Nicolas Laroche

Alexandre Cabagnols

Pascale Hénaut

Pierre-Charles Chinese Romond

Returnee

Chinese Returnee Entrepreneurs: The Essential Human Capital for a Chinese Innovative State

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Chinese Returnee Entrepreneurs: The Essential Human Capital for a Chinese Innovative State

Nicholas Laroche 1, 4 , Alexandre Cabagnols 2 , Pascale Hénaut 3 and P. Romond 1 1 Clermont Université, Université d'Auvergne, Clermont-Ferrand, France 2 Clermont Université, Université Blaise Pascal, Clermont-Ferrand, France 3 Soluscience, Biopole Clermont-Limagne, Saint-Beauzire, France 4 NRS, Clermont-Ferrand, France Abstract: Do Chinese entrepreneurs, who have gained academic and/or professional experience abroad, develop different commercial or technology strategies compared to their competitors? Are they a real enrichment to the Chinese industrial sector? Within a single country -China-we can distinguish two different populations of biotech enterprises: one set up and led by returnees (the majority of whom have been to the USA), the other by "mainland" Chinese entrepreneurs. Returnees are defined as "migrant entrepreneurs" who have left their country of origin for a certain period to then return to it. They have been exposed to an occidental culture (mainly North American and European) and have benefited from an enrichment of their human capital by means of research visits or research fellowships at the best universities worldwide. In our study we compare the technological and product positioning of the two populations of enterprises. We distinguish between the enterprises positioned in mature market segments (who exploit already existing commercial opportunities) and those positioned in emerging market segments, which explore new opportunities. For the present paper we collected data on the Chinese biotech sector. We studied 19 returnees' societies and 23 "mainland Chinese" societies. Based on this sample, we test the influence of the head manager status (returnee/ or not) on the intensity of their companies' innovative behaviour. The results show that the firms managed by returnees are more strongly positioned in emerging markets compared to the firms that are managed by mainland Chinese. We conclude that returnees in the Chinese biotech sector contribute to the diversification of the markets on which the Chinese industry is positioned and to its technological catching-up towards US standards. Keywords: innovativeness / human capital / china / entrepreneurship / biotechnology / international mobility


Debora Amadori
email: debora.amadori@univaq.it

Cleopatra Christoforou
email: christoforou.cleopatra@ucy.ac.cy

Weak solutions with bounded support to an Euler-type flocking model

 on the existence and time-asymptotic flocking of weak solutions to a hydrodynamic model of flockingtype with all-to-all interaction kernel in one-space dimension. An appropriate notion of entropy weak solutions with bounded support is described, to capture the behavior of solutions to the Cauchy problem with any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, a suitable condition on the initial data is provided that allows us to show time-asymptotic flocking for such solutions.

Introduction

This article serves as an overview of the results obtained in [START_REF] Amadori | BV solutions for a hydrodynamic model of flocking type with all-to-all interaction kernel[END_REF] on weak solutions to the hydrodynamic model of flocking-type in one-space dimension that takes the form

∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρv) + ∂ x ρv 2 + p(ρ) = ∫ R K(x, x )ρ(x, t)ρ(x , t) (v(x , t) -v(x, t)) dx (1
) where (x, t) ∈ R×[0, +∞). Here ρ ≥ 0 stands for the mass variable, v for the velocity, p for the pressure and K = K(x, x ) ≥ 0 for a symmetric interaction kernel. Self-organization is an area that has received alot of attention in the research community and especially for systems such as flock of birds, a swarm of bacteria or a school of fish and its study gives rise to new mathematical challenges. Especially, in an effort to understand the emergence of flocking behavior, many mathematical models have been introduced arising from the inspiring work of Cucker and Smale [START_REF] Cucker | Emergent behavior in flocks[END_REF], while led to many subsequent studies; cf. [START_REF] Carrillo | Particle, kinetic, and hydrodynamic models of swarming, In Mathematical modeling of collective behavior in socio-economic and life sciences[END_REF][START_REF] Shvydkoy | Dynamics and Analysis of Alignment Models of Collective Behavior[END_REF] and the references therein. Most results on flocking models involve results at the particle level, the corresponding kinetic equation or its hydrodynamic formulation (cf. [START_REF] Ha | From particle to kinetic and hydrodynamic descriptions of flocking[END_REF][START_REF] Ha | A simple proof of the Cucker-Smale flocking dynamics and mean-field limit[END_REF][START_REF] Motsch | A new model for self-organized dynamics and its flocking behavior[END_REF][START_REF] Karper | Existence of weak solutions to kinetic flocking models[END_REF][START_REF] Ha | A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation[END_REF][START_REF] Ha | A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid[END_REF][START_REF] Ha | Emergent dynamics for the hydrodynamic Cucker-Smale system in a moving domain[END_REF][START_REF] Karper | Hydrodynamic limit of the kinetic Cucker-Smale model[END_REF]) and so far this subject has been investigated mainly in the context of solutions with no discontinuities.

The Euler-type flocking system (1) with pressure

p(ρ) = α 2 ρ , α > 0 . ( 2
)
is rigorously derived as the hydrodynamic limit of the kinetic Cucker-Smale flocking model on (x, t, ω) ∈ (0, T)×R d ×R d by Karper, Mellet and Trivisa [START_REF] Karper | Hydrodynamic limit of the kinetic Cucker-Smale model[END_REF]. In particular, they study the singular limit corresponding to strong noise and strong local alignment with the alignment operator being the usual Cucker-Smale operator, while a diffusion term is present in the kinetic equation and show the convergence of weak solutions to the kinetic equation to strong (suitably smooth) solutions of the Euler system with pressure (2). We remark that, in the literature, hydrodynamic models for flocking are often described by a pressureless Euler system that are obtained from a microscopic description of the particles motion without a stochastic forcing. As a consequence, the system with pressure received less attention than the pressureless one, in particular for weak solutions, that is the class of solutions we aim for. A result on smooth, space-periodic solutions to this model with pressure is established in [START_REF] Choi | The global Cauchy problem for compressible Euler equations with a nonlocal dissipation[END_REF].

In [START_REF] Amadori | BV solutions for a hydrodynamic model of flocking type with all-to-all interaction kernel[END_REF], we assume that the communication rate is of all-to-all type, that is

K(x, x ) ≡ 1 . (3)
Although this assumption, K = 1, simplifies the system and the nonlocal term turns into a local term, this special case still possesses many obstacles in the analysis point of view, in order to capture the existence of solutions and time-asymptotic flocking. In addition, the work on this special case indicates how the mechanism of the dissipative behavior of the solutions works and we expect this to be crucial in the extension of this analysis to general kernels. We consider the Cauchy problem and assign initial data

(ρ, m)(x, 0) = (ρ 0 (x), m 0 (x)) x ∈ R , (4)
for the density ρ = ρ(x, t), and momentum m = ρv. To capture the emergent behavior of self-organized systems, we assume that the initial total mass ρ is confined in a bounded interval, and is uniformly positive in there and the initial momentum m 0 taken to 0 outside that region, i.e. there exist a 0 < b 0 such that

supp{(ρ 0 , m 0 )} ⊂ I 0 = [a 0 , b 0 ] , ess inf I 0 ρ 0 > 0 , (5)
while we use v 0 = m 0 /ρ 0 only in I 0 .

To study weak solutions in the context of flocking, we need to introduce a proper notion of entropy weak solution for which the support is bounded at any given time t > 0 in between the extremal particle paths, that is, there exists two absolutely continuous curves t → a(t), b(t), t ∈ [0, +∞) with

a(0) = a 0 , b(0) = b 0 ; a(t) < b(t) for all t > 0 (6)
and

a (t) = v(a(t)+, t) , b (t) = v(b(t)-, t) for a.e. t > 0 (7) such that supp{(ρ, m)(•, t)} ⊂ I(t) = [a(t), b(t)] , t > 0 . (8)
The following notion of entropy weak solutions with concentration is motivated by the ad-hoc boundary condition:

The vacuum region is connected with the non-vacuum one by a shock discontinuity . This choice is made in order to allow a sharp front with finite speed to arise as expected in flocking. In this way, we exclude the case of a rarefaction connecting a vacuum region with a non-vacuum since in such a case, due to the pressure law (2), the front would not have a proper interpretation in terms of flocking because of the unbounded maximal speed. 1

We assume that

∬ {ρφ t + mφ x } dxdt = 0 , (9)
holds true for all φ ∈ C ∞ 0 (R × (0, ∞)). Therefore, the Rankine-Hugoniot condition [m] = x[ρ] must hold along discontinuities x(t), in particular along the free boundaries a(t) and b(t), which is consistent with [START_REF] Choi | The global Cauchy problem for compressible Euler equations with a nonlocal dissipation[END_REF]. As a consequence, conservation of mass holds:

M(t) = ∫ R ρ(x, t) dx = ∫ I(t) ρ(x, t) dx = ∫ R ρ 0 (x) dx =: M , ∀ t ≥ 0 . ( 10
)
The appropriate definition now follows:

Definition 1 Given the initial data (ρ 0 , m 0 ) ∈ BV(R), together with (5), let (ρ, m) : R × [0, +∞) → R 2 be a function with the following properties:

• the map t → (ρ, m)(•, t) ∈ L 1 loc (R) ∩ BV(R) is continuous in L 1 loc ;
1 For instance, if a rarefaction of the 1 st family connects the two states (ρ , v ) and ( ρ, v( ρ)), then using the pressure term (2), one has

v( ρ) = v - ∫ ρ ρ 1 s p (s) ds = v -α ln ρ ρ , 0 < ρ ≤ ρ . As ρ → 0+, then v( ρ) → ∞. • lim t→0+ (ρ, m)(•, t) = (ρ 0 , m 0 ) in L 1 loc (R);
• there exist two locally Lipschitz curves t → a(t), b(t), t ∈ [0, +∞) and a value ρ in f > 0 such that ( 6), ( 7), ( 8) hold and ess inf

I(t) ρ(•, t) ≥ ρ in f > 0 ∀ t > 0 . (11)
Then (ρ, m) is an entropy weak solution with concentration along a(t) and b(t) of the problem ( 1), ( 4) with ( 2) and ( 3), if (a) the integral identities [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics, Fourth Edition[END_REF] and

∬ mφ t + m 2 ρ + p(ρ) φ x dxdt - ∬ [m M -ρ M 1 (t)] φ dxdt - ∫ ∞ 0 [p(ρ(b(t)-, t))φ(b(t), t) -p(ρ(a(t)+, t))φ(a(t), t)] dt = 0 , (12)
hold true on R × (0, ∞) for all test functions φ ∈ C ∞ 0 (R × (0, ∞)), with M the conserved total mass given at [START_REF] Dafermos | Hyperbolic systems of balance laws with weak dissipation[END_REF],

M 1 (t) = ∫ R m(x, t)dx + P b (t) -P a (t) , (13) and 
          P b (t) := ∫ t 0 e -M(t-s) p(ρ(b(s)-, s)) ds , P a (t) := ∫ t 0 e -M(t-s) p(ρ(a(s)+, s)) ds ; (14)
(b) the solution is entropy weak for every pair of convex entropy-entropy flux (η, q) functions for the system (1) , i.e. the inequality

∂ t η(ρ, m) + ∂ x q(ρ, m) ≤ η m {ρM 1 (t) -mM} holds in the sense of distributions on the open set Ω = {(x, t); t > 0 , x ∈ (a(t), b(t))} ⊂ R × (0, +∞).
As a consequence of the boundary condition above and in conjunction with the above definition of weak solutions, we deduce a conservation of momentum property. More precisely, we introduce a quantity that we call total momentum m, with two delta shocks supported on the free boundaries a(t) and b(t), that is the distribution

m(•, t) := m(•, t) + δ b(t) P b (t) -δ a(t) P a (t) , t > 0 , (15)
where δ a denotes the Dirac delta distribution on R with mass at x = a. This new singularity of the total momentum m along the free boundaries a(t) and b(t) is known as delta shock and references can be found in Dafermos [START_REF] Dafermos | Hyperbolic Conservation Laws in Continuum Physics, Fourth Edition[END_REF], Chapter 9. In what follows, we use the following standard notation < •, • >:

< m(•, t), φ(•, t) >:= ∫ I (t) m(x, t)φ(x, t)dx + P b (t)φ(b(t), t) -P a (t)φ(a(t), t), t > 0
as the value of the functional m over C ∞ 0 , for all test functions φ ∈ C ∞ 0 (R × R + ), noting here that m = 0 for x I(t). Now, by definition [START_REF] Ha | A hydrodynamic model for the interaction of Cucker-Smale particles and incompressible fluid[END_REF], we observe that

P b (t) + MP b (t) = p(ρ(b(t)-, t)) ,
and therefore

∫ ∞ 0 < δ b(t) P b (t), φ t (•, t) -Mφ > dt = - ∫ ∞ 0 p(ρ(b(t)-, t))φ(b(t), t) dt .
Thus, we are led to the identity:

∫ ∞ 0 < m(•, t), φ t (•, t) -Mφ > dt = ∬ Ω m (φ t -Mφ) dxdt - ∫ ∞ 0 [p(ρ(b(t)-, t))φ(b(t), t) -p(ρ(a(t)+, t))φ(a(t), t)] dt .
Hence, the integral identity (12) reduces to:

∫ ∞ 0 < m(•, t), φ t (•, t) > dt + ∬ m 2 ρ + p(ρ) φ x dxdt + ∬ ρ M 1 (t)φ dxdt -M ∫ ∞ 0 < m(•, t), φ > dt = 0 .
If we choose the test function φ(x, t) = φ 1 (x)ψ(t), with φ 1 (x) = 1 for all x ∈ ∪ t ∈[T 1 ,T 2 ] I(t) and ψ(t) = 0 for t [T 1 , T 2 ], with 0 < T 1 < T 2 , and notice that M 1 (t) =< m(•, t), φ 1 >, we get M 1 (T 2 ) -M 1 (T 1 ) = 0. Since this holds true for arbitrary times, 0 < T 1 < T 2 , we deduce the conservation of (extended) momentum M 1 (t). By the time continuity of ∫ m(•, t) dx and the definition ( 14), we conclude that

M 1 (t) = M 1 (0+) that is ∫ I(t) m(x, t) dx + P b (t) -P a (t) = ∫ R m 0 (x) dx =: M 1 ∀ t ≥ 0 . ( 16
)

Theorems

In this section, we state the main results of [START_REF] Amadori | BV solutions for a hydrodynamic model of flocking type with all-to-all interaction kernel[END_REF]. More precisely:

-Theorem 1 in Section 2.1 concerns the global in time existence of solutions obeying the notion introduced in Definition 1 without any restriction on the size of the total variation of the initial data, that conserve mass and momentum and at the same time, the non-vacuum region Ω is separated by the vacuum ones by the two free boundaries alongside delta shocks, which are present in the total momentum; -Theorem 2 in Section 2.2 states that, under an appropriate condition on the initial data, the solution admits time-asymptotic flocking.

Global Existence

First, we state the result on global existence in time of entropy weak solutions with concentration to (1) with bounded support.

Theorem 1 Assume that the initial data (ρ 0 , m 0 ) ∈ BV(R) and satisfy [START_REF] Bressan | Hyperbolic Systems of Conservation Laws -The one-dimensional Cauchy problem[END_REF] with pressure [START_REF] Amadori | Global BV solutions and relaxation limit for a system of conservation laws[END_REF]. Then the Cauchy problem (1), ( 4) with (3), admits an entropy weak solution with concentration (ρ, m) in the sense of Definition 1. Moreover, conservation of mass [START_REF] Dafermos | Hyperbolic systems of balance laws with weak dissipation[END_REF] and of momentum [START_REF] Ha | A simple proof of the Cucker-Smale flocking dynamics and mean-field limit[END_REF] hold true.

From the definition of solution (ρ, m), we clarify that we use variable v only in the support I(t) for all t > 0 satisfying v = m ρ , where it is well defined, while only the variables ρ and m are used in the complement of I(t). Now, by conservation of mass and momentum, we have the values M and M 1 in ( 10) and ( 16) respectively, and the average velocity v defined by

v = M 1 /M . (17)
By means of ( 10) and ( 16), system (1) rewrites as

∂ t ρ + ∂ x (ρv) = 0, ∂ t (ρv) + ∂ x ρv 2 + p(ρ) = -Mρ (v -v) . (18)
Thus, to prove Theorem 1, it is equivalent to establish an entropy weak solution with concentration to [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF] for the initial data assumed in the theorem that conserves mass and momentum. Now, system [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF] belongs to the class of system of balance laws and the Cauchy problem for strictly hyperbolic systems of balance has been studied in [START_REF] Dafermos | A system of hyperbolic conservation laws with frictional damping[END_REF][START_REF] Dafermos | Hyperbolic systems of balance laws with weak dissipation[END_REF] under appropriate conditions. However, the previous results do not apply to system (18) because of large data and the loss of strict hyperbolicity due to the vacuum present in R \ [a 0 , b 0 ]. For the notion of the physical vacuum boundary for [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF], we refer to [START_REF] Liu | Compressible flow with damping and vacuum[END_REF][START_REF] Liu | Compressible flow with vacuum and physical singularity[END_REF]. About the asymptotic behavior of the solutions it is conjectured to obey the porous media equation and for the case of the pressure (2), we refer to [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF], but when the initial density ρ 0 tends to a positive value as x → ±∞, which is not our case. Other works [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF][START_REF] Huang | L 1 convergence to the Barenblatt solution for compressible Euler equations with damping[END_REF] study the system with the pressure p(ρ) = ρ γ with γ > 1 and therefore, their analysis cannot be applied again here. One of the main challenges to construct a convergent approximate sequence to system (18), ( 4) is the loss of strict hyperbolicity. To overcome this, we transform the problem into Lagrangian coordinates in the spirit of Wagner [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]. Recasting system [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF] from Eulerian (ρ(x, t), v(x, t)) into the Lagrangian variables (u(y, t), v(y, t)), we get the equations

∂ τ u -∂ y v = 0, ∂ τ v + ∂ y (α 2 /u) = -M(v -v) ( 19
)
with the domain {(y, t); t ≥ 0 , y ∈ (0, M)} . However, the equivalence of weak solutions between the Eulerian and Lagrangian solutions does not apply from [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF] because of the finite mass condition (5) in our case. Actually, because of this, the problem in Lagrangian coordinates is not Cauchy any more but boundary value problem and further difficulties arise. Now, system [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF] has been studied but mostly in the context of the Cauchy problem or with different boundary conditions, cf. [START_REF] Nishida | Global solution for an initial boundary value problem of a quasilinear hyperbolic system[END_REF][START_REF] Dafermos | A system of hyperbolic conservation laws with frictional damping[END_REF][START_REF] Luo | Global BV solutions to a p-system with relaxation[END_REF][START_REF] Amadori | Global BV solutions and relaxation limit for a system of conservation laws[END_REF][START_REF] Frid | Initial-boundary value problems for conservation laws[END_REF]. Our strategy to prove Theorem 1 is: First, using the Riemann solution to (18) around the vacuum that is admissible for the flocking model and the definition of weak solution with concentration, we recast the problem ( 18), (4) in Lagrangian variables to [START_REF] Huang | Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum[END_REF] with non-reflecting boundary conditions at y = 0, M. Actually, the boundary conditions are expressed by the fact that, when a wave-front reaches the boundary, there is no resulting emitted wave. This is the natural counterpart to the behavior of the free boundaries that, in Eulerian variables, delimit the non-vacuum region. Next, we construct approximate solutions (u ν , v ν ) to ( 18), (4) using the front tracking algorithm (cf. Bressan [START_REF] Bressan | Hyperbolic Systems of Conservation Laws -The one-dimensional Cauchy problem[END_REF] and Holden-Risebro [START_REF] Holden | Front Tracking for Hyperbolic Conservation Laws[END_REF]), define appropriate Lyapunov functionals and show that the total variation in space and time of the approximate sequence remains bounded. Combining these results, we prove the convergence to an entropy weak solution of the system in Lagrangian coordinates. The last step is to transform this analysis to problem [START_REF] Huang | Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum[END_REF] in Eulerian. We work, at the level of the approximate solutions, to show the equivalence between Eulerian and Lagrangian variables in the spirit of [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF] and within the domain

I(t) = [a(t), b(t)]
where no vacuum is present, see Fig. 1. In this way, we construct the approximate Fig. 1 On the left, the domain in Lagrangian variables for (u ν , v ν ) and on the right, the domain in Eulerian variables for (ρ ν , v ν ) solutions (ρ ν , v ν ) to ( 18) that inherits the convergence property from the change of coordinates and it is appropriately extended to the half-plane. Also, this coordinate transformation together with the approximation scheme allow us to pass from the non-reflecting boundary conditions at y = 0, M to the free boundaries a(t) < b(t) in the Eulerian coordinates. It is shown finally that, in the limit, (ρ ν , v ν ) converges to an entropy weak solution with concentration that conserves mass and (extended) momentum.

Time-Asymptotic Flocking

Another important issue we address, except of the global in time existence, is the longtime behavior of the entropy weak solution with concentration to (1), ( 4) with [START_REF] Amadori | Global BV solutions and relaxation limit for a system of conservation laws[END_REF], which is interesting in the context of self-organization. The terminology "flocking" corresponds to the phenomenon in which self-organized individuals, using only limited environmental information and simple rules, get organized into an ordered motion. In the spirit of [START_REF] Ha | From particle to kinetic and hydrodynamic descriptions of flocking[END_REF][START_REF] Ha | A simple proof of the Cucker-Smale flocking dynamics and mean-field limit[END_REF], this behavior is captured in the following definition: Definition 2 We say that the entropy weak solution with concentration (ρ, m)(x, t) to system (1), admits time-asymptotic flocking if:

1. the support I(t) of the solution remains bounded for all times, i.e.

sup 0≤t<∞ {b(t) -a(t)} < ∞ . (20)
2. the velocity satisfies lim t→∞ ess sup

x 1 , x 2 ∈I (t) |v(x 1 , t) -v(x 2 , t)| = 0. (21)
Indeed, condition [START_REF] Huang | L 1 convergence to the Barenblatt solution for compressible Euler equations with damping[END_REF] assures that the support of the solution is uniformly bounded, thus defining the "flock", while condition (21) yields that alignment occurs, i.e. the diameter of the set of velocity states within the support I(t) goes to zero timeasymptotically.

We observe that solutions obtained in Theorem 1 satisfy condition (20) immediately by combining (8), [START_REF] Dafermos | A system of hyperbolic conservation laws with frictional damping[END_REF] and [START_REF] Dafermos | Hyperbolic systems of balance laws with weak dissipation[END_REF]. Therefore, to establish the time-asymptotic flocking property, it suffices to show [START_REF] Holden | Front Tracking for Hyperbolic Conservation Laws[END_REF], or equivalently that lim t→∞ ess sup

x 1 ,x 2 ∈I(t) |v(x 1 , t) -v| = 0
where v is defined at [START_REF] Ha | From particle to kinetic and hydrodynamic descriptions of flocking[END_REF]. Assuming that the initial data satisfy

e 2q M 2 < α max {ρ 0 (a 0 +), ρ 0 (b 0 -)} , (22)
where q stands for the initial bulk:

q := 1 2 TV {ln(ρ 0 )} + 1 2α TV {v 0 } (23)
and TV the total variation in the support I 0 , we show that flocking occurs. In fact, if initial bulk q is controlled by the initial density at the endpoints a 0 and b 0 according to [START_REF] Karper | Existence of weak solutions to kinetic flocking models[END_REF], then velocity decays to v at an exponential in time rate. The result is:

Theorem 2 Let (ρ, m) be the entropy weak solution with concentration to (1), ( 2), ( 4) with (3), the initial data (ρ 0 , m 0 ) ∈ BV(R) satisfying (5) and with q > 0 given in [START_REF] Karper | Hydrodynamic limit of the kinetic Cucker-Smale model[END_REF] as obtained in Theorem 1. Suppose that [START_REF] Karper | Existence of weak solutions to kinetic flocking models[END_REF] holds true, then the solution (ρ, m) admits time-asymptotic flocking. More precisely, the oscillation of the velocity decays exponentially fast, i.e. there exists t 0 > 0 such that ess sup

x 1 , x 2 ∈I (t) |v(x 1 , t) -v(x 2 , t)| ≤ C 2 e -C 1 t , ∀ t ≥ t 0 ( 24
)
for some positive constants C 1 , C 2 .

Our strategy to prove Theorem 2 involves careful analysis of the long time behavior of the approximate solutions constructed in the existence part using the results about the wave strength dissipation for the system of isothermal flow that is, the homogeneous version of ( 1)-( 2) as obtained in [START_REF] Amadori | Global BV solutions and relaxation limit for a system of conservation laws[END_REF][START_REF] Amadori | On a model of multiphase flow[END_REF][START_REF] Amadori | Global weak solutions for a model of two-phase flow with a single interface[END_REF]. These properties are used to provide uniform bounds on the vertical traces of approximate solutions and for the time-asymptotic analysis. In addition, we introduce a new functional, called the total variation that is weighted by generation order

V(t) = k ≥1 ξ k F k (t)
for ξ ≥ 1 taking values in an appropriate interval and F k (t) being the total variation functional of generation k. This allows us to detect a geometric decay property in terms of the number of wave reflections for the homogeneous system and it turns out that we can control under assumption [START_REF] Karper | Existence of weak solutions to kinetic flocking models[END_REF], the possible increase of the functional which is due to the reflections produced by the damping term leading to the timeexponential decay established in [START_REF] Liu | Compressible flow with damping and vacuum[END_REF].

In a work in progress, we are going to relax the sufficient condition [START_REF] Karper | Existence of weak solutions to kinetic flocking models[END_REF] to allow for time-asymptotic flocking with any (ρ 0 , m 0 ) ∈ BV(R) that satisfies [START_REF] Bressan | Hyperbolic Systems of Conservation Laws -The one-dimensional Cauchy problem[END_REF].

Acknowledgements

The research of paper [1] was partially supported by 2020 INdAM-GNAMPA Project "Buona positura, regolarità e controllo per alcune equazioni di evoluzione". The first author kindly acknowledges the hospitality of the University of Cyprus, where this work was started.


M L Walker
email: m.walker@aip.org.au

MeV Mass Gluonic Colour Singlets in QCD

Keywords: QCD, Cho-Faddeev-Niemi-Shabanov decomposition, monopole condensate PACS: 12.38.-t, 12.38.Aw, 12.39.Mk, 14.70.Dj

come

Introduction

Identifying the internal Abelian directions has been of interest to studies of the QCD vacuum since Savvidy's landmark paper [START_REF] Savvidy | Infrared instability of the vacuum state of gauge theories and asymptotic freedom[END_REF] demonstrating the energetic favourability of a magnetic condensate. This led to a long-running controversy surrounding the condensate's stability [START_REF] Nielsen | An unstable yang-mills field mode[END_REF][START_REF] Kondo | Non-abelian stokes theorem and quark confinement in su(n) yang-mills gauge theory[END_REF][START_REF] Honerkamp | The question of invariant renormalizability of the massless yang-mills theory in a manifest covariant approach[END_REF][START_REF] Volker Schanbacher | Gluon propagator and effective lagrangian in QCD[END_REF][START_REF] Dittrich | Effective qcd lagrangian with zeta function regularization[END_REF][START_REF] Flory | Covariant constant chromomagnetic fields and elimination of the one loop instabilities[END_REF], with recent papers [START_REF] Flory | Covariant constant chromomagnetic fields and elimination of the one loop instabilities[END_REF][START_REF] Cho | Monopole condensation in su(2) qcd[END_REF][START_REF] Cho | Monopole condensation and confinement of color in su(2) qcd[END_REF][START_REF] Cho | Stability of monopole condensation in su(2) qcd[END_REF][START_REF] Kondo | Magnetic condensation, abelian dominance and instability of savvidy vacuum[END_REF][START_REF] Kay | Savvidy vacuum in su(2) yangmills theory[END_REF] concluding in the positive.

What concerns this work is the manner in which the necessarily Abelian internal direction(s) of the condensate were identified. Two-colour studies typically assigned the Abelian direction to ê3 [START_REF] Savvidy | Infrared instability of the vacuum state of gauge theories and asymptotic freedom[END_REF][START_REF] Nielsen | An unstable yang-mills field mode[END_REF][START_REF] Honerkamp | The question of invariant renormalizability of the massless yang-mills theory in a manifest covariant approach[END_REF][START_REF] Volker Schanbacher | Gluon propagator and effective lagrangian in QCD[END_REF][START_REF] Hooft | Topology of the gauge condition and new confinement phases in nonabelian gauge theories[END_REF], in a blatant violation of gauge invariance that always left doubts that the calculated effects might be gauge artifacts. A further defect was that these papers were unable to prove that the magnetic background is due to monopoles.

These problems are avoided by the Cho-Faddeev-Niemi-Shabanov decomposition [START_REF] Cho | Colored monopoles[END_REF][START_REF] Faddeev | Decomposing the yang-mills field[END_REF][START_REF] Shabanov | An effective action for monopoles and knot solitons in yang-mills theory[END_REF], which identifies the Abelian directions without choosing a special gauge. It does this by introducing the Cho connection, a topologically generated contribution to the gluon field which represents [START_REF] Cho | A restricted gauge theory[END_REF] a monopole potential. Thus the problems of gauge invariance and demonstrating the magnetic condensate to be of monopole origin are solved simultaneously.

Identifying the Abelian degrees of freedom in a gauge invariant manner allows one to consider them as physical entities, and not as gauge artifacts. Furthermore, these particular physical entities are colour-neutral, and the primary claim of this paper is that they are not confined. We therefore refer to them as Free Abelian Gluons (FAGs).

Section 2 presents the CFNS decomposition for general SU (N ) gauge groups. Section 3 justifies the claim that the Abelian generators are colourless and unconfined, uses the condensate coupling and dimensional arguments to estimate the FAG's mass, and then goes on to discuss other properties such as stability and decay modes. Some experimental signatures are proposed. Since the FAG mass is found to be of the same order as the pion mass, section 4 discusses the ramifications for the internucleon potential. It is suggested that several qualitative features of this potential are easier to understand in terms of a FAG contribution. The paper concludes with a discussion in section 5.

Specifying Abelian Directions

The CFNS decomposition was first presented by Cho [START_REF] Cho | A restricted gauge theory[END_REF], and later by Faddeev and Niemi [START_REF] Faddeev | Decomposing the yang-mills field[END_REF] and by Shabanov [START_REF] Shabanov | An effective action for monopoles and knot solitons in yang-mills theory[END_REF], as a gauge-invariant means of specifying the Abelian dynamics of two-colour QCD. These authors [START_REF] Cho | Colored monopoles[END_REF][START_REF] Faddeev | Decomposing the yang-mills field[END_REF] also applied it to three-colour QCD. In this section we adapt it to general SU (N ), although we are not the first to do so [START_REF] Faddeev | Partial duality in su(n) yang-mills theory[END_REF][START_REF] Li | Decomposition of su(n) connection and effective theory of su(n) qcd[END_REF], and establish our notation.

The Lie group SU (N ) for N -colour QCD has N 2 -1 generators λ (i) , of which N -1 are Abelian generators Λ (i) . For simplicity, we specify the gauge transformed Abelian directions with ni = U † Λ (i) U . Fluctuations in the ni directions are described by c (i) µ . The gauge field of the covariant derivative which leaves the ni invariant is implicitly defined by

gV µ × ni = -∂ µ ni , ( 1
)
for which the general form is

V µ = c (i) µ ni + B µ , B µ = g -1 ∂ µ ni × ni , ( 2
)
where summation is implied over i. We define the covariant derivative

Dµ = ∂ µ + gV µ × . ( 3
)
It is easily shown that the monopole field strength

H µν = ∂ µ B ν -∂ ν B µ + gB µ × B ν , (4)
has only ni components, ie.

H (i) µν ni = H µν , ( 5
)
where

H (i)
µν has the eigenvalue H (i) . Since we are only concerned with magnetic backgrounds, H (i) is considered the magnitude of a background magnetic field H (i) .

X µ contains the dynamical degrees of freedom (DOF) perpendicular to ni , so if A µ is the gluon field then

A µ = V µ + X µ = c (i) µ ni + B µ + X µ , (6)
where

X µ ⊥ ni . ( 7
)
This appears to leave the gluon field with additional DOF due to ni , B µ , but detailed analyses can be found in [START_REF] Cho | Monopole condensation in su(2) qcd[END_REF][START_REF] Kondo | Magnetic condensation, abelian dominance and instability of savvidy vacuum[END_REF][START_REF] Bae | Qcd versus skyrme-faddeev theory[END_REF][START_REF] Kondo | Brst symmetry of su(2) yang-mills theory in cho-faddeev-niemi decomposition[END_REF] demonstrating that these fields are not fundamental, but a compound of dynamic fields. Hence ni , B µ are dynamic but do not constitute extra DOFs.

Substituting the CFN decomposition into the QCD field strength tensor gives

F 2 = (∂ µ c (i) ν -∂ ν c (i) µ ) 2 + (∂ µ B ν -∂ ν B µ + gB µ × B ν ) 2 +2(∂ µ c (i) ν -∂ ν c (i) µ )n i • (∂ µ B ν -∂ ν B µ + gB µ × B ν ) + ( Dµ X ν -Dν X µ ) 2 +2g((∂ µ c (i) ν -∂ ν c (i) µ )n i + ∂ µ B ν -∂ ν B µ + gB µ × B ν ) • (X µ × X ν ) +g 2 (X µ × X ν ) 2 + 2g( Dµ X ν -Dν X µ ) • (X µ × X ν ). ( 8
)
This expression holds for all N -colour QCD except N = 2 where the last term vanishes.

The kinetic terms for c

(i)
µ are unmistakably those of Abelian fields. Eq. ( 8) has its analogue in studies [START_REF] Savvidy | Infrared instability of the vacuum state of gauge theories and asymptotic freedom[END_REF][START_REF] Nielsen | An unstable yang-mills field mode[END_REF][START_REF] Flory | Covariant constant chromomagnetic fields and elimination of the one loop instabilities[END_REF][START_REF] Flyvbjerg | Improved qcd vacuum for gauge groups su(3) and su(4)[END_REF] utilising the maximal Abelian gauge. However, dependence on a particular gauge casts a shadow on any analysis and makes it impossible to consider the corresponding DOFs as physically significant.

The CFN decomposition also introduces additional gauge DOFs, which a proper application must fix. This is done by imposing the condition [START_REF] Flory | Covariant constant chromomagnetic fields and elimination of the one loop instabilities[END_REF] with the gauge condition Dµ X µ = 0, [START_REF] Cho | Monopole condensation and confinement of color in su(2) qcd[END_REF] which also reduces the number of gauge degrees of freedom to that of conventional QCD [START_REF] Kondo | Brst symmetry of su(2) yang-mills theory in cho-faddeev-niemi decomposition[END_REF]. These analyses were performed in two-colour QCD, but their application to N colours is straightforward [START_REF] Walker | Stability of the magnetic monopole condensate in three-and four-colour qcd[END_REF].

The most important advantage of the CFNS decomposition for the purpose of this paper is that the Abelian dynamics can be specified in a gauge-invariant, well-defined manner that makes it physically meaningful to say that the fields c (i) µ describe the Abelian component of the gluon field. This is in contrast to the MAG which is not gauge invariant and where the physical meaning of the Abelian direction is not well-defined. MAG depends on confinement to hide the gauge artifact. The CFNS decomposition has no gauge artifact.

The Properties of FAGs

An Abelian gluon has no colour charge, just as a photon has no electric charge. It therefore feels no confining potential, unlike quarks and the valence gluons X µ , which are coloured. I shall now argue on dimensional grounds that FAGs must be massive, which renders their effects short-range.

The propagation of FAGs outside of a hadron, through the monopole condensate, is like that of photons in a conventional superconductor, which are well-known to gain an effective mass from the same Cooper pairs which restrict magnetic fields to flux tubes. Indeed, eq. ( 8) contains the term

2(∂ µ c (i) ν -∂ ν c (i) µ )n i • (∂ µ B ν -∂ ν B µ + gB µ × B ν ), (10)
clearly indicating that the monopole condensate does indeed act as a sink/source for Abelian gluons. Significantly, there is no corresponding sink/source term for the valence gluons, although it has been argued [START_REF] Kondo | Magnetic condensation, abelian dominance and instability of savvidy vacuum[END_REF][START_REF] Walker | Stability of the magnetic monopole condensate in three-and four-colour qcd[END_REF] that their mass-gap term is generated from the interaction term

(∂ µ B ν -∂ ν B µ + gB µ × B ν ) • (X µ × X ν ) (11)
The corresponding mass-gap can be estimated on dimensional grounds. The monopole condensate neutralises the magnetic component of a FAG, unless it oscillates faster than the characteristic time of the condensate. This characteristic time would of course go to infinity when the condensate vanishes at deconfinement. Hence the lower limit on a FAG's period can be estimated from the QCD critical temperature using a suitable combination of dimensional constants. Recent lattice calculations ( [START_REF] Aoki | The QCD transition temperature: results with physical masses in the continuum limit II[END_REF] and references therein) of T QCD vary between 151 and 195 MeV. This should be compared to the π 0 mass of 135 MeV and the π ± mass of 140 MeV [START_REF] Amsler | Review of particle physics[END_REF].

Hence an Abelian gluon can propagate outside of a hadron only if it has sufficient energy to overcome the mass-gap imposed by the QCD vacuum. It would then have properties very similar to the Z 0 , except that it couples only to quarks.

This coupling to quarks is a source of instability. Any gluon can couple to a quark-antiquark pair, and from there to a photon and an e + -e -pair. Another decay mode is into a π 0 with a photon to conserve angular momentum. A more interesting process is the interception of a FAG by a virtual pion emitted by a hadron. The virtual pion could absorb the FAG and use its energy to become real while emitting a photon. Note that both neutral and charged pions can participate in this reaction, so a proton could stimulate a FAG to become a π 0 and emit a photon, or to become a π + (and emit a photon) while turning itself into a neutron.

While the energy of a FAG is certainly accessible, hadron collisions at this energy are dominated by jets. Indeed, the mass was derived from the deconfinement temperature so one should not expect the necessary energy to be available in a stable hadron. One possible source is a quark-gluon plasma. It must have sufficient energy by definition, so a quark-gluon plasma, surrounded by normal space, could lower its temperature by emitting FAGs.

The internucleon potential

Since hadrons are colour neutral, the inter-hadron effect of FAGs is a massive, ie. exponentially decaying, van der Waals interaction. At the simplest level of understanding, mesons are genuine dipoles in this respect, being quarkantiquark pairs, while baryons are more complicated objects. Nonetheless their colour polarization can be crudely modelled by observing that the combination of any two colours is the opposite of the remaining colour. Hence the baryon contains three axes of polarization, of which only two are linearly independent.

Hence we expect that two baryons in close proximity to each other will attempt to align their polarization axes, as shown schematically in figure [START_REF] Savvidy | Infrared instability of the vacuum state of gauge theories and asymptotic freedom[END_REF]. This can lead to complicated behaviour for several reasons apart from the inherent difficulty of nonperturbative systems. One is that the exchange of valence gluons between quarks will alter a hadron's colour polarizations by swapping colours around. This suggests an additional mechanism for loss of nucleon mass in the atomic nucleus, in addition to sharing virtual pions. An energetically favourable configuration of nucleonic colour polarizations will suppress the valence gluon exchanges which would otherwise disturb it, thus inhibiting a significant contribution to the baryons' mass.

Another source of complicated behaviour is the difficulty of aligning multiple, coupled polarization axes when more than two hadrons are in close proximity. This would obviously contribute to the three-body effects of the internucleon force. While further analysis is needed, colour polarization may explain this naturally.

Discussion

I have made a case for M eV mass gluonic colour singlets. It is based on the observation that two of the eight gluon generators in three-colour QCD are without colour charge, and that it is colour which is confined. The arguments require the CFNS decomposition to identify the Abelian directions in a gauge invariant way. Without this it is impossible to claim that the Abelian degrees of freedom have real physical meaning. It has been noted that the analysis is not sensitive to the number of colours, with one Abelian degree of freedom in the two-colour case and N -1 of them for N -colour QCD.

An attractive feature of the CFNS decomposition, which makes it useful in dual-Meissner effect studies, is that it unambiguously identifies the gluon's monopole degrees of freedom [START_REF] Faddeev | Decomposing the yang-mills field[END_REF][START_REF] Shabanov | An effective action for monopoles and knot solitons in yang-mills theory[END_REF][START_REF] Cho | A restricted gauge theory[END_REF]. It is easily shown furthermore, at least to one-loop order, that the corresponding monopole condensate is non-zero [START_REF] Savvidy | Infrared instability of the vacuum state of gauge theories and asymptotic freedom[END_REF][START_REF] Flyvbjerg | Improved qcd vacuum for gauge groups su(3) and su(4)[END_REF]. Especially important for this work, a term describing the condensate acting as a sink/source appears. This not only allows the condensate to restrict the chromoelectric flux to flux tubes as required by the dual Meissner effect, but also provides a mass gap for unconfined gluon fluctuations.

A generic dimensional analysis then predicts a mass of the same order but slightly higher than that of the pion. Thus one expects Abelian gluon exchange to also contribute to the internucleon potential. The resulting potential is essentially a massive van der Waals interaction, although the form of its repulsive component at short distances can only be guessed at. Such a potential implies a mutual colour polarisation which is expected to be affected by the addition of a third nucleon to the immediate vicinity, which has long been observed in the internucleon potential. While it is to be expected that the pion-nucleon coupling would be complicated, the possibility remains that a FAG contribution to the internucleon potential might simplify its analysis.

While their deconfinement scale mass-gap makes FAG production by hadron scattering unlikely, a quark-gluon plasma of sufficient size could well employ FAG emission as a means of lowering its temperature.

The key signatures marking the existence of FAGs are the catalysis of pion production by hadrons in the vicinity of a quark-gluon plasma.

Figure 1 :

 1 Figure 1: Simplistic representation of induced colour polarization generated by FAG exchange between two hadrons. Note that even if the colours are changed or mixed by a gauge transformation, the degree of polarization is unaffected.

Acknowledgements

The author wishes to thank S. Narison, G. Moultaka and V. Zhakarov for helpful discussions. He thanks, for generous hospitality, the physics department at the University of Montpellier (UM2), the physics department of the University of New South Wales, and the College of Quantum Science at Nihon University.