Ludovic Godard-Cadillac

Vortex collapses for the Euler and Quasi-Geostrophic Models

Keywords: Fluid mechanics, Euler equations, Quasi-geostrophic equations, Point-vortices, Differential equations

This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.

1 Introduction

The inviscid surface quasi-geostrophic equation

We are interested in a model of point-vortices for the inviscid surface quasi-geostrophic equation

∂ t ω + v • ∇ω = 0, v = ∇ ⊥ (-∆) -s ω, (SQG)
where v : R + × R 2 → R 2 is the fluid velocity and ω : R + × R 2 → R is called the active scalar.

The notation ⊥ refers to the counterclockwise rotation of angle π 2 . Typically, the surface quasigeostrophic equation models the dynamic in a rotating frame of the potential temperature for a stratified fluid subject to Brunt-Väisälä oscillations. This is a standard model for geophysical fluids and it is intensively used for weather forecast and climatology. For more details about this physical model, see e.g. [START_REF] Pedlowsky | Geophysical Fluid Dynamics[END_REF] or [START_REF] Vallis | Atmospheric and Oceanic Fluid Dynamics[END_REF]. Mathematically, the quasi-geostrophic equation has many properties in common with the two-dimensional Euler equation written in terms of vorticity

∂ t ω + v • ∇ω = 0, v = ∇ ⊥ (-∆) -1 ω.
(Euler 2D)

The two-dimensional Euler equation can be seen as a particular case of the quasi-geostrophic equation where s is equal to 1. Local well-posedness of classical solutions for (SQG) was established in [START_REF] Constantin | Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar[END_REF], where are also studied the analogies with the two and three-dimensional Euler equation. Solutions with arbitrary Sobolev growth were constructed in [START_REF] Kiselev | A simple energy pump for the surface quasi-geostrophic equation[END_REF] in a periodic setting. So far and contrarily to the two-dimensional Euler equations, establishing global well-posedness of classical solutions for (SQG) is an open problem. Note also that the global existence of weak solutions in L 2 (R 2 ) was established in [START_REF] Resnick | Dynamical problems in non-linear advective partial differential equations[END_REF], but below a certain regularity threshold, these weak solutions show dissipative behaviors and non-uniqueness is possible [START_REF] Buckmaster | Nonuniqueness of weak solutions to the SQG equations[END_REF]. Exhibiting global smooth solutions or patch solutions is a challenging issue as there is no equivalent of the Yudovitch theorem [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF]. A first example was recently provided in [START_REF] Castro | Global smooth solutions for the inviscid SQG equation[END_REF] by developing a bifurcation argument from a specific radially symmetric function. The variational construction of an alternative example in the form of a smooth traveling-wave solution was completed in [START_REF] Gravejat | Smooth travelling-wave solutions to the inviscid surface quasigeostrophic equation[END_REF][START_REF] Godard-Cadillac | Smooth traveling-wave solutions to the generalized inviscid surface quasi-geostrophic equation[END_REF].

Corotating patch solutions with two patches [START_REF] Hmidi | Existence of corotating and counter-rotating vortex pairs for active scalar equations[END_REF] and N patches forming an N -fold symmetrical pattern [START_REF] Garcia | Vortex patches choreography for active scalar equations[END_REF] were recently exhibited with bifurcation argument. The C 1 analogous of these solutions has also been investigated independently recently in [START_REF] Ao | Travelling and rotating solutions to the generalized inviscid surface quasi-geostrophic equation[END_REF]. Another recent independent result [START_REF] Godard-Cadillac | Co-rotating vortices with n fold symmetry for the inviscid surface quasi-geostrophic equation[END_REF] also build corotating solutions with N patches using variational argument. In this last article, the desingularization of the associated point-vortex problem is achieved. The present work aims at developing the understanding of the links between the quasigeostrophic equation and the two-dimensional Euler equation through the study of the pointvortex model. In the case of the two-dimensional Euler equation, the point-vortex model is a system of differential equations for points on R 2 that approximates situations where the vorticity ω is highly concentrated around several points. In such a situation, it is more convenient to see the vorticity as being a sum of Dirac masses evolving in time. This model is widely studied in fluid mechanics of the plane. An extensive presentation of the main results on this system can be found at [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4], completed by [START_REF] Marchioro | Vortex methods in two-dimensionnal fluid mechanics[END_REF]. The desingularization problem which consists in a rigorous derivation of the point-vortex model is a classical issue for the two-dimensional Euler equation [START_REF] Smets | Desingularization of vortices for the Euler equation[END_REF], to our knowledge it is still open for (SQG) vortices although recent results exist [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF][START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF].

This article generalizes several existing results known for the Euler vortices or extends known results for Euler to the quasi-geostrophic case. The first proposition of this article is the generalization of a uniform bound result, Theorem 2.1 in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]. We prove that under the non-neutral cluster hypothesis (defined hereafter) the vortices stays bounded in finite time and this bound does not depend on the singularity of the kernel nor on the initial position of the vortices. We also provide a uniform relative bound for a slight relaxation of the non-neutral cluster hypothesis. In the second part of this work we prove that under the non-neutral cluster hypothesis the trajectories of the vortices for the Euler model are convergent in finite time even in the case of collapses. The quasi-geostrophic case is left open. The third part of this work is devoted to the question of the improbability of collapses. This consists in studying the Lebesgue measure of the initial conditions leading to a collapse, which is expected to be equal to 0. This question has been successfully answered by Theorem 2.2 in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] under the non-neutral cluster hypothesis for the Euler point-vortices. The extension to the quasi-geostrophic case was achieved by [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF] in the case 1/2 < s < 1. We generalize this result to (SQG) for all s ∈ (0, 1] and we weaken the non-neutral cluster hypothesis since we allow the total sum of the intensities of the vortices to be equal to 0.

Presentation of the point-vortex model

The point-vortex model on the plane R 2 consists in assuming that at time t = 0 the vorticity can write as a sum of Dirac masses,

ω(t = 0, x) = N i=1 a i δ x i . (1)
The points x i are the respective position of the vortices a i δ x i and the coefficients a i = 0 are their intensity. The first equation in (SQG) or (Euler 2D) is a transport equation on the vorticity ω.

It is expected that the Dirac masses initially located at x i are left unchanged but transported by the flow. Formally, if we solve the evolution equations (SQG) or (Euler 2D) with initial datum (1), we obtain that the initial speed writes

v(t = 0, x) = N i=1 a i ∇ ⊥ x G |x i -x| , (2)
where G is the profile of the Green function of the fractional Laplace operator (-∆) s in the plane R 2 . Here the parameter s is chosen to be in (0, 1]. In this case, The profiles G : R * + → R are given by:

G 1 (r) := 1 2π log 1 r and G s (r) := Γ(1 -s) 2 2s πΓ(s) 1 r 2(1-s) , (3)
with Γ the classical Gamma function. Nevertheless, a problem arises from the singularity of the speed in [START_REF] Aref | Motion of three vortices[END_REF]. Since the vorticity is concentrated in one point then it is usually assumed that it does not interact with itself but only with the vortices that are at a positive distance. We derive that the differential equation describing the evolution of the position of the vortices is given by

d dt x i (t) = N j=1 j =i a j ∇ ⊥ G |x i (t) -x j (t)| , (4)
where ∇G(|x|) is a shortcut for the gradient of the function x → G(|x|). This useful notation is coming from the reference work [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]. In the other cases and to avoid ambiguity, the gradient of a function will be denoted using the variable in subscript: ∇ x i . We are going to generalize some of the results in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] to more general kernel profiles G that include the quasi-geostrophic case. In the sequel, the function G : R * + → R is assumed to be chosen such that (4) satisfy the hypothesis of the Cauchy-Lipschitz theorem (also known as Picard-Lindelöf theorem) as long as the distances between vortices remain positive. Definition 1.1 (Set of collapses). The set of initial datum such that two or more vortices collapse on the interval of time [0, T ) is defined by

C T := X ∈ R 2N : ∃ T X ∈ [0, T ), lim inf t→T X min i =j |x i (t) -x j (t)| = 0 . (5)
We then set

C := +∞ T =1 C T . (6)
The set C is called the set of collapses and T X is the time of collapse associated to the initial datum X ∈ C. Note that these sets depend on the choice of the kernel G.

The point-vortex differential equation ( 4) is well-defined for all initial datum X ∈ R 2N \ C. If we restrict the analysis to a bounded interval of time [0, T ) then it is well-defined for all initial datum X ∈ R 2N \ C T , which eventually allows us to study a possible vortex collapse at time t = T.

Concerning the point-vortex problem, the main element to point-out about this dynamic is its Hamiltonian nature. The Hamiltonian of the point-vortex system is given by

H : R 2N -→ R, X = (x 1 . . . x N ) -→ i =j a i a j G |x i -x j | . (7)
The system (4) can be rewritten

a i d dt x i (t) = ∇ ⊥ x i H(X). (8)
The first consequence of this Hamiltonian reformulation is the preservation of the Hamiltonian H along the flow S t of (4):

∀ t ∈ [0, T ), d dt H(S t X) = 0. ( 9
)
We recall that the flow of a differential equation is the function S t that maps the position X ∈ R 2N at time t = 0 to the position at time t. In other words, S t X = x 1 (t), . . . , x N (t) solution to (4), with initial positions X = (x 1 , . . . , x N ). Another consequence of the Hamiltonian of the system is the Liouville theorem that ensures the preservation of the Lebesgue measure by the flow. More precisely, if V 0 ⊆ R 2N \ C T is measurable, then we have

∀ t ∈ [0, T ), d dt L 2N (S t V 0 ) = 0, (10)
where L 2N denotes the Lebesgue measure on R 2N . For the proof of the Liouville Theorem, we refer to the one given by Arnold in [3, Part 3]. With the Hamiltonian formulation also comes the Noether theorem [START_REF] Noether | Invariante variationsprobleme[END_REF] that provides the quantities left invariant by the flow corresponding to the geometrical invariances of the Hamiltonian H. The vorticity vector is defined for all initial datum X ∈ R 2N by

M (X) := N i=1 a i x i . (11)
The translations invariance of H implies the conservation of the vorticity vector:

∀ t ∈ [0, T ), d dt M (S t X) = 0. ( 12
)
When the system is non-neutral, meaning that i a i = 0, this lemma implies the preservation of the center of vorticity of the system defined by

B(X) := N i=1 a i -1 N i=1 a i x i . (13)
Similarly, the invariance by the rotations, implies the conservation of the moment of inertia defined by

I(X) := N i=1 a i |x i | 2 . ( 14
)
We have:

∀ t ∈ [0, T ), d dt I(S t X) = 0. ( 15
)
The combination of these two lemmas implies the preservation of

C(X) := N i=1 N j=1 i =j a i a j |x i -x j | 2 . ( 16
)
Indeed, if we expand the square in the right-hand side of [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF], we obtain by a straight-forward calculation

C(X) = 2 N i=1 a i I(X) -2 M (X) 2 . ( 17
)
The preservation of this quantity is referred as a collapse constraint because it is widely used in the study of vortex collapses for small number of vortices [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF][START_REF] Novikov | Vortex collapse[END_REF][START_REF] Aref | Motion of three vortices[END_REF][START_REF] Badin | Collapse of generalized Euler and surface quasi-geostrophic pointvortices[END_REF]. Indeed, a collapse means that |x i -x j | 2 vanishes for some values of i and j. Combined with the preservation of C, this gives a necessary condition for a vortex collapse. For instance, in the case of a collapse for a system of 3 vortices, this gives the constraint C = 0.

2 Main results

Uniform bound results

The uniform bound Theorem

The specific case of the Euler point-vortex system corresponds to the Green function of the Laplacian (3). This particular case is studied in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]. More precisely, they focused on a specific situation for which the intensities for the vortices satisfy

∀ A ⊆ {1 . . . N } s.t. A = ∅, i∈A a i = 0. (18)
A vortex system such that the sum of all the intensities a i is equal to 0 is called in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] a "neutral system". No name to Hypothesis ( 18) is given and we suggest that to call it "nonneutral clusters hypothesis".

The main interest of this hypothesis relies on the preservation of the center of vorticity property [START_REF] Gravejat | Smooth travelling-wave solutions to the inviscid surface quasigeostrophic equation[END_REF]. Under the non-neutral cluster hypothesis, the center of vorticity is well defined, not only for the whole system but also for any subset of vortices. It can be said intuitively that a vortex cluster is expected to "turn around its center of vorticity". More precisely, we provide a bound on the trajectories that is uniform with respect to the initial datum X ∈ R 2N but also with respect to the singularity of the kernel profile G near 0. Proposition 2.1 (Uniform bound on the trajectories). Consider the point-vortex dynamic (4) under the non-neutral clusters hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] with a kernel profile G ∈ C 1,1 loc R * + ∩C 1,1 [1, +∞[ . Then, given any positive time T > 0, there exist a constant C such that for all initial datum

X ∈ R 2N \ C T , sup t∈[0,T ) X -S t G X ≤ C, (19)
where S t G is the (4) flow associated to kernel profile G. Moreover, the constant C depends only on N , the intensities a i , the final time T and on supremum of r → dG dr (r) for r ≥ 1. This constant C does not depend on the initial datum X ∈ R 2N nor on the singularity of the kernel r → G(r) when r → 0.

In [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4], a weaker version of this result is established only for the Euler case. We extend this result to a general case where it holds no matter what the singularity of the kernel G in 0 + is with a proof widely inspired from Theorem 2.1 in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]. We remark that the non-neutral cluster hypothesis (18) is essential. Indeed, the simple situation of a vortex pair with intensities +1 and -1 gives raise to a translation motion along two parallel lines at a speed that blows up as the initial distance between the two vortices goes to 0 + . The trajectories are bounded in finite time but the bound is not uniform, depending on the initial conditions and on the singularity of the kernel near r = 0. We underline that Theorem 2.1 apply to the quasi-geostrophic case given by the Green function of the fractional Laplacian [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF]. In this sense, this theorem is the extension of Theorem 2.1 in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] to the quasi-geostrophic case. Another improvement lays in an explicit computation of the constant appearing in [START_REF] Novikov | Vortex collapse[END_REF] (unlike [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]). It is given by a recursive formula with respect to the number off vortices.

The case of intensities a i all positive

As a further consequence of Theorem 2.1 we can show the the impossibility of collapses in the case where the intensities a i are all positive.

Corollary 2.2. Let G be a profile such that

G(r) -→ +∞ as r → 0. ( 20
)
Assume that the intensities a i are all positive and consider an initial datum X ∈ R 2N for the point-vortex system (4) such that x i = x j for all i = j. Then there is no collapse of vortices at any time.

Contrarily to the existence result given by Theorem 2.2 in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] and its generalization to (SQG) at Theorem 2.7, this result is true for all initial datum and not only for almost every one. Hypothesis (20) may appear a bit restrictive. Indeed, if we consider the kernels

∀ r > 0, dG dr (r) = 1 r α , (21)
with 0 < α < 1 then the associated kernel G does not satisfy [START_REF] Pedlowsky | Geophysical Fluid Dynamics[END_REF]. The possibility to extend Corollary 2.2 to this case is an open problem. Nevertheless, the physical relevant cases are α = 1 for the Euler model or 3 < α < 1 for the quasi-geostrophic model. For these values of α, Corollary 2.2 apply.

The uniform relative bound theorem

A natural question concerning Theorem 2.1 is to ask what this result becomes when the nonneutral cluster hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] ceases to be satisfied. For instance, we consider instead of ( 18) the following hypothesis:

∀ A ⊆ {1 . . . N } s.t. A = ∅ or {1 . . . N }, i∈A a i = 0. ( 22
)
In other words, all the strict sub-clusters must have the sum of their intensities different from 0 but we allow the total sum N i=1 a i to be equal to 0. This situation is achieved for instance by the vortex pair of intensities +1 and -1 that are translating at a constant speed. Proposition 2.3 (Uniform relative bound on the trajectories). For a given set of points noted X = (x 1 . . . x N ) ∈ R 2N , we define the diameter of this set by diam(X) := max

i =j |x i -x j |. ( 23
)
Consider the point-vortex dynamic (4) under hypothesis [START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF] with a kernel profile

G ∈ C 1,1 loc R * + ∩ C 1,1 [1, +∞[ . Let T > 0 the final time. Then for all kernel profile G ∈ C 1,1 loc R * + ∩ C 1,1 [1, +∞[ and for all initial datum X ∈ R 2N that are not leading to collapse on [0, T ), sup t∈[0,T ) diam S t G X ≤ diam(X) + C, (24)
where S t G is the flow associated to (4) with the kernel profile equal to G. Moreover, the constant C depends only on N , the intensities a i , the final time T and on supremum of r → dG dr (r) for r ≥ 1. This constant C does not depend on the initial datum X ∈ R 2N nor on the singularity of the kernel r → G(r) when r → 0. The reasoning is close to the one of Proposition 2.1.

Similarly, the constant is computed explicitly throughout the proof of the theorem.

Convergence result for Euler point-vortices

The systems of vortices that are studied with more details in [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] are the systems for which the non-neutral clusters hypothesis (18) holds. As stated in the previous section, their result concerning uniform bound on the trajectories can be improved to consider a much wider class of point-vortex systems, including the quasi-geostrophic case. Nevertheless, in the particular case of the Euler point-vortex system with the non-neutral clusters hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF], we are also able to write a convergence result. When vortices come to collapse, their speed may become infinite as a consequence of the kernel profile singularity in 0 + . But if their speed blows up, then any pathological behavior near the time of collapse T X is a priori possible. We prove here that the trajectories are actually convergent in the Euler case.

Theorem 2.4 (Convergence for Euler vortices under non-neutral clusters hypothesis). Consider the point-vortex model (4) under hypothesis (18) with a kernel profile G 1 corresponding to the Green function of the Laplacian (3). Let X ∈ C be an initial datum leading to a collapse at time T X . Then, for all i = 1 . . . N , there exists an

x * i ∈ R 2 such that x i (t) -→ x * i as t → T - X . (25)
The first step of the proof consists in the following idea: For a fixed value of t ∈ [0, T X ) define the distribution P t := N i=1 a i δ x i (t) . The point-vortex equation gives that this distribution converges as t → T X . In a second time, it is possible to prove that this convergence is actually stronger and obtain [START_REF] Yudovich | Non-stationary flows of an ideal incompressible fluid[END_REF] by exploiting the non-neutral cluster hypothesis. The proof of this result is specific to the Euler case and we do not know yet whether the conclusion extends to the (SQG) case.

Improbability of collapses for point-vortices

Understanding the sets of collapses C T is an important issue for the study of point-vortices since these sets give the time of existence of the point-vortex system for a given initial datum. Marchioro and Pulvirenti in their study of the point-vortex problem [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF] provided the following improbability result Theorem 2.5 (Improbability of collapses, [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]. Consider the pointvortex problem (4) with kernel profile G 1 , the kernel associated to the Green function of the Laplacian on the plane (3). Assume that the intensities of the vortices satisfy the non neutral cluster hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF].

Then, the set of initial datum for the dynamic (4) that lead to collapses in finite time is a set of Lebesgue measure equal to 0. The non neutral cluster hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] is an important hypothesis for this theorem as it allows us to use Theorem 2.1 that provides a uniform bound. Concerning this result, the most natural question consists in removing this assumption on the intensities of the vortices, which eventually leads to the following conjecture.

Conjecture 2.6. The set of collapses C has a Lebesgue measure 0.

It is not yet possible to prove such a conjecture. The main difficulty lays in the understanding of the situations where some vortices collide in such a way that they go to infinity in finite time, or show an unbounded pathological behavior. The existence of unbounded trajectories in finite time is also an open problem. Although we are not able to answer to Conjecture (2.6), we are able to improve Theorem 2.5 as stated in the following theorem: Theorem 2.7 (improbability of collapses for Euler and SQG vortices). Consider the pointvortex problem (4) with kernel profile G s , the kernel associated to the Green function of the Laplacian or the fractional Laplacian on the plane (3) for s ∈ (0, 1]. Assume that the intensities of the vortices satisfy [START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF].

Then, the set of initial datum leading to collapses has a Lebesgue measure equal to 0.

This theorem is slightly more general than Theorem 2.5 for two reasons. First, it is true both for Euler and for quasi-geostrophic point-vortex models. This aspect was already partially improved by [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF], where they obtained the result for s > 1/2. Indeed, the value s = 1/2 appear to be a critical value for the quasi-geostrophic equations where integrability problems arise. Our arguments manage to pass trough these difficulties and obtain the result for all s ∈ (0, 1]. The second improvement lays in the fact that we managed to replace the non neutral cluster hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] by the weaker hypothesis [START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF]. This weaker hypothesis can be seen at first sight as a small improvement. Nevertheless, it make a quite important difference because the non neutral cluster hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] implies that the trajectories are bounded (Theorem 2.1) whereas hypothesis [START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF] only imply a relative bound (Theorem 2.3). In other words, this improbability result allows a simple unbounded behaviors: the cases where the vortices collectively goes to infinity. In the article [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF], the authors study the evolution in time of the following function:

Φ(X) := N i=1 N j=1 j =i x i -x j -β (26)
when the x i (t) evolves according to the point-vortex equation ( 4) to obtain sufficent conditions for a collapse. Indeed, this function blows up if |x i (t)-x j (t)| → 0 as t → T at a speed depending on the choice of β > 0 and then it is possible to study the collapses as being the set points for wich this function becomes unbounded in time. Unfortunately, the choice of β that must be made for this proof (depending on the value of s) create intergability problems when 0 < s ≤ 1/2. We overcome these problems arising in the proof of [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF] when s ≤ 1/2 by replacing the singularity r -β in the definition of Φ by a regularized version with parameter ε > 0. We then proceed to a reasoning similar to the one of Marchioro and Pulvirenti [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4] and then conclude by letting ε → 0.

Outlines of the proofs

To ease the general reading and understanding of the article, we draw here only the outlines of the proofs of the main Theorems. These larger proofs are decomposed into smaller lemmas stated in this section and forming the main intermediate steps. The links and articulations between the different lemmas are developed in this section but the detailed technical proofs of each different Lemma are all postponed to Section 4. Concerning Propositions 2.1 and 2.3, the proofs are shorter and directly done in section 4.

Outline of the proof for Theorem 2.4

The first part of the proof consists in considering Dirac masses located on the vortices and to prove that this converges in the distributional sense as t → T X , the time of collapse. Lemma 3.1 (Convergence of the Dirac measures). Let X ∈ R 2N . From the vortices x i (t) evolving according to the differential equations (4) with initial datum X, define the distribution

P X (t) := N i=1 a i δ x i (t) , (27)
where δ x denotes the Dirac mass at point x ∈ R 2 . Assume now that the evolution problem associated to the initial datum X is well defined on an interval of time [0, T ) for some T > 0.

Then, there exists

X * ∈ R 2N and b ∈ {0, 1} N such that N i=1 a i δ x i (t) -→ N i=1 a i b i δ x * i in the weak sense of measure as t → T - (28)
and such that

b i = 0 =⇒ sup t∈[0,T [ |x i (t)| = +∞.
Note that the proof makes no use of the non-neutral clusters hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF]. In the particular case where this hypothesis is satisfied, theorem 2.1 implies that the vortices stay bounded on bounded intervals of time. Therefore, in this particular case all the coefficients b i given by this lemma are equal to 1.

The next step of the proof exploits more precisely Hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] and the continuity of the trajectories to obtain that a given vortex x i (t) can only converge as t → T -to one element of the set {x * 1 . . . x * N } given by Lemma 3.1. More precisely, if a given point x i (t) have at least two adherence points, it is possible to extract a third adherence point that does not belongs to {x * 1 . . . x * N } and this provides a contradiction with Lemma 3.1. See Section 4 for a detailed proof.

3.2 Outline of the proof for Theorem 2.7

The modified system

Let i be fixed in {1 . . . N }. The modified system consists in studying the evolution of y ij := x i -x j for j ∈ {1 . . . N } \ {i}. The idea is that knowing the relative position of the vortices (the differences y ij ) is enough to study the problem of collapses. The initial problem (4) implies that

d dt (x i -x j )(t) = k =i a k ∇ ⊥ G s |x i -x k | - k =j a k ∇ ⊥ G s |x j -x k | . (29)
Therefore, the evolution of y ij is given by

d dt y ij = (a i + a j )∇ ⊥ G s |y ij | + k =i,j a k ∇ ⊥ G s |y ik | + ∇ ⊥ G s |y ij -y ik | . (30)
The main interest of this new system is that Theorem 2.7 can be reformulated using only the differences y ij .

Lemma 3.2 (Reformulation of Theorem 2.7). Denote by Y i (t) := y ij (t) j =i the solution to (30) 1) . Assume that for all i ∈ {1 . . . N } and for all T > 0 and ρ > 0 the set

at time t with initial datum Y i ∈ R 2(N -
Y i := (y ij ) j =i ∈ B(0, ρ) 2(N -1) : ∃ T X ∈ [0, T ], lim inf t→T - X min j =i y ij (t) = 0 . ( 31
)
has its Lebesgue measure L 2(N -1) equal to 0. Then in this case the conclusion of Theorem 2.7 holds.

The notation B(x 0 , ρ) refers to the Euclidean ball of R 2 of center x 0 and radius ρ. The rest of the work consists then in studying this system (30) and to establish that (31) does have measure 0. This modified dynamics has many properties in common with the original one. In particular, this new dynamics still satisfies the Liouville property. Define the function H ij : R 2(N -1) → R by

H ij (y ik ) k =i := (a i + a j )G s |y ij | + y ij • k =i,j a k ∇G s |y ik | + k =i,j a k G |y ij -y ik | . ( 32
)
Combining this equation with (30) gives

d dt y ij = ∇ ⊥ y ij H ij (y ik ) k =i . (33)
This equation says that, in a certain sense, the dynamic of the vector Y i := (y ij ) j =i shows an Hamiltonian structure. It is not an Hamiltonian system because the function H ij does depend on j but there is still a structure with an operator ∇ ⊥ . Therefore, the Schwartz theorem gives

div y ij d dt y ij = div y ij ∇ ⊥ y ij H ij (y ik ) k =i = 0. ( 34
)
Since the velocity is divergent-free, a Liouville theorem holds for this dynamics.

Lemma 3.3 (Liouville theorem). The Lebesgue measure on the space R 2(N -1) given by

j =i dy ij ( 35
)
is preserved by the flow S t i associated to (33). A detailed proof of the Liouville theorem in a more general setting can be found at [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF]Part 3] and we refer to it for the proof of Lemma 3.3.

Estimate the collapses

From the kernel G s given at (3), define now the regularized profiles G s,ε for ε ∈ (0, 1]. The objective here is to drop the singularity of the kernel G s near 0 + . We ask G s,ε to be C 1 on R + (until the boundary) and to verify

• G s,ε (q) = G s (q) when ε ≤ q, ( 36
) • |G s,ε (q)| ≤ |G s (q)| for all q ∈ R + , (37)
• d dq G s,ε (q) ≤ d dq G s (ε) for all q ≤ ε, (38)
• |G s,ε (q)| ≤ 2|G s (ε)| for all q ∈ R + . (39)
Since G s,ε is of class C 1,1 on R + , the dynamic defined by (4) for the kernel G s,ε is always well-defined and is Hamiltonian. In the sequel we denote S t i,ε the flow at time t associated to the evolution equation (30) with the kernel profile G s replaced by the regularization G s,ε . The motion induced by the kernel profile G s,ε coincides with the original motion provided that the distances between the vortices remain higher than ε. Theorem 2.7 can be reformulated as follows.

Lemma 3.4 (Reformulation of Theorem 2.7 with ε-Regularized dynamic). Assume that for all T > 0 and for all ρ > 0 we have the following convergence:

L 2(N -1) Y i = (y ij ) j =i ∈ B(0, ρ) 2(N -1) : min j =i inf t∈[0,T ] y ε ij (t) ≤ ε ----→ ε→0 + 0. ( 40
)
Then for all i ∈ {1 . . . N }, the set (31) has Lebesgue measure 0.

This lemma is nothing more than Lemma 3.2 where we added this parameter ε > 0 allowing us to regularize the kernel. Therefore, combined with Lemma 3.2, this lemma gives Theorem 2.7 provided that the convergence (40) holds. Lemma 3.5. Let i ∈ {1 . . . N }, ε > 0 and ρ > 0. Then

L 2(N -1) Y i = (y ij ) j =i ∈ B(0, ρ) 2(N -1) : min j =i inf t∈[0,T ] y ε ij (t) ≤ ε ≤ C    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε 2s
if s < 0.5.

(41) where the constant C only depends on N , |a i |, T and ρ.

The proof of this last lemma is reminiscent from the proof of Theorem 2.5 with some new arguments. The main idea is to rely on a Bienaymé-Tchebycheff inequality applied to a wellchosen function. This last estimate (41) with Lemma 3.4 concludes the proof of Theorem 2.7.

d dt H(S t X) = N i=1 ∇ x i H(S t X) • d dt x i (t) = N i=1 1 a i ∇ x i H(S t X) • ∇ ⊥ x i H(S t X) = 0, ( 42
)
where we used the general fact:

∇ x f (x) • ∇ ⊥ x f (x) = 0.

Proof of the conservation of the vorticity vector (12)

Let X ∈ R 2N \ C T , by direct computation using (4):

d dt M (S t X) = N i=1 a i d dt x i (t) = N i=1 j=1 j =i a i a j ∇ ⊥ G |x i (t) -x j (t)| = 0, ( 43
)
where we used the identity

∇ ⊥ G(| -x|) = -∇ ⊥ G(|x|).

Proof of the conservation of the moment of inertia (15)

Let X ∈ R 2N \ C T , by direct computation using (4):

d dt I(S t X) = 2 N i=1 a i x i (t) • d dt x i (t) = 2 N i=1 N j=1 j =i a i a j x i (t) • ∇ ⊥ G |x i (t) -x j (t)| . (44)
If we proceed to a symmetrization of the double sum above, swapping the indices i ↔ j, and using the identity ∇ ⊥ G(| -x|) = -∇ ⊥ G(|x|), we are led to

d dt I(S t X) = N i=1 N j=1 j =i a i a j x i (t) -x j (t) • ∇ ⊥ G |x i (t) -x j (t)| . (45)
Observing now that the vector ∇ ⊥ G(|x|) is orthogonal to the vector x, we deduce that all the scalar products appearing in the expression above are 0.

Proofs for the uniform bound results

Proof of Propositions 2.1 and 2.3

The two uniform bounds given by Propositions 2.1 and 2.3 are a consequence of the following proposition:

Proposition 4.1. Let N ∈ N with N = 0 and let a i = 0 with i = 1 . . . N . let X ∈ R 2N be an initial datum for the following differential evolution problem for t ∈ [0, T ):

d dt x i (t) = N j=1 j =i a j ∇ ⊥ G |x i (t) -x j (t)| + f i t, x i (t) , (46)
where

G ∈ C 1,1 loc R * + ∩ C 1,1 [1,
+∞[ is the kernel profile and where f i : [0, T ) × R 2 → R 2 are smooth external fields. One makes the assumption that there are no collapses for all t ∈ [0, T ) so that the dynamic (46) is well-defined. One sets:

a := N i=1 |a i |, A 0 := min P⊆{1...N } P =∅, P ={1...N } i∈P a i , A := min P⊆{1...N } P =∅ i∈P a i = min A 0 ; N i=1 a i . (47)
Then there exists a function C : N × (R + ) 5 → R + , given explicitly, that is non-decreasing with respect to any of its 6 variables and such that:

(i) If A 0 = 0, then ∀ t ∈ [0, T ), ∀ i, j ∈ {1 . . . N }, x i (t) -x j (t) -x i (0) -x j (0) ≤ C N, max k f k L ∞ , T, a, 1 A 0 , sup r≥1 dG dr (r) . ( 48
) (ii) If moreover A = 0, then ∀ t ∈ [0, T ), ∀ i ∈ {1 . . . N }, x i (t) -B(t) -x i (0) -B(0) ≤ a A C N, max k f k L ∞ , T, a, 1 A 0 , sup r≥1 dG dr (r) , (49)
where B(t) is the center of vorticity of the system (13).

The first point of this proposition implies the relative uniform bound stated by Proposition 2.3 because the non-neutral sub-clusters hypothesis ( 22) is equivalent to A 0 = 0. Proposition 2.3 is then obtained by choosing f i ≡ 0 for all index i. Similarly, the second point of this proposition implies the uniform bound stated by Proposition 2.1 since the non-neutral clusters hypothesis ( 18) is equivalent to A = 0. Recall that in the case f i ≡ 0, the center of vorticity B is a constant of the movement [START_REF] Gravejat | Smooth travelling-wave solutions to the inviscid surface quasigeostrophic equation[END_REF].

Proof. To start with, the second point of this proposition is a direct consequence of the first one because of the following estimate that holds for all i = 1 . . . N :

x i (t) -B(t) = N j=1 a j x i (t) -x j (t) N j=1 a j ≤ a A max j,k=1...N |x j (t) -x k (t)|. (50)
Then, there remain to prove Proposition 4.1-(i). The function C is constructed using an iterative argument on the number of vortices N > 0 that is similar to the proof of the uniform bound in the book of Marchioro and Pulvirenti [START_REF] Marchioro | Mathematical theory of incompressible nonviscous fluids[END_REF]Chap. 4]. For the iterative arguments, the case N = 1 is straight-forward and gives C(1, . . . ) ≡ 0. Suppose now that the function C(k, . . . ) has been constructed for all k = 1 . . . N -1 with N ≥ 2 and satisfy the announced estimate (48). One is now constructing the function C(N, . . . ). For that purpose, one defines in the view of (47) the following quantities for all P ⊆ {1 . . . N } non empty:

a(P) := i∈P |a i |, A 0 (P) := min Q⊆P Q =∅, Q =P i∈Q a i , A(P) := min Q⊆P Q =∅ i∈Q a i = min A 0 (P); i∈P a i . (51)
The following increasing properties hold:

Q P =⇒ a(Q) < a(P), Q P =⇒ A(Q) ≥ A 0 (P). ( 52
)
There is also, ∀ P ⊆ {1 . . . N }, A 0 (P) ≥ A(P).

Now, let S > 0 be a parameter supposed very large and that is fixed later on. One defines the set

D S := t ∈ [0, T ) : max j,k=1...N |x j (t) -x k (t)| ≥ S , (54)
and t 0 := min D S ∈ [0, T ]. Since the largest distance between vortices is larger or equal to S at time t 0 , then by the triangular inequality the vortices are divided into two nonempty clusters P, Q {1 . . . N } (with P ∪ Q = {1 . . . N } and P ∩ Q = ∅) separated by a distance d that is bounded from below by S/(N -1). Indeed, the least favorable case consists in the situation where all the vortices forms a rectilinear chain made of N points spaced with intervals of same length and that link the two vortices that realize the maximal distance. If the distance d is larger than 1 then the interaction between two vortices that does not belong to the same cluster is bounded by sup r≥1 dG dr (r) . For that purpose, one makes now the assumption that S ≥ N so that d > 1 (this constraint is handled later in the choice of S) n that case, the evolution of the points x i in cluster P, using (46), is given by

d dt x i (t) = j∈P j =i a j ∇ ⊥ G |x i (t) -x j (t)| + f i t, x i (t) , (55)
where,

f i t, x i (t) = f i t, x i (t) + j∈Q j =i a j ∇ ⊥ G |x i (t) -x j (t)| . ( 56
)
In particular,

f i L ∞ ≤ f i L ∞ + a(Q) G , (57)
where we have set the notation:

G := sup r≥1 dG dr (r) . (58)
Then, as long as the distance between the two clusters remain larger than 1, it is possible to apply the result of Proposition 4.1 recursively to (55). Note that A(P) > 0 since A 0 = A 0 ({1 . . . N }) > 0 by hypothesis and P = {1, . . . , N }. It is therefore possible to define the center of vorticity of the cluster P:

B P (t) := j∈P a j -1 j∈P a j x j (t). ( 59
)
Proposition 4.1-(ii) applied recursively to P with a number of vortices #P < N gives that for all index i ∈ P,

∀ t ∈ [t 0 , t 1 ), x i (t) -B P (t) -x i (t 0 ) -B P (t 0 ) ≤ a(P) A(P) C #P, max k∈P f k L ∞ , |t 1 -t 0 |, a(P), 1 A 0 (P) , G , (60)
where

t 1 := sup t ∈ [t 0 , T ) : min i∈P min j∈Q |x i (t) -x j (t)| ≥ 1 . ( 61
)
Note also that t 1 > t 0 since S ≥ N implies that at time t 0 the distance is larger than N/N -1 > 1.

Since the function C is increasing with respect to any of its variables, the estimate (60) with the increasing properties (52) becomes, for all i, j ∈ P, and for all t ∈ [t 0 , t 1 ),

x i (t) -B P (t) -x i (t 0 ) -B P (t 0 ) ≤ a A 0 C N -1, max k f k L ∞ , T, a, 1 A 0 , G , ≤ a A 0 C N -1, max k f k L ∞ + a G , T, a, 1 A 0 , G , (62)
Where ( 57) is used for the second inequality. On the other hand, the center of vorticity is preserved by the flow of the point-vortex dynamics [START_REF] Gravejat | Smooth travelling-wave solutions to the inviscid surface quasigeostrophic equation[END_REF] and therefore B P is only moved by the external field:

∀ t ∈ [t 0 , t 1 ), d dt B P (t) = j∈P a j -1 j∈P a j f t, x j (t) . ( 63
)
Thus,

∀ t ∈ [t 0 , t 1 ), B P (t) -B P (t 0 ) ≤ a(P) A(P) T max k f k L ∞ ≤ a A 0 T max k f k L ∞ + a G , ( 64
)
where was used (57) for the second inequality. One now establishes that t 1 = T if S is chosen large enough. Let t ∈ [t 0 , t 1 ) and let i ∈ P, j ∈ Q. Equations ( 62) and (64) together gives

x i (t) -x j (t) ≥ x i (t 0 ) -x j (t 0 ) -x i (t) -B P (t) -x i (t 0 ) -B P (t 0 ) -B P (t) -B P (t 0 ) -x j (t) -B Q (t) -x j (t 0 ) -B Q (t 0 ) -B Q (t) -B Q (t 0 ) ≥ S N -1 -2 a A 0 C N -1, max k f k L ∞ + a G , T, a, 1 A 0 , G -2 a A 0 T max k f k L ∞ + a G . (65)
If one chooses S big enough so that the last expression above is larger than 2, then the definition of t 1 implies t 1 = T . For that purpose we set the parameter S equal to:

S N := 2(N -1) + 2(N -1) a A 0 C N -1, max k f k L ∞ + a G , T, a, 1 A 0 , G +2(N -1) a A 0 T max k f k L ∞ + a G . (66)
Remark that the constraint S N ≥ N previously required in the case N ≥ 2 is indeed satisfied with such a definition.

It is now possible to establish the announced estimate. First, in the case 0 ≤ t ≤ t 0 , then by definition of t 0 ,

∀ i, j = 1 . . . N, x i (t) -x j (t) -x i (0) -x j (0) ≤ 2S N . ( 67
) Otherwise if t 0 ≤ t < t 1 = T , ∀ i, j = 1 . . . N, x i (t) -x j (t) -x i (0) -x j (0) ≤ x i (t 0 ) -x j (t 0 ) -x i (0) -x j (0) + x i (t) -x i (t 0 ) + x j (t) -x j (t 0 ) (68)
The first term above is estimated similarly as (67). For the other terms, suppose for instance that i ∈ P, then one uses again (62) and (64) to get

x i (t) -x i (t 0 ) ≤ x i (t) -B P (t) -x i (t 0 ) -B P (t 0 ) + B P (t) -B P (t 0 ) ≤ a A 0 C N -1, max k f k L ∞ + a G , T, a, 1 A 0 , G + a A 0 T max k f k L ∞ + a G = S N -(N + 1) 2(N -1) ≤ S N 2 (69)
Thus, gathering the two estimates (67) and (69),

∀ i, j, ∀ t ∈ [0, T ), x i (t) -x j (t) -x i (0) -x j (0) ≤ 3S N . (70)
In view of the definition of S N at (66), it is possible to define the function C(N, . . . ) such that

C N, max k f k L ∞ , T, a, 1 A 0 , G = 3S N (71)
It is a direct computation to check that the function C is increasing with respect to any of its variables and (70) corresponds exactly to (48).

Proof of Corollary 2.2

Since the a i are all positive, the non-neutral clusters hypothesis ( 18) is satisfied and then, as a consequence of Theorem 2.1 the trajectories are bounded by a constant C. Thus,

i =j a i a j G |x i (t) -x j (t)| ≥ a i 0 a j 0 G |x i 0 (t) -x j 0 (t)| + i =j {i,j} ={i 0 ,j 0 } a i a j min 0<r≤C G(r). ( 72
)
where i 0 = j 0 are two fixed indices. The left-hand side in the inequality above is a constant of the motion [START_REF] Geldhauser | Point vortices for inviscid generalized surface quasigeostrophic models[END_REF]. Since by hypothesis G(r) → +∞ as r → 0 + , we conclude that inf t∈[0,T )

|x i 0 (t) -x j 0 (t)| > 0.

Proof of Proposition 2.3

Let S > 0 parameter very large fixed later. Suppose toward a contradiction that there exists an initial datum X ∈ R 2N \ C T , two indices i 0 and j 0 and a time t 0 such that

|x i 0 (t 0 ) -x j 0 (t 0 )| ≥ max i,j |x i -x j | + S. (73)
By continuity of the trajectories, there exists an interval of time [t 1 , t 2 ] such that

|x i 0 (t 1 ) -x j 0 (t 1 )| = max i,j |x i -x j | + S 2 , |x i 0 (t 2 ) -x j 0 (t 2 )| = max i,j |x i -x j | + S, (74)
and such that

∀ t ∈ [t 1 , t 2 ], |x i 0 (t) -x j 0 (t)| ≥ max i,j |x i -x j | + S 2 . ( 75
)
As a consequence of this last property and similarly as in the proof of Theorem 2.1, we can split the system of vortices into two non-empty subsets P and

Q with P ∪ Q = {1 . . . N } such that for all t ∈ [t 1 , t 2 ], min i∈P min j∈Q |x i (t) -x j (t)| ≥ S 2N . (76)
It can be assumed for instance that i 0 ∈ P and j 0 ∈ Q. Therefore, the vortices in the set P evolves according to equation

d dt x i (t) = j∈P j =i a j ∇ ⊥ G |x N i (t) -x N j (t)| + F i (x N i (t), t). (77)
where the external fields F i is the interaction with the vortices that belongs to Q. As a consequence of (76), and if S is chosen large enough, this field satisfies

|F i (x, t)| ≤ sup r≥1 d dr G(r) . (78)
The analogous equation holds for the vortices of the set Q. We now want to apply Proposition 4.1 to the dynamic of the cluster P on [t 1 , t 2 ] given by (77). The non-neutral clusters hypothesis is satisfied for the cluster P as a consequence of ( 22) because P is a strict subset of {1 . . . N }.

We obtain from Proposition 4.1 a constant C such that the dynamic of the cluster P without external field F i is bounded by C. If we add the smooth external field F i (x, t) and since the bound given by Proposition 4.1 is uniform, we end up with sup

t∈[t 1 ,t 2 ] max i∈P |x i (t) -x i (t 1 )| ≤ C + t 2 t 1 sup x∈R 2N |F i (x, t )| dt . (79)
Combining this with (78) gives sup

t∈[t 1 ,t 2 ] max i∈P |x i (t) -x i (t 1 )| ≤ C + T sup r≥1 d dr G(r) . (80)
Doing an analogous argument on the cluster Q gives sup

t∈[t 1 ,t 2 ] max i∈Q |x i (t) -x i (t 1 )| ≤ C + T sup r≥1 d dr G(r) . (81)
On the other hand, the condition (74) implies that

|x i 0 (t 2 ) -x i 0 (t 1 )| ≥ S 4 or |x j 0 (t 2 ) -x j 0 (t 1 )| ≥ S 4 . (82)
The two bounds (80) and (81) are in contradiction with (82) if S is chosen large enough.

4.3 Proofs of the lemmas for Theorem 2. Let P X (t) be the distribution defined by ( 27), assumed well-defined for all t ∈ [0, T [. Let ϕ ∈ D(R 2 ). Then,

d dt P X (t), ϕ D D = d dt N i=1 a i ϕ x i (t) = N i=1 a i ∇ϕ x i (t) • d dt x i (t). ( 83
)
The equations of motion (4) give

d dt P X (t), ϕ D D = i =j a i a j ∇ϕ x i (t) • x j (t) -x i (t) ⊥ |x j (t) -x i (t)| 2 = 1 2 i =j a i a j ∇ϕ x i (t) -∇ϕ x j (t) • x j (t) -x i (t) ⊥ |x j (t) -x i (t)| 2 .
(84) Therefore,

d dt P X (t), ϕ D D ≤ 1 2 i =j |a i a j | ∇ 2 ϕ ∞ . (85)
Then, t → P X (t), ϕ D D is Lipschitz and converges as t → T -. Since this holds for any ϕ ∈ D(R 2 ), this implies that P X (t) converges in the sense of distributions towards some P X ∈ D (R 2 ) as t → T -. There remains to prove that P X is actually a measure that takes the form given by (28). Consider now an increasing sequence (t n ) converging towards T -. We remark first that it is always possible, up to an omitted extraction of the sequence, to reduce the problem to

either x i (t n ) -→ x * i or x i (t n ) -→ +∞ (86)
for some X * ∈ R 2N . Indeed, if x i (t n ) -→ +∞ is not satisfied then there exists an extraction such that x i (t n ) stays bounded. But if it stays bounded then another extraction makes this sequence converge towards some x * i . Repeating this process for i from 1 to N gives (86). Now that (86) holds, define

b i = 0 if x i (t n ) -→ +∞, 1 either. ( 87
)
Therefore it holds

N i=1 a i δ x i (tn) -→ N i=1 a i b i δ x * i as n → +∞, (88)
in the distributionnal sense. By uniqueness of the limit, it is possible to identify

P X = N i=1 a i b i δ x * i . (89)
The fact that the convergence of P X (t) towards P X in D is actually a convergence in the weak sense of measure comes from the fact that the measure P X (t) is bounded by i |a i | for all t.

Proof of Theorem 2.4

• Step 1: Consider the X * ∈ R 2N given by Lemma 3.1. Let z ∈ R 2 such that for all i = 1 . . . N , z = x * i . We are going to prove that for all i = 1 . . . N , lim inf

t→T - |x i (t) -z| > 0. ( 90
)
Suppose toward a contradiction that there exists A ⊆ {1 . . . N } with A = ∅ such that for all i ∈ A, lim inf

t→T - |x i (t) -z| = 0. ( 91
)
This set A can be chosen such that for all i / ∈ A,

lim inf t→T - |x i (t) -z| > 0. ( 92
)
Define

d 1 z := min |x * i -z| : i = 1 . . . N > 0, d 2 z := min lim inf t→T - |x i (t) -z| : i / ∈ A > 0, d * z := min d 1 z , d 2 z > 0, (93)
where by convention, for the definition of d 2 z , the minimum of the empty set is +∞. Let ϕ be a C ∞ function supported on the ball B(z, d * z /2) and equal to 1 on the ball B(z, d * z /4). As a consequence of Lemma 3.1 (and Theorem 2.1 to obtain that b i = 1 for all i) and by definition of d * z we have,

N i=1 a i δ x i (t) , ϕ D ,D -→ N i=1 a i δ x * i , ϕ D ,D = 0 as t → T -. (94)
Using now ( 91) and (92), we infer the existence of an increasing sequence (t n ) n∈N converging towards T -such that for all n ∈ N,

∀ i ∈ A, x i (t n ) ∈ B z, d * z 4 and ∀ i / ∈ A, x i (t n ) / ∈ B z, d * z 2 . ( 95
)
With the definition of ϕ and the definition of d * z given at (93), holds for all n ∈ N,

N i=1 a i δ x i (tn) , ϕ D ,D = i∈A a i . (96)
As a consequence of the non-neutral clusters hypothesis [START_REF] Novikov | Dynamics and statistics of a system of vortices[END_REF] we have i∈A a i = 0. Therefore, Equations ( 94) and (96) are in contradiction and then (90) holds.

• Step 2: Assume now that a given vortex x i (t) has two adherence points. By (90), these two points must be some x * i . For instance, x * j and x * k with j and k such that x * j = x * k . Define the smallest distance between x * j and any other possible adherence point

r * j := min r > 0 : ∃ l = 1 . . . N, |x * j -x * l | = r . (97)
Consider then the circle

S := x ∈ R 2 : |x -x * j (t)| = 1 2 r * j . (98)
Since x * j is inside the ball of radius r * j /2 and x * k is outside, since these two points are adherence points for the dynamics of x i (t) as t → T -and since the trajectories are continuous, there exist an increasing sequence of time (t n ) n∈N converging towards T -such that

∀ n ∈ N, x i (t n ) ∈ S. (99)
By compactness of S, it can be assumed that, up to an extraction,

x i (t n ) → x * ∈ S as n → ∞.
By definition of r * j , for all l = 1 . . . N , x * l / ∈ S. These two facts together are in contradiction with (90) and this concludes the proof of Theorem 2.4. First, the conclusion of Theorem 2.7 can be formulated as follows

L 2N X ∈ R 2N : ∃ T X ∈ R + , lim inf t→T - X min i =j x i (t) -x j (t) = 0 = 0, ( 100
)
where L d refers to the Lebesgue measure of dimension d. It is possible to reduce the problem to bounded intervals of time and bounded regions of space by rewriting (100) as follows.

L 2N +∞ T =1 +∞ ρ=1 X ∈ R 2N : ∃ T X ∈ [0, T ], lim inf t→T - X min i =j x i (t) -x j (t) = 0 and max i =j x i (t = 0) -x j (t = 0) ≤ ρ = 0. (101)
Indeed, we can directly check that the two sets appearing respectively in (100) and (101) are equal. Since the reunion in (101) is a countable reunion, then to conclude that (100) holds it is enough to prove that for all T > 0 and ρ > 0,

L 2N X ∈ R 2N : ∃ T X ∈ [0, T ], lim inf t→T - X min i =j x i (t) -x j (t) = 0 and max i =j x i (t = 0) -x j (t = 0) ≤ ρ = 0. (102)
Now, let T > 0 and ρ > 0. For i fixed in {1 . . . N } denote by T i the isomorphism that gives the position of the point vortices (x k ) N k=1 knowing the position of x i and knowing the differences (y ij ) j =i . In other words define,

T i : R 2 × R 2(N -1) → R 2N (x, Y ) → (x + y i1 . . . x + y i(i-1) , x, x + y i(i+1) . . . x + y iN ). (103)
Thus,

X ∈ R 2N : ∃ T X ∈ [0, T ], lim inf t→T - X min j=1...N j =i
x i (t) -x j (t) = 0 and max i =j

x i (t = 0) -x j (t = 0) ≤ ρ. = N i=1 T i R 2 × Y i := (y ij ) j =i ∈ B(0, ρ) 2(N -1) : ∃ T X ∈ [0, T ], lim inf t→T - X min j =i y ij (t) = 0 . (104)
Using now hypothesis (31) and the Fubini theorem,

L 2N R 2 × Y i := (y ij ) j =i ∈ B(0, ρ) 2(N -1) : ∃ T X ∈ [0, T ], lim inf t→T - X min j =i y ij (t) = 0 = 0. ( 105
) Since T i is a linear map, it is absolutely continuous and therefore maps any sets of Lebesgue measure 0 into sets of Lebesgue measure 0. Therefore, combining this fact with (104) and ( 105) gives (102).

Proof of Lemma 3.4

Let T > 0, ρ > 0 and ε > 0. For i = j, we define the set of initial datum such that occurs an ε-collapse between the two vortices x i and

x j Γ ε,ρ ij := Y i = (y il ) l =i ∈ B(0, ρ) 2(N -1) : ∃ t ∈ [0, T ], y ij (t) ≤ ε . (106)
We also define the time at which occurs the ε-collapse.

Let Y i ∈ j =i Γ ε,ρ ij , T ε Y i := inf t ∈ [0, T ] : min j =i y ij (t) ≤ ε . (107)
We are also interested in the situations where other collapses occur, far from x i . This corresponds to the ε-collapses of vector y jk :=

x j -x k = y ij -y ik . Let k = i, j, define Γ ε,ρ ijk := Y i ∈ Γ ε,ρ ij : ∃ t < T ε Y i , y ij (t) -y ik (t) ≤ ε . (108)
The fact that Γ ε,ρ ijk gives information on whether another ε-collapse occurs far from x i with x j before the expected ε-collapse between x i and x j implies the following inclusion.

Γ ε,ρ ijk ⊆ Γ ε,2ρ kj \ Γ ε,2ρ kji . (109)
This inclusion must be understood as follows. If occurs an ε-collapse between x j and x k before the first ε-collapse between x i and x j (left-hand side of the inclusion above), then in particular we have an ε-collapse between x j and x k (the set Γ ε,2ρ kj in the right-hand side above). Yet, since we do not have an ε-collapse between x j and x i before the ε-collapse between x j and x k , we can remove the set Γ ε,2ρ kji in the right-hand side of the inclusion above. Another important inclusion is

Γ ε,ρ ijk ⊆ Γ ε,2ρ ijk . (110)
These two definitions ( 106) and ( 108) study the ε-collapse on the exact system with kernel G s . We need the same definitions with the regularized kernels G s,ε .

Γ ε,ρ ij := Y i = (y il ) l =i ∈ B(0, ρ) 2(N -1) : ∃ t ∈ [0, T ], y ε ij (t) ≤ ε , T ε Y i := inf t ∈ [0, T ] : min j =i y ε ij (t) ≤ ε , Γ ε,ρ ijk := Y i ∈ Γ ε,ρ ij : ∃ t < T ε Y i , y ε ij (t) -y ε ik (t) ≤ ε . (111)
One remarks now that as long as the quantities y ij and y ij -y ik remain higher than ε for all j = i and k = i, j, then the dynamics of y ij and y ε ij coincide as a consequence of (36). This property implies in particular, using the sets defined at (106), ( 108) and (111),

Γ ε,ρ ij \ k =i,j Γ ε,ρ ijk = Γ ε,ρ ij \ k =i,j Γ ε,ρ ijk . (112)
The hypothesis of Lemma 3.4 can be rephrased as follows: for all ρ > 0, and for all i = j,

L 2N Γ ε,ρ ij -→ 0 as ε → 0 + . (113)
Concerning the conclusion, it is enough to prove that for all ρ > 0, and for all i = j,

L 2N Γ ε,ρ ij -→ 0 as ε → 0 + , (114)
because for all i = 1 . . . N the following equality holds:

Y i := (y ij ) j =i ∈ B(0, ρ) 2(N -1) : ∃ T X ∈ [0, T ], lim inf t→T -
One remarks that it is possible to do the same computation with the remaining term on the very right using again (109) and this gives

Γ ε,2ρ kj \ Γ ε,2ρ kji = Γ ε,2ρ kj \ l =k,j Γ ε,2ρ kjl ∪ l =i,j,k Γ ε,2ρ kjl ⊆ Γ ε,2ρ kj \ l =k,j Γ ε,2ρ kjl ∪ l =i,j,k Γ ε,4ρ lj \ Γ ε,4ρ ljk . (117)
Doing the same computation as above recursively until the residual term is empty and using (110) transforms (116) into

Γ ε,ρ ij ⊆ k =j Γ ε, 2 N ρ kj \ l =j,k Γ ε, 2 N ρ kjl . ( 118
)
Using now (112), we finally get

Γ ε,ρ ij ⊆ k =j Γ ε, 2 N ρ kj \ l =j,k Γ ε, 2 N ρ kjl ⊆ k =j Γ ε, 2 N ρ kj . ( 119
)
Thus the convergences (113) imply the convergences (114) and the lemma is proved.

Proof of Lemma 3.5

Let i ∈ {1 . . . N }, ε > 0 and ρ > 0.

Step 1. Let a > 0. We define a kernel L a by L a (q) := q -2-a .

(120)

and we associate to the kernel L a its ε-regularization L a,ε as defined by (36)-(39). From this we define the function Φ(Y i ) :=

j =i L a,ε |y ij | . ( 121
)
This function is all the most high valued as the system is close to collapse with vortex x i . Denote by S t i,ε the flow of the modified system (30) with the regularized kernel G s,ε . This gives

d dt Φ S t i,ε Y i = j =i ∇L a,ε |y ε ij (t)| • d dt y ε ij (t), = j =i ∇L a,ε |y ε ij (t)| • (a i + a j )∇ ⊥ G s,ε |y ε ij | + k =i,j a k ∇ ⊥ G s,ε |y ε ik | + ∇ ⊥ G s,ε |y ε ij -y ε ik | , = j =i k =i,j a k ∇L a,ε |y ε ij (t)| • ∇ ⊥ G s,ε |y ε ik | + ∇ ⊥ G s,ε |y ε ij -y ε ik | , (122)
where for the last equality we used the identity ∇f • ∇ ⊥ g = 0 that holds for f and g two radial functions. Thus,

d dt Φ S t i,ε Y i ≤ Ψ S t i,ε Y i , (123)
where Ψ Y i :=

Ψ Y i dS -t i,ε Y i dt = B(0,ρ) N -1 Φ(Y i ) dY i + T 0 S t i,ε B(0,ρ) N -1 Ψ Y i dY i dt. (132)
We now make use of hypothesis [START_REF] Rosenzweig | Justification of the point vortex approximation for modified surface quasigeostrophic equations[END_REF] on the intensities of the vorticies. Indeed, this hypothesis allows us to use Theorem 2.3 which states the existence of a constant C independent on ε (but dependent on ρ, T , s and the a i ) such that S t i,ε B(0, ρ) N -1 ⊆ B(0, C ) N -1 .

(133) Thus, the estimate (132) above becomes

B(0,ρ) N -1 sup t∈[0,T ] Φ S t i,ε Y i dY i ≤ B(0,ρ) N -1 Φ(Y i ) dY i + T 0 B(0,C ) N -1 Ψ Y i dY i dt. ≤ Cε -a + C T ε -2-a    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε 2s if s < 0.5. , (134)
where for the last estimate we used (127) and (129).

Step 3. By definition of the function Φ, there exists a constant c > 0 such that, Y i = (y ij ) j =i ∈ B(0, ρ) N -1 : min

j =i inf t∈[0,T ] y ε ij (t) ≤ ε ⊆ Y i = (y ij ) j =i ∈ B(0, ρ) N -1 : sup t∈[0,T ] Φ S t i,ε Y i ≥ c ε -2-a . ( 135
)
Combining this inclusion with the Bienaymé-Tchebycheff inequality gives L 2(N -1) Y i = (y ij ) j =i ∈ B(0, ρ) N -1 : min

j =i inf t∈[0,T ] y ε ij (t) ≤ ε ≤ ε 2+a c B(0,ρ) N -1 sup t∈[0,T ] Φ S t i,ε Y i dY i . (136)
Using now (134) in (136), L 2(N -1) Y i = (y ij ) j =i ∈ B(0, ρ) N -1 : min

j =i inf t∈[0,T ] y ε ij (t) ≤ ε ≤ C    ε if s > 0.5, ε log(1/ε) if s = 0.5, ε 2s
if s < 0.5.

,

(137) where C is a constant that depends on T , ρ, N , s and on the a i . The lemma is proved.

1

 1 Proofs for the Hamiltonian formulation 4.1.1 Proof of the preservation of the Hamiltonian (9) Let X ∈ R 2N \ C T . The Hamiltonian formulation (8) implies

4

 4

4. 3 . 1

 31 Proof of Lemma 3.1

4. 4

 4 Proofs of the lemmas for Theorem 2.

7

 7

4. 4 . 1

 41 Proof of Lemma 3.2

jStep 2 . 1 T 0 Ψ

 210 =i k =i,j a k ∇L a,ε |y ij | ∇G s,ε |y ik | + ∇G s,ε |y ij -y ik | . (124)We now observe, recalling the definition of G s given at[START_REF] Arnold | Mathematical methods of classical mechanics[END_REF], that the ε-regularization (36)-(39) implies, by a direct computation using polar coordinates, ρ) is the Euclidean ball on R 2 . T he constant C depends on ρ and s. Similarly with the definition of the kernel L a at (120) and since a > 0,B(0,ρ) L a,ε |y| dy ≤ Cε -a and B(0,ρ) ∇L a,ε |y| |dy ≤ Cε -1-a . (126)Therefore using (126) with the definition of Φ givesB(0,ρ) N -1 Φ(Y i ) dY i ≤ Cε -a .(127)Similarly, the definition of Ψ given at (124) givesB(0,ρ) N -1 Ψ(Y i ) dY i = B(0,ρ) N -1 j =i k =i,j a k ∇L a,ε |y ij | ∇G s,ε |y ik | + ∇G s,ε |y ij -y ik | using (125) and (126),B(0,ρ) N -1 Ψ(Y i ) dY i ≤ Cε -2-aIt is now possible to integrate Φ along the flow. We obtainB(0,ρ) N -1 sup t∈[0,T ] Φ S t i,ε Y i dY i ≤ B(0,ρ) N -1 Φ(Y i ) dY i + Y i dY i ≤ B(0,ρ) N -1 Φ(Y i ) dY i + B(0,ρ) N -S t i,ε Y i dt dY i . (131)Using the Fubini theorem in (131) and the Liouville theorem 3.3 leads toB(0,ρ) N -1 sup t∈[0,T ] Φ S t i,ε Y i dY i ≤ B(0,ρ) N -1 Φ(Y i ) dY i + T 0 S t i,ε B(0,ρ) N -1

X min j =i y ij (t) = 0 = +∞ n=1 j =i Γ 1 n ,ρ ij . (115)The fact that the convergences (113) imply the convergences (114) is given by the following computations. First, using (109) we getΓ ε,ρ ij = Γ ε,ρ ij \ k =i,j Γ ε,ρ ijk ∪ k =i,j Γ ε,ρ ijk ⊆ Γ ε,ρ ij \ k =i,j Γ ε,ρ ijk ∪ k =i,j Γ ε,2ρ kj \ Γ ε,2ρ kji .(116)

Acknowledgments

I would like to acknowledge my PhD advisors Philippe GRAVEJAT and Didier SMETS for their confidence and their scientific support, constructive criticism and suggestions during all my work on the point-vortex model. I would like also to acknowledge them for their meticulous rereadings and their advises during the redaction of this article.

The author acknowledges grants from the Agence Nationale de la Recherche (ANR), for the project "Ondes Dispersives Aléatoires" (ANR-18-CE40-0020-01). The problem considered in this paper was inspired from the Workshop of this ANR ODA on 27-28 June 2019 in Laboratoire Paul Painlevé (Lille, France).


Yongqiang Tu
email: yongqiang.tu@ensam.eu

Javier A Arrieta-Escobar

Alaa Hassan

Uzair Khaleeq Uz Zaman

Ali Siadat

Gongliu Yang

Optimizing process parameters of direct ink writing for dimensional accuracy of printed layers

Keywords: Extrusion-based 3D printing, Process parameters optimization, Dimensional accuracy, Taguchi method

Direct ink writing (DIW) belongs to extrusion-based 3D printing techniques. The success of DIW process depends on well-printable ink and optimized process parameters. After ink preparation, DIW process parameters considerably affect the parts' dimensional accuracy and process parameters optimization for dimensional accuracy of printed layers is necessary for quality control of parts in DIW. In this study, DIW process parameters were identified and divided into two categories as the parameters for printing a line and the parameter from lines to a layer. Then a two-step method was proposed for optimizing process parameters.

Step 1 was to optimize process parameters for printing a line. In Step 1, continuity and uniformity of extruded filaments and printed rectangular objects were observed in screening experiments to determine printability windows for each process parameter. Then, interaction effect tests were conducted and degree of freedom (DOF) for experiments was calculated followed by orthogonal array (OA) selection for Taguchi design. Next, main experiments of line printing based on Taguchi method were conducted. Signal-to-noise ratio (SNR) calculations and analysis of variance (ANOVA) were performed to find the optimal combination and evaluate the significance respectively. Step 2 was to optimize the parameter from lines to a layer. In

Step 2, the average width of the printed line under optimal condition was measured firstly. Then, single-factor tests of rectangular object printing were conducted to find the optimal parameter from lines to a layer. After these two steps, confirmation results were conducted to verify the reliability of the proposed method and the method robustness on other shapes and other materials; parameter adaptability in 3D parts printing from printed layers' analyses for the proposed method and parameter adaptability in constructs fabricated as 100% infill or with porosities.

Introduction

Direct ink writing (DIW), also known as extrusion-based 3D printing [START_REF] Tarassoli | Searching for the optimal bioink in extrusion-based 3D bioprinting for reconstructive surgery[END_REF], relies on extruding ink-based materials with suitable printability through a nozzle using either mechanical (piston or screw driven) or pneumatic forces. DIW is attracting widespread interest in the fields of tissue engineering [START_REF] Campos | Synchronized Dual Bioprinting of Bioinks and Biomaterial Inks as a Translational Strategy for Cartilage Tissue Engineering[END_REF], industrial applications such as batteries [START_REF] Cao | 3D Printed High-Performance Lithium Metal Microbatteries Enabled by Nanocellulose[END_REF], sensors [START_REF] Kim | 3D Printed Disposable Wireless Ion Sensors with Biocompatible Cellulose Composites[END_REF], fourdimensional (4D) printing [START_REF] Mulakkal | Responsive cellulose-hydrogel composite ink for 4D printing[END_REF], soil science [START_REF] Arrieta-Escobar | 3D printing: An emerging opportunity for soil science[END_REF] and everyday life [START_REF] Park | Conductive Cellulose Composites with Low Percolation Threshold for 3D Printed Electronics[END_REF] because it is capable of printing 3D structures in a flexible, low-cost and well-controlled fashion. The success of DIW process depends on well-printable ink and optimized process parameters.

Recent studies [START_REF] Siqueira | Cellulose Nanocrystal Inks for 3D Printing of Textured Cellular Architectures[END_REF][START_REF] Del-Mazo-Barbara | Rheological characterisation of ceramic inks for 3D direct ink writing: a review[END_REF][START_REF] Román-Manso | Hierarchically Porous Ceramics via Direct Writing of Binary Colloidal Gel Foams[END_REF] focused on inks preparation and rheological analyses as ink rheology plays the most critical role in extrusion process, shrinkage, shape fidelity and structural stability. Meanwhile, DIW process parameters concerned with the equipment considerably affect the parts' dimensional accuracy [START_REF] Luo | Printing single-walled carbon nanotube/Nafion composites by direct writing techniques[END_REF]. In 3D printing, dimensional accuracy of printed layers could affect the whole part quality. At present, monitoring and control of printed layers is a research focus in AM quality control [START_REF] Jin | Autonomous in-situ correction of fused deposition modeling printers using computer vision and deep learning[END_REF][START_REF] Petsiuk | Open source computer vision-based layer-wise 3D printing analysis[END_REF]. He and Huang [START_REF] Luan | Prescriptive Modeling and Compensation of In-Plane Shape Deformation for 3-D Printed Freeform Products[END_REF] predicted the in-plane shape of printed layers to improve the geometric accuracy of AM built products. He et al. [START_REF] He | Profile monitoring based quality control method for fused deposition modeling process[END_REF] captured the image of each printed layer to control the whole process of fused deposition modeling (FDM). Mohammad et al. [START_REF] Bisheh | A layer-by-layer quality monitoring framework for 3D printing[END_REF] proposed a quality monitoring framework for 3D printing through monitoring each layer. Therefore, after well-printable inks are prepared to guarantee shape fidelity and structural stability, process parameters optimization for dimensional accuracy of printed layers is necessary for quality control of parts in DIW. Recently, the selection of process parameters for DIW has been conducted by many scholars [START_REF] Paxton | Proposal to assess printability of bioinks for extrusion-based bioprinting and evaluation of rheological properties governing bioprintability[END_REF][START_REF] Kim | Enhanced rheological behaviors of alginate hydrogels with carrageenan for extrusion-based bioprinting[END_REF][START_REF] Koch | Generic method of printing window adjustment for extrusion-based 3D-bioprinting to maintain high viability of mesenchymal stem cells in an alginate-gelatin hydrogel[END_REF]. However, all these studies just determined a printability window instead of an optimal value and few studies are available for process parameters significance analysis and optimization for DIW. Therefore, it is imperative to investigate the significance of each process parameter and find optimal process parameters setting to improve dimensional accuracy of parts in DIW. Taguchi method [START_REF] Dong | Optimizing process parameters of fused deposition modeling by Taguchi method for the fabrication of lattice structures[END_REF] and machine learning (ML) methods [START_REF] Meißner | Artificial Neural Networks-Based Material Parameter Identification for Numerical Simulations of Additively Manufactured Parts by Material Extrusion[END_REF][START_REF] Li | Improving precision of material extrusion 3D printing by in-situ monitoring & predicting 3D geometric deviation using conditional adversarial networks[END_REF] have been widely used to control process parameters of material extrusion-based 3D printing. ML methods enable the proper parameters selection with little process analyses but need large dataset, additional time and computing resources [START_REF] Goh | A review on machine learning in 3D printing: applications, potential, and challenges[END_REF]. Taguchi technique is more efficient [START_REF] Papazetis | Mapping of deposition-stable and defect-free additive manufacturing via material extrusion from minimal experiments[END_REF] with a significant slash in experimental time and cost [START_REF] Mahmood | Taguchi based process optimization for dimension and tolerance control for fused deposition modelling[END_REF] and it can determine the significance of each parameter by analysis of variance (ANOVA) [START_REF] Zaman | Impact of fused deposition modeling (FDM) process parameters on strength of built parts using Taguchi's design of experiments[END_REF]. The main purpose of this paper is to perform DIW process parameters optimization with minimum material and time. Thus, Taguchi method is adopted in this study. In this paper, we identified DIW process parameters through printing analyses and proposed a two-step optimization method to optimize DIW process parameters for dimensional accuracy of printed layers and analyze the significance of each process parameter. To verify the method, confirmation tests were conducted considering the condition comparison and the method robustness on other shapes and other materials; parameter adaptability in 3D parts printing from printed layers' analyses for the proposed method and parameter adaptability in constructs fabricated as 100% infill or with porosities.

Materials and methods

Materials and DIW process

A piston driven DIW 3D printer TM-081 (Tobeca Company, France) was used as illustrated in Figure 1(a). The piston movement is driven by a mechanical system, which extrudes out ink in the syringe through the nozzle into the air and finally on the substrate. Then, a three-axis movement frame guides the nozzle to move in x and y directions to deposit the extruded filaments on a planar region as nozzle is fixed in z direction. Afterward, the nozzle moves to the next layer and 3D parts is printed layer by layer. Finally, the printed part is solidified through heating-based evaporation [START_REF] Kim | Rubber ink formulations with high solid content for direct-ink write process[END_REF] or ultraviolet (UV) light [START_REF] Arrington | UV-assisted direct ink write printing of fully aromatic Poly(amide imide)s: Elucidating the influence of an acrylic scaffold[END_REF] according to types of printed materials. To verify the method robustness on other materials, a well-printable cellulose-based ink was prepared following the literature [START_REF] Mulakkal | Responsive cellulose-hydrogel composite ink for 4D printing[END_REF]. The ingredients of the cellulose-based ink were: 2.36% cellulose fibers, 0.59% hydroxyethylcellulose (HEC), 1.77% carboxymethyl cellulose (CMC), 0.47% montmorillonite, 0.47% citric acid (CA) and 94.34% demineralized water. The detailed preparation process and rheological characterization of the cellulose-based ink can be found in [START_REF] Mulakkal | Responsive cellulose-hydrogel composite ink for 4D printing[END_REF].

Identification of process parameters

Process parameters were identified through DIW process analysis. As shown in Figure 1(b), DIW process for a printed layer includes two steps.

Step 1 is to print a line by moving nozzle from point A to point B. Then, a layer is printed by arranging lines tightly in Step 2. Thus, DIW process parameters for printed layers are divided into two categories as: 1) Process parameters for printing a line; 2) Extra process parameter from lines to a layer.

Process parameters for printing a line

The process parameters for printing a line is illustrated in Figure 1 In extrusion-based AM, optimal n v depends on p v [START_REF] Comminal | Motion planning and numerical simulation of material deposition at corners in extrusion additive manufacturing[END_REF] and optimal h depends on n D [START_REF] Comminal | Numerical modeling of the strand deposition flow in extrusion-based additive manufacturing[END_REF]. If p v , n v and h were set independently without regard to the relationship between these parameters, the optimization would be inefficient and even incorrect. Thus, process parameters for printing a line were identified as: We supposed that n v and e v should satisfy Eq. ( 1) for optimal process design as follows:

n v , C R (calibrated
ne v R v =. ( 1
)
Where R is the ratio of n v to e v .

The relationship between real piston velocity and e v could be calculated according to conservation of mass. However, real piston velocity is not equal to p v because p v are nominal values of positon velocity set in the G-code. The main sources of errors between p v and real piston velocity came from the uncertainty in the control of the piston velocity. To correct systematic errors in the piston velocity control, the relationship between p v and e v was calculated using Eq. ( 2) according to conservation of mass with a calibration procedure [START_REF] Serdeczny | Experimental validation of a numerical model for the strand shape in material extrusion additive manufacturing[END_REF] as follows:

22 11 44 s p f n e D v C D v   =   . (2)
Where f C is a calibration factor which amends the volume conservation to compensate for sources of errors in the piston velocity due to the uncertainty in the control of the piston velocity.

Eq. ( 1) and Eq. ( 2) were combined and simplified into Eq. (3) as follows: Consequently, DIW process parameters were identified as:

22 22 f nn p n C n ss C DD v v R v R D D =   =   . (3)
1) Process parameters for printing a line:

n v , C R and H ;
2) Extra process parameter from lines to a layer: W R .

Definition and measurement method of dimensional errors

Dimensional errors were divided into dimensional error for printed lines and dimensional error for printed layers according to the two categories of identified process parameters. x-y plane dimensional accuracy is of primary interest and dimensional accuracy in the z axis can be ignored for layer accuracy analyses because previous studies [START_REF] Jin | Autonomous in-situ correction of fused deposition modeling printers using computer vision and deep learning[END_REF][START_REF] Luan | Prescriptive Modeling and Compensation of In-Plane Shape Deformation for 3-D Printed Freeform Products[END_REF][START_REF] Bisheh | A layer-by-layer quality monitoring framework for 3D printing[END_REF] have verified that the use of control charts with image data only considering x-y plane dimensions for each layer could improve quality of parts in 3D printing significantly.

Dimensional error for printed lines and its measurement method

A. Definition of dimensional error for printed lines

As illustrated in Figure 3 w . Thus, totally 29 dimensional errors were used for dimensional accuracy analysis of printed lines. In practice, a ruler was put beside printed lines as reference dimension and photo of printed lines with ruler was taken using a camera (Canon LEGRIA HF R86 Noir, Canon Inc., Japan). Then, photo was imported into MATLAB and widths were measured using virtual ruler of the graphic processing tools in MATLAB.

Dimensional error for printed rectangular objects and its measurement method

A. Definition of dimensional error for printed rectangular objects

As illustrated in Figure 3(c . Average lengths for x and y directions were calculated as follows:

2 2 1 / N i i lx lx N = =  . ( 7
) 3 3 1 / N i i ly ly N = =  . ( 8
)
Designed dimensions for rectangular object in x and y directions were Lx and Ly .

Dimensional errors for the printed rectangular object in x and y directions were defined as follows:

) ii erx lx Lx i N = - = . 2 , ( 1, 2,...,

., )

ii ery ly Ly i N = -= .

B. Measurement method of dimensional error for printed rectangular objects

To reduce measurement error and consider the printing stability in the whole range, measuring points setting for printed rectangular objects was shown in Figure 3(d). Designed dimensions for rectangular object in x and y directions were Lx =30 mm and Ly =20 mm, respectively. In x direction, 10 measuring points were set uniformly as 22 dimensional errors were used for dimensional accuracy analysis of printed rectangular objects. In practice, the lengths measurement method was same as widths measurement for printed lines using ruler beside the printed object and MATLAB.

Orthogonal array selection procedure

Instead of conducting full factorial experiments, Taguchi method based experiments of different process conditions are conducted through an orthogonal array (OA), minimizing the number of experiments while ensuring the validity and robustness of the data [START_REF] Guo | A reinforcement learning decision model for online process parameters optimization from offline data in injection molding[END_REF]. The OA is selected based on degree of freedom (DOF) calculation considering the number of factors, interactions between them and the number of levels of each factor [START_REF] Chockalingam | Optimization of stereolithography process parameters for part strength using design of experiments[END_REF]. The procedure for OA selection is as follows:

(1) Determine the number, levels and interaction effects of factors.

(2) Calculate the total DOF of factors. DOF for a single factor is equal to one less than the number of levels of that factor. DOF for a two-factor interaction effect is equal to ( ) ( ) 11) as follows [START_REF] Li | Energy performance optimisation of building envelope retrofit through integrated orthogonal arrays with data envelopment analysis[END_REF]:

1 total T DOF + . ( 11
)
Where T is number of experiments;

total DOF is total DOF of factors and their interaction effects.

Framework for DIW process optimization

The framework of the two-step process parameters optimization method was shown in Figure 4.

Step 1 was to optimize process parameters for printing a line. In step 1, continuity and uniformity of extruded filaments in the air and on the substrate and printed rectangular objects on the substrate were observed in screening experiments to determine printability windows for each process parameter. Then, interaction effect tests were conducted and DOF for experiments was calculated followed by OA selection for Taguchi design. Next, main experiments of line printing based on Taguchi method were conducted. SNR calculations and ANOVA were performed for Taguchi method based data to find the optimal combination and evaluate the significance respectively.

Step 2 was to optimize extra process parameter from lines to a layer. In Step 2, the average width of printed lines under optimal condition was obtained firstly. Then, single-factor tests of rectangular object printing were conducted to find the optimal W R . Finally, the optimal combination of total identified process parameters was summarized through step 1 and step 2 of the framework. The term 'printability window' is defined as the combination of process parameters required for successful extrusion for DIW [START_REF] Paxton | Proposal to assess printability of bioinks for extrusion-based bioprinting and evaluation of rheological properties governing bioprintability[END_REF]. In experiments, printability window determination criterions include three points: continuity and uniformity of 1) extruded filaments in the air; 2) extruded filaments on the substrate and 3) printed rectangular object on the substrate. Detailed information for screening experiments setting were listed in the Section SI 2 of SI.

Through screening experiments, printability windows of each process parameter for the prepared ink were listed as: (5~11) mm/s for n v ; (1~1.6) for C R ; (0.65~1.05) for H and (0.5~0.9) for W R .

Orthogonal array selection

The lower value, middle value and higher value of printability windows for each factor in step

1 ( n v , C
R and H ) were considered to be level 1, 2 and 3 respectively as listed in Table 1. In this study, only two-factor interaction effects were considered. In interaction effect tests, one factor of n v , C R and H was fixed and the remaining two factors were changed to print lines for interaction effect analysis of the remaining two factors. Average widths of printed lines under each process parameters setting were calculated as indicator to analyze interaction effects. Interaction effect graphs were plotted in Figure 5.

Interaction effect graph is a tool to study whether influence of one factor affects the function of the other factors [START_REF] Senthilkumar | A study of individual and interaction effect of process parameters on friction welded AISI 410 and AISI 430 joint[END_REF]. In the graph, parallel line shows no interaction effect and the intersect or intersect trend between lines means a high degree of interaction [START_REF] Kim | Design of experiment (DOE) method considering interaction effect of process parameters for optimization of copper chemical mechanical polishing (CMP) process[END_REF]. As illustrated in DOF of single factor and their interaction effects were calculated according to Section 2.4 and the total DOF of factors was 18. Thus,

SNR analysis

Main experiments of step 1, containing 27 experimental runs, were conducted as shown in Table 2. Based on the requirements of response, SNR was divided into three categories namely medium-the-better, higher-the-better and lower-the-better [START_REF] Sivaiah | Modeling and optimization of sustainable manufacturing process in machining of 17-4 PH stainless steel[END_REF]. In this study, the dimensional error is the lower-the-better to enhance the printing accuracy. Hence, Eq. ( 12) was used to calculate the SNR; and results were listed in Table 2.

Where  was the average SNR, n was the number of experiments conducted at level i and i y was the evaluation value (dimensional error in this study).

The mean SNR response of each process parameters in each level was calculated and listed in Table 3. Figure 5(b) represented the mean SNR graph. Higher SNR represents the minimum variation difference between the desired output and evaluated output. Therefore, optimal process parameters for high dimensional accuracy using Taguchi method were found as:

n v =5 mm/s, C R =1.3, H =0.65.

ANOVA results

ANOVA is a tool to investigate the significant effect and indicate the contribution of process parameters in machinability researches [START_REF] Geng | Effects of Printing Parameters on the Mechanical Properties of High-Performance Polyphenylene Sulfide Three-Dimensional Printing[END_REF]. ANOVA results for dimensional error of printed lines obtained using Minitab software was shown in Table 4. From Table 4, it was found that dimensional error was significantly influenced by At the beginning of step 2, lines were printed using optimal process parameters obtained in step 1 to get line width under optimum condition. Average width of the optimal printed line was calculated using Eq. ( 5) as 1.31 mm.

Single-factor tests

In order to get a layer from lines, distance between lines l d was calculated by Eq. ( 4). In Eq.

(4), average width of optimal lines O w was obtained at the beginning of step 2; and only single-factor W R determines printing results in step 2. Single-factor tests, containing 5 experimental runs, were conducted as shown in Table 5 to find optimal value of W R . The lower-the-better SNR was used and results were listed in Table 5. Higher SNR represents better dimensional accuracy. Therefore, optimal W R were found as: W R =0.9. Consequently, optimal DIW process parameters for a layer fabricated in the prepared ink were found as:

n v =5 mm/s, C R =1.3, H =0.65, W R =0.9.

Confirmation tests 4.3.1 Confirmation tests for method robustness on shapes

To validate that the optimum conditions can be generalized to layers in other shapes, three shapes as Shape A (disk), Shape 2 (leaf) and Shape C (maple leaf) were printed under optimal condition (achieved by proposed method) and condition A (

n v =8 mm/s, C R =1, H =0.85, W R =0.8
). As shown in Figure 6(a), the dimensional quality under optimal condition was higher than that under condition A, verifying the method robustness in other shapes.

Confirmation tests for method robustness on materials

To verify the method robustness on materials, a well-printable cellulose-based ink was chosen. The optimized process parameters for the cellulose-based ink in DIW obtained using the proposed method were: n v =10 mm/s, C R =1.5, H =0.25, W R =0.5. As shown in Figure 6(a), printed layers in three shapes using the cellulose-based ink have high dimensional accuracy same as Nivea Crème, verifying the method robustness on materials. Moreover, compared with the optimized process parameters for cellulose-based ink (low viscosity) and Nivea Crème (high viscosity), it could be concluded that ink with higher viscosity in DIW needs smaller values for

Confirmation tests for parameter adaptability in 3D parts printing

In order to verify the adaptability of optimized process parameter from printed layers' analyses in 3D parts printing, three 3D models were printed using the same optimized process parameters and layer height was set as h for printed layers in Nivea Crème based DIW. As shown in Figure 6(b), the printed 3D parts had good dimensional quality by comparing printed parts under optimal condition and condition A (

n v =8 mm/s, C R =1, H =0.85, W R =0.8
), verifying that the proposed method has parameter adaptability in 3D parts printing from printed layers.

Confirmation tests for parameter adaptability in constructs with porosities

To verify the adaptability of optimized process parameter for constructs with porosities, confirmation tests were conducted as shown in Figure 7. Two grids named as Grids A and Grids B were designed as shown in Figure 7(a): the total dimensions of grids A and grids B are same in 31.2 mm X 31.2 mm; and the dimensions of porosity in grids A and grids B are 3 mm X 3 mm and 6.8 mm X 6.8 mm, respectively. As shown in Figure 7(b) and Figure 7(c), the printed single-layer and 10-layers grids had good dimensional quality by comparing printed grids under optimal condition and condition A (

n v =8 mm/s, C R =1, H =0.85, W R =0.8),
verifying that the proposed method has parameter adaptability in constructs with porosities.

Contributions and future work

Dimensional accuracy of printed parts in DIW depends both on inks' material properties (especially rheological properties) and process parameters. To fully understand DIW, mathematical and simulation models were established to predict printing resolution using rheological properties and process parameters as input factors [START_REF] Suntornnond | A Mathematical Model on the Resolution of Extrusion Bioprinting for the Development of New Bioinks[END_REF][START_REF] Gala | Theoretical Evaluation of the Melting Efficiency for the Single-Screw Micro-Extrusion Process: The Case of 3D Printing of ABS[END_REF][START_REF] Liu | Assessing the dynamic extrusion-based 3D printing process for power-law fluid using numerical simulation[END_REF]. However, scholars were focusing on inks preparation and rheological analyses to realize a successful DIW and process parameters were determined by simple variable tests. Few work fully analyzed process parameters. It is useful and necessary to analyze process parameters through process analyses and optimize them as process parameters identification and optimization are important for DIW applications. This work focuses on the identification and optimization of process parameters for well-printable inks (such as Nivea Crème and cellulose-based ink) whose rheological properties are already analyzed in the literature. The contributions of this work include: (a) key process parameters identification through process analyses; (b) process parameters optimization through the proposed two-step method.

In the future, a model for DIW using the identified process parameters in this work and inks' rheological properties as input factors through analytical and simulation methods will be established to fully understand the DIW process.

Conclusion

In this study, the DIW process parameters were identified through process analyses and a two-step method was proposed to optimize DIW process parameters for dimensional accuracy and analyze the significance of each process parameter. In the DIW process parameters identification, DIW process parameters for printed layers were divided into two categories and identified by process analysis as: 1) process parameters for printing a line and 2) extra process parameter from lines to a layer. The process parameters identification considered the relationship between intermediate process parameters as well as main sources of errors in the experiments, thus improved the efficiency and accuracy of the optimization process. In the two-step optimization method for identified DIW process parameters, Step 1 was to optimize process parameters for printing a line and Step 2 was to optimize the parameter from lines to a layer. Confirmation results verified the proposed method and the method robustness on other shapes and other materials; parameter adaptability in 3D parts printing from printed layers' analyses for the proposed method and parameter adaptability in constructs fabricated as 100% infill or with porosities. This proposed two-step optimization method satisfies the requirement of high quality DIW process by finding optimum process parameters setting with minimum materials and time. The proposed method will allow a rapid and reproducible process parameters optimization of a wide variety of inks in DIW.

Figure 1 .

 1 Figure 1. Process parameters identification for DIW process: (a) the DIW 3D printer; (b) the DIW process; (c) process parameters setting in Step 1.Nivea Crème (Art. No. 80104) (Beiersdorf Global AG, Germany) was chosen as material to verify the proposed method because Nivea Crème is used as printability reference for inks to test the process parameters in DIW and can represent numerous printable inks in DIW[START_REF] Paxton | Proposal to assess printability of bioinks for extrusion-based bioprinting and evaluation of rheological properties governing bioprintability[END_REF] (many inks in DIW are designed and produced to have the same printability of Nivea Crème). A complete rheological characterization for Nivea Crème was conducted as ink rheology plays the most critical role in DIW. The yield stress value was determined as 534 Pa through shear stress ramp test (Figure2(a)), which means the ink stays rest when the shear stress value is below 534 Pa and starts to flow when the shear stress value exceeds 534 Pa. Figure2(b)shows the shear-thinning characterization of the ink as viscosity decreases with the increase of shear rate. The ink exhibits a solid-like response as its storage modulus (G') exceeds its loss modulus (G'') (Figure2(c)), which means the ink has low shrinkage and good shape fidelity. Moreover, the structural stability of the ink is also good as shown in Figure2(d): from 0-200 s, the viscosity stays stable and in high value when shear rate value is near 0; from 200-300 s, the

Figure 2 .

 2 Figure 2. Rheological characterization of Nivea Crème (Art. No. 80104): (a) Yield stress determination through shear stress ramp test; (b) Viscosity as a function of shear rate; (c) Storage (G') and loss (G'') modulus as a function of frequency; (d) stability analysis through recovery test.

  the main sources of errors in the experiments, improves the efficiency and accuracy of the optimization process.2.2.2 Extra process parameter from lines to a layerAs illustrated in Figure1(b), O w (average width of the optimal printed line) of printed line under optimal condition was measured in Step 1. Then, a rectangular object (reprehensive of the printed layers) was printed by controlling l d (distance between lines) line by line. In extrusion-based AM, optimal l d depends onO w[START_REF] Serdeczny | Numerical simulations of the mesostructure formation in material extrusion additive manufacturing[END_REF]. Thus, extra process parameter from lines to a layer was identified as

R

  mainly reflects the impact of W (the weight of the printed layer) on distortion in x-y plane (detailed relationship between W R and W as well as its derivation process were shown in the Section SI 1 of Supporting Information (SI)).

(a), 1 NFigure 3 . 1 w to 27 w

 13127 Figure 3. Definition and measurement method of dimensional errors: (a) definition of dimensional errors for printed lines; (b) measuring points setting for printed lines; (c) definition of dimensional errors for printed rectangular objects; (d) measuring points setting for printed rectangular objects.B. Measurement method of dimensional error for printed linesAs shown in Figure3(b), to reduce measurement error and consider the printing stability in the whole range, three lines were printed (each line was 100 mm long) and 9 measuring points were set for each line uniformly. 27 measuring points were uniformly distributed along lines as

), 2 N and 3 N

 23 measuring points were set along x and y directions in the printed rectangular object uniformly and lengths on each measuring point along x and y direction were measured as

2 S

 2 are DOF of the two factors. Total DOF of factors is cumulative sum of DOF of all single factors and DOF of all two-factor interaction effects. (3) OA is selected from commonly used standard OAs based on total DOF of factors as number of experiments should require Eq. (

Figure 4 .

 4 Figure 4. Framework of the proposed two-step method for optimizing DIW process parameters for printed layers.

Figure 5 RFigure 5 .

 55 Figure 5, interaction effects of

Figure 6 .

 6 Figure 6. Confirmation tests for the proposed method in (a) verification of method robustness on layer shapes and materials; (b) verification for parameter adaptability in 3D parts printing.

Figure 7 .

 7 Confirmation tests for the proposed method in constructs with porosities: (a) two designed grids: grids A and grids B; (b) printed grids A under optimal condition and condition A in single layer and 10-layers; (c) printed grids B under optimal condition and condition A in single layer and 10-layers.

Table 1 .

 1 Levels of process parameters in step 1.

	Parameter Range	Level 1	Level 2	Level 3
	v	(5~11) mm/s	5	8	11
	n

  experiments of step 1 as shown in Table2, from commonly used standard OAs for three-level Taguchi design including 9

	L , 18 L and

27

L OA ( T L means OA with T experiments) was selected for main 27 L according to Eq.

[START_REF] Luo | Printing single-walled carbon nanotube/Nafion composites by direct writing techniques[END_REF]

.

Table 2 .

 2 Experimental plan and calculated SNR for main experiments of step 1.

		Experimental condition
	Experiment number	v /( mm/s)	R	H	SNR of dimensional errors
		n	C
	N1	5	1	0.65	22.51
	N2	5	1	0.85	18.75
	N3	5	1	1.05	22.73
	N4	5	1.3	0.65	31.60
	N5	5	1.3	0.85	28.22
	N6	5	1.3	1.05	42.12
	N7	5	1.6	0.65	38.58
	N8	5	1.6	0.85	30.33
	N9	5	1.6	1.05	35.91
	N10	8	1	0.65	20.81
	N11	8	1	0.85	20.64
	N12	8	1	1.05	22.07
	N13	8	1.3	0.65	37.32
	N14	8	1.3	0.85	33.76
	N15	8	1.3	1.05	25.78
	N16	8	1.6	0.65	22.82
	N17	8	1.6	0.85	25.37
	N18	8	1.6	1.05	29.11
	N19	11	1	0.65	36.61
	N20	11	1	0.85	24.71
	N21	11	1	1.05	23.74
	N22	11	1.3	0.65	27.22
	N23	11	1.3	0.85	26.96
	N24	11	1.3	1.05	25.94
	N25	11	1.6	0.65	26.78
	N26	11	1.6	0.85	23.98
	N27	11	1.6	1.05	32.92

Table 3 .

 3 Mean SNR response table for dimensional error of main experiments in step 1.

	Process parameters	Mean SNR for dimensional error Level 1 Level 2	Level 3	Max-Min	Rank
	v	30.08	27.65	27.65	3.67	2
	n
	R	23.62	29.53	29.53	7.37	1
	C
	H	29.36	28.92	28.92	3.50	3

Table 4 .

 4 ANOVA of dimensional error for main experiments in step 1.

	and	v on dimensional error was 27.78%, 6.64% and 6.37%, respectively, meaning	R is the
		n						C
	most important process parameters for dimensional error of printed lines. Meanwhile,
	interaction effects of	vR  ,	vH 	and	RH 	influenced dimensional error in 26.65%,
				nC		n		C
	7.01% and 4.41% percentage of contribution respectively. Thus, OA selection for Taguchi
	design should consider interaction effects of	vR  ,	vH 	and	RH 	as interaction
								nC	n	C
	effects of	vR  ,	vH 	and	RH 	had big influences on dimensional error.	R is
			nC	n			C	C
	highly significant in dimensional error at 95% level of confidence as p-value<0.05. Thus,	R
								C
	was principal in single process parameters setting which should be paid more attention in
	DIW.
	Source	DOF	Sum of squares (SS)	Mean squares (MS)	F-ratio	p value	Contribution (%)

C R followed by nC vR  , n vH  , H , n v and C RH  . The contribution percentage of process parameters setting for C R , H

Table 5 .

 5 Single-factor tests for step 2.

	Experiment number	NR1	NR2	NR3	NR4	NR5
		0.5	0.6	0.7	0.8	0.9
	R
	W
	SNR of dimensional error	1.96	0.88	1.73	0.38	16.95

Author Disclosure Statement

No competing financial interests exist.

Funding Information

The authors would like to acknowledge the funding from the China Scholarship Council scholarship (No. 201906020135).