Self-propelled motion of a rigid body inside a density dependent incompressible fluid

This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole $\mathbb{R}^3$. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution.


Introduction
Fluid-structure interaction (FSI) systems are systems which include a fluid and a solid component. The study of motion of small particles in fluids became one of the main focuses in research in the last 50 years. The presence of the particles has influence on the flow of the fluid and the fluid affects the motion of the particles. It implies that the problem of determining the flow characteristic is highly coupled. For such type of everyday phenomena with a wide range of applications we refer to, e.g., [3,4,17] and references therein. Systems that arise from modeling such phenomena are typically nonlinear systems of partial differential equations with a moving boundary or interface. We investigate a system where the rigid body moves in an incompressible Newtonian fluid which fills the whole space. The position of the rigid body at any given time moment is determined by two vectors describing the translation of the center of the mass and the rotation around the center of the mass, respectively. A system of six ordinary differential equations (Euler equations) describing the conservation of linear and angular momentum describes the dynamics of a rigid body. the case of positive density and the case of a viscosity depending on the density and some further remarks and open problems. In the appendix we present a derivation of the weak formulation of the problem.

The mathematical model
For T > 0 given and for any time t ∈ (0, T ) let S(t) ⊂ R 3 denote a closed, bounded and simply connected rigid body. We assume that the rest of the space, i.e., R 3 \ S(t) = F(t) is filled with a viscous incompressible nonhomogeneous fluid. The initial domain of the rigid body is denoted by S 0 and is assumed to have a smooth boundary. Correspondingly, F 0 = R 3 \ S 0 is the initial fluid domain.
Our system of a rigid body moving in a fluid is in the first instance a moving domain problem in an inertial frame in which the velocity of the rigid body is described by for (x, t) ∈ S(t) × (0, T ). (2.1) Here, h (t) is the linear velocity of the centre of mass h and ω(t) is the angular velocity of the body. The solid domain at time t in the same inertial frame is given by where Q(t) ∈ SO(3) is associated to the rotation of the rigid body. Mathematically, it will turn out useful to define U S for all (x, t) ∈ R 3 × (0, T ). Let , U and P be the density, velocity and pressure of the viscous incompressible nonhomogeneous fluid, respectively, which satisfy
Here we consider a self-propelled motion of the body S(t), which we describe by a vectorial flux W at ∂S(t).
In this article we consider Navier type conditions at the fluid-structure interface; we prescribe the normal and tangential parts of the flux by: W · N = 0 on ∂S(t) × (0, T ), where N is the unit outward normal to the boundary of F(t), i.e., directed towards S(t) and α is a given constant. We refer to Remark 5.4 for a discussion of the assumption that the normal component is zero.
For a given viscosity coefficient ν > 0, we set Σ(U, P ) = −P I +2νD(U ) with D(U ) = 1 2 Let S denote the density of the rigid body. Then m = S(t) S (x, t)dx is its total mass, which is constant in time and does not change under the coordinate transformation below. Further, the moment of inertia J(t) is defined by 4Š. NECASOVÁ ET AL.
The ordinary differential equations modeling the dynamics of the rigid body then read We suppose the density and the velocity of the fluid to satisfy (see [24], P. 20): as well as the initial conditions Following [16,28,30], we apply a global change of variables that transforms the system in such a way that it is formulated in a frame which is attached to the rigid body. At time t = 0, the two frames are assumed to coincide with h(0) = 0. Hence the transformed system is posed on the fixed domain F 0 × (0, T ) via the following change of variables: Further, the moment of inertia transforms to By (2.1) and the extension of U S to the whole space, we have that where (t) = Q(t) h (t) is the transformed linear velocity and r(t) = Q(t) ω(t) the transformed angular velocity of the rigid body. The normal and tangential parts of the self-propulsion flux in the new frame are given by: It follows from (2.3) that ρ is bounded and u satisfies u(y, t) → 0 as |y| → ∞.
The system of partial and ordinary differential equations that we investigate in this work consists of the equations (2.7)-(2.13).

Notations and functional framework
Before initiating our analysis, we collect here some basic notation that will be used throughout. The linear space D(Ω) consists of all infinitely differentiable functions that have compact support in Ω. The norm in a Lebesgue space L p (resp. Sobolev space W k,p ) is denoted by · p (resp. · k,p ). We denote by W −1,r (Ω) the dual space of W 1,r 0 (Ω), where 1/r + 1/r = 1. For convenience, we introduce We want to give an appropriate notion of weak solution to system (2.7)-(2.13). In order to do so, following [28], we introduce the space of divergence free vector functions For any φ ∈ H, there exist φ ∈ R 3 and r φ ∈ R 3 such that φ(y) = φ + r φ × y =: φ S (y) for all y ∈ S 0 . (2.14) Note that the tangential component of φ ∈ H is allowed to jump at ∂S 0 , while the normal component has a continuous representative. We remark that, in the integrals on ∂S 0 below, φ denotes the trace from the fluid side whereas φ S denotes the trace from the solid side.
We consider the space L 2 (R 3 ) with the following inner product: This inner product is equivalent to the usual inner product of L 2 (R 3 ). Furthermore, when φ, ψ ∈ H, we have The norm associated with the above inner product is denoted by · = (·, ·) H . Let us define the space Next we introduce some notations of time-dependent functions that we need later to describe the compactness properties. For h > 0, the translated function of a function f denoted as τ h f is given by Let E be a Banach space. For 1 q ∞, 0 < s < 1, Nikolskii spaces are defined by (2.15)

Energy inequality and definition of weak solution
Let u be a smooth solution of (2.7)-(2.13) and φ ∈ C ∞ ([0, T ]; H) such that φ| F0 ∈ C ∞ ([0, T ]; D(F 0 )) . In the appendix we derive the weak form of our system. It reads Remark 2.1. Let us remark that in the definition of a weak solution our weak formulation is introduced separately on F 0 and S 0 and not in the whole Ω. The reason is that we consider the Navier type of boundary condition. Moreover, in (2.16) the integral over S 0 is hidden e.g. in the term m (0) · φ (0) − J 0 r(0) · r φ (0) through the definition of m and J 0 .
This relation helps us to obtain an energy estimate for the system (2.7)-(2.13).
The relation (2.16) motivates us to define weak solutions in the following way.
is a weak solution to system (2.7)-(2.13) if the following conditions hold true: ).
-The equation of continuity (2.7) is satisfied in the weak sense, i.e., . Also, a renormalized continuity equation holds in a weak sense, i.e., ) and for all t ∈ [0, T ], the relation (2.16) holds.
-In Definition 2.3, ρ stands for the following extended version: -In Definition 2.3, due to a density argument (see [28]), we can take less smooth test functions φ in the weak formulation of balance of linear momentum (2.16). In particular, , we can enlarge the space of the test functions by considering the following space -There is no a priori reason that the momentum ρu is continuous in time. We only have that, for any . -The introduction of the renormalized continuity equation in the definition of weak solutions to our system yields that ρ ∈ C([0, T ]; L q loc (R 3 )) for all q ∈ [1, ∞), see below for details. -Let us mention that due to properties of the transport equation and additional assumption on velocity field, see e.g. Lemma 3.1 of [27], the behavior of the initial density ρ 0 yields the behavior of the density for all t ∈ [0, T ).
In the following theorem we state the main result of our paper regarding the global existence of a weak solution to system (2.7)-(2.13).
Theorem 2.5. Let S 0 ⊂ R 3 be a C 1,1 bounded, closed, simply connected set and and that there exist constants c 1 , c 2 > 0 such that Then, for any T > 0 there exists a weak solution (ρ, u) to system (2.7)-(2.13). Moreover, we have and for a.e. t ∈ [0, T ], the energy inequality (2.17) holds.
In order to prove the main result, as a first step, we will prove the existence of a weak solution in a bounded domain B R , where B R = {y ∈ R 3 | |y| < R}, see Section 3. Thereafter, we take R to infinity to establish the existence of a weak solution for the whole space R 3 in Section 4.

Existence of a weak solution in a bounded domain
In this section we consider the system (2.7)-(2.13) in a smooth bounded domain Ω ⊂ R 3 , along with the complementary boundary condition u(y, t) = 0, y ∈ ∂Ω. (3.1) We assume S 0 ⊂ Ω and set F 0 = Ω \ S 0 . Let us define a weak solution to system (2.7)-(2.13) in any bounded domain Ω. To do that, we introduce two spaces H Ω , V Ω for the bounded domain Ω, which are analogous to H, V. We set Definition 3.1. Let T > 0 and let Ω ⊂ R 3 be a smooth bounded domain. A pair (ρ, u) is a weak solution to system (2.7)-(2.13) with (3.1) if the following conditions hold true: -The equation of continuity (2.7) is satisfied in the weak sense, i.e., for any test function φ ∈ D([0, T ) × Ω). Also, a renormalized continuity equation holds in a weak sense, i.e., ) and for all t ∈ [0, T ], the relation (2.16) holds.
Next we assert the existence of a weak solution in a bounded domain.
Theorem 3.2. Let R be sufficiently large and and that there exist constants c 1 , c 2 > 0 such that Then for any time T > 0 there exists a weak solution (ρ R , u R ) to system (2.7)-(2.13) satisfying (3.1) on the time interval (0, T ). Moreover, we have inf and for a.e. t ∈ [0, T ], the energy inequality (2.17) holds for (ρ R , u R ).
Proof. The proof is divided into several steps. At first we construct an approximate solution and then establish the existence of a weak solution to system (2.7)-(2.13) as the limit of this approximation.
Step 1: Construction of N th level approximate solution Since the set If u N (t), z j ∈ X N , then there exist N (t), zj ∈ R 3 and r N (t), r zj ∈ R 3 such that u N (y, t) = N (t) + r N (t) × y, for all y ∈ S 0 , z j (y) = zj + r zj × y, for all y ∈ S 0 .
Let us define the following quantities: We are looking for u N , ρ N , the solution of the approximate problem, such that for some T N > 0, satisfy for all z j ∈ X N : Here u 0N and ρ 0N are functions satisfying We study the local existence of u N , ρ N by similar arguments as in Theorem 9 of [35], or in Chapter VI, Theorem VI.2.1 of [5]. We define the Banach space We will construct a map N : E N → E N which allows to find a fixed point that is a solution to the approximate problem. We do so in three steps. Firstly, let v N ∈ E N be given. We consider We define the trajectory Y N = Y N x,t (s) of a particle located at x at time t as such that ρ N (0) = ρ N 0 . Secondly, after the construction of ρ N , we are looking for u N ∈ E N satisfying for all z j ∈ X N . Now we seek the solution u N of (3.10) in the form , for any j = 1, 2, ..., N.

Let us introduce the matrices and vectors
det(ρ N r zi , z k , z j ) dy + det(m zi , r z k , zj ) + det(J 0 r zi , r z k , r zj ) .
Thus, equation (3.10) can be viewed as 11) where α N ∈ R N is the unknown vector, A N , M N are N × N matrices and B N , C ∈ R N are vectors defined as above. As (·, ·) H is a scalar product on L 2 (R 3 ) and z i is an orthonormal basis, we have that M N is invertible. Hence, by Cauchy-Lipschitz theory, equation (3.11) with the corresponding initial condition has a unique solution in some interval [0, T N ]. As a consequence, there exists a unique u N ∈ E N which satisfies (3.10). Thirdly, given v N ∈ E N , we have obtained a unique couple (ρ N , u N ) that solves (3.9)-(3.10). Now we define the map N : Following the steps of Chapter VI, Theorem VI.2.1 in [5], we obtain that the map N has a fixed point in a suitable subset of E N . This fixed point (denoted by u N ) along with ρ N is the solution of the nonlinear approximate problem (3.3)-(3.5).
Step 3: Compactness argument and convergence properties We show that the approximate solution (ρ N , u N ) constructed in Step 1 has a limit in suitable spaces and its limit is a solution to system (2.7)-(2.13) with (3.1). Let X ⊂ E ⊂ Y be Banach spaces and the imbedding X → E be compact.
We consider equation (3.3) with z j replaced with φ(y, t) = ξ(y)ψ(t) and use the relation which follows similarly as (2.19). We then obtain The main problematic term is the first one on the left hand side. To deal with this term, we integrate from 0 to t and apply the product rule so that the derivative is on the test function φ. In this term as well as in the other terms, we can then pass to the limit as N → ∞ by using the convergence results obtained in Step 3: Finally, we obtain t 0 d ds This equality clearly yields (2.16).

Proof of Theorem 2.5
We prove the existence of a weak solution in the unbounded domain by using Theorem 3.2 (existence of solution in a ball B R ) and letting R → ∞.
Observe that for |y| > R : Thus, by Lebesgue's dominated convergence theorem, we have for any φ ∈ V : and u S,R (y, t) → (t) + r(t) × y = u S (y, t) as R → ∞.
These above mentioned convergence properties allow us to pass to the limit R → ∞ in each term of (4.1) and to obtain the identity (2.16).

Discussion
In this section, we discuss two variants of our system and give a few remarks. Firstly, in the case of a positive initial density we mention the stronger results that can be obtained. Then, we will concentrate on the case when the fluid viscosity depends on the density.

Positive initial density
We can improve the results when the initial density is away from zero (inf ρ 0 > 0), i.e., when the fluid does not contain any vacuum regions. Actually, for the approximate solution in the bounded domain, we obtain -Under the translation operator τ h : f → f (· + h), we can obtain from ( [5], Chapt. VI, Lem. VI.2.5) that i.e., {u N } is bounded in N 1/4,2 (0, T ; L 2 (Ω)).
-As V Ω → L 2 (Ω) is compact and u N ∈ L 2 (0, T ; V Ω ) ∩ N 1/4,2 (0, T ; L 2 (Ω)), we have that, according to Lemma 4(iv) of [35], u N is relatively compact in L 2 (0, T ; L 2 (Ω)). This allows us to achieve the strong convergence of u N to u in L 2 (0, T ; L 2 (Ω)) which is the same as in the case of a homogeneous fluid (i.e., when the fluid has constant density).

Fluid viscosity depends on the density
We discuss how to deal with the case if the fluid viscosity depends on density. Here we consider the fluid viscosity ν as a C 1 function of the fluid density and it satisfies the following: there exists ν 1 , ν 2 > 0 such that ν 1 ν(η) ν 2 for all η ∈ R and ν is bounded. (5.1) In this case, we have the following relation analogous to (2.16): We can also define weak solutions in this case as previously: Definition 5.1. Let T > 0. A pair (ρ, u) is a weak solution to system (2.7)-(2.13) with ν = ν(ρ) if the following conditions hold true: Assume that the self-propelled motion w satisfies (2.5)-(2.6), the fluid viscosity satisfies (5.1) and that there exist constants c 1 , c 2 > 0 such that Then for an arbitrary T > 0, there exists a weak solution (ρ, u) to system (2.7)-(2.13) with ν = ν(ρ). Moreover, we have and for a.e. t ∈ [0, T ], the energy inequality holds.
The proof of this theorem is similar as before. We start with an approximation for the system on a bounded domain; later we pass to the unbounded domain. Additionally to before, we have to justify the passing of the limits in the terms: Now to do this, observe that we already have With the help of hypothesis (5.1) for the fluid viscosity, we have for all q ∈ [1, ∞) : Thus, we have strong convergence of the viscosity The above strong convergence of viscosity and u N u in L 2 (0, T ; V Ω ) weakly a transformation to a fixed domain is not possible since the body may change its shape as time evolves. Hence different methods need to be developed.
Regarding the first term on the right hand side of (A.4), we observe that by (2.10)