t, x), b(t, x)) λ(u(λ 2 t, λx), b(λ 2 t, λx)) and π(t, x) λ 2 π(λ 2 t, λx), (1.5) provided the initial data (u 0 , b 0 ) is rescaled according to (u 0 (x), b 0 (x)) (λu 0 (λx), λb 0 (λx)).

(

One can refer [START_REF] Miao | On the well-posedness of the Cauchy problem for an MHD system in Besov spaces[END_REF] for the well-posedness of (M HD) in critical Besov spaces. But we see the Hall-term in (1.3) breaks the above scaling and system (1.1)-(1.3) does not have any scaling invariance. It is pointed out in [START_REF] Chae | On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics[END_REF] that if we set the fluid velocity u to 0 in (1.3) In other words, ∇b has the same scaling invariance as the fluid velocity u in (M HD).

On the another hand, Danchin and the second author [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF] have transformed the Hall-MHD system into a system having some scaling invariance, they consider the current function J := ∇ × b as an additional unknown. Since b is divergence free, then thanks to the vector identity

∇ × (∇ × v) + ∆v = ∇div v,
we have b = curl -1 J :=(-∆) -1 ∇ × J, where the -1-th order homogeneous Fourier multiplier curl -1 is defined on the Fourier side by

F(curl -1 J)(ξ) := iξ × J |ξ| 2 • (1.9)
With that notation, the system (1.1)-(1.3) can be extended to the following extended Hall-MHD system:

         ∂ t u + u • ∇u -µ∆u + ∇π = b • ∇b, ∂ t b -b • ∇(u -εJ) + (u -εJ) • ∇b -ν∆b = 0, ∂ t J -∇ × ∇ × ((u -εJ) × curl -1 J) -ν∆J = 0, div u = div b = div J = 0. (1.10) with initial data (u(0, x), b(0, x), J(0, x)) = (u 0 (x), b 0 (x), ∇ × b 0 (x)), x ∈ R 3 .
(1.11)

The reason of considering the extended system (1.10) rather than the initial system (1.1)-(1.

3) is that it has a scaling invariance, which is actually the same as that of the incompressible MHD equations. Even better, the quadratic terms in the first two lines of (1.10) are keep the same type with the incompressible MHD equations.

It is thus natural to study whether the above system goes beyond the theory of the generalized Navier-stokes equations as presented in e.g. [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF].

Next, we focus on the large time behavior of the solution in critical Besov spaces by using time-weighted estimates.

Our third purpose is to prove the stability of an a priori global solution to Hall-MHD system. Compared with the classical incompressible Navier-Stokes equations which is semi-linear, the Hall-MHD system is quasi-linear, it may forces us to go beyond the theory of the generalized Navier-Stokes equations, since the differentiation is outside instead of being inside on the curl -1 J in the last line of (1.10). However, in the case µ = ν, it is possible to take advantage of the cancellation property found in [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF] combined with standard energy method to recover the stability results as Navier-Stokes equation (see [START_REF] Gallagher | Asymptotics and stability for global solutions to the Navier-Stokes equations[END_REF]). To this, we have to take p = q = 2.

Throughout this paper, we use C to denote a general positive constant which may different from line to line. And we sometimes write A B instead of

A ≤ CB. Likewise, A ∼ B means that C 1 B ≤ A ≤ C 2 B with absolute constants C 1 , C 2 .
For X a Banach space, p ∈ [1, ∞] and T > 0, the notation L p (0, T ; X) or L p T (X) designates the set of measurable functions f : [0, T ] → X with t → f (t) X in L p (0, T ), endowed with the norm • L p T (X) := • X L p (0,T ) , and agree that C([0, T ], X) denotes the set of continuous functions from [0, T ] to X. Slightly abusively, we will keep the same notations for multi-component functions.

Main results

After the work of [START_REF] Chemin | Remarques sur l'existence globale pour le système de Navier-Stokes incompressible[END_REF], we know that the incompressible Navier-Stokes equations is locally well-posed in all homogeneous Besov spaces Ḃ 3 p -1 p,r with 1 ≤ p < ∞ and 1 ≤ r ≤ ∞, for any initial data, and globally well-posed for small initial data (see [START_REF] Miao | On the well-posedness of the Cauchy problem for an MHD system in Besov spaces[END_REF] for similar results of MHD equations). Once the Hall-MHD system (1.1)-(1.3) has been recast into its extended version as in (1.10), compared to a recent work obtained in [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF], they prove global well-posdedness for initial data

(u 0 , b 0 , J 0 ) ∈ Ḃ 3 p -1 p,1 × Ḃ 3 p -1 p,1 × Ḃ 3 p -1 p,1 with 1 ≤ p < ∞.
It is interesting to consider well-posedness for initial data (u 0 , b 0 , J 0 ) in the general critical homogeneous Besov spaces

Ḃ 3 p -1 p,1 × Ḃ 3 q -1 q,1 × Ḃ 3 q -1 q,1 .
The solution (u, b) thus lies in the space E p × E q , which defines as follow:

E p (T ):={v ∈ C([0, T ], Ḃ 3 p -1 p,1 ), ∇ 2 x v ∈ L 1 (0, T ; Ḃ 3 p -1 p,1 ) and div v = 0} with v Ep(T ) := v L ∞ T ( Ḃ 3 p -1 p,1 ) + v L 1 T ( Ḃ 3 p +1 p,1 )
.

or in its global version, denoted by E p , for solutions defined on R + × R 3 .

Our first result states the global well-posedness of the Hall-MHD system (1.1)-(1.4) for all positive coefficients µ, ν, ε.

Theorem 2.1. Let 1 ≤ p ≤ q < ∞ be such that -min{ 1 3 , 1 2p } ≤ 1 q - 1 p .
(2.1)

Assume that u 0 ∈ Ḃ 3 p -1 p,1 and b 0 , ∇ × b 0 ∈ Ḃ 3 q -1 q,1 .
There exists a positive constant ε 0 depends on µ, ν, ε, p, q such that if

u 0 Ḃ 3 p -1 p,1 + b 0 Ḃ 3 q -1 q,1 + ∇ × b 0 Ḃ 3 q -1 q,1 ≤ ε 0 , (2.2)
then the Cauchy problem (1.1)-(1.4) admits a unique global-in-time solution

(u, b) ∈ E p × E q , and ∇ × b ∈ E q , (2.3)
with u

L ∞ (R+; Ḃ 3 p -1 p,1 ) + µ u L 1 (R+; Ḃ 3 p +1 p,1 )
+ (b, J)

L ∞ (R+; Ḃ 3 q -1 q,1 )
+ ν (b, J)

L 1 (R+; Ḃ 3 q +1 q,1 )
≤ 2ε 0 .

(2.4)

If only ∇ × b 0 fulfills (2.
2) and in addition

- 1 3 < 1 q - 1 p , (2.5)
there exists a time T > 0 such that Hall-MHD system admits a unique local-in-time solution

(u, b) ∈ E p (T ) × E q (T ) with ∇ × b ∈ E q (T ).
Next, we prove that the solution has the following decay estimates.

Theorem 2.2. Let 1 ≤ p ≤ q < ∞ satisfy (2.
D m u Ḃ 3 p -1 p,1 + D m b Ḃ 3 q -1 q,1 ≤ C 0 ε 0 t -m 2 ,
for all t > 0, where

D m u Ḃ 3 p -1 p,1 := sup |α|=m D α u Ḃ 3 p -1 p,1
, and the positive constant C 0 depends on µ, ν, ε, p, q, m.

The following Theorem states the global stability for possible large solutions of Hall-MHD system in critical spaces when µ = ν.

Theorem 2.3. Assume that (u 0,i , b 0,i ) ∈ Ḃ 1 2 2,1 (R 3 ) with div u 0,i = div b 0,i = 0 such that v 0,i ∈ Ḃ 1 2 2,1 (R 3 ), where v 0,i := u 0,i -ε∇ × b 0,i , i = 1, 2.
Suppose in addition that for µ = ν the Cauchy problem (1.1)-(1.4) supplemented with initial data (u 0,1 , b 0,1 ) admits a global solution (u 1 , b 1 ) such that

(u 1 , b 1 , ∇ × b 1 ) ∈ L 1 (R + ; Ḃ 5 2
2,1 (R 3 )). There exist two positive constants η, C depend on µ, ε such that if

(u 0,1 -u 0,2 , b 0,1 -b 0,2 , v 0,1 -v 0,2 ) Ḃ 1 2 2,1 ≤ η, (2.6
)

then (u 0,2 , b 0,2 ) generate a global solution (u 2 , b 2 ) ∈ E 2 ,
and

(u 1 -u 2 , b 1 -b 2 , ∇ × b 1 -∇ × b 2 ) E2 ≤ η exp C (u 1 , b 1 , ∇ × b 1 ) E2 . (2.7)
Remark 1. In this Theorem, we prove that the flow associated to the Hall-MHD system is Lipschitz in critical regularity setting: perturbing a global solution gives again a global solution, which moreover stays close to the given one. It improve the result in [START_REF] Benvenutti | Existence and stability of global large strong solutions for the Hall-MHD system[END_REF].

Remark 2. As proposed by Chae and Lee in [START_REF] Chae | On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics[END_REF], considering the 2 1 2 D flows for the Hall-MHD system, which reads:

∂ t u + u • ∇u + ∇π = b • ∇b + µ ∆u in R + × R 2 , div u = div b = 0 in R + × R 2 , ∂ t b -∇ × ((u -εj) × b) = ν ∆b in R + × R 2 , (u, B)| t=0 = (u 0 , B 0 ) in R 2 ,
where the unknowns u and b are functions from

R + × R 2 to R 3 , u := (u 1 , u 2 ), b := (b 1 , b 2 ), ∇ := (∂ 1 , ∂ 2 ), div := ∇•, ∆ := ∂ 2 1 + ∂ 2 2 and j := ∇ × b =   ∂ 2 b 3 -∂ 1 b 3 ∂ 1 b 2 -∂ 2 b 1   •
After a small diversification, our method may still works for this case. And we shall see that for any initial data (u 0 , 0) ∈ Ḃ0

2,1 (R 2 ), it will generate a global solution (u, 0) for the above system due to the theory of 2 1 2 D Navier-Stokes equation in [START_REF] Majda | Vorticity and Incompressible Flow[END_REF], thus one can conclude that if

(b 0 , ∇ × b 0 ) Ḃ0 2,1
< η then supplemented with data (u 0 , b 0 ) will also generate a global solution.

Preliminaries

In this section, we first recall the Littlewood-Paley decomposition theory, the definition of homogeneous Besov space and some useful properties. More details and proofs may be found in e.g. [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF].

Let ϕ ∈ D(C ) be a smooth function supported in the annulus C = {k ∈ R 3 :

3 4 ≤ |ξ| ≤ 8 3 } and such that j∈Z ϕ(2 -j k) = 1, ∀ξ ∈ R 3 \{0}.
For u ∈ S (R 3 ), the frequency localization operator ∆j and Ṡj are defined by

∀j ∈ Z, ∆j u := ϕ(2 -j D)u and Ṡj u := ≤j-1 ∆ u.
Then we have the formal decomposition

u = j∈Z ˙ j u, ∀ u ∈ S h (R 3 ) := S (R 3 )/P[R 3 ].
where P[R 3 ] is the set of polynomials (see [START_REF] Peetre | New thoughts on Besov spaces[END_REF]). Moreover, the Littlewood-Paley decomposition satisfies the property of almost orthogonality:

∆j ∆k u = 0, if |j -k| ≥ 2, ∆j (S k-1 u ∆k u) = 0, if |j -k| ≥ 5.
We now recall the definition of homogeneous Besov spaces from [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF].

Definition 3.1. Let s be a real number and (p, r) be in [1, ∞] 2 , we set u Ḃs p,r :=    2 js ∆j u L p (R d ) r (Z) for 1 ≤ r < ∞, sup j∈Z 2 js ∆j u L p for r = ∞.
The homogeneous Besov space Ḃs p,r := {u ∈ S h (R 3 ), u Ḃs p,r < ∞}.

Next, we recall some basic facts on Littlewood-Paley theory and Besov spaces, one may check [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF][START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] for more details. Proposition 3.2. Fix some 0 < r < R. A constant C exists such that for any nonnegative integer k, any couple (p, q) in [1, ∞] 2 with q ≥ p ≥ 1 and any function u of L p with Supp u ⊂ {ξ ∈ R d , |ξ| ≤ λR}, we have

D k u L q ≤ C k+1 λ k+d( 1 p -1 q ) u L p . If u satisfies Supp u ⊂ {ξ ∈ R d , rλ ≤ |ξ| ≤ Rλ}, then we have C -k-1 λ k u L p ≤ D k u L p ≤ C k+1 λ k u L p . Lemma 3.3. Let C be a ring of R 3 , if the support of û is included in λC.
Then, there exist two positive constants c and C such that for all 1 ≤ p ≤ ∞,

e t∆ u L p ≤ Ce cλ 2 t u L p ,
where e t∆ denotes the heat semi-group operator. Proposition 3.4. Let 1 ≤ p ≤ ∞. Then there hold:

• for all s ∈ R and 1 ≤ p, r ≤ ∞, we have

D k u Ḃs p,r ∼ u Ḃs+k p,r .
• for any θ ∈ (0, 1) and s < s, we have

u Ḃθs+(1-θ)s p,1 u θ Ḃs p,1 u 1-θ Ḃs p,1 . Lemma 3.5. Let f be a smooth function on R 3 \ {0} which is homogeneous of degree m. Let 1 ≤ p, r ≤ ∞. Assume that s -m < 3 p , or s -m = 3 p and r = 1. Define f (D) on S h (R 3 ) by F(f (D)u)(ξ):=f (ξ)Fu(ξ),
and assume that f

(D) maps S h (R 3 ) to itself. Then f (D) is continuous from Ḃs p,r (R 3 ) to Ḃs-m p,r (R 3
). In the next, we shall need to use Bony's decomposition from [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF] in the homogeneous context:

u v = T u v + T v u + R(u, v) with T u v := q Ṡq-1 u ∆q v, and R(u, v) := q |q -q|≤1 ∆q u ∆q v.
The above operator T is called the "paraproduct" whereas R is called the "remainder".

Lemma 3.6. Let (s, p, r) ∈ R × [1, ∞] 2 and t < 0, there exists a constant C such that T u v Ḃs p,r ≤ C u L ∞ v Ḃs p,r and
T u v Ḃs+t p,r ≤ C u Ḃt ∞,∞ v Ḃs p,r • For any (s 1 , p 1 , r 1 ) and (s 2 , p 2 , r 2 ) in R × [1, ∞] 2 there exists a constant C such that if s 1 + s 2 > 0, 1 p ≤ 1 p1 + 1 p2 ≤ 1 and 1 r ≤ 1 r1 + 1 r2 ≤ 1 then R(u, v) Ḃs p,r ≤ C u Ḃs 1 p 1 ,r 1 v Ḃs 2 p 2 ,r 2 , with s := s 1 + s 2 -3( 1 p1 + 1 p2 -1 p
), provided that s < 3 p or s = 3 p and r = 1. As an application of the above basic facts on Littlewood-Paley theory, the following product laws in Besov spaces will play a crucial role in the sequel . Lemma 3.7. (see [START_REF] Paicu | Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system[END_REF]) Let q ≥ p ≥ 1, and

s 1 ≤ 3 p , s 2 ≤ 3 q with s 1 + s 2 ≥ 3 min{0, 1 p + 1 q -1}. Let a ∈ Ḃs1 p,1 (R 3 ), b ∈ Ḃs2 q,1 (R 3 ). Then a b ∈ Ḃs1+s2-3 p q,1 (R 3 ), and
a b Ḃs 1 +s 2 -3 p q,1 a Ḃs 1 p,1 b Ḃs 2 q,1 • (3.1) Lemma 3.8. Let q ≥ p ≥ 1 and 1 q - 1 p ≥ -min{ 1 3 , 1 2p }. Assume θ satisfies 3 p - 3 q ≤ θ ≤ 1. Let a, b ∈ Ḃ 3 q -θ q,1 (R 3 ) ∩ Ḃ 3 q +θ q,1 (R 3 ). Then a b ∈ Ḃ 3 p p,1 (R 3 ), and a b Ḃ 3 p p,1 a Ḃ 3 q -θ q,1 b Ḃ 3 q +θ q,1 + a Ḃ 3 q +θ q,1 b Ḃ 3 q -θ q,1 • (3.2) Let a, b ∈ Ḃ 6 q -3 p q,1 (R 3 ) ∩ Ḃ 3 q q,1 (R 3 ). Then a b ∈ Ḃ 3 q p,1 (R 3 )
, and a b

Ḃ 3 q p,1 a Ḃ 3 q q,1 b Ḃ 6 q -3 p q,1 + a Ḃ 6 q -3 p q,1 b Ḃ 3 q q,1 • (3.3)
Proof. The proof is standard, we follow the method of [START_REF] Paicu | Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system[END_REF] and only give a proof of (3.2). In fact, we shall focus on the case q > p, since when q = p, it is obvious. By Bony's decomposition, we can write

ab = T a b + T b a + R(a, b).
By Hölder's inequality and Proposition 3.2, under p < q ≤ 2p we have

2 3 p j S j-1 a ∆j b L p (2 ( 3 q +θ)j ∆j b L q ) k≤j-2 2 ( 3 p -3 q -θ)k ∆k a L pq q-p 2 ( 3 p -3 q -θ)(j-k) (2 ( 3 q +θ)j ∆j b L q ) k≤j-2 2 ( 3 q -θ)k ∆k a L q 2 ( 3 p -3 q -θ)(j-k) .
Thus by Young's inequality with

3 p -3 q ≤ θ T a b Ḃ 3 p p,1 a Ḃ 3 q -θ q,1 b Ḃ 3 q +θ q,1 • Similarly, T b a Ḃ 3 p p,1 a Ḃ 3 q +θ q,1 b Ḃ 3 q -θ q,1
• Thanks to Lemma 3.6, we have R(a, b)

Ḃ 3 p p,1 a Ḃ 3 p -3 q -θ pq q-p ,∞ b Ḃ 3 q +θ q,1 a Ḃ 3 q -θ q,1 b Ḃ 3 q +θ q,1
.

Lemma 3.9. Let 1 ≤ q < ∞. For any homogeneous function σ of degree -1 smooth outside of 0, there hold:

-let a ∈ Ḃ 3 q -1 q,1 and b ∈ Ḃ 3 q +1 q,1 , then (σ(D)a) • ∇b Ḃ 3 q q,1 a Ḃ 3 q -1 q,1 b Ḃ 3 q +1 q,1 . (3.4) -let a ∈ Ḃ 3 q q,1 and b ∈ Ḃ 3 q q,1 , then a • ∇(σ(D)b) Ḃ 3 q q,1 a Ḃ 3 q q,1 b Ḃ 3 q q,1 . (3.5)
Proof. Thanks to Lemma 3.7, we know that when 1 ≤ q < ∞, Ḃ 3 q q,1 is an algebra, thus by Lemma 3.5, we have

(σ(D)a) • ∇b Ḃ 3 q q,1 σ(D)a Ḃ 3 q q,1 ∇b Ḃ 3 q q,1 a Ḃ 3 q -1 q,1 b Ḃ 3 q +1 q,1
.

Similarly,

a • ∇(σ(D)b) Ḃ 3 q q,1 a Ḃ 3 q q,1 ∇(σ(D)b) Ḃ 3 q q,1 a Ḃ 3 q q,1 σ(D)∇b Ḃ 3 q q,1 a Ḃ 3 q q,1 b Ḃ 3 q q,1
.

The basic heat equation reads:

∂ t u -µ∆u = f in R + × R 3 , u| t=0 = u 0 in R 3 . (3.6)
Then, it is classical that for all u 0 ∈ S (R d ) and f ∈ L 1 loc (R + ; S (R d )), the heat equation (3.6) has a unique tempered distribution solution, which is given by the following Duhamel's formula:

u(t) = e µt∆ u 0 + t 0 e (t-s)µ∆ f (τ ) ds, t ≥ 0. (3.7)
The following fundamental results to heat semi-group has been first proved in [START_REF] Chemin | Remarques sur l'existence globale pour le système de Navier-Stokes incompressible[END_REF].

Lemma 3.10. Let s > 0, 1 ≤ p < ∞. Assume that u 0 ∈ Ḃs p,1 , then for any ρ ∈ [1, ∞), lim T →0 e tµ∆ u 0 L ρ T ( Ḃs+ 2 ρ p,1 ) = 0. Lemma 3.11. Let T > 0, s ∈ R and 1 ≤ p ≤ ∞. Assume that u 0 ∈ Ḃs p,1 and f ∈ L 1 T ( Ḃs p,1 ). Then (3.6) has a unique solution u in C([0, T ]; Ḃs p,1 ) ∩ L 1 (0, T ; Ḃs+2 p,1
) and there exists a constant C such that

u L ∞ T L Ḃs p,1 + µ u L 1 T ( Ḃs+2 p,1 ) ≤ C u 0 Ḃs p,1 + f L 1 T ( Ḃs p,1 ) . (3.8)
4. The proof of Theorem 2.1

In this section, we shall give the proof of Theorem 2.1. We only look at the case p < q, since the case p = q is been shown in [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF].

By means of the Leray projector P := Id -∇(-∆) -1 div and the fact that u, b are divergence free vector fields, we can rewrite the system (1.10) as (see [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF]):

     ∂ t u -µ∆u = Q a (b, b) -Q a (u, u), ∂ t b -ν∆b = Q b (u -εJ, b), ∂ t J -ν∆J = ∇ × Q b (u -εJ, curl -1 J), (4.1)
and supplemented with divergence free initial data

(u(0, x), b(0, x), J(0, x)) = (u 0 , b 0 , J 0 ). (4.2)
Where bi-linear forms

Q a (v, w):= 1 2 P(div(v ⊗ w) + div(w ⊗ v)), Q b (v, w):= div(v ⊗ w) -div(w ⊗ v),
and div (v ⊗ w)

j := 3 k=1 ∂ k (v j w k ).
Define free solution u L := e µt∆ u 0 , b L := e νt∆ b 0 , J L := e νt∆ J 0 .

Let u 0 ∈ Ḃ 3 p -1 p,1 and b 0 , J 0 ∈ Ḃ 3 q -1
q,1 , it is easy to find that by Lemma 3.11 that

(u L , b L , J L ) ∈ E p × E q × E q
and there holds

u L L ∞ ( Ḃ 3 p -1 p,1 ) + µ u L L 1 ( Ḃ 3 p +1 p,1 ) ≤ C u 0 Ḃ 3 p -1 p,1 , (4.3
) b L L ∞ ( Ḃ 3 q -1 q,1 ) + ν b L L 1 ( Ḃ 3 q +1 q,1 ) ≤ C b 0 Ḃ 3 q -1 q,1 , (4.4)
J L L ∞ ( Ḃ 3 q -1 q,1 ) + ν J L L 1 ( Ḃ 3 q +1 q,1 ) ≤ C J 0 Ḃ 3 q -1 q,1 . (4.5) Define (ū, b, J) := (u -u L , b -b L , J -J L ). Then (u, b, J
) is a solution of (4.1) if and only if (ū, b, J) satisfies the following system:

                       ∂ t ū -µ∆ū = Q a ( b, b) + Q a (b L , b) + Q a ( b, b L ) + Q a (b L , b L ) -Q a (ū, ū) -Q a (u L , ū) -Q a (ū, u L ) -Q a (u L , u L ), ∂ t b -ν∆ b = Q b (ū -ε J, b) + Q b (u L -εJ L , b) + Q b (ū -ε J, b L ) + Q b (u L -εJ L , b L ), ∂ t J -ν∆ J = ∇ × Q b (ū -ε J, curl -1 J) + Q b (u L -εJ L , curl -1 J) + Q b (ū -ε J, curl -1 J L ) + Q b (u L -εJ L , curl -1 J L ). (4.6)
In what follows, we will employing the iterative method to prove that there exists an unique solution (ū, b, J) to the system (4.6). More precisely, the iterating approximate system is constructed as follows:

                     ∂ t ūn -µ∆ū n = Q a ( bn-1 , bn-1 ) + Q a (b L , bn-1 ) + Q a ( bn-1 , b L ) + Q a (b L , b L ) -Q a (ū n-1 , ūn-1 ) -Q a (u L , ūn-1 ) -Q a (ū n-1 , u L ) -Q a (u L , u L ), ∂ t bn -µ∆ bn = Q b (ū n-1 -ε Jn-1 , bn-1 ) + Q b (u L -εJ L , bn-1 ) + Q b (ū n-1 -ε Jn-1 , b L ) + Q b (u L -εJ L , b L ), ∂ t Jn -µ∆ Jn = Q b (ū n-1 -ε Jn-1 , curl -1 Jn-1 ) + Q b (u L -εJ L , curl -1 Jn-1 ) + Q b (ū n-1 -ε Jn-1 , curl -1 J L ) + Q b (u L -εJ L , curl -1 J L ).
(4.7) We start the approximate system with (ū 0 , b0 , J0 )(t, x) = (0, 0, 0) for all t ≥ 0, and assume the initial data of the iterative approximate system (4.7) satisfied for all n ∈ N (ū n , bn , Jn )(0, x) = (0, 0, 0).

(4.8)

In the arguments proving the convergence (n → ∞) of the iterative approximate solutions of (4.7)-(4.8), it is essential to obtain uniform estimates for it.

4.1. Uniform boundedness of (ū n , bn , Jn ). We claim that there exists a positive constant M such that for all n ∈ N,

ūn L ∞ ( Ḃ 3 p -1 p,1 ) + µ ūn L 1 ( Ḃ 3 p +1 p,1 )
+ bn

L ∞ ( Ḃ 3 q -1 q,1 ) + ν bn L 1 ( Ḃ 3 q +1 q,1 ) + Jn L ∞ ( Ḃ 3 q -1 q,1 ) + ν Jn L 1 ( Ḃ 3 q +1 q,1 )
≤ M.

(4.9)

Obviously, (4.9) is satisfied when n = 0. Assume the claim (4.9) holds true for n -1, i.e., ūn-1

L ∞ ( Ḃ 3 p -1 p,1 )
+ µ ūn-1

L 1 ( Ḃ 3 p +1 p,1 )
+ bn-1

L ∞ ( Ḃ 3 q -1 q,1 ) + ν bn-1 L 1 ( Ḃ 3 q +1 q,1 ) + Jn-1 L ∞ ( Ḃ 3 q -1 q,1 ) + ν Jn-1 L 1 ( Ḃ 3 q +1 q,1 ) ≤ M.
With smallness condition (2.2), we now devote to the proof of (4.9) through finding some suitable M. Firstly, we need to state the following product laws for quadratic terms Q a , Q b , it will play a significant role in the later parts. If (2.1) is assumed, by using Lemma 3.7, Lemma 3.9, direct calculation tells us that:

Q a (v, w) Ḃ 3 p -1 p,1 v ⊗ w Ḃ 3 p p,1 v Ḃ 3 p p,1 w Ḃ 3 p p,1 , (4.10)
and

Q b (v, w) Ḃ 3 q -1 q,1 v ⊗ w Ḃ 3 q q,1 v Ḃ 3 q q,1 w Ḃ 3 q q,1 , (4.11)
and

∇ × Q b (v, curl -1 w) Ḃ 3 q -1 q,1 Q b (v, curl -1 w) Ḃ 3 q q,1 v • ∇(curl -1 w) Ḃ 3 q q,1 + (curl -1 w) • ∇v Ḃ 3 q q,1 v Ḃ 3 q q,1 w Ḃ 3 q q,1 + v Ḃ 3 q +1 q,1 w Ḃ 3 q -1 q,1 , (4.12)
and

∇ × Q b (v, curl -1 w) Ḃ 3 q -1 q,1 Q b (v, curl -1 w) Ḃ 3 q q,1 v • ∇(curl -1 w) Ḃ 3 q q,1 + (curl -1 w) • ∇v Ḃ 3 q q,1 v Ḃ 3 p p,1 w Ḃ 3 q q,1 + v Ḃ 3 p +1 p,1 w Ḃ 3 q -1 q,1 . (4.13)
Thanks to (3.2) in Lemma 3.8 with θ = 3 p -3 q ≤ 1, we have

Q a (v, w) Ḃ 3 p -1 p,1 v ⊗ w Ḃ 3 p p,1 v Ḃ 6 q -3 p q,1 w Ḃ 3 p q,1 + w Ḃ 6 q -3 p q,1 v Ḃ 3 p q,1 . (4.14)
Notice that from Lemma 3.11, there exists a constant C such that + µ ūn

u L L 1 ( Ḃ 3 p +1 p,1 ) + u L L 2 ( Ḃ 3 p p,1 ) ≤ Cε 0 (1 + 1 µ ), (4.15)
(b L , J L ) L 1 ( Ḃ 3 q +1 q,1 ) + (b L , J L ) L 2 ( Ḃ 3 q q,1 ) + b L L 2pq pq-3q+3p Ḃ 6 q -3 p q,1 ) + b L L 2pq pq+3q-3p ( Ḃ 3 p q,1 ) ≤ Cε 0 (1 + 1 ν ).
L 1 ( Ḃ 3 p +1 p,1 ) ≤C Q a ( bn-1 , bn-1 ) + Q a (b L , bn-1 ) + Q a ( bn-1 , b L ) + Q a (b L , b L ) L 1 ( Ḃ 3 p -1 p,1 ) + C Q a (ū n-1 , ūn-1 ) + Q a (u L , ūn-1 ) + Q a (ū n-1 , u L ) + Q a (u L , u L ) L 1 ( Ḃ 3 p -1 p,1 ) ≤C ∞ 0 bn-1 Ḃ 6 q -3 p q,1 bn-1 Ḃ 3 p q,1 + b L Ḃ 6 q -3 p q,1 bn-1 Ḃ 3 p q,1 + + b L Ḃ 3 p q,1 bn-1 Ḃ 6 q -3 p q,1 + b L Ḃ 6 q -3 p q,1 b L Ḃ 3 p q,1 + ūn-1 Ḃ 3 p p,1 ūn-1 Ḃ 3 p p,1 + 2 u L Ḃ 3 p p,1 ūn-1 Ḃ 3 p p,1 + u L Ḃ 3 p p,1 u L Ḃ 3 p p,1 (t) dt ≤C M 2 + ε 0 M + ε 2 0 1 + 1 ν + 1 µ 2 ,
where we have used the facts

∞ 0 b L Ḃ 6 q -3 p q,1 bn-1 Ḃ 3 p q,1 (t)dt ≤C b L L 2pq pq-3q+3p ( Ḃ 6 q -3 p q,1 ) bn-1 L 2pq pq+3q-3p ( Ḃ 3 p q,1 )
, and

∞ 0 bn-1 Ḃ 6 q -3 p q,1 b L Ḃ 3 p q,1 (t)dt ≤C bn-1 L 2pq pq-3q+3p ( Ḃ 3 q +1 q,1 ) b L L 2pq pq+3q-3p ( Ḃ 3 p q,1 )
.

Again, taking advantage of (4.10), (4.11), interpolation and Hölder inequality, it follows from Lemma 3.11 that bn

L ∞ ( Ḃ 3 q -1 q,1 ) + ν bn L 1 ( Ḃ 3 q +1 q,1 ) ≤C Q b (ū n-1 -ε Jn-1 , bn-1 ) + Q b (u L -εJ L , bn-1 ) L 1 ( Ḃ 3 q -1 q,1 ) + C Q b (ū n-1 -ε Jn-1 , b L ) + Q b (u L -εJ L , b L ) L 1 ( Ḃ 3 q -1 q,1 ) ≤C ∞ 0 bn-1 Ḃ 3 q q,1 ūn-1 Ḃ 3 p p,1 + bn-1 Ḃ 3 q q,1 Jn-1 Ḃ 3 q q,1 + u L Ḃ 3 p p,1 bn-1 Ḃ 3 q q,1 + bn-1 Ḃ 3 q q,1 J L Ḃ 3 q q,1 + ūn-1 Ḃ 3 p p,1 b L Ḃ 3 q q,1 + Jn-1 Ḃ 3 q q,1 b L Ḃ 3 q q,1 + u L Ḃ 3 p p,1 b L Ḃ 3 q q,1 + J L Ḃ 3 q q,1 b L Ḃ 3 q q,1 (t)dt ≤C M 2 + ε 0 M + ε 2 0 1 + 1 ν + 1 µ 2 ,
and Jn

L ∞ ( Ḃ 3 p -1 p,1 ) + ν Jn L 1 ( Ḃ 3 p +1 p,1 ) ≤C Q b (ū n-1 -ε Jn-1 , curl -1 Jn-1 ) + Q b (u L -εJ L , curl -1 Jn-1 ) L 1 ( Ḃ 3 q q,1 ) + C Q b (ū n-1 -ε Jn-1 , curl -1 J L ) + Q b (u L -εJ L , curl -1 J L ) L 1 ( Ḃ 3 q q,1 ) ≤C ∞ 0 Jn-1 Ḃ 3 q q,1 Jn-1 Ḃ 3 q q,1 + ūn-1 Ḃ 3 p p,1
Jn-1

Ḃ 3 q q,1 + Jn-1 Ḃ 3 q +1 q,1 Jn-1 Ḃ 3 q -1 q,1 + ūn-1 Ḃ 3 p +1 p,1 Jn-1 Ḃ 3 q -1 q,1 + u L Ḃ 3 p p,1
Jn-1

Ḃ 3 q q,1 + u L Ḃ 3 p p,1 J L Ḃ 3 q q,1 + Jn-1 Ḃ 3 q -1 q,1 u L Ḃ 3 p +1 p,1 + Jn-1 Ḃ 3 q -1 q,1 J L Ḃ 3 q +1 q,1 + ūn-1 Ḃ 3 p p,1 J L Ḃ 3 q q,1 + Jn-1 Ḃ 3 q q,1 J L Ḃ 3 q q,1 + Jn-1 Ḃ 3 q +1 q,1 J L Ḃ 3 q -1 q,1 + ūn-1 Ḃ 3 p +1 p,1 J L Ḃ 3 q -1 q,1 + J L Ḃ 3 q q,1 J L Ḃ 3 q q,1 + J L Ḃ 3 q q,1 u L Ḃ 3 p p,1 + J L Ḃ 3 q -1 q,1 J L Ḃ 3 q +1 q,1 + u L Ḃ 3 p +1 p,1 J L Ḃ 3 q -1 q,1 (t)dt ≤C M 2 + ε 0 M + ε 2 0 1 + 1 ν + 1 µ 2 .
By choosing ε 0 , M sufficiently small such that

M ≤ 1 9C 1 + 1 µ + 1 ν 2 , ε 0 ≤ 1 9C 1 + 1 µ + 1 ν 2 , (4.17)
then one find that ūn

L ∞ T ( Ḃ 3 p -1 p,1 )
+ µ ūn

L 1 T ( Ḃ 3 p +1 p,1 )
+ bn

L ∞ T ( Ḃ 3 q -1 q,1 )
+ ν bn

L 1 T ( Ḃ 3 q +1 q,1 )
+ Jn

L ∞ T ( Ḃ 3 q -1 q,1 ) + ν Jn L 1 T ( Ḃ 3 q +1 q,1 ) ≤ M.
Arguing by induction, we conclude that (4.9) holds true for all n ∈ N.

Once the uniform bounds is established for (ū n , bn , Jn ), we shall use compactness arguments to prove convergence. 4.2. Convergence of (ū n , bn , Jn ). We claim that (ū n , bn , Jn ) n∈N is a Cauchy sequence in

L ∞ ( Ḃ 3 p -1 p,1 ) ∩ L 1 ( Ḃ 3 p +1 p,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1
q,1 ) .

For all n ∈ N, let us consider the difference (δ ūn , δ bn , δ Jn ) := (ū n+1 , bn+1 , Jn+1 ) -(ū n , bn , Jn ), and define

δ n := δ ūn L ∞ ( Ḃ 3 p -1 p,1 )
+ µ δ ūn

L 1 ( Ḃ 3 p +1 p,1 ) + δ bn L ∞ ( Ḃ 3 q -1 q,1 ) + ν δ ūn L 1 ( Ḃ 3 q +1 q,1 ) + δ Jn L ∞ ( Ḃ 3 q -1 q,1 ) + ν δ Jn L 1 ( Ḃ 3 q +1 q,1 )
.

Then thanks to Lemma 3.11, we only need to estimate the following terms

I n 1 := Q a (δ bn-1 , bn ) L 1 ( Ḃ 3 p -1 p,1 ) + Q a ( bn-1 , δ bn-1 ) L 1 ( Ḃ 3 p -1 p,1 ) + Q a (b L , δ bn-1 ) L 1 ( Ḃ 3 p -1 p,1 ) + Q a (δ bn-1 , b L ) L 1 ( Ḃ 3 p -1 p,1 )
+ Q a (δ ūn-1 , ūn )

L 1 ( Ḃ 3 p -1 p,1 )
+ Q a (ū n-1 , δ ūn-1 )

L 1 ( Ḃ 3 p -1 p,1 )
+ Q a (u L , δ ūn-1 )

L 1 ( Ḃ 3 p -1 p,1 ) + Q a (δ ūn-1 , u L ) L 1 ( Ḃ 3 p -1 p,1 )
,

I n 2 := Q b (δ ūn-1 -εδ Jn-1 , bn ) L 1 ( Ḃ 3 q -1 q,1 ) + Q b (ū n-1 -ε Jn-1 , δ bn-1 ) L 1 ( Ḃ 3 q -1 q,1 ) + Q a (u L -εJ L , δ bn-1 ) L 1 ( Ḃ 3 q -1 q,1 ) + Q a (δ ūn-1 -εδ Jn-1 , b L ) L 1 ( Ḃ 3 q -1 q,1 )
,

I n 3 := Q b (δ ūn-1 -εδ Jn-1 , curl -1 Jn ) + Q b (ū n-1 -ε Jn-1 , curl -1 δ Jn-1 ) L 1 ( Ḃ 3 q q,1 ) + Q a (u L -εJ L , curl -1 δ Jn-1 ) + Q a (δ ūn-1 -εδ Jn-1 , curl -1 J L ) L 1 ( Ḃ 3 q q,1 )
.

Along extremely similar calculations as previous subsections, thanks to (4.17) and uniform bounds (4.9), one can get

δ n ≤ C(I n 1 + I n 2 + I n 3 ) ≤ Cδ n-1 1 + 1 µ + 1 ν 2 M + ε 0 ≤ 1 2 δ n-1 .
Thus, we know that (ū n , bn , Jn ) n∈N is a Cauchy sequence in the space

L ∞ ( Ḃ 3 p -1 p,1 ) ∩ L 1 ( Ḃ 3 p +1 p,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 )
and there exists a triplet (ū, b, J) such that

ūn → ū in L ∞ ( Ḃ 3 p -1 p,1 ) ∩ L 1 ( Ḃ 3 p +1 p,1 ), bn → b in L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 ), Jn → J in L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1
q,1 ).

By the product laws that we used frequently before, it is not difficult to prove the convergence of non-linear terms in (4.7), by an example, we show that

Q b (ū n -ε Jn , curl -1 Jn ) -Q b (ū -ε J, curl -1 J) L 1 T ( Ḃ 3 q q,1 ) ≤ Q b (ū n -ū + ε Jn -ε J, curl -1 Jn ) L 1 ( Ḃ 3 q q,1 ) + Q b (ū -ε J, curl -1 J -curl -1 Jn ) L 1 ( Ḃ 3 q q,1 ) M ūn -ū L 2 T ( Ḃ 3 p p,1 )∩L 1 ( Ḃ 3 p +1 p,1 ) + ε Jn -ε J L ∞ ( Ḃ 3 q -1 q,1 )∩L 1 ( Ḃ 3 q +1 q,1 )
.

Hence, we conclude that (ū, b, J) is indeed a solution of (4.1). This implies that (u, b, J)

= (u L + ū, b L + b, J L + J) is a solution of (1.10) in L ∞ ( Ḃ 3 p -1 p,1 ) ∩ L 1 ( Ḃ 3 p +1 p,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 ) × L ∞ ( Ḃ 3 q -1 q,1 ) ∩ L 1 ( Ḃ 3 q +1 q,1 )
and satisfies (2.4). The continuity of (u, B, J) is straightforward. Indeed, the right-hand sides of (4.1) belong to

L 1 T ( Ḃ 3 p -1 p,1 ), L 1 T ( Ḃ 3 q -1 q,1 ), L 1 T ( Ḃ 3
q -1 q,1 ) respectively.

4.3. Uniqueness. Let (u 1 , b 1 , J 1 ) and (u 2 , b 2 , J 2 ) be two solutions of (4.1) in

E p × E q × E q with same initial data (u 0 , b 0 , J 0 ) ∈ Ḃ 3 p -1 p,1 × Ḃ 3 q -1 q,1 × Ḃ 3 q -1 q,1 .
Without loss of generality, we assume (u 2 , b 2 , J 2 ) is the solution that constructed in the previous steps (in fact, one only needs J 2 is small in E q ).

Set

δu := u 2 -u 1 , δb := b 2 -b 1 , and δJ := J 2 -J 1 , we see that (δu, δb, δJ) satisfies            ∂ t δu -µ∆δu = Q a (δb, b 1 ) + Q a (b 2 , δb) -Q a (δu, u 1 ) -Q a (u 2 , δu), ∂ t δb -ν∆δb = Q b (u 1 -εJ 1 , δb) + Q b (δu -εδJ, b 2 ), ∂ t δJ -ν∆δJ = ∇ × Q b (u 1 -εJ 1 , curl -1 δJ) + ∇ × Q b (δu -εδJ, curl -1 J 2 ),
(δu(0, x), δb(0, x), δJ(0, x)) = (0, 0, 0).

With our assumptions on two solutions, one can verify that the right-hand sides of above system belong to L 1 ( Ḃ 3

p -1 p,1 ), L 1 ( Ḃ 3 q -1 q,1 ), L 1 ( Ḃ 3 q -1
q,1 ) respectively, thus by Lemma 3.11, Lemma 3.8 (take θ = 3 p -3 q ) and product laws (4.10)-(4.13), one has δu(t)

Ḃ 3 p -1 p,1 + (δb, δJ)(t) Ḃ 3 q -1 q,1 + t 0 δu(t) Ḃ 3 q +1 p,1 + (δb, δJ)(t) Ḃ 3 q +1 q,1 dτ t 0 (b 1 , b 2 ) Ḃ 3 p q,1 δb Ḃ 6 q -3 p q,1 + (b 1 , b 2 ) Ḃ 6 q -3 p q,1 δb Ḃ 3 p q,1 + (u 1 , u 2 ) Ḃ 3 p p,1 δu Ḃ 3 p p,1 + ( u 1 Ḃ 3 p p,1 + J 1 Ḃ 3 q q,1
) δb

Ḃ 3 q q,1 + b 2 Ḃ 3 q q,1 ( δu Ḃ 3 p p,1 + δJ Ḃ 3 q -1 q,1
)

+ ( u 1 Ḃ 3 p p,1 + (J 1 , J 2 ) Ḃ 3 q q,1
) δJ

Ḃ 3 q q,1 + ( u 1 Ḃ 3 p +1 p,1 + (J 1 , J 2 ) Ḃ 3 q +1 q,1
) δJ

Ḃ 3 q -1 q,1 + δu Ḃ 3 p p,1 J 2 Ḃ 3 q q,1 + δu Ḃ 3 p +1 p,1 J 2 Ḃ 3 q -1 q,1 dτ, t 0 Ω(τ ) δu(τ ) Ḃ 3 p -1 p,1 + (δb, δJ)(τ ) Ḃ 3 q -1 q,1 dτ,
where

Ω(τ ) := (u 1 , u 2 )(τ ) Ḃ 3 p -1 p,1 + 1 (u 1 , u 2 )(τ ) Ḃ 3 p +1 p,1 + (b 1 , b 2 , J 1 , J 2 )(τ ) Ḃ 3 q -1 q,1 + 1 (b 1 , b 2 , J 1 , J 2 )(τ ) Ḃ 3 q +1 q,1
.

It is clear that our assumptions ensure Ω ∈ L 1 (R + ), Gronwall lemma then enables us to conclude that (δu, δb, δJ

) ≡ 0 on R + × R 3 .
For completing the proof of the existence for the original Hall-MHD system, we have to check that J 0 = ∇ × b 0 implies J = ∇ × b, so that (u, b) is indeed a distributional solution of (1.1)-(1.3). Actually, it is easy to see that ∇ × b L = J L , thus we only need to check J = ∇ × b. Noticing that

(∂ t -∆)(∇ × b -J) = ∇ × Q b (u -εJ, curl -1 (∇ × b -J)).
Hence, using Lemma 3.11 and product law (4.11), one gets for all t ≥ 0,

(∇ × b -J)(t) Ḃ 3 q -2 q,1 + t 0 ∇ × b -J Ḃ 3 q q,1 dτ ≤C t 0 u -εJ Ḃ 3 q q,1 ∇ × b -J Ḃ 3 q -1 q,1
dτ.

Then (2.3) combined with interpolation inequality and Gronwall lemma enables that ∇ × b -J ≡ 0 on R + × R 3 .

A slight modification on the proof could yields local well-posedness by assuming only

∇ × b 0 Ḃ 3 q -1 q,1
is small enough and in addition

- 1 3 < 1 q - 1 q .
In fact, by Lemma 3.10 one can guarantee (4.15) and (4.16) are satisfied with sufficient small time T > 0. And Lemma 3.11 implies that

J L L ∞ T ( Ḃ 3 q -1 q,1 ) ∇ × b 0 Ḃ 3 q -1 q,1 ε 0 .
We omit another details here and thus the proof of Theorem 2.1 is completed.

The proof of Theorem 2.2

In this section, we devote to proving the decay estimates of the solution provided by Theorem 2.1. Started with the data (u 0 , b 0 ) satisfies (2.2), we know that the solution (u, b) ∈ E p × E q such that ∇ × b ∈ E q and satisfies (2.4).

For any fixed m ≥ 1, let T ≥ 0 be the largest t such that

W (t) := sup 0≤τ ≤t τ m 2 D m u(τ ) Ḃ 3 p -1 p,1 + D m b(τ ) Ḃ 3 q -1 q,1 ≤ C 0 ε 0 ,
where C 0 will be chosen later.

∂ t ∆j D α x u -µ∆ ∆j D α x u = ∆j D α x Pdiv (b ⊗ b) -∆j D α x Pdiv (u ⊗ u). Then ∆j D α x u = e tµ∆ ∆ j D α x u 0 + t 0 e (t-s)µ∆ P ∆j D α x div (b ⊗ b) -∆j D α x div (u ⊗ u) ds. Lemma 3.3 thus implies that ∆j D α x u L p ≤Ce -cµ2 2j t ∆j D α x u 0 L p + C t 0 e -cµ2 2j (t-s) ∆j D α x Pdiv (b ⊗ b) L p + ∆j D α x Pdiv (u ⊗ u) L p ds ≤Ce -cµ2 2j t ∆j D α x u 0 L p + A 1 + A 2 + A 3 , (5.1)
where

A 1 :=C t 2 0 e -cµ2 2j (t-s) ∆j D α x Pdiv (b ⊗ b) L p + ∆j D α x Pdiv (u ⊗ u) L p ds, A 2 :=C t t 2 e -cµ2 2j (t-s) 2 j ∆j D α-1 x Pdiv (b ⊗ b) L p ds, A 3 :=C t t 2 e -cµ2 2j (t-s) 2 j ∆j D α-1 x div (u ⊗ u) L p ds.
Noticing there exists a constant c > 0 such that e -cµ2 2j t 2 jk ≤ e -cµ2 2j t t -k 2 , for any k ≥ 0.

(5.2)

By employing Proposition 3.2, a straightforward calculation shows that

A 1 ≤Ct -m 2 t 2 0 e -cµ2 2j (t-s) ∆j div (b ⊗ b) L p + ∆j div (u ⊗ u) L p ds ≤Cg j t -m 2 2 -( 3 p -1)j div (b ⊗ b) L 1 ( Ḃ 3 p -1 p,1 ) + div (u ⊗ u) L 1 ( Ḃ 3 p -1 p,1 ) ≤Cg j t -m 2 2 -( 3 p -1)j u 2 Ep + b 2 Eq ≤Cg j t -m 2 2 -( 3 p -1)j (1 + 1 µ + 1 ν )ε 2 0 , (5.3)
where {g j } j∈Z ∈ 1 and {g j } 1 ≤ 1.

Thanks to (3.3) , we have ∆j

D α-1 x Pdiv (b ⊗ b) L p ≤C2 j ∆j (D α-1 x b ⊗ b) L p ≤Cg j 2 -( 3 q -1)j D α-1 x b ⊗ b Ḃ 3 q p,1 ≤Cg j 2 -( 3 q -1)j D α-1 x b Ḃ 6 q -3 p q,1 b Ḃ 3 q q,1 + Cg j 2 -( 3 q -1)j D α-1 x b Ḃ 3 q q,1 b Ḃ 6 q -3 p q,1
.

By means of interpolation , we get

D α-1 x b Ḃ 6 q -3 p q,1 b Ḃ 3 q q,1 + D α-1 x b Ḃ 3 q q,1 b Ḃ 6 q -3 p q,1 D α x b 1-r Ḃ 3 q -1 q,1 b r Ḃ 3 q -1 q,1 D α x b 1 m Ḃ 3 q -1 q,1 b 1-1 m Ḃ 3 q -1 q,1 + D α x b Ḃ 3 q -1 q,1 D α x b 1 m -r Ḃ 3 q -1 q,1 b 1-1 m +r Ḃ 3 q -1 q,1 D α x b 1+ 1 m -r Ḃ 3 q -1 q,1 b 1-1 m +r Ḃ 3 q -1 q,1
,

where r := 3 m ( 1 p -1 q ) ≤ 1 m . Since 3 p -3 q < 1, (5.
2) and Hölder inequality imply that

A 2 ≤Cg j 2 -( 3 p -1)j t t 2 (t -s) -1 2 (1+ 3 p -3 q ) D α x b 1+ 1 m -r Ḃ 3 q -1 q,1 b 1-1 m +r Ḃ 3 q -1 q,1 ds ≤Cg j 2 -( 3 p -1)j ( t 2 ) -m 2 (1+ 1 m -r) W 1+m-r ε 1-1 m +r 0 t t 2 (t -s) -1 2 (1+ 3 p -3 q ) ds ≤2 m 2 +1 Cg j 2 -( 3 p -1)j t -m 2 W 1+ 1 m -r ε 1-1 m +r 0 .
(5.4) Thanks to (4.10), we have ∆j D

α-1 x Pdiv (u ⊗ u) L p ≤C2 j ∆j (D α-1 x u ⊗ u) L ∞ (L p ) ≤Cg j 2 -( 3 p -1)j D α-1 x u ⊗ u Ḃ 3 p p,1 ≤Cg j 2 -( 3 p -1)j D α-1 x u Ḃ 3 p p,1 u Ḃ 3 p p,1
.

By means of interpolation,

D α-1 x u Ḃ 3 p p,1 u Ḃ 3 p p,1 D α x u Ḃ 3 p -1 p,1 D α x u 1 m Ḃ 3 p -1 p,1 u 1-1 m Ḃ 3 p -1 p,1 .
Thus, (5.2) and Hölder inequality imply that

A 3 ≤Cg j 2 -( 3 p -1)j t t 2 (t -s) -1 2 D α x u 1+ 1 m Ḃ 3 p -1 p,1 u 1-1 m Ḃ 3 p -1 p,1 ds ≤Cg j 2 -( 3 p -1)j ( t 2 ) -m+1 2 W 1+ 1 m ε 1-1 m 0 t t 2 (t -s) -1 2 ds. ≤2 m 2 +1 Cg j 2 -( 3 p -1)j t -m 2 W 1+ 1 m ε 1-1 m 0 .
(5.5) Putting (5.3), (5.4) and (5.5) together, one has

t m 2 D m u Ḃ 3 p -1 p,1 ε 0 + (1 + 1 µ + 1 ν )ε 0 2 + W 1+ 1 m -r ε 0 1-1 m +r + W 1+ 1 m ε 0 1-1 m .
(5.6) 5.2. Decay estimates for Magnetic fields. Applying ∆j to equation (1.3) and taking D α on the resulting equation leads to

∂ t ∆j D α b -ν∆ ∆j D α b = ∆j D α ∇ × ((u -εJ) × b). (5.7) Then ∆j D α x b = e tν∆ ∆ j D α x b 0 + t 0 e (t-s)ν∆ ∆j D α x ∇ × (u × b) -ε ∆j D α x ∇ × (J × b) ds. Lemma 3.3 thus implies that ∆j D α x b L q ≤Ce -cν2 2j t ∆j D α x b 0 L q + C t 0 e -cν2 2j (t-s) ∆j D α x ∇ × (u × b) L p -∆j D α x ∇ × (εJ × b) L q ds ≤Ce -cν2 2j t ∆j D α x b 0 L q + A 4 + A 5 + A 6 , (5.8)
where

A 4 :=C t 2 0 e -cν2 2j (t-s) ∆j D α x ∇ × (u × b) L q + ∆j D α x ∇ × (εJ × b) L q ds, A 5 :=C t t 2 e -cν2 2j (t-s) 2 j ∆j D α-1 x ∇ × (u × b) L q ds, A 6 :=C t t 2 e -cν2 2j (t-s) 2 j ∆j D α-1 x ∇ × (εJ × b) L q ds.
Similar with the method of getting estimates (5.3) and (5.5), one can easily show that

A 4 ≤Ct -m 2 t 2 0 e -cµ2 2j (t-s) ∆j ∇ × (u × b) L q + ∆j ∇ × (εJ × b) L q ds ≤Cg j t -m 2 2 -( 3 q -1)j ∇ × (u × b) L 1 ( Ḃ 3 q -1 q,1 ) + ∇ × (εJ × b)) L 1 ( Ḃ 3 q -1 q,1 ) ≤Cg j t -m 2 2 -( 3 q -1)j u 2 Ep + (b, J) 2 Eq ≤Cg j t -m 2 2 -( 3 q -1)j (1 + 1 µ + 1 ν )ε 2 0 , (5.9)
and

A 5 ≤Cg j 2 -( 3 q -1)j t t 2 (t -s) -1 2 D α-1 x (u × b) Ḃ 3 q q,1 ds ≤Cg j 2 -( 3 q -1)j t t 2 (t -s) -1 2 D α-1 x u × b Ḃ 3 q q,1 + u × D α-1 x b Ḃ 3 q q,1 ds ≤Cg j 2 -( 3 q -1)j W 1+ 1 m ε 1-1 m 0 t t 2 (t -s) -1 2 s -1 2 (1+m) ds ≤2 m 2 +1 Cg j 2 -( 3 q -1)j t -m 2 W 1+ 1 m ε 1-1 m 0 .
(5.10)

Because div b = 0, one can rewrite

∇ × (εJ × b) = ε∇ × div (b ⊗ b) -∇( |b| 2 2 ) = ε∇ × div (b ⊗ b) ,
then, Hölder inequality yields

A 6 ≤Cε t t 2 e -cν2 2j (t-s) 2 3 j ∆j D α-1 x b ⊗ b L q ds ≤Cε2 -( 3 q -1)j g j t t 2 e -cν2 2j (t-s) 2 2j D α-1 x (b ⊗ b) Ḃ 3 q q,1 ds ≤Cε2 -( 3 q -1)j g j t t 2 e -cν2 2j (t-s) 2 2j D α x b Ḃ 3 q -1 q,1 ∇ × b Ḃ 3 q -1 q,1 ds ≤Cεε 0 2 -( 3 q -1)j g j ( t 2 ) -m 2 t t 2 e -cν2 2j (t-s) 2 2j s m 2 D α x b Ḃ 3 q -1 q,1 ds ≤2 m 2 +1 C ε ν 2 -( 3 q -1)j g j t -m 2 W (T )ε 0 , (5.11)
here we use the fact that

t t 2 e -cν2 2j (t-s) 2 2j ds ≤ 2 cν .
Putting (5.9), (5.10) and (5.11) together, one has

t m 2 D m b Ḃ 3 q -1 q,1 ε 0 + (1 + 1 µ + 1 ν )ε 0 2 + W 1+ 1 m ε 0 1-1 m + ε ν W ε 0 .
This combine with (5.6) implies that

W (T ) ≤ Cε 0 + C 1 + 1 µ + 1 ν + ε ν C 0 + C 1+ 1 m -r 0 + C 1+ 1 m 0 ε 2 0 .
Thanks to (4.17), one can take suitable C 0 such that W (T ) < 1 2 C 0 ε 0 . By the continuous induction, we have W (t) ≤ C 0 ε 0 for all t ≥ 0. It completes the proof of Theorem 2.2.

The proof of Theorem 2.3

In order to prove the theorem ( µ = ν is assumed), we need to notice from [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF] that if (u, b) is a solution of Hall-MHD system (1.1)-(1.3) in the sense of distribution, then the so-called velocity of electron v := u -εJ satisfies:

∂ t v -µ∆v = P div (b ⊗ b) -div (u ⊗ u) -ε∇ × ((∇ × v) × b) + ∇ × (v × u) + 2ε∇ × (v • ∇b).
(6.1)

The equation (6.1) is still quasi-linear compare to the equation of current J. However, owing to

((∇ × (∇ × v)) × b, v) L 2 = ((∇ × v) × b, ∇ × v) L 2 = 0,
the most nonlinear term cancels out when performing an energy method. Thus, contrasting to the uniqueness part of Theorem 2.1, it will help us to release the smallness assumption on current J. We now focusing to the proof of Theorem 2.3. Since (u 0,2 , b 0,2 ) satisfies the initial conditions in [START_REF] Danchin | On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces[END_REF] -div (u 1 ⊗ u) ,

d 2 := ∇ × (-v × b + v 1 × b + v × b 1 ), d 3 := -ε∇ × (-(∇ × v) × b + (∇ × v 1 ) × b + (∇ × v) × b 1 ), d 4 := ε∇ × (-v × u + v 1 × u + v × u 1 ), d 5 := 2ε∇ × (-v • ∇ b + v 1 • ∇ b + v • ∇b 1 ).
We know that ( u, b, v) ∈ L ∞ t ( Ḃ 1 2,1 ) firstly, which is quite easy. A standard energy method gives that for all t ∈ [0, T * ), ( u, B, v)(t)

∼ (u, v) Ḃ 1 2 2,1
, ∇b

Ḃ 3 2 2,1 ∼ (u, v) Ḃ 3 2 2,1
.

one has

d 1 Ḃ 1 2 2,1 b 2 Ḃ 3 2 2,1 + b Ḃ 3 2 2,1 b 1 Ḃ 3 2 2,1 + u 2 Ḃ 3 2 2,1 + u Ḃ 3 2 2,1 u 1 Ḃ 3 2 2,1 , d 2 Ḃ 1 2 2,1 v Ḃ 3 2 2,1 b Ḃ 3 2 2,1 + v 1 Ḃ 3 2 2,1 b Ḃ 3 2 2,1 + v Ḃ 3 2 2,1 b 1 Ḃ 3 2 2,1 , d 4 Ḃ 1 2 2,1 v Ḃ 3 2 2,1 u Ḃ 3 2 2,1 + v 1 Ḃ 3 2 2,1 u Ḃ 3 2 2,1 + v Ḃ 3 2 2,1 u 1 Ḃ 3 2 2,1 , d 5 Ḃ 1 2 2,1 v Ḃ 3 2 2,1 ∇ b Ḃ 3 2 2,1 + v Ḃ 3 2 2,1 ∇b 1 Ḃ 3 2 2,1 + v 1 Ḃ 3 2 2,1 ∇ b Ḃ 3 2 2,1 v Ḃ 3 2 2,1 ( u, v) Ḃ 3 2 2,1 + ( u, v) Ḃ 3 2 2,1 (u 1 , v 1 ) Ḃ 3 2 2,1
,

  , then we get the following Hall equation for b : (Hall) ∂ t b + ε∇ × ((∇ × b) × b) = ν∆b, b| t=0 = b 0 , which is invariant by the rescaling b(t, x) b(λ 2 t, λx), (1.7) provided the data b 0 is rescaled according to b 0 (x) b 0 (λx). (1.8)

(4. 16 )

 16 Combining (4.10),(4.14), take use of interpolation inequality in Proposition 3.4 and Hölder inequality, remember the "norm" of free solution (u L , b L , J L ) is small thanks to (4.15) and (4.16), it follows from Lemma 3.11 that ūn

5. 1 .

 1 Decay estimates for velocity fields. Applying ∆j to the equation (1.1) and taking D α x (|α| = m) on the resulting equation leads to

∂

  Theorem 2.2 about the local well-posedness of Hall-MHD.Thus, supplemented with initial data (u 0,2 , b 0,2 ) there exists a solution (u 2 , b 2 ) on the maximal time interval [0, T * ) fulfilling(u 2 , b 2 , ∇ × b 2 ) ∈ E 2 (t), for all t < T * . Define v i := u i -εJ i , (i = 1, 2). It is then convenient to consider the difference ( u, b, v) := (u 1 -u 2 , b 1 -b 2 , v 1 -v 2 ), which satisfies: t u -µ∆ u := d 1 , ∂ t b -µ∆ b := d 2 , ∂ t v -µ∆ v := d 1 + d 3 + d 4 + d 5 , ( u, B, v)| t=0 =(u 0,1 -u 0,2 , b 0,1 -b 0,2 , v 0,1 -v 0,2 ), (6.2)whered 1 := P -div ( b ⊗ b) + div ( b ⊗ b 1 ) + div (b 1 ⊗ b) + div ( u ⊗ u) -div ( u ⊗ u 1 )

2 2, 1 ) ∩ L 1 t ( Ḃ 5 2 2, 1 ) 1 2 2 , 1 ,

 2121121 since both (u i , b i , v i ) belong to that space. Now, we shall estimate the difference ( u, b, v) in the space Ḃ one thus has to verify d 1 to d 5 live in the space L 1

(d 1 1 + 2 [

 112 , d 2 , d 4 , d 5 ) ∇ × ((∇ × v 1 ) × b) ∆j , (b 1 + b)×](∇ × v) L 2 dτ. (6

Acknowledgment. Part of this paper was discussed when the first author visit Université Paris-Est. The authors express much gratitude to Prof. Raphaël Danchin and Prof. Weixi Li for their supports. The first author thank to Wuhan university's financial supports to visit Université Paris-Est. The second author is supported by the PhD fellowship from Université Paris-Est.

and

∇ × ((∇ × v 1 ) × b)

.

Thanks to the commutator estimate (see [START_REF] Chae | On the blow-up criterion and small data global existence for the Hallmagnetohydrodynamics[END_REF])

.

Hence, by interpolation and Young's inequality, inequality (6.3) becomes

with Ω(t

• Now, one needs to prove the following bootstrap argument (see similar result in [START_REF]The Global Solvability Of The Hall-magnetohydrodynamics System In Critical Sobolev Spaces[END_REF]). Lemma 6.1. Let X, D, W be three nonnegative measurable functions on [0, T ]. Assume that there exists a nonnegative real constant C such that for any t ∈ [0, T ],

If, in addition,

then, for any t ∈ [0, T ], one has

Then, (6.5) implies that for all t ∈ [0, T ], we have

By Gronwall lemma, this yields for all t ∈ [0, T ],

Hence, it is clear that if (6.6) is satisfied, then (6.8) is satisfied with a strict inequality. A continuity argument thus ensures that we must have T = T and thus (6.7) on [0, T ].

Noticing our assumptions on (u 1 , b 1 ) ensure that Ω ∈ L 1 (R + ). By virtue of (2.6), let η satisfies 2Cη exp Ω L 1 (R+) < µ, and apply Lemma 6.1 to inequality (6.4), we have for any t ∈ [0, T * ), ( u, B, v)(t)

The above inequality ensures that (

2,1 ) and so does (u 2 , b 2 , v 2 ), thus we conclude by classic arguments that (u 2 , b 2 , v 2 ) can continued beyond T * , which finally implies that T * = ∞. This completes the proof of Theorem 2.3.


Pascal Audigane

Florence Quesnel

Hélène Tissoux

Delphine Allier

Justine Briais

Juliette Stephan-Perrey

RGF-BP, the French Geological Reference Platform of the Paris Basin: Status and Perspective

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RGF-BP, the French Geological Reference Platform of the Paris Basin: Status and Perspective

Audigane, P. 1 , Quesnel, F. 1 , Tissoux, H. 1 , Briais, J. 1 , Allier, D. 1 , Hertout, A 1 .

1 BRGM, Orléans, France Contact: p.audigane@brgm.fr

___________________________________________________________________________ <= 250 words

The RGF-BP is the Paris Basin work programme of the French Geological Reference Platform (see Calcagno et al., same session) designed by the BRGM as a national programme for the acquisition and management of geological data. In the continuation of the 1:50000 scale geological maps, covering the whole country, the RGF-BP intends to improve the knowledge of the Paris Basin geology with specific focuses on: (i) the Grand Paris field work conducted for the development of new underground transport lines (metro and RER), (ii) the Tertiary and Quaternary sedimentary formations with the resolution of the geological maps and (iii) the deeper formations from Cretaceous to the Trias which reaches the crystalline basement formations. The main objectives of the RGF-BP are to obtain harmonized numerical geological maps of the Paris Basin, correlated 3D geometries of the sedimentary formations and 3D models populated with geotechnical, hydrogeological, geothermal and geochemical properties, all freely accessible to the public. This large piece of work involved the collection of existing and new data to be validated and incorporated into a common numerical referential (common lithological referential, update of geological maps, correlation transect, datation horizon) validated by the data providers, industries and the scientific community involved in the project via pHD ad Masters.

The new knowledge on subsoils developed by the RGF-BP intends to support innovative responses to a wide range of issues facing our society, e.g. surface and subsurface planning, water and mineral resources management, energy prospection and storage, and risk mitigation.

Please visit http://rgf.brgm.fr for more information.