Global solutions to coupled (Navier-)Stokes Newton systems in R 3

Abstract

We consider the motion of spherical particles in the whole space $R^{3}$ filled with a viscous fluid. We show that, when modelling the fluid behavior with an incompressible Stokes system, solutions are global and no collision occurs between the spheres in finite time.

Keywords

Fluid-solid systems Stokes equations contact issue global-in-time existence of solutions

1. Introduction

In this paper we tackle global-in-time existence of solutions to the system modelling the motion of several spherical particles in $R^{3}$ . The system under consideration reads as follows: $\begin{matrix} (1) & \{\begin{matrix} Re (\partial_{t} u + u \cdot \nabla u) = div Σ (u, p), \\ div u = 0, \end{matrix} on F (t), \end{matrix}$ where $F (t) = R^{3} ∖ ⋃_{i = 1}^{N} B_{i} (t)$ is the whole space $R^{3}$ deprived from a finite number of moving spheres $B_{i} (t)$ of center $X_{i} (t) \in R^{3}$ and same radius $R > 0$ . The unknowns $u : F (t) \to R^{3}$ and $p : F (t) \to R$ stand for the velocity-field and pressure of the ambient fluid. The symbols Re and Σ denote respectively the Reynolds number and the fluid stress tensor given by the Newton law: $\begin{matrix} Σ (u, p) = 2 D (u) - p I_{3}, D (u) = \frac{\nabla u + \nabla u^{⊤}}{2} . \end{matrix}$

The motion of the spheres is also unknown and integrated by solving the equations of solid dynamics. We have first that the motion of the spheres reduces to the combination of a translation and a rotation. Assuming no-slip of the fluid on solid boundaries, we have: $\begin{matrix} (2) & u_{|_{\partial B_{i} (t)}} = V_{i} + ω_{i} \times (x - X_{i}), for i = 1, \dots, N, \end{matrix}$ where × denotes the standard vector product and $(V_{i}, ω_{i}) \in R^{3} \times R^{3}$ with $\begin{matrix} (3) & {\dot{X}}_{i} = V_{i} for i = 1, \dots, N . \end{matrix}$ We have then the Newton laws for the linear and angular momentums: $\begin{matrix} (4) & \{\begin{matrix} m {\dot{V}}_{i} = - \int_{\partial B_{i} (t)} Σ (u, p) n d σ + F_{i}, \\ J {\dot{ω}}_{i} = - \int_{\partial B_{i} (t)} (x - X_{i}) \times Σ (u, p) n d σ + T_{i}, \end{matrix} for i = 1, \dots, N . \end{matrix}$ The symbols m, J stand for the mass and inertia of the spheres and n is the normal to $\partial B_{i} (t)$ directed toward the interior of $B_{i} (t)$ . For simplicity, we assume here that all spheres are homogeneous with the same density. But this asssumption could be relaxed. The symbols ${(F_{i}, T_{i})}_{i = 1, \dots, N}$ take into account forces and torques that might act on the spheres. In contrast, we do not consider any source term in (1). This could also be relaxed. But, what we have in mind here is that the forcing terms is due to gravity. In that case, we can normalize the pressure so that there is no forcing term in the fluid equations and the term $F_{i}$ is computed via archimedes law while $T_{i}$ also vanishes. Finally, the system is completed by requiring that the fluid is at rest at infinity: $\begin{matrix} (5) & lim_{| x | \to \infty} | u (t, x) | = 0 . \end{matrix}$

The above system is meaningful only if the spheres do not overlap and are away from contact, meaning that the set of centers belong to $\begin{matrix} O^{R} : = {{(X_{i})}_{i = 1, \dots, N} s.t. {min}_{i \neq j} | X_{i} - X_{j} | > 2 R} . \end{matrix}$ Here and below, we name indifferently collision or contact the event that the distance between two spheres vanishes. In case of contact, a supplementary equation should be added to prescribe the contact dynamics. In case $Re = 0$ , the system (1)–(5) reduces to a differential system in ${(X_{i}, ω_{i}, V_{i})}_{i = 1, \dots, N}$ where the forcing term appearing in (4) is computed by solving the partial differential equations in (1). In particular, initial conditions reduce to fixing the initial centers and velocities: $\begin{matrix} (6) & X_{i} (0) = X_{i}^{0}, V_{i} (0) = V_{i}^{0}, ω_{i} (0) = ω_{i}^{0}, for i = 1, \dots, N . \end{matrix}$

The system (1)–(5) has been extensively studied in the two-dimensional as well as the three-dimensional setting, when $Re > 0$ and the spheres are restricted to move in a bounded domain. Firstly, existence of classical solutions is obtained by looking at the problem via a change of geometry which fixes the fluid domain [9,21]. Compactness methods are also adapted to this moving-domain setting to obtain existence of weak solutions before [1,3,4,16] and then regardless contacts [6,20]. We emphasize that these latter references extend existence of weak solutions beyond a contact time, if any, but do not consider the contact occurence issue. Existence of solutions prior to contact is also obtained for solid bodies moving in the whole space [2,18,22]. Existence of contacts is discussed in the case of particular geometries: a sphere falling above a ramp [12] or a more complicated boundary [13].

In this paper, we focus on the case $Re = 0$ , our main objective is to prove that solutions to (1)–(5) are global and, in particular, that no contact between the spheres occurs in finite time. The result would hold for the full Navier Stokes system. However, we restrict to the case $Re = 0$ for several reasons. Firstly, our motivation to tackle this problem is to discuss the influence of complex geometries on the contact issue thus extending the previous results in [12,13]. In these previous contributions, it is shown that the convective term does not modify the occurence or not of collisions. Secondly, the case $Re = 0$ is of particular interest for tackling homogenization problems [5,14,15]. Finally, tackling the contact occurence in case $Re = 0$ enables to write a global existence and uniqueness result for system (1)–(5).

Our main result regarding the existence of solution to the differential system (4)–(5) satisfied ${(X_{i}, ω_{i}, V_{i})}_{i = 1, \dots, N}$ with initial data satisfying (6) and forcing terms computed through (17) then reads

Theorem 1.
Assume that $\begin{matrix} (F_{i}, T_{i}) \in L_{loc}^{2} ([0, \infty); R^{3} \times R^{3}) \forall i = 1, \dots, N . \end{matrix}$ Given initial data ${(X_{i}^{0})}_{i = 1, \dots, N} \in O^{R}$ and ${(V_{i}^{0}, ω_{i}^{0})}_{i = 1, \dots, N} \in {(R^{3} \times R^{3})}^{N}$ , there exists a unique global solution ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ to ( 1 )–( 5 ) completed with initial condition ( 6 ). In particular, there holds: $\begin{matrix} {(X_{i} (t))}_{i = 1, \dots, N} \in O^{R} \forall t \in [0, \infty) . \end{matrix}$

Since the fluid unknowns $(u, p)$ are implicitly involved in the computations, we do not make precise their regularity in the above result. Straightforward analysis would yield that these fluid unknowns are smooth in space (in the fluid domain) and, broadly, as smooth as solid trajectories, in time. The content of the paper is as follows. In the next section, we recall the construction of local-in-time solutions for (1)–(5) when $Re = 0$ and we show that finite-time blow-up reduces to possible contacts between the spheres. Global existence is then obtained by a contradictory argument. Broadly, we assume that contact holds in $t = T_{}$ and we show then that, for t close to $T_{}$ the minimal distance between particles: $\begin{matrix} d (t) = min_{i \neq j} dist (B_{i} (t), B_{j} (t)), \end{matrix}$ solves a differential inequality (see Proposition 9): $\begin{matrix} (7) & \frac{| \dot{d} (t) |}{d (t)} ≲ f + | ln (d (t)) |, \end{matrix}$ with an explicit source term f bounded by initial data only. We conclude then by a Gronwall argument that there is a contradiction. The differential inequality (7) is obtained by measuring the repulsion force exerted by the fluid in the gap between two spherical particles. In particular, such repulsion forces occur as soon as there is a small gap between two particles. It is then necessary to analyse the contact geometry to prevent the repulsion forces to compensate. In this respect, the key geometric argument in our proof is that there is at least one colliding particle at the contact time with contact points strictly on one hemisphere. This geometrical analysis and the full contradictory argument is detailed in Section 3. This strategy involves to multiply equation (1) with a suitable test-function. The construction of this test-function and its analysis follows previous constructions of the first author (see [8,11–13]). In Section 3, we explain how to adapt the construction to the context under consideration here and recall the main properties at stake for our purpose.

Our result extends straightforwardly to the alternative context in which bodies have no inertia either ( $m = 0$ , $J = 0$ ). In this case, our result extends the previous analysis of [14] in which the author assumes further that the solid phase is dilute in the fluid. However, despite considering several solid bodies in a container, our no-collision result remains restricted to spherical solid bodies and containers with simple geometries. The result or techniques do not seem to extend straightforwardly in a more general context. Firstly, with more general geometries, contact occurence may divide the fluid domain into several connected components. This may happen in the case of one disk moving in a square in 2D or a sphere moving in a hourglass in 3D. The particular 2D case mentioned here is investigated by the second author in [19]. She shows there that the test-function method that we use herein does not apply since the test-function that we construct locally in the neighbourhood of each contact points cannot be extended to the whole fluid domain into a divergence free vector field for a flux argument. However, no contact still holds by a simpler functional inequality relating the dissipation of the solid velocity with the fluid dissipation. Secondly, our key geometrical argument is not valid in full generality. Particular examples in 2D [10] and 3D [13] showed already that there exist configurations with exterior boundaries which make this argument to fail. Finally, we restrict ourselves to the case of spherical particles as a paradigm for bodies with smooth boundaries. The roughness-induced effect on the collision process has been studied in [8].
2. Local-in-time Cauchy theory and blow up analysis

In this section, we prove existence and uniqueness of a non-extendable solution and analyze the scenarios of blow-up. To yield the first part, we study local-in-time existence and uniqueness of solutions. Our result reads as follows:

Proposition 2.
Assume that $\begin{matrix} (F_{i}, T_{i}) \in L_{loc}^{2} ([0, \infty); R^{3} \times R^{3}), \forall i = 1, \dots, N . \end{matrix}$ Given initial data ${(X_{i}^{0})}_{i = 1, \dots, N} \in O^{R}$ and ${(V_{i}^{0}, ω_{i}^{0})}_{i = 1, \dots, N} \in {(R^{3} \times R^{3})}^{N}$ , there exists $T_{0} > 0$ and a unique solution ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ to ( 1 )–( 5 ) on $(0, T_{0})$ completed with initial condition ( 6 ).

For completeness, we give in this section a brief proof of this result by relying on the same approach as for the case of Navier Stokes equations. We point out again that our system can be seen as a differential system with source depending on the unknows through solving the fluid pde. With this approach, the key difficulty would be to analyze the regularity of the fluide forces and torques depending on the position of the bodies. We priviledge here another approach closer to the classical construction of solutions to the Navier Stokes equations in time-dependent domain. We transform the Stokes system on the moving domain into a system in the cylindrical domain $F (0) \times (0, T)$ . For this reason, we use a non-linear and local change of coordinates ϕ which only acts on a neighbourhood of the spheres and maps $F (0)$ into $F (t)$ . This change of variable is initially introduced by Inoue and Wakimoto in [17]. For simplicity, $F (0)$ will be denoted throughout this section by $F_{0}$ and $B_{i} (0)$ by $B_{i}$ .
2.1. The change of variables

Let us fix N couples $\begin{matrix} {(V_{i}, ω_{i})}_{i = 1, \dots, N} \in H^{1} (0, T, R^{3} \times R^{3}), \end{matrix}$ and consider $X = {(X_{i}^{0})}_{i = 1, \dots, N} \in O^{R}$ . Moreover, we fix in the sequel $ε > 0$ such that $\begin{matrix} inf_{i \neq j} | X_{i}^{0} - X_{j}^{0} | - 2 R > ε . \end{matrix}$ Consider then a family of smooth functions ${(ψ_{i})}_{i = 1, \dots, N}$ , such that $supp ψ_{i} \subset B (X_{i}^{0}, R + \frac{ε}{2})$ , $0 ⩽ ψ_{i} (x) ⩽ 1$ and $ψ_{i} \equiv 1$ in $\overline{B} (X_{i}^{0}, R + \frac{ε}{4})$ . We define the mapping $Λ : R^{3} \times [0, T] \to R^{2}$ as follows: $\begin{matrix} (8) & Λ (x, t) = \frac{1}{2} \sum_{i = 1}^{N} \nabla \times (ψ_{i} (x) V_{i} (t) \times x) . \end{matrix}$ Since the cut-off functions ${(ψ_{i})}_{i = 1, \dots, N}$ are smooth, it follows that for all $t \in [0, T]$ the function $Λ (t, \cdot) \in C^{\infty} (R^{3}; R^{3})$ . Moreover, for all $x \in R^{3}$ , the function $Λ (\cdot, x)$ is of class $H^{1} (0, T; R^{3})$ as ${(V_{i})}_{i = 1, \dots, N} \in H^{1} (0, T, R^{3})$ . We state now two lemmas whose proofs are left to the reader (see [21, Section 4] for more details in the 2D case). Fristly, we have:

Lemma 3.
The mapping Λ defined in ( 8 ) satisfies the following properties:
$\exists r_{0} > 0$ , such that $Λ \equiv 0$ outside $K_{0} : = \overline{B (0, r_{0})}$ , for all $t \in [0, T]$ ,

$\nabla \cdot Λ = 0$ in $R^{3} \times [0, T]$ ,

$Λ (x, t) = V_{i} (t)$ on $\overline{B} (X_{i}^{0}, R + \frac{ε}{4})$ .

The mapping ϕ is then defined as the solution of the following Cauchy problem: $\begin{matrix} (9) & \{\begin{matrix} \frac{\partial ϕ}{\partial t} (y, t) = Λ (ϕ (y, t), t), t \in] 0, T], \\ ϕ (y, 0) = y \in R^{3} . \end{matrix} \end{matrix}$ Secondly, by applying Cauchy–Lipschitz–Picard theorem, we obtain: Lemma 4.
For all $y \in R^{3}$ , the initial-value problem ( 9 ) admits a unique solution $ϕ (y, \cdot) : [0, T] \to R^{3}$ , which is $C^{1}$ on $[0, T]$ . Moreover, we have the following properties:
For all $t \in [0, T]$ , the mapping $ϕ (\cdot, t)$ is a $C^{\infty}$ -diffeomorphism from $R^{3}$ onto itself and from $B_{i}$ onto $B_{i} (t)$ whenever $B_{i} (t) \subset B (X_{i}^{0}, R + \frac{ε}{4})$ .

Let $ϕ^{- 1} (\cdot, t)$ be the inverse mapping of $ϕ (\cdot, t)$ . Then, for all $x \in R^{3}$ , the mapping $t \to ϕ^{- 1} (x, t)$ is a $C^{1}$ function in $[0, T]$ .

For all $(y, t) \in R^{2} \times [0, T]$ ; the determinant of the jacobian matrix $J_{ϕ}$ of the mapping $ϕ (\cdot, t)$ is one.

2.2. Equations in cylindrical domain

Now, we rewrite system (1)–(6) in the cylindrical domain $F_{0} \times (0, T)$ . With the definitions of the previous subsection, we set $\begin{matrix} (10) & \begin{matrix} U (y, t) = J_{ϕ^{- 1}} (ϕ (y, t), t) u (ϕ (y, t), t), \\ P (y, t) = p (ϕ (y, t), t), \forall (y, t) \in F_{0} \times [0, T], \end{matrix} \end{matrix}$ where $J_{ϕ}$ and $J_{ϕ^{- 1}}$ are respectively the Jacobian matrix of the diffeomorphism mapping $ϕ (\cdot, t)$ and its inverse.

Formal computations show that $(U, P, {(V_{i}, ω_{i})}_{i = 1, \dots, N})$ satisfies the following set of equations when $Re = 0$ : $\begin{matrix} (11) & \{\begin{matrix} - [L U] + [G P] = 0, & in F_{0} \times] 0, T [, \\ \nabla \cdot U = 0, & in F_{0} \times] 0, T [, \\ U (y, t) = V_{i} (t) + ω_{i} (t) \times (y - X_{i}^{0}), & in \partial B_{i} \times [0, T [, \end{matrix} \end{matrix}$ and for all $i \in {1, \dots, N}$ , we have: $\begin{matrix} (12) & \{\begin{matrix} m {\dot{V}}_{i} (t) = - \int_{\partial B_{i}} Σ (U, P) n d σ + F_{i}, & t \in] 0, T [, \\ J {\dot{ω}}_{i} (t) = - \int_{\partial B_{i}} (y - X_{i}^{0}) \times Σ (U, P) n d σ + T_{i}, & t \in] 0, T [, \end{matrix} \end{matrix}$ with $\begin{matrix} (13) & {\dot{X}}_{i} (t) = V_{i} (t), \forall t \in] 0, T [. \end{matrix}$ Here the operators $[L U]$ and $[G P]$ are defined as follows: $\begin{array}{l} {[L U]}_{i} = \sum_{j, k = 1}^{2} \frac{\partial}{\partial y_{j}} (g^{j k} \frac{\partial U_{i}}{\partial y_{k}}) + 2 \sum_{j, k, ℓ = 1}^{2} g^{k ℓ} Γ_{j, k}^{i} \frac{\partial U_{j}}{\partial y_{ℓ}} \\ (14) & + \sum_{j, k, ℓ = 1}^{2} {\frac{\partial}{\partial y_{k}} (g^{k ℓ} Γ_{j, ℓ}^{i}) + \sum_{m = 1}^{2} g^{k ℓ} Γ_{j, ℓ}^{m} Γ_{k, m}^{i}} U_{j}, \\ (15) & {[G P]}_{i} = \sum_{j = 1}^{2} g^{i j} \frac{\partial P}{\partial y_{j}}, \end{array}$ with the metric contravariant tensor $g^{i j}$ , metric covariant tensor $g_{i j}$ and the Christoffel symbol $Γ_{i, j}^{k}$ are given by: $\begin{matrix} g^{i j} = \sum_{k = 1}^{2} \frac{\partial ϕ_{i}^{- 1}}{\partial x_{k}} \frac{\partial ϕ_{j}^{- 1}}{\partial x_{k}}, g_{i j} = \sum_{k = 1}^{2} \frac{\partial ϕ_{k}}{\partial y_{i}} \frac{\partial ϕ_{k}}{\partial y_{j}}, Γ_{i, j}^{k} = \frac{1}{2} \sum_{ℓ = 1}^{2} g^{k ℓ} {\frac{\partial g_{i ℓ}}{\partial y_{j}} + \frac{\partial g_{j ℓ}}{\partial y_{i}} - \frac{\partial g_{i j}}{\partial y_{ℓ}}} . \end{matrix}$ With these notations at-hand, we recall the following equivalence result.

Lemma 5.
Assume $T_{0} > 0$ is chosen such that $\begin{matrix} B_{i} (t) \subset B (X_{i}^{0}, R + \frac{ε}{2}), \forall t \in [0, T_{0}] . \end{matrix}$ Then ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ is a solution to ( 1 )–( 5 ) on $(0, T_{0})$ if and only if ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ is a strong solution of ( 11 )–( 12 )–( 13 ) on $(0, T_{0})$ .

The proof of this lemma readily adapts from the Navier Stokes case. We refer the reader to see Propositions 4.5 and 4.6 in [21] for more details.
2.3. Reformulation of problem (11)–(12)

To simplify the approach and possibly reuse former computations on similar topics, we propose to solve the differential system (1)–(5) by running explicitly the fixed-point argument on its rephrasing (11)–(12)–(13). To this end, we rewrite (11) as follows: $\begin{matrix} (16) & \{\begin{matrix} - Δ U + \nabla P = f, & in F_{0} \times] 0, T [, \\ \nabla \cdot U = 0, & in F_{0} \times] 0, T [, \\ U (y, t) = V_{i} (t) + ω_{i} (t) \times (y - X_{i}^{0}), & in \partial B_{i} \times [0, T [, \end{matrix} \end{matrix}$ where $f = f_{1} [U] + f_{2} [P]$ with $\begin{matrix} f_{1} [U] = [L U] - Δ U, f_{2} [P] = \nabla P - [G P] . \end{matrix}$ To study the above problem, we deal with f as a source term which acts as a perturbation in small time since the g tensors are both the identity at $t = 0$ .

Next, we state the following theorem that is a direct consequence of Theorem V.2.1 in [7] and [7, Lemma IV.6.1] up to a localizing argument:

Lemma 6.
Let $F_{0}$ be the whole space $R^{3}$ deprived from N spheres $B_{1}, \dots, B_{N}$ with respective centers $X_{i}^{0}$ . Given $\bar{f} \in L^{2} (F_{0})$ with compact support and ${({\bar{V}}_{i}, {\bar{ω}}_{i})}_{i = 1, \dots, N} \in R^{6}$ , there exists a unique pair $(U, P) \in H_{loc}^{2} (F_{0}) \times H_{loc}^{1} {(F_{0})}_{/ R}$ solving $\begin{matrix} (17) & \{\begin{matrix} - Δ U + \nabla P = \bar{f}, & on F_{0}, \\ \nabla \cdot U = 0, & on F_{0}, \\ U (y) = {\bar{V}}_{i} + {\bar{ω}}_{i} \times (y - X_{i}^{0}), & on \partial B_{i}, for i = 1, \dots, N, \\ | U (y) | \to 0 & when | y | \to \infty . \end{matrix} \end{matrix}$ Moreover, there exist constants $C_{0}$ and $C_{K}$ , depending on the relatively compact subset $K \subset R^{3}$ containing all the $B_{i}$ , such that: $\begin{array}{l} (18) & ‖ \nabla U ‖_{L^{2} (F_{0})} ⩽ C_{0} (‖ \bar{f} ‖_{L^{2} (F_{0})} + \sum_{i = 1}^{N} | {\bar{V}}_{i} | + | {\bar{ω}}_{i} |), \\ (19) & ‖ U ‖_{H^{2} (K ∖ ⋃_{i = 1}^{N} {\overline{B}}_{i})} + ‖ P ‖_{H^{1} (K ∖ ⋃_{i = 1}^{N} {\overline{B}}_{i})} ⩽ C_{K} (‖ \bar{f} ‖_{L^{2} (F_{0})} + \sum_{i = 1}^{N} | {\bar{V}}_{i} | + | {\bar{ω}}_{i} |) . \end{array}$

This motivates the definition of the solution operator of problem (17): $\begin{matrix} (20) & U (\bar{f}, {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}) : = U, P (\bar{f}, {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}) : = P . \end{matrix}$ Setting $Z (t) = (Z_{1} (t), \dots, Z_{N} (t))$ where $Z_{i} (t) = (X_{i} (t), V_{i} (t), ω_{i} (t)) \in O^{R} \times R^{3} \times R^{3}$ , we obtain that Z solves the following Cauchy problem: $\begin{matrix} (21) & \{\begin{matrix} \frac{d Z}{d t} (t) = A^{- 1} S [Z] (t), \\ Z (0) = Z^{0}, \end{matrix} \end{matrix}$ where A is a diagonal matrix whose diagonal entries are respectively $1, m, J, \dots, 1, m, J$ , $S = (S_{1}, \dots, S_{N})$ with: $\begin{matrix} S_{i} [Z] (t) = (\begin{matrix} V_{i} \\ - \int_{\partial B_{i}} Σ (U, P) n d σ + F_{i} \\ - \int_{\partial B_{i}} Σ (U, P) n \times (y - X_{i}^{0}) d σ + T_{i} \end{matrix}) \end{matrix}$ in which the vector field $(U, P)$ is a fixed point of: $\begin{matrix} (22) & \begin{matrix} U (f_{1} [U] + f_{2} [P], V_{1}, ω_{1}, \dots, V_{N}, ω_{N}) = U, \\ P (f_{1} [U] + f_{2} [P], V_{1}, ω_{1}, \dots, V_{N}, ω_{N}) = P, \end{matrix} \end{matrix}$ and $Z^{0} = (Z_{1}^{0}, \dots, Z_{N}^{0})$ with $Z_{i}^{0} = (X_{i}^{0}, V_{i}^{0}, ω_{i}^{0})$ for $i = 1, \dots, N$ . We point out that we denoted $S_{i} [Z]$ on purpose since the operator $S_{i}$ depends on the whole past of the function Z through the computations of L and G. We cannot write the equations satisfied by ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ as a simple differential system where $S$ would depend on $Z (t)$ only.

We construct a fixed-point of (21)–(22) via a standard contraction argument. We introduce the mapping $S_{T}$ of $\begin{matrix} H^{1} {((0, T); R^{9})}^{N} \times L^{2} (0, T; {\dot{H}}^{1} (F_{0}) \cap H^{2} (K_{0} \cap F_{0})) \times L^{2} (0, T; H^{1} (K_{0} \cap F_{0}) \end{matrix}$ defined as follows: $\begin{matrix} S_{T} : (\bar{Z}, \bar{U}, \bar{P}) \mapsto S_{T} (\bar{Z}, \bar{U}, \bar{P}) = (Z, U, P), \end{matrix}$ with: $\begin{matrix} \{\begin{matrix} Z (t) = Z_{0} + \int_{0}^{t} A^{- 1} S [\bar{Z}] (s) d s, \\ U (t, \cdot) = U (f_{1} [\bar{U} (t, \cdot)] + f_{2} [\bar{P} (t, \cdot)], {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}), \\ P (t, \cdot) = P (f_{1} [\bar{U} (t, \cdot)] + f_{2} [\bar{P} (t, \cdot)], {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}) |_{K_{0}} . \end{matrix} \end{matrix}$ We recall here that $K_{0}$ is defined in item i. of Lemma 3. To analyse the properties of this mapping, we fix $O_{T}^{0} [M]$ the set of triplets $\begin{matrix} (Z, U, P) \in (H^{1} {((0, T); R^{9})}^{N} \times L^{2} (0, T; {\dot{H}}^{1} (F_{0}) \cap H^{2} (K_{0} \cap F_{0})) \times L^{2} (0, T; H^{1} (K_{0} \cap F_{0})), \end{matrix}$ such that $\begin{array}{l} (23) & {‖ X_{i} - X_{i}^{0} ‖}_{L^{\infty} (0, T)} < \frac{ε}{4}, V_{i} (0) = V_{i}^{0}, ω_{i} (0) = ω_{i}^{0}, for i = 1, \dots, N, \\ (24) & \sum_{i = 1}^{N} ‖ V_{i} ‖_{H^{1} (0, T)}^{2} + ‖ ω_{i} ‖_{H^{1} (0, T)}^{2} ⩽ 4 \sum_{i = 1}^{N} {| X_{i}^{0} |}^{2} + {| V_{i}^{0} |}^{2} + {| ω_{i}^{0} |}^{2} + ‖ F_{i} ‖_{L^{2} (0, T)}^{2} + ‖ T_{i} ‖_{L^{2} (0, T)}^{2}, \\ (25) & \int_{0}^{T} {‖ \nabla U (t, \cdot) ‖}_{L^{2} (F_{0})}^{2} + {‖ U (t, \cdot) ‖}_{H^{2} (K_{0} \cap F_{0})}^{2} + {‖ P (t, \cdot) ‖}_{H^{1} (K_{0} \cap F_{0})}^{2} ⩽ M^{2} . \end{array}$ We emphasize that $O_{T}^{0} [M]$ is a convex set that is complete when endowed with the distance associated to the norm: $\begin{array}{rcl} {‖ (Z, U, P) ‖}^{2} & = & \sum_{i = 1}^{N} (‖ X_{i} ‖_{L^{\infty} (0, T)}^{2} + ‖ V_{i} ‖_{H^{1} (0, T)}^{2} + ‖ ω_{i} ‖_{H^{1} (0, T)}^{2}) \\ + \int_{0}^{T} {‖ \nabla U (t, \cdot) ‖}_{L^{2} (F_{0})}^{2} + {‖ U (t, \cdot) ‖}_{H^{2} (K_{0} \cap F_{0})}^{2} + {‖ P (t, \cdot) ‖}_{H^{1} (K_{0} \cap F_{0})}^{2} . \end{array}$

It is also non-empty under the condition that T is sufficiently small with respect to ${(X_{i}^{0})}_{i = 1, \dots, N}$ (so that condition (23)–(24) is satisfied by a constant extension of initial data). We also point out that, under the condition (23) there holds: $\begin{matrix} ‖ V_{i} ‖_{L^{\infty} (0, T)} ⩽ | V_{i}^{0} | + \sqrt{T} ‖ V_{i} ‖_{H^{1} (0, T)}, \\ ‖ ω_{i} ‖_{L^{\infty} (0, T)} ⩽ | ω_{i}^{0} | + \sqrt{T} ‖ ω_{i} ‖_{H^{1} (0, T)}, \end{matrix} \forall i = 1, \dots, N,$ so that we have $\begin{array}{l} \sum_{i = 1}^{N} ‖ V_{i} ‖_{L^{\infty} (0, T)}^{2} + ‖ ω_{i} ‖_{L^{\infty} (0, T)}^{2} \\ (26) & ⩽ 10 \sum_{i = 1}^{N} {| X_{i}^{0} |}^{2} + {| V_{i}^{0} |}^{2} + {| ω_{i}^{0} |}^{2} + 8 \sum_{i = 1}^{N} ‖ F_{i} ‖_{L^{2} (0, T)}^{2} + ‖ T_{i} ‖_{L^{2} (0, T)}^{2}, \end{array}$ when $T < 1$ .

With the construction up to now and the previous remarks, the proof of Proposition 2 reduces eventually to the following lemma.
Lemma 7.
Given $M > 0$ sufficiently large, there exists $T_{M} > 0$ such that, for $T < T_{M}$ , $S_{T}$ realizes a contraction on $O_{T}^{0} [M]$ .

What remains of this part is devoted to the proof of this lemma. In the following computations, we fix M and T and point out the restrictions when necessary.

Firstly, we recall that, by reproducing the computations of [21] we have that whenever $\bar{Z} \in H^{1} {((0, T); R^{9})}^{N}$ satisfies (24), there exists a constant $C_{M}$ depending only on M (and ε) so that: $\begin{matrix} (27) & \{\begin{matrix} ‖ (L - Δ) U ‖_{L^{2} (F_{0})} ⩽ C_{M} \sqrt{T} ‖ U ‖_{H^{2} (K^{0} \cap F^{0})}, \\ ‖ (G - \nabla) P ‖_{L^{2} (F_{0})} ⩽ C_{M} \sqrt{T} ‖ P ‖_{H^{1} (K^{0} \cap F^{0})}, \end{matrix} \forall t \in (0, T) . \end{matrix}$ Consequently, given $(\bar{Z}, \bar{U}, \bar{P}) \in O_{T}^{0} [M]$ and $(Z, U, P) = S_{T} (\bar{Z}, \bar{U}, \bar{P})$ we first apply Lemma 6 to yield that for arbitrary $t \in [0, T]$ $\begin{array}{l} ‖ \nabla U ‖_{L^{2} (F_{0})} + ‖ U ‖_{H^{2} (K_{0} \cap F_{0})} + ‖ P ‖_{H^{1} (K_{0} \cap F_{0})} \\ ⩽ (C_{0} + C_{K_{0}}) (C_{M} \sqrt{T} (‖ \bar{U} ‖_{H^{2} (F_{0} \cap K_{0})} + ‖ \bar{P} ‖_{H^{1} (F_{0} \cap K_{0})}) + \sum_{i = 1}^{N} | {\bar{V}}_{i} | + | {\bar{ω}}_{i} |) . \end{array}$ After integration in time, we obtain: $\begin{array}{l} \int_{0}^{T} (‖ \nabla U ‖_{L^{2} (F_{0})}^{2} + ‖ U ‖_{H^{2} (K_{0} \cap F_{0})}^{2} + ‖ P ‖_{H^{1} (K_{0} \cap F_{0})}^{2}) \\ (28) & ⩽ C \sqrt{T} (M + 10 \sum_{i = 1}^{N} {| X_{i}^{0} |}^{2} + {| V_{i}^{0} |}^{2} + {| ω_{i}^{0} |}^{2} + 8 \sum_{i = 1}^{N} ‖ F_{i} ‖_{L^{2} (0, T)}^{2} + ‖ T_{i} ‖_{L^{2} (0, T)}^{2},) . \end{array}$ where we applied that $(V_{i}, ω_{i}) \in L^{\infty} (0, T)$ with the above remark (26). In particular, we obtain (25) for T sufficiently small and M sufficiently large. By a standard trace estimate, we have then that $\begin{matrix} ‖ V_{i} ‖_{H^{1} (0, T)}^{2} ⩽ {| V_{i}^{0} |}^{2} + C \int_{0}^{T} (‖ U ‖_{H^{2} (K_{0} \cap F_{0})}^{2} + ‖ P ‖_{H^{1} (K_{0} \cap F_{0})}^{2}) + ‖ F_{i} ‖_{L^{2} (0, T)}^{2} . \end{matrix}$ We have also a similar equation for $ω_{i}$ . Summing the various inequalities, recalling (28) and choosing T sufficiently small, we obtain (24). Finally, the initial conditions $V_{i} (0) = V_{i}^{0}$ and $ω_{i} (0) = ω_{i}^{0}$ are matched by construction. It remains to note that: $\begin{matrix} {‖ X_{i} - X_{i}^{0} ‖}_{L^{\infty} (0, T)} ⩽ T ‖ {\bar{V}}_{i} ‖_{L^{\infty} (0, T)}, \end{matrix}$ so that we obtain (23) up to assume that T is sufficiently small. Eventually, $S_{T}$ maps $O_{T}^{0} [M]$ into itself.

We analyze now the lipschitz properties of this mapping. Let $({\bar{Z}}^{(a)}, {\bar{U}}^{(a)}, {\bar{P}}^{(a)})$ and $({\bar{Z}}^{(b)}, {\bar{U}}^{(b)}, {\bar{P}}^{(b)})$ be two sets of datas and $(Z^{(a)}, U^{(a)}, P^{(a)})$ (resp. $(Z^{(b)}, U^{(b)}, P^{(b)})$ ) their images through $S_{T}$ . Corespondingly, we denote $(\bar{Z}, \bar{U}, \bar{P})$ and $(Z, U, P)$ their differences.

Firstly, by linearity, we have: $\begin{array}{l} U (t, \cdot) = U (f_{1}^{(a)} [{\bar{U}}^{(a)} (t, \cdot)] - f_{1}^{(b)} [{\bar{U}}^{(a)} (t, \cdot)] + f_{2}^{(a)} [{\bar{P}}^{(a)} (t, \cdot)] \\ - f_{2}^{(b)} [{\bar{P}}^{(b)} (t, \cdot)], {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}), \\ P (t, \cdot) = P (f_{1}^{(a)} [{\bar{U}}^{(a)} (t, \cdot)] - f_{1}^{(b)} [{\bar{U}}^{(a)} (t, \cdot)] + f_{2}^{(a)} [{\bar{P}}^{(a)} (t, \cdot)] \\ - f_{2}^{(b)} [{\bar{P}}^{(b)} (t, \cdot)], {\bar{V}}_{1}, {\bar{ω}}_{1}, \dots, {\bar{V}}_{N}, {\bar{ω}}_{N}) |_{K_{0}}, \end{array}$ where $\begin{matrix} {\bar{V}}_{i} = {\bar{V}}_{i}^{(a)} - {\bar{V}}_{i}^{(b)} and {\bar{ω}}_{i} = {\bar{ω}}_{i}^{(a)} - {\bar{ω}}_{i}^{(b)}, \forall i = 1, \dots, N . \end{matrix}$ In these expressions, we emphasized with exponents $(a)$ and $(b)$ that the geometric terms in $f_{1}$ and $f_{2}$ are computed with respect to ${\bar{Z}}^{(a)}$ and ${\bar{Z}}^{(b)}$ . Let for instance consider the $f_{1}$ source terms. Keeping the convention that exponents $(a)$ (resp. $(b)$ ) stands for geometrical terms computed with respect to ${\bar{Z}}^{(a)}$ (resp. ${\bar{Z}}^{(b)}$ ), we note that: $\begin{matrix} f_{1}^{(a)} [{\bar{U}}^{(a)} (t, \cdot)] - f_{1}^{(b)} [{\bar{U}}^{(a)} (t, \cdot)] = (L^{(a)} - L^{(b)}) {\bar{U}}^{(a)} + (L^{(b)} - Δ) \bar{U} . \end{matrix}$

Following once again the lines of the proof Corollary 6.6 and using similar estimates as those on Lemma 6.3, Lemma 6.4 and Corollary 6.5 on [21], we obtain $\begin{matrix} {‖ [(L^{(b)} - Δ) \bar{U}] ‖}_{L^{2} (F_{0})} ⩽ C_{0} T^{1 / 2} ‖ \bar{U} ‖_{H^{2} (F_{0})}, \end{matrix}$ Moreover, by drawing similar estimates similar to those on Lemmata 6.10-6.13 in [21], we get $\begin{matrix} {‖ [(L^{(a)} - L^{(b)}) {\bar{U}}^{(a)}] ‖}_{L^{2} (F_{0})} ⩽ C_{0} T^{1 / 2} \sum_{j = 1}^{N} ‖ {\bar{V}}_{j} ‖_{H^{1} (0, T, R^{3})} {‖ {\bar{U}}^{(a)} ‖}_{H^{2} (F_{0})} . \end{matrix}$ Combining the two latter estimates we obtain finally that $\begin{array}{l} {‖ f_{1}^{(a)} [{\bar{U}}^{(a)} (t, \cdot)] - f_{1}^{(b)} [{\bar{U}}^{(a)} (t, \cdot)] ‖}_{L^{2} (F_{0})} \\ ⩽ C_{0} T^{1 / 2} (‖ \bar{U} ‖_{H^{2} (F_{0})} + \sum_{j = 1}^{N} ‖ {\bar{V}}_{j} ‖_{H^{1} (0, T)} {‖ {\bar{U}}^{(a)} ‖}_{H^{2} (F_{0})}) . \end{array}$ We proceed similarly with the $f_{2}$ terms and apply Lemma 6 to yield that: $\begin{array}{l} \int_{0}^{T} (‖ \nabla U ‖_{L^{2} (F_{0})}^{2} + ‖ U ‖_{H^{2} (K_{0} \cap F_{0})}^{2} + ‖ P ‖_{H^{1} (K_{0} \cap F_{0})}^{2}) \\ (29) & ⩽ C T (\int_{0}^{T} [‖ \bar{U} ‖_{H^{2} (F_{0} \cap K_{0})}^{2} + ‖ \bar{P} ‖_{H^{1} (K_{0} \cap F_{0})}^{2}] + M^{2} \sum_{j = 1}^{N} ‖ {\bar{V}}_{j} ‖_{H^{1} (0, T)}^{2}) . \end{array}$ We obtain the contraction for T small enough in terms of the variables U, P. The contraction in terms of the variable Z follows with standard trace estimates like in the previous part. This end the proof of Proposition 2.
2.4. Blow-up alternative

From the above existence result, we obtain classically that, for any initial data there exists a unique non-extendable solution defined on a time-span $(0, T_{*})$ . We have furthermore the blow-up alternative:

either $T_{*} = \infty$ ;

either $T_{*} < \infty$ and we have at least one of the two possibilities: $\begin{matrix} \underset{t \to T_{*}}{lim inf} min_{i \neq j} dist (B_{i} (t), B_{j} (t)) = 0, \\ \underset{t \to T_{*}}{lim sup} max {| X_{i} (t) |, | V_{i} (t) |, | ω_{i} (t) |, i = 1, \dots, N} = \infty . \end{matrix}$

We prove here that if

T_{*} < \infty

then we must have the first possibility:

\begin{matrix} (30) & \underset{t \to T_{*}}{lim inf} min_{i \neq j} dist (B_{i} (t), B_{j} (t)) = 0 . \end{matrix}

Since

{\dot{X}}_{i} = V_{i}

this reduces to obtaining an a priori bound on

(V_{i}, ω_{i})

. This is the content of the next proposition.

Proposition 8.
Given initial data ${(X_{i}^{0})}_{i = 1, \dots, N} \in O^{R}$ , ${(V_{i}^{0}, ω_{i}^{0})}_{i = 1, \dots, N} \in {[R^{6}]}^{N}$ and source term ${(F_{i}, T_{i})}_{i = 1, \dots, N} \in L^{2} ([0, T]; {[R^{6}]}^{N})$ , assume that ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ is a solution to ( 1 )–( 6 ) on $(0, T_{0})$ . Then there exists a universal constant C depending only on R such that: $\begin{array}{l} \frac{1}{2} sup_{[0, T_{0}]} (\sum_{i = 1}^{N} m | V_{i} |^{2} + J | ω_{i} |^{2}) + \frac{1}{2} \int_{0}^{T_{0}} \int_{R^{3}} | \nabla u |^{2} \\ (31) & ⩽ \frac{1}{2} (\sum_{i = 1}^{N} m {| V_{i}^{0} |}^{2} + J {| ω_{i}^{0} |}^{2}) + C \int_{0}^{T} \sum_{i = 1}^{N} (| F_{i} |^{2} + | T_{i} |^{2}) . \end{array}$

We point out here that in the above statement, we extended u on the $B_{i}$ by setting $u (x) = V_{i} + ω_{i} \times (x - X_{i})$ . Such an extension preserves the ${\dot{H}}^{1}$ -regularity. We use this extension without mention below.

The proof of the above inequality is a standard energy estimate that we recall for completeness. By taking the inner product of the first line of (1) by u, integrating over $F (t)$ and integrating by parts over $F (t)$ , we get $\begin{matrix} 2 \int_{F (t)} {| D [u] |}^{2} d x = \int_{\partial F (t)} Σ (u, p) n \cdot u d σ . \end{matrix}$ Taking now the inner product of the first line (3) by $V_{i}$ and that of the second line by $ω_{i}$ and using the no-slip condition (2), we obtain $\begin{matrix} \frac{1}{2} \frac{d}{d t} (\sum_{i = 1}^{N} m {| V_{i} (t) |}^{2} + J {| ω_{i} (t) |}^{2}) = - \int_{\partial F (t)} Σ (u, p) n \cdot u d σ + \sum_{i = 1}^{N} (F_{i} \cdot V_{i} (t) + T_{i} \cdot ω_{i} (t)) . \end{matrix}$ By combining the two identities, we get that $\begin{matrix} \frac{1}{2} \frac{d}{d t} (\sum_{i = 1}^{N} m {| V_{i} (t) |}^{2} + J {| ω_{i} (t) |}^{2}) + 2 \int_{F (t)} {| D [u] |}^{2} d x = \sum_{i = 1}^{N} (F_{i} \cdot V_{i} (t) + T_{i} \cdot ω_{i} (t)) . \end{matrix}$ At this point, we recall that, by standard integration by parts, we have the Korn identity: $\begin{matrix} 2 \int_{F (t)} {| D [u] |}^{2} d x = \int_{R^{3}} | \nabla u |^{2} d x . \end{matrix}$ Moreover, because of the embedding ${\dot{H}}^{1} (R^{3}) \subset L^{6} (R^{3}) \subset L_{loc}^{2} (R^{3})$ we infer that there is a constant $C > 0$ (depending only on R) for which: $\begin{matrix} \sum_{i = 1}^{N} ({| V_{i} (t) |}^{2} + {| ω_{i} (t) |}^{2}) ⩽ C \int_{R^{3}} | \nabla u |^{2} . \end{matrix}$ Hence, with a standard Cauchy–Schwarz inequality, we conclude that: $\begin{matrix} \frac{1}{2} \frac{d}{d t} (\sum_{i = 1}^{N} m {| V_{i} (t) |}^{2} + J {| ω_{i} (t) |}^{2}) + \frac{1}{2} \int_{R^{3}} | \nabla u |^{2} d x ⩽ \frac{C}{2} \sum_{i = 1}^{N} | F_{i} |^{2} + | T_{i}^{2} | . \end{matrix}$ The expected result then yields by standard integration in time.
3. Proof of Theorem 1

In this section, we detail the contradiction argument which rules out possible contacts in finite time and thus possible blow-up. So, we fix initial data $\begin{matrix} {(X_{i}^{0})}_{i = 1, \dots, N} \in O^{R}, {(V_{i}^{0}, ω_{i}^{0})}_{i = 1, \dots, N} \in {[R^{3} \times R^{3}]}^{N}, \end{matrix}$ and we assume that the associated non-extendable solution ${(X_{i}, V_{i}, ω_{i})}_{i = 1, \dots, N}$ to (1)–(5) with initial data (6) blows up in $T_{*} < \infty$ . We also denote by $(t, x) \mapsto (u (t, x), p (t, x))$ the associated solution to (1) on $[0, T_{*})$ . According to the computations in the previous section (see (30) in subsection 2.4), blow-up in $T_{*}$ means that $\begin{matrix} (32) & \underset{t \to T_{*}}{lim inf} min_{i \neq j} (| X_{i} (t) - X_{j} (t) |) = 2 R . \end{matrix}$ According to (31), we have that $X_{i} \in W^{1, \infty} (0, T_{*}) \subset C ([0, T_{*}])$ . Consequently, our non-extendable solution converges to a configuration with contact in $T_{*}$ . Let denote with stars this configuration: $\begin{matrix} X_{i}^{*} = X_{i} (T_{*}) B_{i}^{*} = B_{i} (X_{i}^{*}, R) for i = 1, \dots, N . \end{matrix}$

For each $k \in {1, \dots, N}$ we denote $I_{k}$ the set of indices of the spheres different to $B_{k}^{*}$ that are in contact with $B_{k}^{*}$ . Obviously, $k \notin I_{k}$ by construction and, due to (32), there exists at least one k such that $I_{k}$ is not empty. We can even go one step further as in [18]. Since the set of $(X_{i}^{*} - X_{j}^{*}) / | X_{i}^{*} - X_{j}^{*} | \in S^{2}$ ( $1 ⩽ i < j ⩽ N$ , such that $I_{i} \neq \emptyset$ and $I_{j} \neq \emptyset$ ) is finite, we can find $e \in S^{2}$ which is not orthogonal to this set and look at an index k such that $X_{k}^{*} \cdot e$ is maximal, we obtain that: $\begin{matrix} ({Cond}_{e}) & (X_{j}^{*} - X_{k}^{*}) \cdot e < 0 \forall j \in {1, \dots, N} ∖ {k} . \end{matrix}$ Note that this implies that for such a e the index k is unique. In what follows we assume e and k are constructed as above. Up to renumbering the particles, we assume that $k = N$ . For $i \in I_{N}$ we denote then: $\begin{matrix} \{\begin{matrix} d_{i} (t) : = dist (B_{i} (t), B_{N} (t)), \\ cos (θ_{i} (t)) : = \frac{(X_{N} (t) - X_{i} (t)) \cdot e}{| X_{N} (t) - X_{i} (t) |} . \end{matrix} \end{matrix}$ We note that all the $d_{i}$ are $W^{1, \infty} ([0, T_{*}))$ and that (32) means that they all vanish in $T_{*}$ . Moreover, for t close to $T_{*}$ we have $\begin{matrix} (cos \min_{e}) & cos (θ_{i} (t)) ⩾ {cos}_{min} > 0, \end{matrix}$ as this property holds in $T_{*}$ by construction.

Our contradiction then follows from the following proposition:

Proposition 9.
Let $\begin{matrix} C_{N} (t) : = c_{0} \sum_{i \in I^{N}} cos (θ_{i} (t)) | ln (d_{i} (t)) |, \forall t \in [0, T_{}), \end{matrix}$ with a constant* $c_{0} \in (0, \infty)$ to be fixed later on. For t close to $T_{}$ there exists two positive constants* $K_{}$ and* $K_{0}$ such that: $\begin{matrix} (33) & {\dot{C}}_{N} (t) ⩽ K_{0} C_{N} (t) + K_{} ‖ \nabla u ‖_{L^{2} (R^{3})} + K_{0} - m {\dot{V}}_{N} (t) \cdot e, \end{matrix}$ Moreover* $K_{}$ (resp.* $K_{0}$ ) depends only on physical parameters R, m, J and our construction (resp. depends moreover on initial data and source terms through ( 31 )).

Indeed, once the above proposition holds true, we have by multiplying both sides of (33) by $e^{- K_{0} t}$ that $\begin{matrix} \frac{d}{d t} (C_{N} (t) exp (- K_{0} t)) ⩽ (K_{} ‖ \nabla u ‖_{L^{2} (R^{3})} + K_{0}) exp (- K_{0} t) - m ({\dot{V}}_{N} (t) \cdot e) exp (- K_{0} t) . \end{matrix}$ It follows by integrating the above inequality on $[t_{0}, T_{})$ with $t_{0}$ sufficiently close to $T_{}$ that $\begin{array}{rcl} C_{N} (t) & ⩽ & C_{N} (t_{0}) exp (K_{0} (t - t_{0})) + \int_{t_{0}}^{t} (K_{} {‖ \nabla u (s) ‖}_{L^{2} (R^{3})} + K_{0}) exp (K_{0} (t - s)) d s \\ - m \int_{t_{0}}^{t} ({\dot{V}}_{N} (s) \cdot e) exp (K_{0} (t - s)) d s . \end{array}$ Thus, $\begin{array}{rcl} C_{N} (t) & ⩽ & C_{N} (t_{0}) exp (K_{0} t) + \int_{t_{0}}^{t} (K_{} {‖ \nabla u (s) ‖}_{L^{2} (R^{3})} + K_{0}) exp (K_{0} (t - s)) d s \\ - m ({[(V_{N} \cdot e) exp (K_{0} (t - s))]}_{s = t_{0}}^{t} + K_{0} \int_{t_{0}}^{t} (V_{N} (s) \cdot e) exp (K_{0} (t - s)) d s), \end{array}$ where the right-hand side remains bounded on $[t_{0}, T_{}]$ thanks to energy estimate (31). In particular, we infer that $C_{N}$ remains bounded on $[t_{0}, T_{}]$ . The fact that $C_{N}$ does not blow up entails then that all the $| ln (d_{i} (t)) |$ remain bounded as ( cos min e ) condition holds and that the $d_{i} (T_{})$ do not vanish. We obtain a contradiction.
3.1. Proof of Proposition 9

The proof of Propositon 9 is based on the use of a suitable test function in (1). We provide the extensive arguments in the case where the force $F_{i}$ is identically 0 in the first equation of (4). The general case follows straightforwardly. Thus, we get $\begin{matrix} (34) & m {\dot{V}}_{N} \cdot e = - \int_{\partial B_{N}} Σ (u, p) n \cdot e d σ \cdot \end{matrix}$ We interpret here the vector e as the boundary value of a smooth divergence-free w. Precisely, for t fixed close to $T_{*}$ , we construct in the next subsection $w \in C^{\infty} (R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t))$ such that:

w has bounded support (independent of t);

$div w = 0$ on $R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)$ ;

$w = e$ on $\partial B_{N} (t)$ and $w = 0$ on $\partial B_{k} (t)$ for $k \neq N$ .

Doing so, we may rephrase (34) by integrating by parts:

\begin{matrix} m {\dot{V}}_{N} \cdot e = 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (u) : D (w) . \end{matrix}

Below, we shall adapt the constructions of [11–13] to the specific geometric setting under consideration herein. Namely, we construct w such that:

\begin{matrix} w = w_{reg} + \sum_{i \in I_{N}} w_{sing}^{(i)}, \end{matrix}

where

w_{reg}

and

{(w_{sing}^{(i)})}_{i \in I_{N}}

have bounded support independent of t and enjoy the following properties. Firstly, we have:

\begin{matrix} (35) & ‖ \nabla w_{reg} ‖_{L^{2} (R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t))} ⩽ K \sum_{i \in I_{N}} \sqrt{| ln (d_{i} (t)) |} . \end{matrix}

for some K depending only on R. Secondly, we have, for arbitrary

i \in I_{N}

there exists strictly positive constants $c_{0}$ and $K_{*}$ independent of t for which, given any divergence-free $v \in H_{loc}^{1} (R^{3})$ such that $v = 0$ on $B_{N} (t)$ and $v = W + ω \times (x - X_{i})$ on $B_{i} (t)$ , for some constant vectors W and ω of $R^{3}$ , we have:

\begin{array}{l} 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (w_{sing}^{(i)}) : D (v) = c_{0} \frac{cos (θ_{i} (t))}{| X_{i} - X_{N} |} \frac{W \cdot (X_{i} - X_{N})}{d_{i} (t)} + R_{i}, \\ (36) & where | R_{i} | ⩽ K_{*} ‖ \nabla v ‖_{L^{2} (Supp (w_{sing}^{(i)}))} + K_{0} | ln (d_{i} (t)) | . \end{array}

With this construction, we split then

\begin{array}{rcl} m {\dot{V}}_{N} \cdot e & = & 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (u - u_{N}) : D (w_{reg}) \\ (37) & + \sum_{i \in I_{N}} 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (u - u_{N}) : D (w_{sing}^{(i)}), \end{array}

where

u_{N} (x) = V_{N} + ω_{N} \times (x - x_{N})

We proceed with extracting the leading and remainder terms. In (37) we bound the first term on the right-hand side by (35) since $w_{reg}$ has bounded support. This yields by using Cauchy–Schwartz inequality there exists a constant K for which: $\begin{array}{l} 2 | \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (u - u_{N}) : D (w_{reg}) | & ⩽ K \sum_{i \in I_{N}} \sqrt{| ln (d_{i} (t)) |} {‖ \nabla (u - u_{N}) ‖}_{L^{2} (R^{3})} \\ ⩽ K \sum_{i \in I_{N}} \sqrt{| ln (d_{i} (t)) |} (‖ \nabla u ‖_{L^{2} (R^{3})} + | ω_{N} |) \\ ⩽ K \sum_{i \in I_{N}} cos (θ_{i} (t)) | ln (d_{i} (t)) | + K ‖ \nabla u ‖_{L^{2} (R^{3})}^{2} + K_{0} . \end{array}$ As for the second term on the right-hand side, we apply (36) with $v = u - u_{N}$ using (31) to bound $‖ \nabla v ‖_{L^{2} (Supp (w_{sing}^{(i)}))}$ . We point out that, on $B_{i}$ there holds: $\begin{matrix} u (x) - u_{N} (x) = V_{i} - V_{N} + ω_{N} \times (X_{i} - X_{N}) + (ω_{i} - ω_{N}) \times (x - X_{i}) \end{matrix}$ so that we replace in (36): $\begin{matrix} “ W \cdot (X_{i} - X_{N}) ” = (V_{i} - V_{N}) \cdot (X_{i} - X_{N}) = | X_{i} - X_{N} | \dot{d_{i}} (t) . \end{matrix}$ Eventually, we obtain, with similar computations as above: $\begin{array}{l} \sum_{i \in I_{N}} 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (u - u_{N}) : D (w_{sing}^{(i)}) = c_{0} \sum_{i \in I_{N}} \frac{cos (θ_{i} (t)) {\dot{d}}_{i} (t)}{d_{i}} + R_{max} \\ where | R_{max} | ⩽ K_{0} | ln (d_{i} (t)) | + K_{0} + K_{*} ‖ \nabla u ‖_{L^{2} (R^{3})} . \end{array}$ Combining these computations, we obtain a constant $K_{0}$ for which: $\begin{matrix} (38) & c_{0} \sum_{i \in I_{N}} \frac{cos (θ_{i} (t)) {\dot{d}}_{i} (t)}{d_{i}} ⩾ m {\dot{V}}_{N} \cdot e - K_{0} C_{N} (t) - R_{max} . \end{matrix}$ To conclude, we mention that, for t close to $T_{*}$ we have $| ln (d_{i} (t)) | = - ln (d_{i} (t))$ and by using (31), we have also $\begin{matrix} | \frac{d}{d t} cos (θ_{i} (t)) | ⩽ \frac{| V_{i} | + | V_{N} |}{R} ⩽ K_{0} . \end{matrix}$ Hence, we infer with (31) again that: $\begin{matrix} c_{0} \sum_{i \in I_{N}} \frac{cos (θ_{i} (t)) {\dot{d}}_{i} (t)}{d_{i}} = - {\dot{C}}_{N} (t) + {\tilde{R}}_{max} where | {\tilde{R}}_{max} | ⩽ K_{0} C_{N} (t) . \end{matrix}$ Plugging this latter identity in (38), we obtain (33).

To complete the proof of Proposition 9, it remains to explain the construction of $w_{reg}$ and $w_{sing}^{(i)}$ . This is the content of the next section.

3.2. Construction of w and analysis of remainder terms

For the construction, we need to analyze the geometry of $F (t)$ when t is close to $T_{*}$ . This closeness will be fixed later on. Firstly, we introduce $(e_{1}, e_{2}, e_{3})$ a frame and $(x_{1}, x_{2}, x_{3})$ a system of coordinates associated with this frame and centred in $X_{N}$ .

Fig. 1.

Geometry in the local coordinates.

For each $i \in I_{N}$ we introduce a system of coordinates $({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}, {\tilde{x}}_{3}^{(i)})$ centered in $X_{i}$ and along a (direct) frame $({\tilde{e}}_{1}^{(i)}, {\tilde{e}}_{2}^{(i)}, {\tilde{e}}_{3}^{(i)})$ such that $\begin{matrix} {\tilde{e}}_{3}^{(i)} = \frac{X_{N} - X_{i}}{| X_{N} - X_{i} |}, e \in ⟨ {\tilde{e}}_{1}^{(i)}, {\tilde{e}}_{3}^{(i)} ⟩ . \end{matrix}$ With these notations and condition ( Cond e ), we have then that there exists $θ_{i} \in [0, π / 2)$ for which: $\begin{matrix} e = sin (θ_{i}) {\tilde{e}}_{1}^{(i)} + cos (θ_{i}) {\tilde{e}}_{3}^{(i)} . \end{matrix}$ We point out that these constructions depend a priori on time t. However, we do not stress this dependence since it has no influence on the computations in the Stokes case under consideration. Associated with this frame, we plit the ∇ operator into the tangential “nabla” operator ${\tilde{\nabla}}_{/ /}^{(i)}$ that contains the differentials parallel to the directions ${\tilde{e}}_{1}^{(i)}$ and ${\tilde{e}}_{2}^{(i)}$ while we denote by ${\tilde{\partial}}_{3}^{(i)}$ the differential along the third direction. We can also parametrize $\partial B_{N}$ and $\partial B_{i}$ in a neighborhood $V^{(i)}$ of the segment joining $X_{i}$ to $X_{N}$ as follows: $\begin{matrix} x \in V^{(i)} \cap \partial B_{N} \Leftrightarrow {| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | < R and {\tilde{x}}_{3}^{(i)} = 2 R + d_{i} - γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)}, \\ x \in V^{(i)} \cap \partial B_{i} \Leftrightarrow {| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | < R and {\tilde{x}}_{3}^{(i)} = γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)} \end{matrix},$ where $| \cdot |$ stands for the euclidean norm and: $\begin{matrix} γ (s) = \sqrt{R^{2} - s^{2}}, \forall s < R . \end{matrix}$ For $ℓ \in (0, R)$ , we construct then $\begin{array}{rcl} G_{ℓ}^{(i)} : = {x \in R^{3} s.t. | ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | < ℓ and γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) < {\tilde{x}}_{3}^{(i)} < 2 R + d_{i} - γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)} . \end{array}$ The symbol $G$ stands for “gap” since the domain $G_{ℓ}^{(i)}$ stands for the cylindrical gap of width ℓ between the spheres $B_{i}$ and $B_{N}$ (see Fig. 1).

We state now the restriction on time t. We note that, for $i \in I_{N}$ we have $d_{i} (T_{*}) = 0$ while for $j \notin I_{N}$ there holds $d_{j} (T_{*}) ⩾ d_{*} > 0$ . Hence, we can construct $T^{*}$ sufficiently close to $T_{*}$ such that for $t \in [T^{*}, T_{*}]$ , there holds: $\begin{matrix} \{\begin{matrix} d_{i} (t) < R / 8, & \forall i \in I_{N}, \\ d_{j} (t) > d_{*} / 2, & \forall j \notin I_{N} . \end{matrix} \end{matrix}$ We assume from now on that $t \in [T^{*}, T_{*}]$ . This entails by a geometrical argument that, setting $ℓ_{0} : = R / 2$ we have that $G_{ℓ_{0}}^{(i)} \subset F (t)$ (since no $B_{k}$ can intersect $G_{ℓ_{0}}^{(i)}$ ). We have then a constant $d_{0}$ depending on $d_{*}$ and R such that, outside all the $G_{ℓ_{0} / 2}^{(i)}$ ( $i \in I_{N}$ ) the $d_{0}$ exterior set of $B_{N}$ is in the fluid domain. We mean here that, for arbitrary direction $e_{*} \in S^{2}$ either $X_{N} + {Re}_{*} \in \partial G_{ℓ_{0} / 2}^{(i)}$ for some $i \in I_{N}$ or the segment joining $X_{N} + {Re}_{*}$ to $X_{N} + (R + d_{0}) e_{*}$ lies in $F (t)$ . Eventually, given a non negative distance $d_{i}$ , we set: $\begin{array}{rcl} A^{(i)} & : = & {x \in R^{3} s.t. ℓ_{0} / 2 < | ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | < ℓ_{0} \\ and 2 R + d_{i} - γ (ℓ_{0} / 2) < {\tilde{x}}_{3}^{(i)} < 2 R + d_{i} - γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)} \end{array}$ for $i \in I^{N}$ . We point out that these $A^{(i)}$ are all lipschitz domains that are isometric to a same “triangular” annulus circumventing the south pole of the sphere of radius R at distance $ℓ_{0}$ of the vertical axis. They are independent of the relative positions between the spheres (since $ℓ_{0}$ and γ are fixed with respect to R only).

These domains and parameters being fixed, we can now proceed with the construction of $w_{reg}$ and $w_{sing}^{(i)}$ . For this, we introduce a truncation function $ζ \in C^{\infty} (R)$ such that: $\begin{matrix} 𝟙_{(- \infty, 1 / 2]} ⩽ ζ ⩽ 𝟙_{(- \infty, 1]} . \end{matrix}$

Firstly, we set: $\begin{matrix} A_{reg}^{(0)} (x) : = ζ (\frac{| x | - R}{d_{0}}) \frac{e \times x}{2} . \end{matrix}$ We recall that $x = (x_{1}, x_{2}, x_{3})$ is the system of coordinates associated with the frame $(e_{1}, e_{2}, e_{3})$ centered in $X_{N}$ . By standard arguments, the vector field $\nabla \times A_{reg}^{(0)}$ is divergence-free, matches the vector field e on $\partial B_{N}$ and vanishes at a distance larger than $d_{0}$ of $B_{N}$ . In particular, it matches the required boundary condition on $B_{j}$ for $j \notin I_{N}$ but not on $B_{i}$ for $i \in I_{N}$ inside $G_{ℓ_{0} / 2}^{(i)}$ . For $i \in I_{N}$ , we build on the former computations of [11–13], and we introduce:, $\begin{matrix} A^{(i)} (x) : = (\begin{matrix} - {\tilde{x}}_{2}^{i} cos (θ_{i}) / 2 \\ {\tilde{x}}_{1}^{(i)} cos (θ_{i}) / 2 \\ {\tilde{x}}_{2}^{(i)} sin (θ_{i}) \end{matrix}) P (\frac{{\tilde{x}}_{3}^{(i)} - γ (| ({\tilde{x}}_{1}^{i}, {\tilde{x}}_{2}^{(i)}) |)}{2 R + d_{i} - 2 γ (| ({\tilde{x}}_{1}^{i}, {\tilde{x}}_{2}^{(i)}) |)}), \forall x \in G_{ℓ_{0}}^{(i)} . \end{matrix}$

Here, the vector decomposition must be understood in the frame $({\tilde{e}}_{1}^{(i)}, {\tilde{e}}_{2}^{(i)}, {\tilde{e}}_{3}^{(i)})$ and P is the explicit polynomial $P (s) = 3 s^{2} - 2 s^{3}$ . This particular choice is guided by the condition: $\begin{matrix} P (0) = P^{'} (0) = P^{'} (1) = 0, P (1) = 1 . \end{matrix}$ Beyond the fact that we need a polynomial of order 3 to match all these boundary conditions simultaneously, we point out that the fact that the third order derivative of P is constant will have a key-role below. The order 3 has also variational origins that are detailed further in [8]. At this point, we simply mention that we have $\begin{matrix} \{\begin{matrix} A^{(i)} = \nabla \times A^{(i)} = 0 & on \partial B_{i} \cap \partial G_{ℓ_{0}}^{(i)}, \\ \nabla \times A^{(i)} = e & on \partial B_{N} \cap \partial G_{ℓ_{0}}^{(i)} . \end{matrix} \end{matrix}$ The argument of P is an affine (in ${\tilde{x}}_{3}^{(i)}$ ) mapping which maps $G_{ℓ_{0}}^{(i)}$ into $[0, 1]$ . In particular: $\begin{matrix} h_{i} (s) = 2 R + d_{i} - 2 γ (s) \end{matrix}$ stands for the “vertical” distance between $\partial B_{i}$ and $\partial B_{N}$ on any vertical line at distance s of the vertical axis $R {\tilde{e}}_{3}^{(i)}$ . It matches the value $d_{i}$ when $s = 0$ . The above value for the test-function potential will be used afterwards in $G_{ℓ_{0}}^{(i)}$ only. Hence, we do not need to create an extension of the full $R^{3}$ . In what follows, we shall use systematically that, in the gap $G_{ℓ_{0}}^{(i)}$ , differentiating the formula for $A^{(i)}$ can be seen as multiplication operators in the following way: $\begin{array}{l} | {({\tilde{\partial_{1}}}^{(i)})}^{α_{1}} {({\tilde{\partial_{2}}}^{(i)})}^{α_{2}} {({\tilde{\partial}}_{3}^{(i)})}^{β} A^{(i)} (x) | \\ (39) & ≲ \frac{| A^{(i)} (x) |}{(h_{i} (| {({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |))}^{\frac{α_{1} + α_{2}}{2} + β}}, \forall x \in G_{ℓ_{0}}^{(i)}, \forall (α_{1}, α_{2}, β) \in N^{3} . \end{array}$ Such analogies can be made rigorous via tedious but straightforward computations (see [11] for example).

We fix then: $\begin{matrix} w (x) = \nabla \times A + \sum_{i \in I_{N}} {\tilde{w}}_{reg}^{(i)} (x), \end{matrix}$ where $\begin{matrix} A (x) = \{\begin{matrix} (1 - ζ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) / ℓ_{0})) A_{reg}^{(0)} (x) + ζ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) / ℓ_{0}), A^{(i)} (x) & in G_{ℓ_{0}}^{(i)} for i \in I_{N}, \\ A_{reg}^{(0)} (x), & else . \end{matrix} \end{matrix}$ and ${\tilde{w}}_{reg}^{(i)}$ , is the unique (weak) solution to the Stokes problem: $\begin{matrix} \{\begin{matrix} - Δ u + \nabla p = 0, \\ div u = 0, \end{matrix} on A^{(i)}, \end{matrix}$ with boundary conditions: $\begin{matrix} \{\begin{matrix} u = \nabla ζ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | / ℓ_{0})) \times (A^{(i)} - A_{reg}^{(0)}), & on \partial A^{(i)} \cap \partial B_{N}, \\ u = 0, & on \partial A^{(i)} ∖ \partial A^{(i)} \cap \partial B_{N} . \end{matrix} \end{matrix}$

Some comments are in order. In a first step, we construct w in the form of the curl of a potential vector A to ensure the divergence-free condition. The potential vector $A^{(i)}$ suits the shape of the gap $G_{ℓ_{0}}^{(i)}$ . So, A is constructed by interpolating $A^{(i)}$ with $A_{reg}^{(0)}$ in $G_{ℓ_{0}}^{(i)} ∖ G_{ℓ_{0} / 2}^{(i)}$ and along the tangential coordinates $({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)})$ to avoid introducing new singularities. However, since $A^{(i)} \neq A_{reg}^{(0)}$ on $\partial B_{N}$ , the interpolation induces a boundary error. So, in a second step, we correct with ${\tilde{w}}_{reg}^{(i)}$ this error term.

We point out that the lifting operator – on which the construction of ${\tilde{w}}_{reg}^{(i)}$ is based – is the resolution of a Stokes problem on a lipschitz domain that is isometric to a reference domain independent of the distance between the spheres or the direction of the axis linking $X_{i}$ to $X_{N}$ . Since $A^{(i)}$ is not singular on $G_{ℓ_{0}}^{(i)} ∖ G_{ℓ_{0} / 2}^{(i)}$ we infer that – extending ${\tilde{w}}_{reg}^{(i)}$ by 0 on the fluid domain – we obtain a divergence-free vector-field such that $\begin{matrix} (40) & {‖ {\tilde{w}}_{reg}^{(i)} ‖}_{H^{1} (F)} ⩽ K, \forall i \in I_{N} . \end{matrix}$ for some constant K depending on R only. From now on, we denote with ≲ an inequality which involves a multiplicative constant K which depends only on R. For instance, this latter inequality reads: $\begin{matrix} {‖ {\tilde{w}}_{reg}^{(i)} ‖}_{H^{1} (F)} ≲ 1 \forall i \in I_{N} . \end{matrix}$

In $A^{(i)}$ we also identify that there is a singular and a regular term. The singular term is responsible for the dominating asymptotics when $d_{i} \to 0$ while we gather lower order terms in notations with the “reg” index. Indeed, we split $A^{(i)} = A_{reg}^{(i)} + A_{sing}^{(i)}$ with: $\begin{matrix} A_{sing}^{(i)} = (\begin{matrix} - {\tilde{x}}_{2}^{(i)} cos (θ_{i}) / 2 \\ {\tilde{x}}_{1}^{(i)} cos (θ_{i}) / 2 \\ 0 \end{matrix}) P (\frac{{\tilde{x}}_{3}^{(i)} - γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)}{2 R + d_{i} - 2 γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)}), \end{matrix}$ and $\begin{matrix} A_{reg}^{(i)} = (\begin{matrix} 0 \\ 0 \\ {\tilde{x}}_{2}^{(i)} sin (θ_{i}) \end{matrix}) P (\frac{{\tilde{x}}_{3}^{(i)} - γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)}{2 R + d_{i} - 2 γ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |)}) . \end{matrix}$ We remark that (in the local frame) there holds: $\begin{matrix} \nabla \times A_{reg}^{(i)} = (\begin{matrix} {[{\tilde{\nabla}}_{/ /}^{(i)} A_{reg, 3}^{(i)}]}^{⊥} \\ 0 \end{matrix}), \end{matrix}$ where ⊥ denotes the (2 dimensional) rotation of angle $π / 2$ . So, the most singular term of $\nabla [\nabla \times A_{reg}^{(i)}]$ is ${\tilde{\partial}}_{3}^{(i)} {\tilde{\nabla}}_{/ /}^{(i)} A_{reg}^{(i)}$ which diverges like $\begin{matrix} \frac{| {\tilde{x}}_{2}^{(i)} |}{| h_{i} (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) |^{\frac{3}{2}}} . \end{matrix}$ Consequently, we obtain that: $\begin{matrix} {‖ \nabla [\nabla \times A_{reg}^{(i)}] ‖}_{L^{2} (G_{ℓ_{0}}^{(i)})}^{2} ≲ \int_{G_{l_{0}}^{(i)}} \frac{| {\tilde{x}}_{2}^{(i)} |^{2}}{| h_{i} (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) |^{3}} d {\tilde{x}}_{1}^{(i)} d {\tilde{x}}_{2}^{(i)} d {\tilde{x}}_{3}^{(i)} . \end{matrix}$ Going to cylindrincal coordinates $(r, ϕ, z)$ and remarking that $\begin{matrix} h_{i} (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) |) ⩾ d_{i} + r^{2} / R, \end{matrix}$ we infer that: $\begin{matrix} (41) & {‖ \nabla [\nabla \times A_{reg}^{(i)}] ‖}_{L^{2} (G_{ℓ_{0}}^{(i)})}^{2} ≲ \int_{0}^{ℓ_{0}} \frac{r d r^{2}}{(d_{i} + r^{2} / R)} ≲ | ln (d_{i}) | . \end{matrix}$

Eventually, we split: $\begin{matrix} w = w_{reg} + \sum_{i \in I_{N}} w_{sing}^{(i)}, \end{matrix}$ where: $\begin{matrix} w_{sing}^{(i)} (x) = \nabla \times [ζ ((| {\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)} |) / ℓ_{0}) A_{sing}^{(i)} (x)], \forall i \in I_{N} . \end{matrix}$ This entails that: $\begin{matrix} w_{reg} (x) = \nabla \times A_{reg} + \sum_{i \in I_{N}} {\tilde{w}}_{reg}^{(i)} (x), \end{matrix}$ where: $\begin{matrix} A_{reg} (x) = \{\begin{matrix} (1 - ζ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | / ℓ_{0})) A_{reg}^{(0)} (x) + ζ (| ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}) | / ℓ_{0}) A_{reg}^{(i)} (x), & in G_{ℓ_{0}}^{(i)}, for i \in I_{N}, \\ A_{reg}^{(0)} (x), & else. \end{matrix} \end{matrix}$ Combining (40) and (41) with the remark that the truncation operator is bounded in sobolev spaces with norm bounded in terms of R, we conclude that $w_{reg}$ has support in the $d_{0}$ neighborood of $B_{N}$ and satisfies (35).

It remains to show that $w_{sing}^{(i)}$ satisfies (36) for arbitrary $i \in I_{N}$ . For this, we construct a suitable pressure $p_{sing}^{(i)}$ following the previous computations of [13]. In what follows the index i plays no role and the change of coordinates $(x_{1}, x_{2}, x_{3}) \to ({\tilde{x}}_{1}^{(i)}, {\tilde{x}}_{2}^{(i)}, {\tilde{x}}_{3}^{(i)})$ is a standard isometry that does not modify the computation of the Stokes operator (up to compute the vector fields in the corresponding frame). So, we drop exponent $(i)$ and tildas in notations having in mind that all computations below must be understood in the local frame. We introduce $a_{1}$ , $a_{2}$ so that $w_{sing}$ reads $\nabla \times (a_{1} e_{1} - a_{2} e_{2})$ . We have then: $\begin{matrix} Δ w_{sing} = (\begin{matrix} \partial_{311} a_{2} + \partial_{322} a_{2} + \partial_{333} a_{2} \\ \partial_{311} a_{1} + \partial_{322} a_{1} + \partial_{333} a_{1} \\ - \partial_{1} Δ a_{2} - \partial_{2} Δ a_{1} \end{matrix}) . \end{matrix}$ At this point, we construct a pressure in order to annihilate the most diverging terms, namely, the ones with the most number of derivatives along $e_{3}$ . For this, we remark that: $\begin{matrix} (\begin{matrix} \partial_{333} a_{2} \\ \partial_{333} a_{1} \\ 0 \end{matrix}) = \frac{6 cos (θ) ζ (| (x_{1}, x_{2}) | / ℓ_{0})}{2 R + d_{i} - 2 γ (| (x_{1}, x_{2})) |} (\begin{matrix} x_{1} \\ x_{2} \\ 0 \end{matrix}) . \end{matrix}$ In the above formula, we have no dependencies in $x_{3}$ since the third order derivative of P is constant. Consequently, we set: $\begin{matrix} p_{/ /} (r) = 6 cos (θ) \int_{ℓ_{0}}^{r} \frac{ζ (s / ℓ_{0}) s}{{(2 R + d_{i} - 2 γ (s))}^{3}} d s, p_{⊥} = - \partial_{13} a_{2} - \partial_{23} a_{1}, \\ p_{sing} (x_{1}, x_{2}, x_{3}) = p_{/ /} (| (x_{1}, x_{2}) |) + p_{⊥} (x_{1}, x_{2}, x_{3}) . \end{matrix}$ We point out that this pressure vanishes on $| (x_{1}, x_{2}) | = ℓ_{0}$ so that we can extend the value of $p_{sing}$ by 0 outside the gap. With these particular formulas, we see that: $\begin{matrix} div Σ (w_{sing}, p_{sing}) = (\begin{matrix} 2 \partial_{311} a_{2} + \partial_{322} a_{2} + \partial_{123} a_{1} \\ \partial_{311} a_{1} + 2 \partial_{322} a_{1} + \partial_{123} a_{2} \\ - \partial_{111} a_{2} - \partial_{122} a_{2} - \partial_{211} a_{1} - \partial_{222} a_{1} \end{matrix}) = : err . \end{matrix}$

In particular, for any divergence-free $v \in H^{1} (R^{3})$ we have by integrating by parts: $\begin{matrix} (42) & \begin{matrix} 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (w_{sing}) : D (v) = & \int_{\partial B_{N}} Σ (w_{sing}, p_{sing}) n \cdot v + \int_{\partial B_{i}} Σ (w_{sing}, p_{sing}) n \cdot v \\ - \int_{G_{ℓ_{0}}^{(i)}} err \cdot v . \end{matrix} \end{matrix}$ Under the further assumption that v vanishes on $\partial B_{N}$ , and $v = W + ω \times (x - X_{i})$ on $B_{i}$ , the above identity yields: $\begin{matrix} 2 \int_{R^{3} ∖ ⋃_{k = 1}^{N} B_{k} (t)} D (w_{sing}) : D (v) = \int_{\partial B_{i}} Σ (w_{sing}, p_{sing}) n \cdot (W + ω \times (x - X_{i})) + R_{i}^{b}, \end{matrix}$ where: $\begin{matrix} R_{i}^{b} = - \int_{G_{ℓ_{0}}^{(i)}} err \cdot v . \end{matrix}$ To bound $R_{i}^{b}$ we construct $\begin{matrix} Err (x_{1}, x_{2}, x_{3}) = - \int_{x_{3}}^{2 R + d_{i} - γ (| (x_{1}, x_{2}) |)} err (x_{1}, x_{2}, z) d z . \end{matrix}$ in $G_{ℓ_{0}}^{(i)}$ so that: $\begin{array}{l} | R_{i}^{b} | & = | \int_{G_{ℓ_{0}}^{(i)}} Err \cdot \partial_{3} v | \\ ⩽ ‖ Err ‖_{L^{2} (G_{ℓ_{0}}^{(i)})} ‖ \nabla v ‖_{L^{2} (R^{3})} \\ ⩽ {‖ h_{i} (| (x_{1}, x_{2}) |) err ‖}_{L^{2} (G_{ℓ_{0}}^{(i)})} ‖ \nabla v ‖_{L^{2} (R^{3})} . \end{array}$ We point out that we obtain the first line by integration by parts in the $e_{3}$ direction, we apply then a standard Cauchy–Schwarz inequality to reach the second line. To reach the last line, we finally use a Poincaré inequality (for fixed $(x_{1}, x_{2})$ ) using that the optimal constant is lower than the distance $h_{i} (| (x_{1}, x_{2}) |)$ . At this point, we remark that, from (39), there holds: $\begin{matrix} | h_{i} (| (x_{1}, x_{2}) |) err | ⩽ \frac{| x_{1} | + | x_{2} |}{h_{i} (| (x_{1}, x_{2}) |)} . \end{matrix}$ With similar computations as previously to bound the right-hand side of this latter inequality, we obtain eventually $\begin{matrix} | R_{i}^{b} | ≲ ‖ \nabla v ‖_{L^{2} (R^{3})} . \end{matrix}$ To complete the proof, we compute now: $\begin{matrix} F^{(i)} : = \int_{\partial B_{i}} Σ (w_{sing}, p_{sing}) n, T^{(i)} : = \int_{\partial B_{i}} (x - X_{i}) \times Σ (w_{sing}, p_{sing}) n . \end{matrix}$ Since $w_{sing}^{(i)}$ is invariant by rotation around $e_{3}$ , we have: $\begin{matrix} F^{(i)} = (\int_{\partial B_{i}} Σ (w_{sing}, p_{sing}) n \cdot e_{3}) e_{3}, and T^{(i)} = 0 . \end{matrix}$ It follows that for all $v \in H^{1} (R^{3})$ such that $v = W + ω \times (x - X_{i})$ on $B_{i}$ and vanishes on $B_{N}$ , we have: $\begin{matrix} (43) & 2 \int_{G_{ℓ_{0}}^{(i)}} D (w_{sing}) : D (v) = F^{(i)} \cdot W + R_{i}^{b}, \end{matrix}$ where: $\begin{matrix} R_{i}^{b} ≲ ‖ \nabla v ‖_{L^{2} (R^{3})} . \end{matrix}$

To compute the last component of $F^{(i)}$ we use conservation and variational arguments that we depict now. Firstly, we integrate the Stokes equation satisfied by $w_{sing}$ on $G_{ℓ_{0}}^{(i)}$ to transform our integral on $B_{i}$ into an integral on $B_{N}$ . Since $Σ (w_{sing}, p_{sing})$ vanishes on the lateral boundaries of $G_{ℓ_{0}}^{(i)}$ and remains bounded outside $G_{ℓ_{0} / 2}^{(i)}$ we have: $\begin{matrix} | \int_{\partial B_{i}} Σ (w_{sing}, p_{sing}) n - \int_{\partial B_{N} \cap \partial G_{ℓ_{0} / 2}^{(i)}} Σ (w_{sing}, p_{sing}) n | ≲ ‖ err ‖_{L^{1} (G_{ℓ_{0}}^{(i)})} . \end{matrix}$ where, with similar computations as previously: $\begin{matrix} ‖ err ‖_{L^{1} (G_{ℓ_{0}}^{(i)})} ≲ \int_{0}^{ℓ_{0}} \frac{r^{2}}{d_{i} + r^{2} / R} d r ≲ 1 . \end{matrix}$

We remark then that $w_{sing}$ is divergence-free, vanishes on $\partial B_{i}$ , is equal to $e_{3} cos (θ_{i})$ on $\partial B_{N} \cap \partial G_{ℓ_{0} / 2}^{(i)}$ and remains bounded outside. This entails: $\begin{matrix} | \int_{\partial B_{N}} Σ (w_{sing}, p_{sing}) n \cdot w_{sing} - cos (θ_{i}) \int_{\partial B_{N} \cap \partial G_{ℓ_{0} / 2}} Σ (w_{sing}, p_{sing}) n \cdot e_{3} | ≲ 1, \end{matrix}$ and $\begin{array}{rcl} - \int_{G_{ℓ_{0}}^{(i)}} div Σ (w_{sing}, p_{sing}) \cdot w_{sing} & = & - \int_{\partial B_{N} \cap \partial G_{ℓ_{0}}^{(i)}} Σ (w_{sing}, p_{sing}) n \cdot w_{sing} \\ (44) & + 2 \int_{G_{ℓ_{0}}^{(i)}} {| D (w_{sing}) |}^{2} . \end{array}$ By applying Holder’s inequality and noting that $\begin{matrix} | \partial_{3} w_{sing} | ≲ \frac{(x_{1}^{2} + x_{2}^{2}) cos θ_{i}}{{(2 R + d_{i} - 2 γ (| (x_{1}, x_{2}) |))}^{\frac{3}{2}}}, \end{matrix}$ we get $\begin{array}{rcl} | \int_{G_{ℓ_{0}}^{(i)}} div Σ (w_{sing}, p_{sing}) \cdot w_{sing} | & = & | \int_{G_{ℓ_{0}}^{(i)}} err \cdot w_{sing} | \\ = & | \int_{G_{ℓ_{0}}^{(i)}} Err \cdot \partial_{3} w_{sing} | \\ ⩽ & ‖ Err ‖_{L^{2} (G_{ℓ_{0}}^{(i)})} ‖ \partial_{3} w_{sing} ‖_{L^{2} (G_{ℓ_{0}}^{(i)})} \\ ≲ & ‖ \partial_{3} w_{sing} ‖_{L^{2} (G_{ℓ_{0}}^{(i)})} \\ ≲ & cos θ_{i} | ln (d_{i}) | \end{array}$ Moreover, as the most divergent term of $| D [w_{sing}] |^{2}$ is $| \partial_{23} a_{1} |^{2} + | \partial_{23} a_{2} |^{2}$ which – up to a strictly positive multiplicative constant– diverges like: $\begin{matrix} \frac{| x_{1} |^{2} + | x_{2} |^{2}}{{(h_{i} (| (x_{1}, x_{2}) |))}^{3}} cos {(θ_{i})}^{2}, \end{matrix}$ we get that $\begin{matrix} \int_{G_{ℓ_{0}}^{(i)}} {| D (w_{sing}) |}^{2} d x \sim \frac{cos {(θ_{i})}^{2}}{d_{i}} . \end{matrix}$

Combining the last two inequalities with (44) we get that there exists a constant $c_{0}$ for which: $\begin{matrix} | \int_{\partial B_{N} \cap \partial G_{ℓ_{0}}^{(i)}} Σ (w_{sing}, p_{sing}) n \cdot w_{sing} - \frac{c_{0} cos {(θ_{i})}^{2}}{d_{i}} | ≲ | ln (d_{i}) | cos (θ_{i}) . \end{matrix}$

By remarking that $cos (θ_{i})$ does not vanish, we obtain finally $\begin{matrix} | \int_{\partial B_{i} \cap \partial G_{ℓ_{0} / 2}^{(i)}} Σ (w_{sing}, p_{sing}) n \cdot e_{3} - \frac{c_{0} cos (θ_{i})}{d_{i}} | ≲ | ln (d_{i}) | . \end{matrix}$ Eventually, we obtain $\begin{matrix} F^{(i)} = c_{0} cos (θ_{i}) / d_{i} + R_{i}^{s}, \end{matrix}$ where $| R_{i}^{s} | ≲ | ln (d_{i}) |$ . Combining the above estimate with (43), we get (36) by setting $R_{i} = R_{i}^{b} + R_{i}^{s}$ .

Footnotes

Acknowledgement

The first author acknowledges support of the Institut Universitaire de France and project “SingFlows” ANR-grant number: ANR-18-CE40-0027. This paper was written while M.H. was benefiting a “subside à savant” from Université Libre de Bruxelles. He would like to thank the mathematics department at ULB for its hospitality.

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