Jacek Jendrej
email: jendrej@math.univ-paris13.fr

KINK NETWORKS FOR SCALAR FIELDS IN DIMENSION 1 + 1

We consider a scalar eld equation in dimension 1+1 with a positive external potential having non-degenerate isolated zeros. We construct weakly interacting pure multi-solitons, that is solutions converging exponentially in time to a superposition of Lorentz-transformed kinks, in the case of distinct velocities. We nd that these solutions form a 2K-dimensional smooth manifold in the space of solutions, where K is the number of the kinks. We prove that this manifold is invariant under the transformations corresponding to the invariances of the equation, that is space-time translations and Lorentz boosts. each of which is parallel to the linear space E := H 1 (R) × L 2 (R) which we call the energy space, see Section 2.1.

∂ 2 t φ(t, x) -∂ 2 x φ(t, x) + W (φ(t, x)) = 0, (t, x) ∈ R × R, φ(t, x) ∈ R.
We study (1.1) for potentials W satisfying the following conditions: (A1) W (φ) ≥ 0 for all φ ∈ R, (A2) Ω := W -1 (0) ⊂ R has no accumulation points, (A3) W (ω) > 0 for all ω ∈ Ω.

(A4)

∞ 0 W (φ) dφ = 0 -∞ W (φ) dφ = ∞.
Any ω ∈ Ω is called a vacuum. We sort them in the increasing order, so that Ω = {ω n } n∈I , I ⊂ Z, ω n < ω n+1 for all n, n + 1 ∈ I. Typical examples include the φ 4 model W (φ) = (1 -φ 2 ) 2 , the φ 6 model W (φ) = φ 2 (1 -φ 2 ) 2 , and the sine-Gordon model W (φ) = 1 -cos(φ).

We denote (φ 0 , φ0 ) a generic element of the phase space. The potential energy E p , the kinetic energy E k and the total energy E are given by

E p (φ 0 ) = R 1 2 (∂ x φ 0 ) 2 + W (φ 0 ) dx, E k ( φ0 ) = R 1 2 ( φ0 ) 2 dx, E(φ 0 , φ0 ) = R 1 2 ( φ0 ) 2 + 1 2 (∂ x φ 0 ) 2 + W (φ 0 ) dx.
The set of nite energy states (φ 0 , φ0 ) is a union of |I| 2 ane spaces, called sectors, (

E m,n := {(φ 0 , φ0 ) : E(φ 0 , φ0 ) < ∞ and lim

x→-∞ φ 0 (x) = ω m , lim x→∞ φ 0 (x) = ω n },
Equation (1.1) admits static solutions. They are the critical points of the potential energy. The trivial ones are the vacuum elds φ(t, x) = ω n for some n ∈ I. The solution φ(t, x) = ω n has zero energy and is the ground state in E n,n . There are also non-constant static solutions φ(t, x) connecting two consecutive vacua, that is One can discribe all these solutions. For all n, n ∈ I with |n -n | = 1 there exists a function H n,n (x) such that all the static solutions satisfying (1.3) are φ(t, x) = H n,n (x -a) for some a ∈ R.

(1.3)
These solutions are the ground states in E n,n :

(1.4) inf (φ 0 , φ0 )∈E n,n E(φ 0 , φ0 ) = E p (H n,n ).

If n = n + 1, we call these static solutions the kinks. If n = n -1, we call them antikinks. Remark 1.1. There is in general no canonical choice of H n,n among the family of its space translates. For every pair n, n ∈ I such that |n -n | = 1, we make an arbitrary choice.

An important property of the equation (1.1) is the invariance by space-time translations and Lorentz transformations (1.5) (t, x) = t 0 + γ(t + vx ), x 0 + γ(x + vt ) ⇔ ⇔ (t , x ) = γ(t -t 0 -v(x -x 0 )), γ(x -x 0 -v(t -t 0 )) ,

where (t 0 , x 0 ) ∈ R 2 , -1 < v < 1 and γ := (1 -v 2 ) -1/2 , that is, if ψ(t , x ) is a solution of (1.1) in some region of the (t , x )-space-time, then φ(t, x) := ψ(t , x ) is a solution of (1.1) as well, in the corresponding region of the (t, x)-space-times. Applying this transformation to the static solutions found above, we obtain the moving kinks and antikinks φ(t, x) = H n,n (γ(x -vt -a)), |n -n | ≤ 1.

One can easily check that these are all the travelling waves, that is all the nite-energy solutions of (1.1) such that φ(t, x) = ψ(x -vt) for some function ψ : R → R and some v ∈ R.

Denition 1.2. Let K ∈ {0, 1, 2, . . .}. If n = (n 0 , n 1 , . . . , n K ) ∈ I K+1 is a sequence such that |n k-1 -n k | = 1 for all k ∈ {1, . . . , K}, we say that n is a chain of vacua. We say that

S (K) := {v = (v 1 , . . . , v K ) ∈ R K : -1 < v 1 < . . . < v K < 1}
is the set of admissible velocities.

Fix K ∈ {0, 1, 2, . . .} and a chain of vacua n.

Given v = (v 1 , . . . , v K ) ∈ S (K) and a = (a 1 , . . . , a K ) ∈ R K , we set H(v, a; t, x) := ω n 0 + K k=1 H n k-1 ,n k (γ k (x -v k t -a k )) -ω n k-1 , where γ k := (1 -v 2 k ) -1 2
is the Lorentz factor. Thus, for t large, H(v, a) is a superposition of translated and Lorentz-transformed kinks, separated by large distances.

1.2. Statement of the results and comments. Our rst goal is to construct smooth multisoliton solutions, that is solutions converging to H(v, a; t, x) in the energy space. The function space relevant for us is the space of functions dened for large times whose energy decays exponentially.

For T ∈ R and δ > 0, we set

ψ 2 H T,δ := sup t>T e 2δt R |∂ t ψ(t, x)| 2 + |∂ x ψ(t, x)| 2 + |ψ(t, x)| 2 dx < ∞.
We can state our result on the existence and uniqueness of solutions which converge to multi-kinks exponentially in time.

Theorem 1.3. For all K ∈ N 0 , chain of vacua n, v ∈ S (K) and a ∈ R K , there exist

T 0 ∈ R, δ 0 > 0 and Ψ(v, a; •, •) ∈ H T 0 ,δ 0 such that φ(t, x) := H(v, a; t, x) + Ψ(v, a; t, x) is a solution of (1.1). If T ∈ R, δ > 0 and ψ ∈ H T,δ is such that φ(t, x) = H(v, a; t, x) + ψ(t, x) is a solution of (1.1), then ψ(t, x) = Ψ(v, a; t, x).
Next, we consider the action of the transformations (1.5) on the set of the constructed solutions. This is not a trivial task, since there is no absolute time, hence functions decaying exponentially in time t could have some other behaviour with respect to another time t . It is easy to check that, under this transform, H(v, a) is mapped to H(v , a ), where (1.6)

v j := v j -v 1 -v j v , a j := (γ j ) -1 γ j (a j + v j t 0 -x 0 )
(the formula for v j is the well-known velocity-addition formula in Special Relativity).

Theorem 1.4. The set of the solutions constructed in Theorem 1.3 is invariant under the transformations (1.5): if (t, x) and (t , x ) are related by (1.5), then

H(v, a; t, x) + Ψ(v, a; t, x) = H(v , a ; t , x ) + Ψ(v , a ; t , x ),
where v and a are given by (1.6).

Finally, we address the question of smoothness of these objects.

Theorem 1.5. If K ∈ N 0 , n is a chain of vacua and A ⊂⊂ S (K) × R K is an open set with compact closure, then (T 0 , δ 0 ) in Theorem 1.3 can be chosen uniformly for all (v, a) ∈ A. The function

Ψ : A × (T 0 , ∞) × R → R constructed in Theorem 1.3 is of class C ∞
and all its partial derivatives decay exponentially in t → ∞.

Remark 1.6. Uniqueness of pure multi-kink solutions could also be proved, modulo minor technicalities, under the assumption of convergence faster than any power of t (instead of the exponential convergence). Uniqueness under no assumptions on the rate of convergence remains an interesting open problem. Note that in the elliptic case this was achieved by del Pino, Kowalczyk and Pacard [START_REF] Del Pino | Moduli space theory for the Allen-Cahn equation in the plane[END_REF], and for KdV and generalised KdV by Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF].

Remark 1.7. We can say that S (K) × R K is the parameter manifold of multikinks in the forward time direction for a given chain of vacua (under the assumption of exponential in time convergence), and Φ denes the solution map. The Lorentz transforms act on this parameter manifold, and the quotient manifold parametrises the multikink solutions, up to invariances of the equation. It has dimension 0 for K = 1 and dimension 2K -

3 if K ≥ 2.
Remark 1.8. In the statement of Theorem 1.4, Ψ(v, a; t, x) and Ψ(v , a ; t , x ) are, a priori, welldened only if both t and t are large enough, but in fact, under our assumptions, the problem is globally well-posed, so that the Lorentz transform is well-dened.

Remark 1.9. A well-known open problem is to describe the collisions, that is the behaviour of a given multikink as t → -∞, see for example [START_REF] Goodman | Kink-antikink collisions in the φ 4 equation: the n-bounce resonance and the separatrix map[END_REF] for a mathematically non-rigorous discussion. It is expected that, for generic W , the collisions are inelastic, meaning that for t → -∞ the solution has a dierent behaviour than for t → ∞. In this paper, we only consider the evolution in one time direction.

1.3. Comparison with previous works. Several multi-soliton constructions followed the paper by Martel [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] cited above, see [START_REF] Combet | Multi-soliton solutions for the supercritical gKdV equations[END_REF][START_REF] Ming | Multi-solitons and related solutions for the water-waves system[END_REF], and in particular [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF][START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF] devoted to the Klein-Gordon equations, very similar to (1.1). These works focused on the questions of existence, smoothness and uniqueness of multi-solitons for given velocity and shift parameters.

Our contribution is to study the solution map, which involves the examination of smooth dependence of solutions with respect to the velocities and shifts. The action of Lorentz transforms on multi-solitons for wave equations was not claried in the existing literature either.

All of these works followed the scheme introduced in [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] and [START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF], based on energy estimates, weak compactness and weak continuity of the ow. Here, we propose an alternative approach, based on the Contraction Principle, which we believe is a convenient way of proving smooth dependence with respect to velocities and shifts. Most likely, the weak compactness approach of [START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF][START_REF] Merle | Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity[END_REF] could also be adopted. We point out, however, that proving smooth dependence on the parameters is a more dicult task than proving smoothness of each multi-soliton with respect to t and x. Indeed, the latter can be easily deduced from the regularising eect of the equation. In contrast, we believe that smoothness with respect to the velocities and shifts cannot be directly inferred from the arguments of [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF][START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF].

1.4. Acknowledgements. J.Jendrej was supported by ANR-18-CE40-0028 project ESSED.

We would like to thank Michaª Kowalczyk for enlightening discussions at an early stage of this project and for his detailed explanations about the paper [START_REF] Del Pino | Moduli space theory for the Allen-Cahn equation in the plane[END_REF].

We also thank the referees for suggesting many improvements in the submitted version of the manuscript.

2. Kinks, multikinks and their coercivity properties 2.1. Finite energy sectors. We prove that any nite energy state lies in one of the sectors dened by (1.2).

Proposition 2.1. If φ 0 : R → R is a measurable function which satises E p (φ 0 ) < ∞, then

φ 0 ∈ C(R) and there exist m, n ∈ I such that lim x→-∞ φ 0 (x) = ω m , lim x→∞ φ 0 (x) = ω n .
Proof. The proof is based on the so-called Bogomolny trick. Consider the auxiliary function

Γ(φ) := φ 0 2W (y) dy.
Then Γ is a strictly increasing C 1 function and assumption (A4) yields lim φ→±∞ Γ(φ) = ±∞.

Since ∂ x φ 0 ∈ L 2 (R), it is clear that φ 0 ∈ C(R). Fix x 1 , x 2 ∈ R, x 1 < x 2 . We will check that (2.1) |Γ(φ 0 (x 2 )) -Γ(φ 0 (x 1 ))| = x 2 x 1 1 2 (∂ x φ 0 ) 2 + W (φ 0 ) dx - 1 2 x 2 x 1 |∂ x φ 0 | -2W (φ 0 ) 2 dx ≤ x 2 x 1 1 2 (∂ x φ 0 ) 2 + W (φ 0 ) dx.
By approximation, we can assume φ 0 ∈ C 1 ([x 1 , x 2 ]). The chain rule yields

|∂ x Γ(φ 0 (x))| = |∂ x φ 0 (x)| 2W (φ 0 (x)) = 1 2 (∂ x φ 0 (x)) 2 + W (φ 0 (x)) - 1 2 |∂ x φ 0 | -2W (φ 0 ) 2 ,
and it suces to integrate in x.

Since E p (φ 0 ) < ∞, we deduce that lim x→∞ Γ(φ 0 (x)) exists and is nite. This implies, in turn, that lim x→∞ φ 0 (x) exists and is nite. It is immediate from the denition of E p that E p (φ 0 ) < ∞ implies lim x→∞ φ 0 (x) ∈ Ω. The situation is analogous for x → -∞.

We skip the discussion of the Cauchy theory for (1.1). By well-established methods, one can obtain that (1.1) is locally well-posed in each sector E m,n and the energy is conserved. The computation above shows, using again the assumption (A4), that a solution of nite energy is bounded, which implies the absence of blow-up, hence global well-posedness in each sector. We thus have the following result, which we state without proof. Proposition 2.2. For all m, n ∈ I, (φ 0 , φ0 ) ∈ E m,n and t 0 ∈ R there exists a unique solution

(φ, ∂ t φ) ∈ C(R, E m,n ) of (1.1) such that (φ(t 0 ), ∂ t φ(t 0 )) = (φ 0 , φ0 ).
H n,n+1 (x) = G -1 n (x), with G n (ψ) := ψ ψ n,n+1 dy 2W (y) for all ψ ∈ (ω n , ω n+1 ),
where ψ n,n+1 ∈ (ω n , ω n+1 ) is chosen arbitrarily. Since, by assumption, ω n and ω n+1 are nondegenerate zeros of W , we see that

lim ψ→ωn G n (ψ) = -∞ and lim ψ→ω n+1 G n (ψ) = ∞, thus H n,n+1 is a well-dened smooth increasing function, lim x→-∞ H n,n+1 (x) = ω n and lim x→∞ H n,n+1 (x) = ω n+1 .
It satises the dierential equation

(2.3) ∂ x H n,n+1 (x) = 2W (H n,n+1 (x)).
We also dene the antikink

H n+1,n (x) := H n,n+1 (-x),
which satises the dierential equation

∂ x H n+1,n (x) = -2W (H n+1,n (x)).
We denote m n := W (ω n ) > 0 the mass corresponding to the vacuum ω n . The exponential decay of H n,n (x) and its derivatives for |x| large will be essential for our analysis.

Proposition 2.3. Let n, n ∈ I with |n -n | = 1. The function H n,n is of class C ∞ . For any k ∈ {0} ∪ N there exists C = C(n, n , k) > 0 such that ∂ k x H n,n (x) -δ 0,k ω n ≤ Ce mnx , for all x ≤ 0, (2.4)
∂ k x H n,n (x) -δ 0,k ω n ≤ Ce -m n x , for all x ≥ 0.
where δ 0,0 = 1 and δ 0,k = 0 for k > 0.

Proof. We consider the case n = n + 1 and x ≤ 0; the remaining cases are reduced to this one using the symmetries of the problem.

Since 2W (y) = m 2 n (y -ω n ) 2 (1 + O(y -ω n )) as y > ω n and y → ω n , we have (2W (y)) -1/2 = (m n (y -ω n )) -1 + O(1), and (2.2) yields G n (ψ) = 1 m n log(y -ω n ) + O(1), y > ω n , y → ω n .
Thus,

1 m n log(H n,n+1 (x) -ω n ) = x + O(1), x → -∞, which implies (2.4) for k = 0.
The bound for ∂ x H n,n+1 (x) follows from (2.3) and the fact that 2W (ψ) is a locally Lipschitz function. For k ≥ 2, (2.4) is proved by induction, dierentiating k -2 times W (H n,n+1 (x)) using the chain and Leibniz rules. Proposition 2.4. All the nite-energy stationary solutions of (1.1) are

• the vacua φ(t, x) = ω n for some n ∈ I, • the kinks φ(t, x) = H n,n+1 (x -a) for some n, n + 1 ∈ I and a ∈ R, • the antikinks φ(t, x) = H n+1,n (x -a) for some n, n + 1 ∈ I and a ∈ R. Proof. A stationary eld φ(t, x) = ψ(x) is a solution of (1.1) if and only if (2.5) ∂ 2 x ψ(x) = W (ψ(x)), for all x ∈ R.
We seek solutions of (2.5

) such that E p (ψ) < ∞, in particular ψ ∈ C(R), so (2.5) and W ∈ C ∞ (R) yield ψ ∈ C ∞ (R). Multiplying (2.5) by ∂ x ψ we get ∂ x 1 2 (∂ x ψ) 2 -W (ψ) = ∂ x ψ ∂ 2 x ψ -W (ψ) = 0, so 1 2 (∂ x ψ(x)) 2 -W (ψ(x)) = k is a constant. But then E p (ψ) < ∞ implies k = 0.
We obtain rst order autonomous equations, called the Bogomolny equations,

(2.6)

∂ x ψ(x) = 2W (ψ(x)) or ∂ x ψ(x) = -2W (ψ(x)), for all x ∈ R. If there exists x 0 ∈ R such that ψ(x 0 ) = ω ∈ Ω, then ψ(x) = ω for all x ∈ R. We thus assume ψ(x) /
∈ Ω for all x ∈ R. We now argue that the sign in (2.6) is always the same. Indeed, the smoothness of ψ implies that the sets {x :

∂ x ψ(x) = 2W (ψ(x))} and {x : ∂ x ψ(x) = -2W (ψ(x))} are closed, and ψ(x) /
∈ Ω implies that they are disjoint. Thus one of them is empty.

We consider the case ∂ x ψ(x) = 2W (ψ(x)) for all x ∈ R, the other case being analogous. In particular, ψ is an increasing function.

If ψ(0) = ψ 0 ∈ (ω n , ω n+1 ) for some n, n+1 ∈ I, then there exists a ∈ R such that H n,n+1 (a) = ψ 0 , and the uniqueness of solutions of ODEs yields ψ(x) = H n,n+1 (x -a) for all x ∈ R, thus ψ is a kink.

If ψ(0) < min Ω, then we could not have lim x→-∞ ψ(x) ∈ Ω, since ψ is increasing. Similarly, ψ(0) > max Ω is impossible.

2.3. Coercivity near a static kink. Let n, n + 1 ∈ I and let φ 0 : R → R be a state such that

E p (φ 0 ) < ∞, lim x→-∞ φ 0 (x) = ω n and lim x→+∞ φ 0 (x) = ω n+1 . Letting x 1 → -∞ and x 2 → ∞ in (2.1), we obtain E p (φ 0 ) ≥ Γ(ω n+1 ) -Γ(ω n ) = ω n+1 ωn 2W (y) dy.
Inspecting the proof of (2.1), we see that equality holds if φ 0 = H n,n+1 , so that

E p (H n,n+1 ) = ω n+1 ωn 2W (y) dy, E p (φ) ≥ E p (H n,n+1
).

The case lim x→-∞ φ(x) = ω n+1 and lim x→+∞ φ(x) = ω n is analogous, and we obtain (1.4).

Let n, n ∈ I such that |n -n | = 1. Writing φ = H n,n + v, we have the Taylor expansion

E p (v) = E p (H n,n ) + 1 2 v, L n,n v ,
where (here and later) •, • denotes the L 2 (R) inner product and

(2.7)

L n,n := -∂ 2 x + W (H n,n ) = -∂ 2 x + V n,n , with V n,n (x) := W (H n,n (x)). Note that |V n,n (x)-m 2 n | e mnx as x → -∞, and |V n,n (x)-m 2 n | e -m n x as x → ∞.
Spectral information on L n,n is obtained using standard arguments. For the convenience of the Reader, we state without proof a minor modication of Lemma 2.6 from [START_REF] Kowalczyk | A sucient condition for asymptotic stability of kinks in general (1+1)-scalar eld models[END_REF].

Proposition 2.5. [7, Lemma 2.6] The operator L n,n dened by (2.7) has the following properties:

(1) L n,n is self-adjoint on L 2 (R), with domain H 2 (R), ( 2
) ker(L n,n ) = span(∂ x H n,n ) and spec(L) ⊂ {0} ∪ [λ, +∞) for some λ > 0, ( 3
) for any Z ∈ L 2 (R) such that Z, ∂ x H n,n = 0 there exists λ 0 > 0 such that for all g ∈ H 1 (R) g, L n,n g ≥ λ 0 g 2 H 1 - 1 λ 0 Z, g 2 .
2.4. Coercivity near a moving kink. The Lorentz transformation corresponding to the velocity v ∈ (-1, 1) followed by a shift in the x direction by a ∈ R is given by

x = γ(x -a -vt), t = γ(t -v(x -a)), γ := (1 -v 2 ) -1/2 .
In the sequel, given space-time variables (t, x), we will use (t , x ) to denote their transform as above.

The inverse transform is

x = a + γ(x + vt ), t = γ(t + vx ).
Let n, n satisfy |n -n | = 1, -1 < v < 1 and a ∈ R. We would like to understand the linearised ow around the moving kink solution

φ(t, x) := H n,n (x ) = H n,n (γ(x -vt -a)).
It is convenient to formulate (1.1) and its linearisation as rst order in time systems. If φ = (φ, φ), then (1.1) can be written as

∂ t φ(t, x) = J DE(φ(t, x))
where

J := 0 1 -1 0 , DE(φ) := -∂ 2 x φ + W (φ) φ
are the standard symplectic matrix and the Fréchet derivative of the energy. The linearisation

around the moving kink H n,n (γ(x -vt -a)) is (2.8) ∂ t h(t, x) = J L v,n,n (vt + a) h(t, x)
where

L v,n,n (a) = -∂ 2 x + V n,n (γ(• -a)) 0 0 1 . The one-dimensional kernel of L n,n induces a two-dimensional iterated kernel of L v,n,n . We dene Y 0 n,n (v; x) := ∂ x H n,n (γx), -γv∂ 2 x H n,n (γx) , (2.9) Y 1 n,n (v; x) := -vx∂ x H n,n (γx), γ∂ x H n,n (γx) + γv 2 x∂ 2 x H n,n (γx) , ψ 0 n,n (v; x) := J Y 0 n,n (v; x), ψ 1 n,n (v; x) := J Y 1 n,n (v; x).
The signicance of these objects for the dynamics of is explained by the next lemma, which we state without proof.

Lemma 2.6. The following functions are solutions of (2.4):

h(t, x) = Y 0 n,n (v; x -vt -a), h(t, x) = Y 1 n,n (v; x -vt -a) + γ((1 -v 2 )t -va)Y 0 n,n (v; x -vt -a).
If h(t, x) is any solution of (2.4), then

d dt ψ 0 n,n (v; • -vt -a), h(t) = 0, d dt ψ 1 n,n (v; • -vt -a), h(t) = - 1 γ ψ 0 n,n (v; • -vt -a), h(t) . Proof. See [1, Lemma 4.3]. Here, •, • denotes the inner product in L 2 (R) × L 2 (R).
The quadratic form associated to the linear equation (2.4) above is

Q v,n,n (a; h 0 , h 0 ) = 1 2 ( ḣ0 ) 2 + 2v ḣ0 ∂ x h 0 + (∂ x h 0 ) 2 + V n,n (γ (• -a)) h 2 0 dx.
By the Lorentz transform, we can translate Proposition 2.5 to a coercivity property around L v,n,n , which yields the following result.

Proposition 2.7. For any n, n such that |n -n | = 1 and -1 < v < 1, there exists λ 0 > 0 such that for all h 0 and a ∈ R the following bound holds:

Q v,n,n (a; h 0 , h 0 ) ≥ λ 0 h 0 2 E - 1 λ 0 ψ 0 n,n (v; • -a) , h 0 2 + ψ 1 n,n (v; • -a) , h 0 2 .
Proof. For a proof in a slightly more general setting, see [START_REF] Chen | Lyapunov-type characterisation of exponential dichotomies with applications to the heat and Klein-Gordon equations[END_REF]Lemma 4.4], or the earlier work [4, Proposition 1] for a dierent proof of a very similar result.

2.5. Coercivity near a multikink. In this subsection, we collect the coercivity properties for the linear operator around a multikink. Let K ∈ N, n be a chain of vacua, v ∈ S (K) and a ∈ R K . We set γ j := (1 -v 2 j ) -1/2 for 1 ≤ j ≤ K. In general, various constants in the estimates below depend on v and a, but can be chosen uniformly for (v, a) ∈ A ⊂⊂ S (K) × R K . We are interested in the linear equation (2.10)

∂ t h(t, x) = J L(v, a; t, x)h(t, x), where L(v, a; t, x) = -∂ 2 x + V (v, a; t, x) 0 0 1 with (2.11) V (v, a; t, x) := V n 0 ,n 1 (γ 1 (x -v 1 t -a 1 )) + K-1 j=1 V n j ,n j+1 (γ j+1 (x -v j+1 t -a j+1 )) -m 2 n j .
Using notations from (2.9), for 1

≤ j ≤ K we dene Y 0 j (t, x) := Y 0 n j-1 ,n j (v j ; x -v j t -a j ) , Y 1 j (t, x) := Y 1 n j-1 ,n j (v j ; x -v j t -a j ) , and
ψ 0 j (t, x) := J Y 0 j (t, x), ψ 1 j (t, x) := J Y 1 j (t,
d dt ψ 1 j (t), h(t) + 1 γ j ψ 0 j (t), h(t) e -ηt h(t) E .
Proof. See the proof of Lemma 4.8 in [START_REF] Chen | Lyapunov-type characterisation of exponential dichotomies with applications to the heat and Klein-Gordon equations[END_REF].

Remark 2.9. Since, for any v, a and t, L(v, a; t) is a self-adjoint operator on L 2 (R) × L 2 (R) and J is a skew-adjoint operator on the same space, (2.12) can be equivalently written

∂ t ψ 0 j (t), h(t) + ψ 0 j (t), J L(v, a; t)h(t) = ∂ t ψ 0 j (t) -J L(v, a; t)ψ 0 j (t), h(t) e -ηt h(t) E ,
in other words (2.14)

∂ t ψ 0 j (t) -J L(v, a; t)ψ 0 j (t) E * e -ηt .
Analogously, (2.13) can be equivalently written (2.15)

∂ t ψ 1 j (t) -J L(v, a; t)ψ 1 j (t) + γ -1 j ψ 0 j (t) E * e -ηt .
Following [START_REF] Côte | Multi-solitons for nonlinear Klein-Gordon equations[END_REF][START_REF] Côte | Multi-travelling waves for the nonlinear Klein-Gordon equation[END_REF], as well as earlier papers [START_REF] Martel | Stability and asymptotic stability for subcritical gKdV equations[END_REF][START_REF] Martel | Asymptotic N -soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations[END_REF] in the KdV setting, we construct a quadratic functional allowing to control the error in a neighborhood of a multi-soliton. The point is that the quadratic form Q v,n,n (a; h 0 , h 0 ) dened above contains a term depending on v. In order to study a neighbourhood of a multikink, we need a functional which looks like Q v,n j-1 ,n j (v j t + a j ; h 0 , h 0 ) near the j-th kink. Set

χ j (t, x) := χ x -v j t -a j ρt ,
where ρ is a small positive number, χ is a smooth bump function such that χ(x) = 1 for x ∈ [-1, 1] and χ(x) = 0 for x ∈ R\ [-2, 2].

Using the notation (2.11), we dene the quadratic form

Q v,a (t; h 0 , h 0 ) = 1 2 R ( ḣ0 ) 2 + (∂ x h 0 ) 2 + 2 K j=1 χ j (t)v j ḣ0 ∂ x h 0 + V (t)h 2 0 dx.
Proposition 2.10. Fix K ∈ N and a chain of vacua n. For any A ⊂⊂ S (K) × R K there exist

T 0 ∈ R and λ 0 > 0 such that for all (v, a) ∈ A, h 0 ∈ E and t ≥ T 0 Q v,a (t; h 0 , h 0 ) ≥ λ 0 h 0 2 E - 1 λ 0 K j=1 ψ 0 j (t), h 0 2 + ψ 1 j (t), h 0 2 .
Proof. See [1, Lemma 4.6].

Proofs of the main results

3.1. Function spaces. For given

T ∈ R, we denote C ∞ 0 ([T, ∞) × R) the set of functions [T, ∞) × R → R which are restrictions to [T, ∞)×R of functions in C ∞ 0 (R 2 ). Similarly, if A ⊂⊂ S (K) ×R K is an open set with compact closure, we denote C ∞ 0 (A×[T, ∞)×R) the set of functions A×[T, ∞)×R → R which are restrictions to A × [T, ∞) × R of functions in C ∞ 0 (R 2K+2 ).
Fix K ∈ N, a chain of vacua n and (v, a) ∈ S (K) × R K . For T ≥ 0, δ > 0 and s ∈ N, we consider the space H s T,δ of functions g : (T, ∞) × R → R, which is dened as the completion of the set

C ∞ 0 ([T, ∞) × R) for the norm g 2 H s T,δ := sup t>T e 2δt R r+u≤s (∂ r t ∂ u x g(t, x)) 2 dx.
In other words, H s T,δ is the space of functions whose partial derivatives of order < s belong to H T,δ = H 1 T,δ . The denition also makes sense for s = 0 and gives a weighted L ∞ L 2 space, which we denote L 2

T,δ = H 0 T,δ . If b ∈ L ∞ ((T, ∞) × R), then (3.1) bh L 2 T,δ ≤ b L ∞ t,x h L 2 T,δ , for all b, h : (T, ∞) × R → R.
By the Chain Rule and the embedding H 1 (R) ⊂ L ∞ (R), we also obtain

(3.2) bh H T,δ b X h H T,δ .
where the norm X is dened as

(3.3) b X := b L ∞ t,x + ∂ t b L ∞ t L 2 x + ∂ x b L ∞ t L 2
x .

It is clear that X is an algebra. Finally, again using

H 1 (R) ⊂ L ∞ (R), we have (3.4) gh H T,δ 1 +δ 2 g H T,δ 1 h H T,δ 2 .
Note that, if ρ is a standard smooth space-time approximation of identity and g ∈ H s T,δ , then lim →0 + ρ * g -g H s T,δ = 0. If, instead of (v, a) being xed, they are allowed to vary in an open set with compact closure A ⊂⊂ S (K) × R K , we consider the space C k (A, H s T,δ ) of functions g : A × (T, ∞) × R → R, which is dened as the completion of the set C ∞ 0 (A × [T, ∞) × R), for the norm

g 2 C k (A,H s T,δ ) := sup (v,a)∈A sup t>T e 2δt R |p|+|q|≤k r+u≤s ∂ p v ∂ q a ∂ r t ∂ u x g(v, a, t, x) 2 dx,
in other words the space of functions whose partial derivatives of order < s belong to H T,δ , and depend in a C k way on the parameters (v, a) ∈ A. Here, p, q ∈ N K are multi-indices. Note that, if g ∈ H s T,δ , then

(3.5) lim t→∞ e 2δt R r+u≤s (∂ r t ∂ u x g(t, x)) 2 dx = 0. Similarly, if g ∈ C k (A, H s T,δ ), then (3.6) lim t→∞ sup (v,a)∈A e 2δt R |p|+|q|≤k r+u≤s (∂ p v ∂ q a ∂ r t ∂ u x g(v, a, t, x)) 2 dx = 0.
If E is a normed vector space, we denote B(E) the unit ball in E.

In the sequel, we adopt the convention of dropping arguments of functions from right to left, that is, if f : X × Y → Z is a function, then for x ∈ X, f (x) denotes the function Y → Z given by f (x)(y) := f (x, y), and similarly for more arguments, thus for example the symbole H can designate the function (v, a; t, x) → H(v, a; t, x). φ := H(v, a) + g solves (1.1). In the next lemma, we prove energy estimates for the linearised problem.

Lemma 3.1. For all K ∈ N, a chain of vacua n, A ⊂⊂ S (K) × R K and δ 0 suciently small (depending on A), there exists T 0 such that if T ≥ T 0 , then for any (v, a) ∈ A and f ∈ L 2 T,δ 0 there exists a unique solution R(v, a)f := h ∈ H T,δ 0 of the equation

(3.8) ∂ 2 t h -∂ 2 x h + V (v, a)h = f, f, h : (T, ∞) × R → R.
Moreover, R(v, a) is a bounded operator L 2 T,δ 0 → H T,δ 0 , whose norm stays bounded when T ≥ T 0 and (v, a) ∈ A.

Proof.

Step 1 (a priori estimates). Let f ∈ L 2 T,δ and assume h ∈ H T,δ is a weak solution of (3.8).

We will show that (3.9)

h H T,δ ≤ C δ f L 2 T,δ
, where C δ depends on δ, but not on T, v or a. Since, in this proof, v and a are considered as xed, we will write V instead of V (v, a).

Without loss of generality, we can assume h and f are smooth functions. Indeed, if ρ is a standard smooth approximation of identity in dimension 2, then

∂ 2 t (ρ * h) -∂ 2 x (ρ * h) + V (ρ * h) = ρ * f -ρ * (V h) + V (ρ * h).
If (3.9) holds for smooth functions, then we obtain

ρ * h H T,δ ≤ C δ ρ * f -ρ * (V h) + V (ρ * h) L 2 T,δ
, for all > 0. Both ρ * (V h) and V ρ * h converge to V h in L 2 T,δ as → 0, hence we get (3.9) in the limit.

We now prove (3.9) for smooth functions by means of an energy estimate, that is, we compute the time-derivatives of Q (t; h(t), h(t)) and of the projections of h on the iterated kernel. One additional diculty is caused by the fact that V is not a sum of localised terms, and approaches distinct values

m 2 n 0 , m 2 n 1 , . . . , m 2 n K
in the regions between the kinks. In order to deal with this issue, we decompose V in the following way. Let x 0 > 0 to be chosen later.

Let V 1 be a smooth function such that V 1 (x) = m 2 n 0 for x ≤ -x 0 and V 1 (x) = m 2 n 1 for x ≥ x 0 . For 2 ≤ j ≤ K, let V j be a smooth function such that V j (x) = 0 for x ≤ -x 0 and V j (x) = m 2 n j -m 2 n j-1 for x ≥ x 0 . We require ∂ x V j L ∞ x -1 0 for all 1 ≤ j ≤ K. We set V 1 (x) := V n 0 ,n 1 (γ 1 x) -V 1 and, for 2 ≤ j ≤ K, V j (x) := V n j-1 ,n j (γ j x) -m 2 n j-1 -V j .
Then V 1 , . . . , V K are smooth exponentially decaying functions and, see (2.11),

V (t, x) = K j=1 V j (x -v j t -a j ) + K j=1 V j (x -v j t -a j ).
In the remaining part of this proof, we call negligible the terms which can be made h(t) 2 E + h(t) E f (t) L 2 upon taking T 0 and x 0 large enough. The sign means equality up to negligible terms. We have

d dt Q (t; h(t), h(t)) = R ∂ t h∂ 2 t h + ∂ x h∂ t ∂ x h + V h∂ t h dx + d dt R J j=1 χ j (t)∂ t hv j ∂ x h dx + 1 2 R h 2 ∂ t V dx. Note that R h 2 ∂ t V dx = R h 2 ∂ t K j=1 V j (• -v j t -a j ) + K j=1 V j (• -v j t -a j ) dx = - K j=1 R h 2 v j ∂ x V j (• -v j t -a j ) dx - K j=1 R h 2 v j ∂ x V j (• -v j t -a j ) dx. Since ∂ x V j L ∞ x -1
0 , the rst sum is negligible. Integrating by parts the terms in the second sum, we obtain

1 2 R h 2 ∂ t V dx K j=1 R v j h∂ x hV j (• -v j t -a j ) dx.
We also have

d dt K j=1 χ j ∂ t hv j ∂ x h dx = K j=1 R ∂ t χ j ∂ t hv j ∂ x h dx (3.10) + K j=1 R χ j ∂ 2 t hv j ∂ x h dx + K j=1 R χ j ∂ t hv j ∂ t ∂ x h dx. Since ∂ t χ j L ∞ t -2
, the rst term of the right hand side is negligible. In the second term of the right-hand side above, we replace ∂ 2 t h by ∂ 2 x h -h -V h + f . Integrating by parts, we see that ∂ 2 x h and h yield negligible terms (whenever the dierentiation hits χ j , it produces a 1 t factor). By the same argument, the third term of the right hand side of (3.10) is negligible. Consider the term

R v j χ j V h∂ x h dx = R v j χ j h∂ x h K i=1 V j (• -v j t -a j ) + K i=1 V j (• -v j t -a j ) dx.
Since V j are exponentially decaying, the terms V i for i = j can be neglected. Integrating by parts, we see that the terms V i can be neglected as well, and we are left with

R v j χ j V h∂ x h dx R v j h∂ x hV j (• -v j t -a j ) dx.
Note that in the third term of (3.10), we perform integration by parts

J j=1 χ j ∂ t hv j ∂ t ∂ x h dx = - 1 2 J j=1 ∂ x χ j v j (∂ t h) 2 dx.
Combining the estimates above, we obtain

(3.11) d dt Q (t; h(t), h(t)) c 0 h(t) 2 E + f (t) L 2 h(t) E ,
where c 0 can be made arbitrarily small by taking T 0 and x 0 large.

Next, we compute the time derivatives of the projections on the iterated kernel. Using the notation from Section 2.5, we can rewrite (3.8) as

∂ t h(t) = J L(v, a; t)h(t) + (0, f (t)).
Using the self-adjointness of L(v, a; t) and the skew-adjointness of J , we have

d dt ψ 0 j (t), h(t) = ∂ t ψ 0 j (t), h(t) + ψ 0 j (t), ∂ t h(t) = ∂ t ψ 0 j (t) -J L(v, a; t)ψ 0 j (t), h(t) + ψ 0 j (t), (0, f (t)) .
Using (2.14), we infer

d dt ψ 0 j (t), h(t) e -ηt h(t) E + f (t) L 2 .
Similarly, (2.15) yields

d dt ψ 1 j (t), h(t) ψ 0 j (t), h(t) + e -ηt h(t) E + f (t) L 2 .
Therefore, integrating from ∞, for zero modes, we obtain that

ψ 0 j (t), h(t) e -(η+δ)t η + δ h H T 0 ,δ + 1 δ e -δt f L 2 T 0 ,δ and ψ 1 j (t), h(t) e -(η+δ)t (η + δ) 2 + e -(η+δ)t η + δ h H T 0 ,δ + 1 δ + 1 δ 2 e -δt f L 2 T 0 ,δ
.

For the quadratic form part, using (3.11) and the bootstrap assumptions, we integrate from ∞ and conclude that

|Q (t; h(t), h(t))| c 0 e -2δt 2δ h 2 H T 0 ,δ + e -2δt 2δ f L 2 T 0 ,δ h H T 0 ,δ .
Therefore by the coercivity, one has

h(t) 2 E e -(η+δ)t (η + δ) 2 + e -(η+δ)t η + δ 2 h 2 H T 0 ,δ + 1 δ + 1 δ 2 2 e -2δt f L 2 T 0 ,δ + c 0 e -2δt 2δ h 2 H T 0 ,δ + e -2δt 2δ f L 2 T 0 ,δ h H T 0 ,δ ,
where c 0 can be made arbitrarily small by taking T 0 and x 0 large, yielding (3.9).

Step 2 (existence and uniqueness of solutions). Let f ∈ L 2 T,δ . Take T ≥ T and let f T (t) := χ(t -T )f (t), where χ is a smooth decreasing function such that χ(t) = 1 for t ≤ -1 and χ(t) = 0 for t ≥ 0. Then lim T →∞ f T -f L 2 T,δ = 0, see (3.5). Let h T (t, x) be the solution of (3.12)

∂ 2 t h T -∂ 2 x h T + V (t)h T = f T such that h T ( T ), ∂ t h T ( T ) = (0, 0)
. This solution exists by the standard local Cauchy theory. Since f (t) = 0 for all t ≥ T , we have h(t) = 0 for all t ≥ T , in particular h ∈ H T,δ .

By

Step 2, we have that (h T ) T satises the Cauchy condition in H T,δ as T → ∞, hence it converges. Let h ∈ H T,δ be the limit. Passing to the limit in (3.12), we obtain that (3.8) holds in the sense of distributions.

Uniqueness follows directly from the a priori estimate.

In the next lemma, we are interested in the dependence on the parameters, thus V should be seen again as a real-valued function dened on A × [T 0 , ∞) × R. Lemma 3.2. For K ∈ N, a chain of vacua n, A ⊂⊂ S (K) × R K , δ 0 suciently small (depending on A) and δ 1 ∈ (0, δ 0 ), there exists T 0 such that if l ∈ N 0 , s ∈ N, T ≥ T 0 , then for any b ∈

B(C 0 (A, H T,δ 0 )) ∩ C l (A, H s T,δ 0 ), f ∈ C l (A, H s-1 T,δ ) there exists a unique solution h ∈ C l (A, H s T 0 ,δ 1 )
of the equation

∂ 2 t h -∂ 2 x h + (V + b)h = f, f, b, h : A × (T 0 , ∞) × R → R.
Proof.

Step 1 (existence of solutions). Let l = 0 and s = 1. For given (v, a) ∈ A, we rewrite the equation as follows:

(3.13) h(v, a) = R(v, a) f (v, a) -b(v, a)h(v, a) .
The embedding H 1 (R) ⊂ L ∞ (R) implies, by taking T 0 large enough, that b L ∞ ((T 0 ,∞)×R) can be made as small as we wish. Thus, by (3.1) and Lemma 3.1, (3.13) denes a contraction in H 1

T 0 ,δ 1 .
Step 2 (continuity with respect to the parameters). For each (v, a) ∈ A, let h(v, a) be the function constructed in Step 1. We will prove that h ∈ C 0 (A, H 1 T 1 ,δ 0 ). To this end, take (v, a), (v , a ) ∈ A and denote, for the sake of brevity, h :

= h(v , a ), f := f (v , a ), V := V (v , a ) and b := b(v , a ), as well as h := h(v, a), f := f (v, a), V := V (v, a) and b := b(v, a). We have ∂ 2 t (h -h) -∂ 2 x (h -h) + V (h -h) = f -f -(V -V )h -b (h -h) + (b -b) h.
By the a priori estimate, it suces to check that lim

(v ,a )→(v,a) f -f L 2 T 0 ,δ 0 + (V -V )h L 2 T 0 ,δ 0 + (b -b) h L 2 T 0 ,δ 0 = 0 (indeed, since b L ∞ t,x
is small, the term b (h -h) can be absorbed). As for f -f , this is clear.

Continuity of b implies (b -b) h L 2
T 0 ,δ 0 → 0. Finally, V -V is bounded and converges to 0 locally in t, so (3.6) implies the desired bound.

Step 3 (dierentiability). We prove regularity of h by induction on s, and then on l. First, we x l = 0. If s = 1, there is nothing to do. Let s ≥ 2. We prove existence of the partial derivatives. Dene h t as the solution of

∂ 2 t h t -∂ 2 x h t + (V + b)h t = ∂ t f -(∂ t V + ∂ t b)h.
By the induction hypothesis, h t ∈ C 0 (A, H s-1 T 0 ,δ 1 ). We prove that ∂ t h = h t in the weak sense. For xed (v, a), we consider r (t, x) := h(v, a; t + , x) -h(v, a; t, x) -h t (v, a; t, x).

It satises the equation

∂ 2 t r -∂ 2 x r + (V + b)r = f -f -(V + b -V -b)h -∂ t f + (∂ t V + ∂ t b)h = (f -f -∂ t f ) -(V + b -V -b -(∂ t V + ∂ t b))h -(∂ t V + ∂ t b)(h -h),
where we have denoted V (v, a; t, x) := V (v, a; t + , x), and similarly for f , h and b . We need to check that the right hand side is o( ) in L 2 T 0 ,δ 1 as → 0. For the rst two terms, this follows from the dierentiability of V, b and f , and the uniform boundedness of h in L 2 T 0 ,δ 1 . For the third term, we use the continuity of h with respect to t. By the a priori bound, r H T 0 ,δ 1 , implying

∂ t h = h t . Analogously, one proves that ∂ x h ∈ H s-1 T 0 ,δ 1 , thus h ∈ H s T 0 ,δ 1
, nishing the induction step with respect to s.

The induction step with respect to l is similar, so we only sketch the argument for ∂ v k h. Dene h v k as the solution of

∂ 2 t h v k -∂ 2 x h v k + (V + b)h v k = ∂ v k f -(∂ v k V + ∂ v k b)h. By the induction hypothesis, h v k ∈ C 0 (A, H s-1
T 0 ,δ 1 ) (note that it is here that the loss in the exponential convergence rate occurs, due to the polynomial growth of ∂ v k V ). We prove that ∂ v k h = h v k in the weak sense. For xed (v, a), we consider r (t, x)

:= h(v + e k , a; t, x) -h(v, a; t, x) -h v k (v, a; t, x),
where e k := (0, . . . , 1, . . . , 0) is the k-th element of the standard basis of R K . It satises the equation

∂ 2 t r -∂ 2 x r + (V + b)r = f -f -(V + b -V -b)h -∂ v k f + (∂ v k V + ∂ v k b)h = (f -f -∂ v k f ) -(V + b -V -b -(∂ v k V + ∂ v k b))h -(∂ v k V + ∂ v k b)(h -h),
where we have denoted V (v, a; t, x) := V (v + e k , a; t, x), and similarly for f , h and b . We need to check that the right hand side is o( ) in L 2 T 0 ,δ 1 . For the rst two terms, this follows from the dierentiability of V, b and f , and the uniform boundedness of h in L 2 T 0 ,δ 1 . For the third term, we

know from the Step 2 that lim →0 h -h L 2 T 0 ,δ 0 = 0, hence lim →0 (∂ v k V + ∂ v k b)(h -h) L 2 T 0 ,δ 1 = 0
for any δ 1 < δ 0 .

3.3. Formulation as a xed point problem and conclusions. In order to make the formulas which follow shorter, we denote

H k (v, a; t, x) := H n k-1 ,n k (γ k (x -v k t -a k ))
the k-th kink. Plugging the decomposition (3.7) into (1.1), we obtain the following equation for the error term g = g(v, a):

∂ 2 t g(v, a) -∂ 2 x g(v, a) + V (v, a)g(v, a) = N (v, a; g(v, a)),
where we dene

N (v, a; g) := -W (H(v, a) + g) + K k=1 W (H k (v, a)) + V (v, a)g.
We are going to reformulate this equation as a xed point problem for a contraction.

We need to study some properties of composition with a smooth function in the relevant functional spaces. Before stating the next lemma, we recall that B(E) denotes the unit ball in a normed space E.

Lemma 3.3. Let F ∈ C ∞ (R).
For any C 0 ≥ 0, T 0 ≥ 0 and δ 0 > 0 there exists C 1 ≥ 0 such that

F (φ + g) -F (φ) H T 0 ,δ 0 ≤ C 1 g H T 0 ,δ 0 , (3.14) F (φ + g) -F (φ) -F (φ)g H T 0 ,2δ 0 ≤ C 1 g 2 H T 0 ,δ 0 . (3.15)
for all g ∈ B(H T 0 ,δ 0 ) and φ satisfying φ X ≤ C 0 .

Proof. The bound (3.14) follows from (3.15) and (3.2).

In order to prove (3.15), we write

F (φ + g) -F (φ) -F (φ)g = g 2 1 0
(1 -σ)F (φ + σg) dσ. For any A ⊂⊂ S (K) × R K and l, s ∈ N, there exist T 0 ∈ R and δ 0 > 0 such that (3.16)

F (φ + σg) L ∞ t,x + ∂ t (F (φ + σg)) L ∞ t L 2 x + ∂ x (F (φ + σg)) L ∞ t L 2
F (H) -F (ω n 0 ) + K k=1 F (H k ) -F (ω n k-1 ) ∈ B(C l (A, H s T 0 ,δ 0 ))

and

(3.17)

(F (H) -F (H j ))∂ x H j ∈ B(C l (A, H s T 0 ,δ 0 )),
for all 1 ≤ j ≤ K.

Moreover, if 0 < δ 1 ≤ δ 0 and g ∈ C k (A, H s T 0 ,δ 1 ), then (3.18)

F (H + g) -F (H) ∈ C k (A, H s T 0 ,δ )
for any 0 < δ < δ 1 .

Proof.

Step 1. We prove that, if j < k, then

(H j -ω n j )(H k -ω n k-1 ) ∈ B(C k (A, H s T 0 ,δ 0 )), (3.19)
(H j -ω n j )∂ x H k ∈ B(C k (A, H s T 0 ,δ 0 )), (3.20)
(H k -ω n k-1 )∂ x H j ∈ B(C k (A, H s T 0 ,δ 0 )).
|∂ p v ∂ q a ∂ r t ∂ u x (H j (v, a; t, x) -ω n j )| ≤ e -δ(x-v j t-a j ) + . Indeed, if |p| + |q| + r + u > 0 then ∂ p v ∂ q a ∂ r t ∂ u
x (H j (v, a; t, x) -ω n j ) is the sum of a nite number of terms, each of which is a polynomial in (t, x) with bounded coecients depending on (v, a), multiplied by ∂ n x H j for some n ∈ N. Similarly,

|p|+|q|≤l r+u≤s |∂ p v ∂ q a ∂ r t ∂ u x (H j (v, a; t, x) -ω n j )| ≤ e -δ(v k t+a k -x) + .
Taking logarithms, we see that e -δ(x-v j t-a j ) + e -δ(v k t+a k -x) + e -δ 0 t , if T 0 is large enough and δ 0 small enough.

The proofs of (3.20) and (3.21) are very similar, so we skip them.

Step 2. We proceed by induction with respect to K. If K = 1, the left hand side is identically 0,

so let K ≥ 2.
Note the formula

F (φ 1 + φ 2 -φ 3 ) -F (φ 1 ) -F (φ 2 ) + F (φ 3 ) = (φ 1 -φ 3 )(φ 2 -φ 3 ) 1 0 1 0 F (σ 1 φ 1 + σ 2 φ 2 + (1 -σ 1 -σ 2 )φ 3 ) dσ 1 dσ 2 ,
obtained by applying the Fundamental Theorem twice. Applying it with φ 2 := H K , φ 3 := ω n k-1

and

φ 1 := H := F (ω n 0 ) + K-1 k=1 F (H k ) -F (ω n k-1 ) ,
Lemma 3.8. Let Ψ : A × (T 0 , ∞) × R → R be the function constructed in Proposition 3.6. Then ∂ a k Ψ exists for all (v, a) ∈ A and solves the equation

∂ 2 t ∂ a k Ψ -∂ 2 x ∂ a k Ψ + W (H + Ψ)∂ a k Ψ = -(W (H + Ψ) -W (H k ))∂ a k H k . Moreover, ∂ x Ψ = -K k=1 ∂ a k Ψ and ∂ t Ψ = -K k=1 v k ∂ a k Ψ.
Futhermore, ∂ v k Ψ exists for all (v, a) ∈ A and solves the equation

∂ 2 t ∂ v k Ψ -∂ 2 x ∂ v k Ψ + W (H + Ψ)∂ v k Ψ = -(W (H + Ψ) -W (H k ))∂ v k H k . Finally, ∂ a k Ψ, ∂ v k Ψ ∈ C(A, H T 0 ,δ ) if δ > 0 is small enough.
Proof. We rst prove that v, Ψ(v, a) is continuous with respect to (v, a) ∈ A as a H T 0 ,δ -valued map for any 0 < δ < δ 0 . Let (v, a), (v , a ) ∈ A. We have

N (v, a ; Ψ(v , a )) -N (v, a; Ψ(v, a)) H T 0 ,δ ≤ N (v , a ; Ψ(v , a )) -N (v, a ; Ψ(v , a )) H T 0 ,δ + N (v, a ; Ψ(v , a )) -N (v, a; Ψ(v , a )) H T 0 ,δ + N (v, a; Ψ(v , a )) -N (v, a; Ψ(v, a)) H T 0 ,δ .
By (3.27), the rst term converges to 0 as (v , a ) → (v, a). By (3.26), the second term as well. The third term is estimated using (3.25), which yields the bound c 0 Ψ(v , a )) -Ψ(v, a)) H T 0 ,δ where c 0 can be made arbitrarily small. By (3.2), we also have

lim (v ,a )→(v,a) (V (v , a ) -V (v, a))Ψ(v , a ) H T 0 ,δ = 0, so the right hand side of (∂ 2 t -∂ 2 x + V (v, a))(Ψ(v , a ) -Ψ(v, a)) = N (v , a ; Ψ(v , a )) -N (v, a; Ψ(v, a)) -(V (v , a ) -V (v, a))Ψ(v , a )
is bounded in H T 0 ,δ by o(1) + c 0 Ψ -Ψ H T 0 ,δ , and Lemma 3.1 allows to absorb the second term, thus nishing the proof of lim (v ,a )→(v,a) Ψ(v , a ) -Ψ(v, a) H T 0 ,δ = 0.

Let k ∈ {1, . . . , K} and dene Θ : A × (T 0 , ∞) × R → R as the solution of (3.28)

∂ 2 t Θ -∂ 2 x Θ + W (H + Ψ)Θ = -(W (H + Ψ) -W (H k ))∂ a k H k , given by Lemma 3.2. Set b := W (H + Ψ) -V . Since Ψ ∈ C 0 (A, H T 0 ,δ ), (3.18) yields W (H + Ψ) - W (H) ∈ C 0 (A, H T 0 ,δ ) (upon diminishing δ if necessary). By (3.16), we also have W (H) -V ∈ C 0 (A, H T 0 ,δ ), thus b ∈ B(C 0 (A, H T 0 ,δ )) (again modifying δ and T 0 if necessary). By Lemma 3.7 and Lemma 3.2, Θ ∈ C 0 (A, H T 0 ,δ ). Let (v, a) ∈ A. We will verify that if T 0 is large enough (depending on A), then ∂ a k Ψ(v, a; t, x) exists for all (t, x) ∈ (T 0 , ∞) × R and ∂ a k Ψ(v, a) = Θ(v, a). To this end, for 0 < | | small enough, consider Ξ : (T 0 , ∞) × R → R given by Ξ (t, x) := Ψ(v, a + e k , t, x) -Ψ(v, a, t, x) -Θ(v, a, t, x),
where e k := (0, . . . , 1, . . . , 0) is the k-th element of the standard basis of R K . Our goal is to verify

that lim →0 | | -1 Ξ H T 0 ,δ = 0.
In the computation below, we denote a := a + e k and Ψ (v, a, t, x) := Ψ(v, a , t, x). We consider (v, a) as being xed, but below we omit the argument (v, a) for the sake of brevity, thus we write H instead of H(v, a) etc. We observe that Ξ solves the equation

(∂ 2 t -∂ 2 x + V )Ξ = (∂ 2 t -∂ 2 x + V )(Ψ -Ψ -Θ) = (∂ 2 t -∂ 2 x )Ψ -(∂ 2 t -∂ 2 x )Ψ + V (Ψ -Ψ) -(∂ 2 t -∂ 2 x -V )Θ = -W (H + Ψ ) + W (H + Ψ) + K j=1 (W (H j ) -W (H j )) + V (Ψ -Ψ) + (W (H + Ψ) -V )Θ + (W (H + Ψ) -W (H k ))∂ a k H k ,
where in the last equality we use (3.28).

We claim that the right hand side is bounded in H T 0 ,δ by o(| |) +c 0 Ξ H T 0 ,δ , with c 0 small (upon adjusting T 0 and δ). Lemma 3.1 will then allow to conclude.

We further rearrange the terms of the right hand side above as follows, using the fact that H j = H j for j = k:

(3.29) -W (H + Ψ ) -W (H + Ψ) -W (H + Ψ)(Ψ -Ψ) -W (H + Ψ ) -W (H + Ψ ) + W (H k ) -W (H k ) + (W (H + Ψ ) -W (H k ))∂ a k H k -(W (H + Ψ ) -W (H + Ψ))∂ a k H k -W (H + Ψ) -V (Ψ -Ψ -Θ).
Consider the last line, noting that Ψ -Ψ -Θ = Ξ . As in the proof of Lemma 3.5, we write

-W (H + Ψ) + V = -W (H + Ψ) + W (H) + -W (H) + V .
Using Lemmas 3.3 and 3.4, we obtain a bound in H T 0 ,δ 3 for some δ 3 > 0. Enlarging T 0 if necessary and using (3.4), we conclude that

-W (H + Ψ) + V Ξ H T 0 ,δ ≤ c 0 Ξ H T 0 ,δ .
Regarding the fourth line, using (3.14) together with the fact that lim →0 Ψ -Ψ H T 0 ,δ = 0, we obtain the bound o( ) in H T 0 ,δ . Consider now the rst line of (3.29). By (3.15), we have

W (H + Ψ ) -W (H + Ψ) -W (H + Ψ)(Ψ -Ψ) H T 0 ,δ Ψ -Ψ 2 H T 0 ,δ Ψ -Ψ H T 0 ,δ = Θ + Ξ H T 0 ,δ + Ξ H T 0 ,δ .
Using the Fundamental Theorem and H -H = H k -H k , we rewrite the second line of (3.29) as (3.30) In the case of space-time translations, the claim is clear, so let us consider a Lorentz boost with velocity v, (t, x) = (γ(t + vx ), γ(x + vt )), (t , x ) = (γ(t -vx), γ(x -vt)). Without loss of generality, we can assume v ∈ (0, 1). Let g := Ψ(v, a), φ := H(v, a) + g, φ(t , x ) := φ(t, x), h := φ -H(v , a ). Recall that g is dened for all (t, x) ∈ R 2 thanks to the global well-posedness. Since H(v , a ; t , x ) = H(v, a; t, x), we have h(t , x ) = g(t, x). We wish to prove that R (∂ t h(t , x )) 2 + (∂ x h(t , x )) 2 + h(t , x ) 2 dx e -δt for some δ > 0. Equivalently, observing that γ(t + vx ) = γ -1 t + vx, where γ := (1 -v 2 ) -1/2 , R (∂ t g(γ -1 t + vx, x)) 2 + (∂ x g(γ -1 t + vx, x)) 2 + g(γ -1 t + vx, x) 2 dx e -δt .

(H k -H k ) 1 0 -W (H + Ψ + σ(H k -H k )) + W (H k + σ(H k -H k )) dσ = -(H k -H k )(H -H k + Ψ )
We consider separately the regions x ≥ -t 2γv and x ≤ -t 2γv . In the rst region, it suces to use the exponential in time decay of g and its derivatives, uniformly in x, which follows from Theorem 1.5 and the embedding H 1 (R) ⊂ L ∞ (R).

Consider the second region. Set (t 0 , x 0 ) := (t /(2γ), -t /(2vγ)) and let ∆ be the cone in the (t, x) plane R × R with vertex at (t 0 , x 0 ), delimited by the half-lines {(t 0 , x) : x ≤ x 0 } and {(t 0 + v(xx 0 ), x) : x ≤ x 0 }. An elementary computation shows that (t, x) ∈ ∆ implies x -v k t -a k x, thus |H(v, a; t, x) -ω n 0 | + |∂ t H(v, a; t, x)| + |∂ x H(v, a; t, x)| e -δx , 21 uniformly for (t, x) ∈ ∆, with a constant independent of t 0 . In particular, proving that x 0 -∞ (∂ t g(γ -1 t + vx, x)) 2 + (∂ x g(γ -1 t + vx, x)) 2 + g(γ -1 t + vx, x) 2 dx e -δt . is equivalent, since W (ω n 0 + g) g 2 for |g| small, to verifying that (3.31)

x 0 -∞ 1 2
(∂ t φ(γ -1 t + vx, x)) 2 + 1 2 (∂ x φ(γ -1 t + vx, x)) 2 + W (φ(γ -1 t + vx, x)) dx e -δt .

For the same reason, the exponential decay in time of g implies (3.32)

x 0 -∞ 1 2 (∂ t φ(t 0 , x)) 2 + 1 2 (∂ x φ(t 0 , x)) 2 + W (φ(t 0 , x)) dx e -δt .

By the Green's theorem applied for the smooth vector eld ( 1 2 (∂ t φ) 2 + 1 2 (∂ x φ) 2 + W (φ), -∂ t φ∂ x φ), in the region ∆, (3.32) implies (3.31).

1 .

 1 Setting of the problem. We study scalar elds in dimension 1 + 1. Let W : R → [0, +∞) be a function of class C ∞ and consider the Lagrangian action φ) 2 -W (φ) dx dt, for real-valued functions φ = φ(t, x). The Euler-Lagrange equation associated with L is the non-

  x) = ω n , lim x→∞ φ(t, x) = ω n , for some n, n ∈ I and |n -n | = 1.

3. 2 .

 2 Energy estimates. For given v = (v 1 , . . . , v K ) ∈ S (K) and a = (a 1 , . . . , a K ) ∈ R K , we seek g = g(v, a) ∈ H s T,δ such that (3.7)

x 1 ,

 1 so (3.2) and (3.4) yield the claim. Lemma 3.4. Let F ∈ C ∞ (R).

3 ,

 3 together with the Chain and Leibniz Rules, |p|+|q|≤l r+u≤s

Proof of Theorem 1 . 5 . 2 and Lemma 3. 8 .

 1528 Induction with respect to s and l. The induction step follows from Lemma 3.Proof of Theorem 1.4. Fix (v, a) ∈ S (K) × R K . We should prove that, applying a space-time translation or a Lorentz transform to H(v, a) + Ψ(v, a) yields a function converging exponentially in time to H(v , a ), where (v , a ) is given by(1.6). The uniqueness part of Theorem 1.3 will then yield the conclusion.

  The functions H n,n are studied in detail in Section 2.2.It is not dicult to see that for n -n ≥ 2 we have

	inf (φ 0 , φ0 )∈E n,n	E(φ 0 , φ0 ) =	inf

(φ 0 , φ0 )∈E n ,n E(φ 0 , φ0 ) = n -1 l=n E p (H l,l+1 ),

but the inmum is not attained, and there is no ground state in E n,n or E n ,n . Constant solutions, kinks and antikinks are all the stationary states of (1.1), see Section 2.2.

  2.8. For any chain of vacua n, v ∈ S (K) and a ∈ R K there exist T 0 ∈ R and η > 0 such that if h(t, x) is a solution of (2.10), then for all t ≥ T 0

	(2.12)	d dt	ψ 0 j (t), h(t)	e -ηt h(t) E ,
	(2.13)

x).

Lemma 2.6, combined with the exponential decay from Proposition 2.3, leads to the following result. Proposition

  W (H k + τ (H -H k + Ψ ) + σ(H k -H k )) dτ dσ. H T 0 ,δ . Using the exponential decay of ∂ a k H k and its derivatives for |x -a k -v k t| 1, we also obtain (H k -H k )(H -H k ) H T 0 ,δ . Moreover, since H k -H k X , 20another application of the Fundamental Theorem givesW (H k + τ (H -H k + Ψ ) + σ(H k -H k )) -W (H k + τ (H -H k + Ψ )) X .Thus, up to terms bounded in H T 0 ,δ by O(2), (3.30) is the same as-(H k -H k )(H -H k + Ψ )Adding the third line of (3.29), we obtain-(H k -H k -∂ a k H k )(H -H k + Ψ ) W (H k + τ (H -H k + Ψ )) dτ. -H k -∂ a k H k X = 2 1 0 (1 -σ)∂ 2 a k H k (v, a + σ e k ) dσ 3.2) yields (H k -H k -∂ a k H k )Ψ H T 0 ,δ 2 . Using the exponential decay of ∂ 2 a k H k and its derivatives for |x -a k -v k t| 1, we also obtain (H k -H k -∂ a k H k )(H -H k ) H T 0 ,δ 2 .This nished the proof of the required estimate of (3.29).The proofs of existence of the other partial derivatives are similar, so we skip them. The formulas for ∂ x Ψ and ∂ t Ψ follow from the fact that ∂x H = -K k=1 ∂ a k H and ∂ t H = -K k=1 v k ∂ a k H.

		1
		0	W (H 1
			0
			2 ,
			X
	1	1
	0	0

Since H k (v, a ) -H k (v, a) X = 1 0 ∂ a k H k (v, a + σ e k ) dσ X , (3.2) yields (H -H)Ψ k + τ (H -H k + Ψ )) dτ. Since H k (

we obtain

The last integral is bounded, and all its derivatives as well. Observing that H -ω n K-1 = K-1 j=1 (H jω n j ) and using K -1 times (3.19), we obtain (3.16).

Step 3. In order to prove (3.17), since ∂ x H j and all its derivatives in v, a, t, x are bounded by a polynomial in t, it suces to check that

But we observe that

hence it suces to prove that Step 4. Finally, in order to prove (3.18), we write

In order to prove continuity, we observe that F (H + σg) is continuous from A to the space X dened by (3.3), uniformly in σ, so it suces to apply (3.2).

Next, we dierentiate under the integral sign in (3.24), using again the fact that we can absorb polynomials in t by diminishing δ. Lemma 3.5. For any A ⊂⊂ S (K) × R K , δ 0 small enough (depending on A) and 0 < δ < δ 0 , there

We rewrite the integrand as

By Lemma 3.3, the rst term is uniformly bounded in H T,δ 0 . By Lemma 3.4 applied with F = W , the second term is uniformly bounded in H T,δ 3 for some δ 3 > 0. Hence, (3.25) follows from (3.4).

We have

Similarly as above, Lemmas 3.3 and 3.4 yield uniform boundedness of the right hand side in H T,δ 0 , proving (3.26). Analogously, ∂ v k N (v, a; g) is uniformly bounded in H T,δ , which proves (3.27). Proposition 3.6. For any A ⊂⊂ S (K) × R K , there exist T 0 , δ 0 > 0 such that the following is true.

Proof. Fix (v, a) ∈ S (K) × R K , and let R(v, a) be given by Lemma 3.1.

Given T ∈ R, δ > 0 and g ∈ B(H T,δ ), we dene Φ(g) := R(v, a)N (v, a; g).

It follows that φ solves (1.1) if and only if g = Φ(g).

By Lemmas 3.1 and 3.5, if T is large enough, then Φ is a contraction in B(H T,δ ). Since the constants in Lemmas 3.1 and 3.5 are uniform in (v, a) ∈ A, T 0 and δ 0 can be chosen uniformly for (v, a) ∈ A.

Proof of Theorem 1.3. The existence part follows from Proposition 3.6, applied for A being the singleton {(v, a)}.

Let ψ ∈ H T,δ be such that φ = H(v, a) + ψ solves (1.1). Without loss of generality, we can assume T ≥ T 0 and δ ≤ δ 0 , so that Ψ(v, a) ∈ B(H T,δ ). Upon modifying δ and T , we can assume ψ ∈ B(H T,δ 0 ).

If T is large enough, then Φ is a contraction on this set, implying uniqueness.

Lemma 3.7. For any A ⊂⊂ S (K) × R K , l, s ∈ N, T 0 suciently large, δ 0 suciently small (both

and

for all 0 < δ < δ 0 .

Proof. By Lemma 3.4,

By (3.18), we have

Taking the sum, we obtain the conclusion.


Ploypailin Romphophak

Claude Le Men

Arnaud Cockx

Carole Coufort-Saudejaud

Pisut Painmanakul

Alain Liné
email: alainliné@insa-toulouse.fr

Analysis of flocculation in a jet clarifier. Part 1 -Global and local hydrodynamic analysis

Keywords: Jet clarifier, hydrodynamics, particle image velocimetry, velocity gradient

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

2 from 4 to 1 hour. Based on the hydrodynamic analysis, the parameter Gt is shown to be constant around 30,000 for different jet flow rates. Efficiency of such jet clarifier can thus be foreseen.

Introduction

In general, the target of surface water treatment consists in removing suspended particles that cause turbidity in water. Jet clarifier is the one type of solid contact clarifier, coupling flocculation and clarification (sedimentation zone) in a single unit [START_REF] Degremont | Water treatment handbook[END_REF][START_REF] Pani | Single-basin jet flocculators[END_REF][START_REF] Romphophak | Study of Flow Pattern in Jet Clarifier for Removal of Turbidity by Residence Time Distribution Approach[END_REF]. In many research, the performance of the flocculator reactors were measured by residual turbidity [START_REF] Garland | Revisiting Hydraulic Flocculator Design for Use in Water Treatment Systems with Fluidized Floc Beds[END_REF]. Normally, the turbidity removal efficiency of the free jet flocculator is about 70 to 75% [START_REF] Randive | Study Of Jet Mixing In Flocculation Process[END_REF]. Nonetheless, the performance of the jet clarifier which is investigated in this study is about 80%. This jet clarifier was designed based on the criteria such as residence time, sedimentation and surface loading rate. Afterward, the trial and error method was used to reach the appropriate operating conditions by varying the inlet flow rates, the height of the sludge blanket, the gap ratio between the sedimentation wall section and the flocculation, and the diameter of the cone base diameter of truncated of flocculation zone; some of the appropriate conditions of this reactor's geometry were presented in [START_REF] Romphophak | Study of Flow Pattern in Jet Clarifier for Removal of Turbidity by Residence Time Distribution Approach[END_REF],. Indeed, the scale of the jet clarifier presented in the previous paper is 4 times larger than the present scale, but the performance of the reactor in different scales is the same. However, the key phenomena that control the flocculation mechanisms are not clearly understood. Sobrinho et al. (1996) also mentioned that the effluent residual turbidity was essentially independent of the flow rate and associated this result to a nearly constant value of Gt. The possible reasons were proposed without any evidence or proof.

For example, Gt may be constant because when the flow increases through the chamber reactor, the mixing intensity increases, and the retention time decreases. The higher mixing intensity (gradient velocity) may be balanced by the shorter retention time, resulting in a nearly constant GT value. Furthermore, the authors based their analysis on head loss to estimate the velocity gradient. In the present study, the floc size distributions are independent of the flow rate too (see part 2 of this paper).

The present analysis of the hydrodynamics of a jet clarifier is aimed at better understanding the physical phenomena explaining these independencies.

Commonly, the hydraulic flocculators have been designed based on a global parameter G t where G stands for the global velocity gradient and t stands for contact time [START_REF] Camp | Velocity Gradients and Internal Work in Fluid Motion[END_REF]Pedochi & Piedra-Cueva, 2005;[START_REF] Garland | Revisiting Hydraulic Flocculator Design for Use in Water Treatment Systems with Fluidized Floc Beds[END_REF][START_REF] Marques | Flocculation kinetics of low-turbidity raw water and the irreversible floc breakup process[END_REF]. In flocculation, the Camp and Stein criteria G t recommended to achieve efficient flocculation is usually in the following range:

10 < < 10 (1)
The analysis of G t in clear water and in the flocculation zone of a jet clarifier will constitute the final outcome of this paper; the global criteria G t will be highlighted by a local analysis of G t, where G will be the local shear rate and t is a hydrodynamic characteristic time.

Modelling of aggregation and break-up of flocs is out of the scope of this paper. However, it is important to recall how flocculation is related to hydrodynamics. One of the first attempts to quantify the collision rate of particles induced by hydrodynamics was proposed by Smoluchowski (1917) in terms of local velocity gradient. Then, [START_REF] Camp | Velocity Gradients and Internal Work in Fluid Motion[END_REF] extended Smoluchowski's model of velocity gradient G. They also introduced a global parameter, the root mean square velocity gradient GRMS, in terms of power input by unit mass of liquid (W/kg). As reported by [START_REF] Pedocchi | Camp and Stein's Velocity Gradient Formalization[END_REF], "the denomination velocity gradient has created some confusion about the physical interpretation of this parameter". Indeed, the work of Camp and Stein has been revisited by many workers (among them [START_REF] Cleasby | Is Velocity Gradient a Valid Turbulent Flocculation Parameter[END_REF][START_REF] Clark | Critique of camp and Stein's RMS velocity gradient[END_REF][START_REF] Kramer | Influence of strain-rate on coagulation kinetics[END_REF]. It is now accepted that the velocity gradient is defined as the square root of the viscous dissipation rate of kinetic energy (W/kg) divided by the kinematic viscosity. It is thus identical to the local shear rate and is defined as:

= = = ̅ + ′ (2)
Where S is the symmetric part of the velocity gradient tensor. Here, is an invariant. The first term on the r.h.s. is related to the square of mean velocity gradients whereas the second one stands for the average of the square of the fluctuating (turbulent) velocity gradients. These two terms are respectively related to the viscous dissipation of the mean flow kinetic energy and to the viscous dissipation of the turbulent kinetic energy. In turbulent flow, the first one is negligible compared to the viscous dissipation of the turbulent kinetic energy. Averaged over the whole tank or clarifier, the dissipated power is equal to the power input.

The present study focuses on hydrodynamics, in terms of local and instantaneous velocity field, induced by the jet in the flocculation zone of a jet clarifier. The two key words being "jet" and "clarifier", the bibliography analysis will consider these two terms. However, in terms of application in chemical engineering and water treatment, the flocculation-clarifier issue will be predominant.

Jet flow

Jets are used in many industrial or environmental applications. In the past, jet flow hydrodynamics was addressed theoretically, experimentally and numerically. Schlichting (1933) was the pioneer to study jets. [START_REF] Bickley | The plane jet[END_REF] derived analytical solutions of jet flows; he demonstrated that the developing jet flow entrains external fluid, increasing the flow rate and decreasing the axial velocity, thus preserving constant momentum. Based on experiments, [START_REF] Miller | Static pressure distribution in a free turbulent jet[END_REF] showed that the jet decreases axially as the square root of the axial position along the jet (the origin being at the orifice outlet) and the jet size enlargement was shown to increase linearly with the axial position. These hydrodynamic phenomena will be investigated in our jet clarifier.

A jet is usually characterized by the Reynolds number at the injection. The Reynolds number is classically defined as:

= 〈 〉 (3)
Where 〈!〉 is the cross-averaged discharge velocity from the nozzle (m/s), d is the circular orifice nozzle internal diameter (m) and ν is the kinematic viscosity of the fluid (m²/s). Referring to [START_REF] Pearce | Critical Reynolds Number for Fully-Developed Turbulence in Circular Submerged Water Jets[END_REF] conclusion, there is no turbulence below 500 and fully turbulent jet starts at 3000. Since in our study, the Reynolds number vary between 1000 and 4000, it corresponds to the transition from laminar to turbulent jet flow. Both jet structure and stability aspects of transition flows have also been reviewed

by [START_REF] Mollendorf | An experimental and numerical study of the viscous stability of a round laminar vertical jet with and without thermal buoyancy for symmetric and asymmetric disturbances[END_REF]. A submerged liquid jet from a circular orifice nozzle into a similar liquid exhibits three characteristics regions: (1) a developing flow region: about 10 nozzle diameters long; this region is called potential conic region;

(2) a developed flow region: up to 100 nozzle diameters from the orifice; (3) a terminal region: above 100 nozzle diameters from the orifice.

It was reported that instabilities appear in the sheared layers induced by the submerged liquid jet.

Downstream, mixing is controlled by the entrainment of surrounding liquid in the decelerating jet velocity region. In the developed flow region, the jet structure weakly depends on inlet conditions, in particular on discharge velocity profile. In our study, the discharge flow corresponds to laminar to turbulent flow pattern in the circular nozzle. In the developed flow region, the liquid flow induced by the jet exhibits radial enlargement. This was first addressed by [START_REF] Lee | General integral formulation of turbulent buoyant jets in cross-flow[END_REF], who assumed that the jet radial size increase was proportional to the discharge jet velocity. This gradual enlargement is related to a decrease of the mean velocity in the jet and to the entrainment of external fluid; thus, the analysis of the axial evolution of the jet radial size will be investigated.

Clarifier

The first issue of clarification is related to flocculation. Indeed, flocculation efficiency is related to mixing in the jet clarifier. The bibliographic analysis must thus focus on mixing induced by jets, in different geometries. In terms of mixing, [START_REF] Fossett | The Application of Free Jets to the Mixing of Fluids in Bulk[END_REF] and [START_REF] Fossett | The action of free jets in mixing of fluids[END_REF] reported inclined side-entry jet mixing of free turbulent jets in cylindrical tanks. [START_REF] Fox | Single-phase blending of liquids[END_REF] investigated both laminar and turbulent inclined side-entry jet regimes and concluded that the main phenomena controlling the mixing time was the momentum source injected by the jet in the tank. In terms of vertical jet mixer, studies were reported by [START_REF] Hiby | Experiments on jet agitation, 6th CHISA congress[END_REF] and by Lane andRice (1981, 1982) in a hemi-spherical base, reporting shorter mixing times compared to flat base cylindrical tank. [START_REF] Maruyama | Jet mixing of fluids in tanks[END_REF] found that the mixing time in jet flow tank depended on the liquid depth, nozzle height, and nozzle angle, and the mixing time is a consequence of jet axis length. [START_REF] Maruyama | An investigation of bubble plume mixing by comparison with liquid jet mixing[END_REF] reviewed mixing induced in different geometries using horizontal, inclined and vertical jets. However, although global circulation was presented and global mixing time were determined, there was neither data nor information on the local phenomena controlling mixing. [START_REF] Grenville | A New Theory Improves the Correlation of Blend Time Data from Turbulent Jet Mixed Vessels[END_REF] studied the free jet mixing time of the tank with H/D ≤ 1 where H is the fluid depth and D is the vessel diameter. They proposed that the mixing time had been correlated by turbulent kinetic energy dissipation rate (or power per unit mass). The turbulent kinetic energy dissipation rate at the end of the jet's free path can be used to estimate the mixing rate and it controlled the mixing rate for the whole vessel. Then, [START_REF] Grenville | Turbulence of flow as a predictor of blend time in turbulent jet mixed vessels[END_REF] proffered the correlation based on the jet nozzle angle and compared their model with the circulation time model.

They found that both models can be used to predict accurate mixing time in the tank and their previous model presented in 1996. Further, Grenville and Tilton (2010) continued their work by studying the mixing time in various tank geometries and found that their jet turbulence model fitted in the range of 0.2 < H/D < 3 and the ratio of mixing time to circulation time is not constant but rather depending on the ratio of fluid depth to diameter of the vessel. [START_REF] Jayanti | Hydrodynamics of jet mixing in vessels[END_REF] reported that the position of the "eye" of the circulation pattern induced by a jet is a key parameter for mixing and it depends on the tank geometry. Jayanti compared hemi-spherical base, ellipsoidal base, conical base with a half cone angle of 31° and conical base with a half cone angle of 58°. The best shape was found to be conical base with a half cone angle of 31°. In this case, the "eye" of the recirculation pattern is half the overall height, the recirculation is quite strong and there is no low velocity region. This conclusion probably contributes to explain the efficiency of the present jet [START_REF] Garland | Revisiting Hydraulic Flocculator Design for Use in Water Treatment Systems with Fluidized Floc Beds[END_REF] analysed the effects of Gt on turbidity removal by hydraulic flocculator, indicating better performance when a floc blanket had been formed. They concluded that appropriate mixing time is a factor that can be used to limit the size of the clarifier. In our paper, since only clear water hydrodynamics is investigated, floc blanket will not be accounted for.

In order to investigate the hydrodynamics of the new jet clarifier, PIV experiments will be presented and discussed, both in terms of jet characteristics and in terms of flow structure leading to the estimation of Gt criteria.

Material and methods

Set up

The laboratory pilot was designed to investigate local hydrodynamic of the jet clarifier. A pseudo-twodimensional jet clarifier was designed with 56 cm. high, 95 cm. long, and 10 cm. thickness, which is named as a flat quasi-bidimensionnal (Q2D) clarifier. The Q2D jet clarifier was constructed using Plexiglass (PMMA) (1 cm thick) enabling velocity measurement with especially the flocculation zone, a cone-shape, inside the reactor. The liquid volume in the pilot is 42 litres. The liquid flows upward to the outlets (6).

As we will see, there is a strong recirculation induced by the jet in the divergent, creating long residence and efficient mixing. In this paper, we will focus on the hydrodynamics in the flocculation zone (mainly fields 1 and 2 of figure 2, and partially field 3).

PIV technique

The PIV system used in this study is the commercial system from Dantec Dynamics Co. (Denmark). The absorption takes place at 532 nm, while the emission is in the red range with a maximum around 590nm. Hence, a laser light sheet in the green range (537nm) was used and a high pass light filter (> 570nm) was added to the camera. Thus, any light disturbances may be reduced as far as possible.

This technique enables acquisition of an instantaneous two-dimensional velocity field in a vertical plane. The vertical plane investigated in this study was fixed in the plane of symmetry of the pilot (at a position Z = 5 cm, half the depth of the quasi-2D pilot). For the jet clarifier, the hydrodynamics were studied in four fields to measure the velocity field in the whole flocculation zone, illustrated in Figure 2, but in this paper, the hydrodynamic study focuses on these fields corresponding to the jet mixing flocculation zone.

The size of each PIV image was fixed to 200 mm. Each velocity field, measured over a square 200 x 200 mm², is composed of the two components (U horizontal and V vertical) of the instantaneous velocity on a 127 x 127 squared matrix. Thus, the distance between two vectors (so called PIV filter) is 1.57 mm (16 pixels). Figure 3 shows that 3000 instantaneous velocity field are sufficient to derive statistically converged mean and rms velocity components.

Global analysis

Before performing local analysis of the hydrodynamic based on PIV, it is necessary to investigate global parameters. Three flow rates were investigated in our study. They are given Table 1, as well as the Reynolds number at the outlet of the orifice nozzle. The flow pattern in the 4 mm internal diameter circular orifice nozzle is laminar for the two lower flow rates and slightly turbulent for the larger one.

Considering the downstream liquid jet, based on the Reynolds number at the outlet of the injection tube, and referring to [START_REF] Pearce | Critical Reynolds Number for Fully-Developed Turbulence in Circular Submerged Water Jets[END_REF] the flow pattern of the liquid jet is laminar for Re smaller than 500 and fully turbulent for Re larger than 3000.

Two characteristic inlet parameters can be quantified. The flux of momentum at the injection is defined as:

" # = ∬ % ! & &' = % () 〈! 〉 = % () * 〈!〉 (4)
Where SCS is the cross-section of the circular orifice nozzle. Since the discharge velocity profile is not uniform, C2 has been introduced to relate the average of the square of the velocity 〈! 〉 to the square of the mean discharge velocity 〈!〉.

The supplied power can be derived from the flux of kinetic energy at the injection:

+ = ∬ % ! , & &' = % () 〈! , 〉 = % () * , 〈! -〉 , (5)
Here again, C3 has been introduced to relate the average of the cubic of the velocity 〈! , 〉 to the cubic of the mean discharge velocity 〈!〉.

Assuming laminar flow in the injection tube, one can express the radial discharge velocity profile in terms of parabolic profile:

. /01 = 1 -3 . 4 5 (6)
Where the maximum velocity is twice the mean discharge velocity 〈!〉. One can thus easily derive analytically the two coefficients C2 and C3 in laminar flow as:

* = 〈 6 〉 〈 〉 6 = , * , = 〈 7 〉 〈 〉 7 = 2 (7)
In turbulent flow, a power law velocity profile is assumed:

-.

/01 = 31 -. 4 5 9 :

(8)

Where the power 1/6 depends on the Reynolds number (Re close to 4300 in this work), following Schlichting book.

One can thus easily derive analytically the two coefficients C2 and C3 in turbulent flow as:

* = 〈 6 〉 〈 〉 6 = 1.03 * , = 〈 7 〉 〈 〉 7 = 1.077 (9)
The numerical values of C2 and C3 are reported in Table 1 for the 3 flow rates. The averaged dissipated power per unit mass is defined as:

〈>〉 = ? @ = ) AB CDE * , 〈! -〉 , ( 10
)
Where m is the mass of liquid (kg) in the pilot and Vol is the volume of water in the pilot (Vol = 42 L).

One can finally derive the volume averaged velocity gradient G or volume averaged shear rate:

= 〈F〉 (11)
The residence time in the whole clarifier is simply given by the volume of liquid in the pilot divided by the discharge flow rate Q.

= CDE G (12)
From the velocity gradient G and the residence time t, one can estimate the Camp and Stein parameter:

= C ) AB * , √ (13)
Where, as previously, C ) AB * , ranges between 1850 and 1350. G t is thus between 25,000 and 45,000; recall that in flocculation, G t is usually in the following range 10 < < 10 (eq. 1). Results concern the hydrodynamics of the flocculation zone (estimated to 7 litres volume), located in the vertical divergent of the jet clarifier (figure 1). In this section, results are organised as follows:

velocity fields in zones 1 and 2 of the pilot (figure 2) are plotted, exhibiting a large circulation.

Circulation flow rates are estimated as well as circulation time that are compared to residence time in this zone. The characteristic shape of the jet is also investigated, in terms of vertical distribution of its width. Then the outflow is analysed, and the outward flow rate is shown to be close to the inlet flow rate.

Mean velocity fields are plotted on figure 4 for the 3 flow rates and the 2 measurement fields 1 and 2;

only the left-hand side of the PIV fields is plotted. These velocity fields exhibit similar circulation loops generated by the vertical jets. The eyes of circulations are located at Y = 340, 360 and 380 mm for the respective flow rates 11, 19 and 49 LPH. The differences between the 3 flow fields seem thus to be weak.

Each horizontal profile of vertical mean velocity is processed. The vertical velocity is positive in the jet plume (upward flow) and negative outside (downward flow). The width of the jet can thus be estimated at each vertical position Y. In order to quantify the similarity of these flow patterns, the vertical profiles of plume width are plotted on figure 5(a). A gradual enlargement is observed and can be related to a decrease of the mean velocity in the jet and to the entrainment of external fluid. The evolutions of the jet plume widths for the 3 flow rates are very similar. As reported by [START_REF] Lee | General integral formulation of turbulent buoyant jets in cross-flow[END_REF], the jet plume width increases with increasing distance from the nozzle (located at Y = 0). The estimation of the width increase with the distance is around 60 mm for 300 mm from the jet, it gives an angle for the jet development close to 10 degrees, much smaller than the geometrical angle between the two internal baffles.

The 3 eyes of circulations being located roughly in the same zones (-80 < X < -60 mm and 340 < Y < 380 mm), the horizontal profiles of vertical mean velocity at the location of the eye of circulation have been plotted on figure 5(b). The vertical velocity profiles are normalised by their maximum velocity, the three profiles can thus be superimposed.

From figure 5(b), it is possible to estimate the vertical flow rate, which will be considered as circulating flow rate Qc, by integrating the horizontal profile vertical velocity between the axis of the pilot and the lateral position of the circulation eye. The width of the half jet plume being close to 50 mm, the thickness of the pilot being 100 mm, an axisymmetric jet could be assumed. Thus, 3 flow rates Qc, called circulation flow rates have been determined. The values of the circulation flow rates Qc are reported in table 2. They range between 12 and 15 times the inlet flow rates, indicating a huge entrainment and a strong recirculation in the flocculation zone.

Clearly, the structure of the flow slightly depends on the injected flow rate, indicating that the circulation loops are similar for the three injected flow rates. Qtotal-laminar (m 3 /s) Total flow rate above the wall assuming laminar velocity profile along Z 4.05 10 -6 4.5 10 -6 6.1 10 -6

Qtotal-laminar / (Q /2) Ratio of total flow rate above the wall and half the injected flow rate 2.6 1.7 0.95

In conclusion, the jet induces a strong circulation loop inside the flocculation zone (vertical divergent).

Whatever the flowrate, 30 cm above the nozzle the angle of the jet development is close to 10°. The presence of the 2 inclined baffles (37°) has clearly an influence on the development of the jet since the circulation patterns are similar for the three flow rates. Characteristic time scales of the circulation have been estimated in Table 2. One can now investigate the distributions of velocity gradients.

Discussion on hydrodynamics for flocculation

In the section, both local-instantaneous and global (time and space averaged) velocity gradient are addressed.

On figure 7, the vertical profiles of four characteristic variables are plotted: the local mean vertical velocity, the viscous dissipation rate of kinetic energy, the Kolmogorov scale and the velocity gradient (shear rate). Figure 7(a) corresponds to the vertical profile of mean velocity along the axis (X=0), normalised by the inlet velocity. Clearly the profiles are identical, except in the lower zone, closer to the injection nozzle where the circulation around the internal baffle should modify the total flow rate that enter in the flocculation zone. This plot confirms that the global hydrodynamics induced by the jet is similar for the three flow rates and the vertical mean velocity is simply proportional to the injected flow rate.

From the instantaneous velocity fields, it is possible to estimate the local viscous dissipation rate of the mean flow kinetic energy and the local viscous dissipation rate of the turbulent kinetic energy.

These dissipation rates are based on the 2D velocity measurement. Thus, they are estimated following the expressions: Here again, its average value along the width of the divergent is derived. In order to present nondimensional velocity gradient, the horizontally averaged velocity gradient is multiplied by the depth of the pilot (h = 100 mm) and divided by the inlet velocity. The choice of inlet velocity as velocity scale is straightforward, but the choice of pilot depth (100 mm) as length scale is arbitrary. It leads to nondimensional velocity gradients close to unity. Distinctly, the three curves of such normalised velocity gradients are identical, indicating that the velocity gradient is proportional to the inlet velocity. The different results are given in Table 4. The flow rates, inlet velocities and residence times in the flocculation zone are recalled. Then the viscous dissipation rate of total kinetic energy is averaged in the plane of measurement in both horizontal and vertical directions. This is done in the two fields (figure 2), field 1 and field 2 and we observe that the viscous dissipation rate is 10 times higher close to the jet inlet where velocity gradient are stronger. The associated global velocity gradients are estimated and are √10 times greater in the bottom field and are proportional to the flowrate. Then, the velocity gradient averaged over the whole flocculation zone (once again in the vertical plane of PIV measurement) is calculated. G(< ε Floc_zone >) ranges between 3 and 13 s -1 , whereas the global values estimated initially (Table 1) were in the range 2-14 s -1 . If we multiply these velocity gradients averaged over the whole flocculation zone by the residence time in the flocculation zone G(< εc>) x tRE, we obtain an almost constant value close to 7000 for the three flow rates. This is a very interesting result that will be exploited to explain the efficiency of such a jet clarifier in terms of flocculation. Recall that during the residence time in the flocculation zone, there is a loop of circulation with a circulation time 10 times smaller than the flocculation residence time. The flocculation zone is thus a mixing zone very efficient to perform floc aggregation, followed by a clarification zone where the velocity gradient decrease progressively and the residence time increase linearly with the velocity reduction due to the geometrical enlargement.

> @IJK LEDM = N

Conclusion

In order to understand the good efficiency of a jet clarifier, a hydrodynamic study was led. In order to use PIV technique for local analysis, a quasi-bidimensional pilot was designed. Three flow rates were investigated which correspond to residence time from 1h to 4h. The local hydrodynamic analysis was limited to the flocculation zone (fields 1 & 2 of figure 2) near the jet inlet where the global velocity gradient G is higher. Results concerning the hydrodynamics of the flocculation zone reveal that the velocity fields exhibit a large circulation loop (figure 4). Circulation flow rates are estimated as well as circulation time that are 10 times larger at least than the residence time in this zone. The characteristic shape of the jet is also investigated, in terms of vertical distribution of its width. Then the outflow is analysed, and the outward flow rate is shown to be close to the inlet flow rate. These features are similar for the three flow rates. Vertical distributions of both the jet width and the vertical velocity divided by the inlet velocity are superimposed, confirming that the velocity field in the flocculation zone only depends on the inlet velocity. The flow structures (circulation) are similar and the amplitude of the velocities are proportional to the inlet (jet) velocity. During the residence time in the flocculator zone, there is a loop of circulation with a circulation time 10 times smaller than the residence time.

The flocculation zone is thus a mixing zone very efficient to perform floc aggregation.

In the discussion, both local and global (time and space averaged) velocity gradients were addressed.

First local viscous dissipation rates of turbulent kinetic energy were derived from PIV data; thus, the local velocity gradients, G, were estimated. Then, these local values of dissipation rate have been averaged in space along the width of the divergent; afterward, they have been averaged along Y (vertical average); thus, the horizontal average of velocity gradients G (horizontal average, figure 7-d)

and their global average were estimated (Table 4)., in order to get global velocity gradients. The vertical profiles of horizontal average velocity gradients normalised by the inlet velocity are superimposed.

Finally, the velocity gradients averaged over the whole flocculation zone were calculated. They range between 3 and 13 s -1 (Table 4), whereas the global values estimated initially (Table 1) were in the range 2-14 s -1 . Increasing the inlet jet flow rate, the global velocity gradients increase linearly with the jet flow rate whereas the residence time decreases linearly. Consequently, the global product G t remains constant in this region where 10 loops are followed during the flocculation process; such circulation may contribute to the strength of the flocs. This is a very interesting result that will be exploited to explain the efficiency of such a jet clarifier in terms of flocculation.

  clarifier since flocculation zone corresponds to a divergent (2D cone).Wasewar (2006) investigated design of jet mixing tank. His review summarizes different studies of jet mixed tank parameters (tank geometry, jet configuration, jet velocity, jet diameter, jet flow rate and fluid properties) to get an optimum design. He pointed out that mixing time is an important parameter to design jet tank devices.[START_REF] Perumal | Experimental Investigation on Mixing time Analysis of Jet Mixer[END_REF] and[START_REF] Randive | Study Of Jet Mixing In Flocculation Process[END_REF] investigated jet mixing; they pointed out that the difference between jet and bulk liquid velocity creates a turbulent mixing zone along the jet boundary. In this mixing zone, some part of the surrounding fluid is circulated at high velocity and create a circulation loop, thus leading to mix the bulk of the liquid. This kind of circulation loop induced by the jet will be investigated in this paper.[START_REF] Randive | Study Of Jet Mixing In Flocculation Process[END_REF] reviewed the jet mixing in the flocculation process and summarized several models to estimate the mixing time in terms of other parameters such as jet velocity, jet diameter, jet path length, and tank diameter and height.Kennedy et al. (2017) studied the effect of the distance between injection and suction ports on the control mixing time of submerged recirculation jets. They found that the distance between the ports can be used to control mixing time at the same value of injection velocity and an empirical correlation to predict the mixing time under short-circuiting conditions of the flow is dominant, which retains the same dependence of mixing time on the injection velocity and the tank diameter.

Figure 1 .

 1 Figure 1. Q2D jet clarifier experimental setup

  required basic elements include a double-pulsed Nd: YAG laser (big sky laser of 30 mJ.) operated at a trigger rate close to 10 Hz. Each pulsation was controlled by the trigger rate between 3 -20 ms depending on the flow velocity. A CCD camera (Flow sense EO, Dantec Dynamic) was used to record the flow at each flash laser. The resolution of each recorded field is 200 × 200 mm 2 with a scale close to 0.1 mm/pixels (2048 × 2048 pixels 2 ). In this work, the Rhodamine B suspensions are well mixed in the water preparation tank by an agitator with the motor before feeding into the flat jet clarifier. The seeding particles are covered with the fluorescent dye Rhodamine B, for which maximum light

Figure 2 .

 2 Figure 2. PIV windows in the vertical plane of symmetry of the jet clarifier (lengths are expressed in

Figure 4 .

 4 Figure 4. Mean velocity field for each injected flow rate

Figure 6

 6 Figure 6 Characteristic flow above the internal wall bounding the flocculation zone.

  Figure 7(b) presents the vertical profiles of the horizontal averaged (along the width of the divergent)

  Figure 7. Vertical profiles of (a) mean velocity components along jet axis (X = 0 mm), (b) viscous

Table 1 .

 1 Global hydrodynamic characteristics for the 3 flow rates

	Q Flow rate (L/h)	11	19	49
	U injection (m/s)	0.24	0.42	1.08
	Re tube	970	1680	4330
	C3 = <U 3 >/<U> 3	2	2	1,077
	Supplied power (mW) 0.36	1.86	17.2
	〈>〉 (W/kg)	4.3 e-6	2.2 e-5	2 e-4

Table 2

 2

		. Processed hydrodynamic characteristics for the 3 flow rates
	Q (L/h)	Injected flow rate	11	19	49
	U (m/s)	Injection velocity	0.24	0.42	1.08
	tRF (min)	Residence time in flocculation zone	38.16	22.08	8.58

Table 3 .

 3 Characteristic data for the flow outside the flocculation zone

	20

Table 4

 4 Estimation of global velocity gradients in flocculation zone, based on PIV data processing Since the velocity gradients in the flocculation zone evolve always more or less linearly with the inlet flow, as the residence time is inversely proportional to the flow rate, the Camp and Stein criteria Gt parameter recommended to achieve efficient flocculation will be almost constant.Finally, the results on velocity gradients are summarized in Table5for the whole jet clarifier. Initially, global estimations of velocity gradients were obtained from global analysis where the total dissipation rate is calculated from a balance with the supplied power at the injection. In table 5, they are compared to the PIV measured velocity gradients, averaged over the 4 PIV planes and derived from local estimation of the shear rate (in terms of local viscous dissipation rate of kinetic energy, given by the sum of equations 16 and 17). On the whole jet tank again, the global velocity gradient increases linearly with the jet flow rate whereas the residence time decreases linearly. Consequently, the product G t remains constant close to 30000 for the 4 PIV fields whereas it was close to 7000 for first two fields near in the flocculation zone. In the same time, the residence time in increased 6 times (the ratio of the total volume divided buy the flocculation volume), it means that the average velocity gradient

	Q (L/h)		Injected flow rate		11	19	49
	U (m/s)		Injection velocity		0.24	0.42	1.08
	tRF (h)	Residence time in flocculation zone	0.636	0.368	0.143
	< εField_1 > (m 2 /s 3 )		Average ε in field 1 zone		1.8 10 -6	4.0 10 -6	3.6 10 -5
	G(< εField_1>) (s -1 )		Velocity gradient in field 1 zone	1.34	2.	6
	< εField_2> (m 2 /s 3 )		Average ε in field 2 zone		1.7 10 -5	5 10 -5	3 10 -4
	G(< εField_2>) (s -1 )		Velocity gradient in field 2 zone	4.1	7.1	17.2
	< εFloc_zone> (m 2 /s 3 )		Average ε in flocculation zone	9.3 10 -6	2.7 10 -5	1.7 10 -4
	G(< ε Floc_zone >) (s -1 )	Velocity gradient in flocculation zone	3.06	5.2	12.9
	G(< εc>) x tRE (-)	First non-dimensional velocity gradient	7010	6890	6640
	G(< εc>) / (U / h)		Second non-dimensional velocity	1.28	1.24	1.19
			gradient
	As shown in the last lines of table 4, two non-dimensional velocity gradients are defined: (1) global G
	(square root of space average dissipation divided by the viscosity) times flocculation zone residence
	time or (2) global G (square root of space average dissipation divided by the viscosity) times (depth/U
	injection); whatever the definition, the non-dimensional global velocity gradient give constant values
	for the different flow rates. Table 5 Values of velocity gradients in the whole jet clarifier
	Q (L/h) Injected flow rate	11	19		49
	〈 〉 (s -1 )		2.1	4.7		14.3

  Sobrino, J.A.H., Thiem, L.T., Alkhatib, E., 1996. Optimizing submerged jet flocculator performance.Journal American Water Works Association, 88, 81 -92. doi:https://10.1002/J.1551-8833.1996.TB06602.X M.R. von Smolan Smoluchowski, Drei Vorträge über Diffusion,Brownsche MolekularBewegung undKoagulation von Kolloidteilchen, Physikaliche Zeitschrift, Jg.17, p. 557-571, 585-599 (1916) Wasewar, K. L., 2006. A Design of Jet Mixed Tank. Chem. Biochem. Eng. Q., 20(1), 31 -46.

	SCS	(m²)	nozzle cross-section
	t	(s)	contact time
	tC	(s)	circulation time
	tRE	(s)	residence time in flocculation zone
	U	(m s -1 )	axial or vertical velocity
	UMAX (m s -1 )	maximum axial or vertical velocity
	u'	(m s -1 )	axial or vertical turbulent velocity
	Nomenclature
	Vol	(m 3 )	volume
	C2, C3 (-)	constant
	V	(m s -1 )	radial or horizontal velocity
	d	(m)	nozzle diameter
	G v'	(s -1 ) (m s -1 )	velocity gradient axial or vertical turbulent velocity
	W	(m s -1 )	velocity component
	h	(m)	depth of the pilot
	J0 w'	(kg m s -2 ) (m s -1 )	flux of momentum turbulent velocity component
	X	(m)	horizontal coordinate
	m	(kg)	mass
	Y	(m)	vertical coordinate
	P	(W)	supplied power
	Q	(m 3 s -1 )	injected flow rate
	QC ε	(m 3 s -1 ) (m² s -3 or W/kg) viscous dissipation of kinetic energy circulation flow rate
	r η	(m) (m)	radial position Kolmogorov scale
	R	(m) (s -1 )	nozzle radius shear rate
	Re ν	(-) (m² s -1 )	Reynolds number kinematic viscosity
	S ρ	(s -1 ) (kg s -3 )	symmetric part of the velocity gradient tensor density

Acknowledgements

This work was supported by Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0152/2558) under Thailand Research Fund (TRF) and French Embassy in Thailand. Funding last year of Ploypailin Romphophak PhD was supported by the team Transfer-Interface-Mixing from Toulouse Biotechnology Institute in Toulouse.