On the trajectory of a light small rigid body in an incompressible viscous fluid

In this paper we study the dynamics of a small rigid body in a viscous incompressible fluid in dimension two and three. More precisely we investigate the trajectory of the rigid body in the limit when the its mass and its size tend to zero. We show that the velocity of the center of mass of the rigid body coincides with the background fluid velocity in the limit. We are able to consider the case where the density of the small rigid body is uniformly bounded respect to its size.


Introduction
In this paper we study the interaction of a small light rigid body with a incompressible viscous fluid in dimension two and three.The system fluid plus rigid body occupies the domain R d for d = 2, 3.The unknowns of the problem are the position of the rigid body S(t) ⊂ Ω and the velocity of the fluid u F which is defined on the fluid domain F (t) = R d \ F (t) with values in R d .Moreover the equations that u F satisfies, are the incompressible Navier-Stokes equation where u S is the velocity of the rigid body and ν > 0 is the viscosity coefficient.Regarding the rigid body we assume that it occupies the position S in at initial time and it is a connected, simply-connected subset of R d with smooth boundary and that it has constant density ρ S ∈ R with ρ S ≥ C > 0.
The evolution of the rigid body is completely determined by the dynamic of the center of mass and of the angular rotation.Recall that S in has mass and center of mass defined respectively by m = S in ρ S dx and h in = 1 m S in ρ S x dx.
If we denote by h(t) the position of the center of mass at time t and Q(t) the rotation around the center of mass respect to the initial configuration then the position of the rigid body at time t is S(t) = y such that Q T (t)(y − h(t)) ∈ S in and in dimension d = 3, its velocity is (2) where we denote ℓ(t) = h ′ (t).Moreover from the fact that Q(t) is a rotation matrix, Q ′ (t)Q T (t) is skew symmetric and it can be identify with a vector ω(t) ∈ R 3 through the relation where x ∈ R 3 .The evolution of ℓ and ω follows the Newton's laws that write where Σ is the stress tensor Σ(u, p) = 2νD(u) − pI where D(u) = ∇u + (∇u) T 2 and J is the inertia momentum and it is given through the formula Finally the initial conditions are such that they satisfy the compatibility conditions div (u in F ) = 0 in F in and u in Moreover without loss of generality we have h(0) = 0 and Q(0) = 0I.
Let now introduce a small parameter ε > 0 and for x ∈ R d , let S in ε ⊂ B ε (x) a sequence of initial positions of the rigid body.In this paper we study the dynamics of the rigid body when ε converges to zero.Let us recall that it has already been shown that the fluid does not interact with a small rigid body see for instance [20], [18], [19] and [11] under some mild assumptions on m ε and S in ε .Similar results are available also for compressible fluid see [3], [10] and Section 6 of [11].Moreover we proved in [4] that under some lower bounds on the masses the small rigid body will move with constant initial velocity.In this paper we show that under the assumptions and appropriate convergence of the initial data, the small rigid body will follow the fluid velocity.This result show a different behaviour respect to the case of an inviscid incompressible fluid where the small rigid body is not accelerate by the fluid, see Section 1.4 of [14].In the proof of our result we take advantage of the viscous term in an essential way.The idea is to use a relative energy inequality where we estimate where R ε is a restriction operator that maps solenoidal vector fields to solenoidal vector fields that are rigid in S ε , i.e. with zero symmetric gradient is the part of the domain occupied by the rigid body, and Rest ε a small reminder.
To estimate it is not clear how to close the estimates.Instead we notice that in dimension three and we look for uniform bounds for the term . Let now introduce the definition of weak solutions and the main result.

Definition of weak solutions and main result
In this section we recall the concept of finite energy weak solutions for the system (1)-( 3)-( 4).Then we state the main result of the paper.
Let introduce some notations from [19].Let denote by which is the extension by 1 of the density of the rigid body.Here for a set A ⊂ R d , we denote by χ A the indicator function of A, more precisely χ A (x) = 1 for x ∈ A and 0 elsewhere.Similarly we define the global velocity Notice that if u in ∈ L 2 (F (0)), then the compatibility conditions (5) on the initial data imply that div (u in ) = 0 in an appropriate weak sense.After all this preliminary we introduce the definition of Hopf-Leray type weak solutions for the system (1)-( 3)-(4).
Definition 1.Let S in and ρ in S the initial position and density of the rigid body, let (u in F , ℓ in , ω in ) satisfying the hypothesis (5) and such that u in ∈ L 2 (R d ).Then a triple (u F , ℓ, ω) is a Hopf-Leray weak solution for the system(1)-( 3)-( 4) associated with the initial data S in , ρ in S , u in F , ℓ in and ω in , if • the functions u F , ℓ and ω satisfy • the vector field u is divergence free in R d with D(u) = 0 in S(t); • the vector field u satisfies the equation in the following sense: • The following energy inequality holds for almost any time t ∈ R + .
The existence of weak solutions for the system (1)-( 3)-( 4) is now classical and can be found for example in [8].
Theorem 1.For initial data S in , ρ in S , u in F , ℓ in and ω in satisfying the hypothesis (5) and such that u in ∈ L 2 (R d ), there exist a Hopf-Leray weak solution (u F , ℓ, ω) of the system (1)-( 3)-( 4).
Let now introduce a small parameter ε > 0 that control the size of the rigid body.We assume that S in ε ⊂ B 0 (ε).We study the dynamics of the rigid body as ε goes to zero for solutions of the system (1)-( 3)-( 4) under some assumptions on the initial data ρ S,ε , u in F ,ε , ℓ in ε and ω in ε and in particular we show that in the limit the small rigid body follows the fluid.This result can be resumed as follows.

Then up to subsequence
Let us notice that in dimension two the time of existence of regular solution is T = +∞.In this case the convergence of h ε and ℓ ε holds in any compact interval.In dimension three the existence of global regular solutions is an open problem but there exist local in time solutions and they are global in time for small initial data.

Proof of the main result
The prove of Theorem 2 is based on a relative energy inequality that it is stated in Lemma 2. We present the proof of Lemma 2 in a separate section because it is technical.The plan for this section is to recall the definition of restriction operator R ε , to state Lemma 2 and prove Theorem 2. In the remaining part of the paper we set ν = 1 to simplify the notation.

The restriction operator
Let us introduce a restriction operator introduced in [10 The restriction operator is defined as where To simplify the notation for any regular enough function ϕ, we denote by by φε (t) = ϕ(t, h ε (t)).This allows us to rewrite the restriction operator in the more compact form In the following lemma we resume some properties of R ε .
Lemma 1.The restriction operator R ε has the following properties.For any ϕ ∈ C 0 (R d ) such that div (ϕ) = 0 and for p ∈ Proof.Let us show estimates ( 7)-( 8).In the following let us use the notation Using the definition of R ε and the fact that 1 − η ε and B ε are supported in a ball of radius 2ε, we have Using the fact that ϕ is divergence free we can rewrite This allows us to deduce Similarly From ( 10)-( 11) and interpolation inequality, we deduce (9).To show estimate (8), it is enough to show Notice that and The above computation and estimate allow us to deduce Remark 1.Let us notice that in the proof of the above lemma we show also inequality for any ϕ ∈ C 0 (R d ) such that div (ϕ) = 0 and for p ∈ [1, +∞] and estimate (12 for any ϕ ∈ W 1,∞ x (R d ) such that div (ϕ) = 0 and if p ∈ (1, +∞).
We will now present the relative energy inequality.

Relative energy inequality
We will use the above restriction operator to deduce a relative energy inequality.
Lemma 2. Under the hypothesis of Theorem 2, we have with C independent of ε.
The above lemma is the key estimate to deduce Theorem 2.

Proof of Theorem 2
Let us show Theorem 2 with the help of Lemma 2.
Proof of Theorem 2. Let us show Theorem 2 in dimension three and explain at the end how to adapted in dimension two.First of all let notice that )) , where we used that ρ ε is constant in S ε for any fixed ε to deduce the first inequality.We can rewrite the above inequality as From Lemma 2, Grömwall's inequality implies that Notice that where we used some Hölder inequalities and ( 7) for d = 3, p = 2 and ϕ = u in .By hypothesis we have and by Lemma 2 The estimate (15) together with ( 16) and (17) ensure that in particular it converge strong in C 0 t .We deduce that (15) together with ( 16) imply t .Finally we pass to the limit in the equation Let now move to the case of dimension two.First of all let notice that ), where 1/p + 1/q = 1/2 and we used Lemma 4 for the last inequality.We deduce that ), for any p < ∞.
Choose now 1/p = δ.Using Lemma 2 we deduce that Grömwall's inequality implies that Using the assumptions (6) and following the same strategy as in the case of dimension three, we prove the theorem.

Proof of Lemma 2
Let us now show Lemma 2.
Proof of Lemma 2. By the hypothesis of Theorem 2 that u in ∈ H k for k > d/2 + 1, there exists a local solution in dimension three and a global solution in dimension two of the Navier-Stokes equations such that With this choice of k we have also the bounds x ) ≤ C. The above bound will be implicitly used in many of the estimates to prove this lemma.
To deduce the relative energy inequality, let start by computing where in the inequality we use the energy inequality for u ε , the weak formulation satisfied by u ε and the energy inequality satisfied by u.After bringing on the left hand side some terms involving Du ε and DR ε (u) , we deduce where It remains to estimate | Rest ε |.To do that, we decompose the remainder We start by estimating the terms To tackle the last term on the right hand side, we notice that R ε (u(t, .)) . The two above observations allow us to estimate ))u(t, .) We estimate the third and fourth terms of Rest 2 analogously and we deduce We are left with the estimate of Collecting ( 18)-( 19)-( 20), we have We now consider the more difficult term Rest 1 ε , which reads To tackle this term we compute the time derivative of R ε (u) and use the equation satisfied by u.
Let us notice that in the above expression there is not a time derivative of η ε inside B ε because the B ε follow the rigid body as η ε .Let rewrite where To tackle the term I 1 , we use the equation satisfied by u.So let start by bounding the other terms.From now on the estimates depend on the dimension so let us focus on the case of dimension three and in the end we explain how to adapted in dimension two.We have that The term I 3 is the only one which is not zero in S ε .We have To tackle the right hand side, we notice that and similarly Equality (23) and estimates ( 24)-( 25)- (26), implies that For the term I 4 we proceed as follows.
Let now tackle I 5 .
We will now consider I 1 .Recall that u is a regular solution, we rewrite where pε = p(t, h ε (t)).We can now rewrite the remainder using the above computations and deduce where Inequalities ( 22)-( 27)-( 28)-( 29) imply that To tackled J 2 , we start by rewriting it as Before estimate the right hand side of the above equality, let us recall from ( 13) and ( 14) the following bounds holds Using equality (32), we deduce After applying some Hölder inequality, the J 3 term is bounded as follows.
We now tackle J 4 .
The bound of all the other terms follows similarly.
Proof.The estimate is well-known if in the right hand side we replace u L 2 (Fε(t)) by u L 2 (R 2 ) .Let us show that To see this let recall that S ε (t) ⊂ B ε (h ε (t)).By translation invariant of the norms we can assume h ε (t) = 0. Introduce the space The Poincaré inequality in X implies the existence of a constant C X such that We deduce that which implies (38).