Measure-Valued Solutions and Weak–Strong Uniqueness for the Incompressible Inviscid Fluid–Rigid Body Interaction

Abstract

We consider a coupled system of partial and ordinary differential equations describing the interaction between an incompressible inviscid fluid and a rigid body moving freely inside the fluid. We prove the existence of measure-valued solutions which is generated by the vanishing viscosity limit of incompressible fluid–rigid body interaction system under some physically constitutive relations. Moreover, we show that the measure-valued solution coincides with strong solution on the interval of its existence. This relies on the weak–strong uniqueness analysis. This is the first result of an existence of measure-valued solution and weak–strong uniqueness in measure-valued sense in the case of inviscid fluid–structure interaction.

Appendix

1.1 Fundamental Theorem on Young Measure

In the following, with a slightly different notation, we recall the Fundamental Theorem on Young measure discussed in [37]. Further, ${\mathcal {M}}({\mathbb {R}}^d)$ denotes the space of Radon measures on ${\mathbb {R}}^d$.

Theorem 4.1

Let $A \subset {\mathbb {R}}^n$ be a measurable set of finite measure and let $f_j : A \rightarrow {\mathbb {R}}^d$ be a sequence of measurable functions. Then there exists a subsequence $f_{j_{k}}$ and a weak – $*$ measurable map $Y : A \rightarrow {\mathcal {M}}({\mathbb {R}}^d)$, such that the following holds

• 1. $Y_x \ge 0$, $\Vert Y_x \Vert _{{\mathcal {M}}({\mathbb {R}}^d)}=\int _{{\mathbb {R}}^d} dY_x \le 1,$ for a.a. $x\in A$;

• 2. For every $g \in C_c({\mathbb {R}}^d)$, we have

  $$\begin{aligned} g(f_{j_{k}}) \rightarrow {\overline{g}} \ \ weakly{\hbox {-}}* \ in \ L^\infty (A),\\ {\overline{g}}(x) = \langle Y_x,g\rangle = \int _{{\mathbb {R}}^d} g dY_x. \end{aligned}$$

• 3. Furthermore, if

  $$\begin{aligned} \lim _{m \rightarrow \infty } \sup _k |\{ |f_{j_{k}}|\ge m \}| = 0 \end{aligned}$$

  (4.1)

  we have

  $$\begin{aligned} \Vert Y_x \Vert _{{\mathcal {M}}({\mathbb {R}}^d)}=1. \end{aligned}$$

As remarked in [37, Remark 2.9, Chapter 4], if $f_{j_{k}}$ are uniformly bounded in $L^p(A)^d$ for $p \in [1,\infty )$, the condition (4.1) is satisfied.

1.2 Reynolds Transport Theorem

For completeness of presentation we state here the Reynolds transport theorem which is one of the key ingredients in analysis of fluid on moving domains.

Theorem 4.2

Let f be a function such that all integrals in the formula below are well defined. Let $\mathbf{u} _{{\mathcal {B}}}$ be a rigid velocity field describing the motion of the body ${{\mathcal {B}}}(t)$. Then the following formula for time derivative of an integral over the fluid domain holds.

$$\begin{aligned} \frac{d}{dt}\int _{{{\mathcal {F}}}(t)} f \,\mathrm{d}{{\mathbf {x}}}= \int _{{{\mathcal {F}}}(t)} \partial _t f \,\mathrm{d}{{\mathbf {x}}}+ \int _{\partial {{\mathcal {B}}}(t)} f \mathbf{u} _{{\mathcal {B}}}\cdot \mathbf{n} \, \mathrm{d} {S}\end{aligned}$$

(4.2)

1.3 Weak Formulation of the Momentum Equation

In this section, we present the calculation behind the weak formulation of the momentum equation (2.6).

Let all the functions be sufficiently smooth so that we can do integration by parts. We multiply equation (1.2)$_1$ by test function $\pmb {\varphi }\in V_T$ and integrate over $\mathcal {F}(t)$ to obtain

$$\begin{aligned}&\frac{d}{dt} \int _{\mathcal {F}(t)} \mathbf{u} _{\mathcal {F}}\cdot \pmb {\varphi }_{\mathcal {F}} - \int _{\mathcal {F}(t)} \mathbf{u} _{\mathcal {F}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {F}} - \int _{\mathcal {F}(t)} (\mathbf{u} _{\mathcal {F}} \otimes \mathbf{u} _{\mathcal {F}}) : \nabla \pmb {\varphi }_{\mathcal {F}}\nonumber \\&\quad = -\int _{\partial \Omega } p_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}} - \int _{\partial \mathcal {B}(t)} p_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}}. \end{aligned}$$

(4.3)

As $\pmb {\varphi }\in V_T$, we have $\pmb {\varphi }_{\mathcal {F}} \cdot \mathbf{n} =0$ on $\partial \Omega$. Using the Reynolds transport theorem 4.2 and the mass transport of the body we have

$$\begin{aligned}&\frac{d}{dt}\int _{\mathcal {B}(t)}\varrho _{{\mathcal {B}}}\mathbf{u} _{{\mathcal {B}}}\cdot \pmb {\varphi }_{\mathcal {B}} = \int _{\mathcal {B}(t)} \frac{\partial }{\partial t} (\varrho _{{\mathcal {B}}}\mathbf{u} _{{\mathcal {B}}}\cdot \pmb {\varphi }_{\mathcal {B}}) + \int _{\mathcal {B}(t)} \mathrm{div}\left( \varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}(\mathbf{u} _{{\mathcal {B}}}\cdot \pmb {\varphi }_{\mathcal {B}})\right) \\&\quad = \int _{\mathcal {B}(t)} \varrho _{{\mathcal {B}}}\mathbf{u} _{{\mathcal {B}}}\cdot \frac{\partial }{\partial t} \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}(t)} \varrho _{{\mathcal {B}}}\Big (\frac{\partial }{\partial t} \mathbf{u} _{{\mathcal {B}}} + \mathbf{u} _{{\mathcal {B}}} \cdot \nabla \mathbf{u} _{{\mathcal {B}}}\Big )\cdot \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}(t)} \rho _{{{\mathcal {B}}}} (\mathbf {u}_{\mathcal {B}} \otimes \mathbf{u} _{{\mathcal {B}}}) : \nabla \pmb {\varphi }_{\mathcal {B}}. \end{aligned}$$

As ${\mathbb {D}}(\pmb {\varphi }_{\mathcal {B}})=0$, we have $(\mathbf {u}_{\mathcal {B}} \otimes \mathbf{u} _{{\mathcal {B}}}) : \nabla \pmb {\varphi }_{\mathcal {B}}=0$. Using the rigid body equations (1.4) and the boundary condition $\pmb {\varphi }_{{{\mathcal {F}}}} \cdot \mathbf{n} = \pmb {\varphi }_{{{\mathcal {B}}}} \cdot \mathbf{n}$ on $\partial {\mathcal {B}}(t)$, we obtain

$$\begin{aligned} - \int _{\partial \mathcal {B}(t)} p_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}} = -\frac{d}{dt}\int _{\mathcal {B}(t)}\varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}(t)} \varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \frac{\partial }{\partial t} \pmb {\varphi }_{\mathcal {B}} . \end{aligned}$$

(4.4)

Thus by combining the above relations (4.3)–(4.4) and then integrating in time, we have

$$\begin{aligned} - \int _0^\tau \int _{\mathcal {F}(t)} \mathbf{u} _{\mathcal {F}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {F}} - \int _0^\tau \int _{\mathcal {B}(t)}\varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {B}} - \int _0^\tau \int _{\mathcal {F}(t)} (\mathbf{u} _{\mathcal {F}} \otimes \mathbf{u} _{\mathcal {F}}) : \nabla \pmb {\varphi }_{\mathcal {F}} \end{aligned}$$

$$\begin{aligned} = \int _{\mathcal {F}_0} (\mathbf{u} _{\mathcal {F}}\cdot \pmb {\varphi }_{\mathcal {F}})(0) - \int _{\mathcal {F}_\tau } (\mathbf{u} _{\mathcal {F}}\cdot \pmb {\varphi }_{\mathcal {F}})(\tau ) + \int _{\mathcal {B}_0} (\varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}})(0) - \int _{\mathcal {B}_\tau } (\varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}})(\tau ) \end{aligned}$$

(4.5)

for $\tau \in [0,T]$.

1.4 Derivation of (3.26)

We multiply Eq. (3.17)$_1$ by test function $\pmb {\varphi }\in V_T$, integrate over $\mathcal {F}_1(t)$ and use the Reynolds transport theorem 4.2 to obtain

$$\begin{aligned}&\frac{d}{dt} \int _{\mathcal {F}_1(t)} \mathbf{U}^s_{{{\mathcal {F}}}}\cdot \pmb {\varphi }_{\mathcal {F}} - \int _{\mathcal {F}_1(t)} \mathbf{U}^s_{{{\mathcal {F}}}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {F}}\nonumber \\&\qquad - \int _{\mathcal {F}_1(t)}\Big ( \langle Y_{t,{{\mathbf {x}}}},\mathbf{u} _{1\mathcal {F}}\rangle \otimes \mathbf{U} ^s_{\mathcal {F}}) : \nabla \pmb {\varphi }_{\mathcal {F}} - (\mathbf {U}^s_\mathcal {F} - \langle Y_{t,{{\mathbf {x}}}},\mathbf{u} _{1\mathcal {F}}\rangle )\cdot \nabla \mathbf {U}^s_\mathcal {F}\cdot \pmb {\varphi }_{\mathcal {F}} \Big )\nonumber \\&\quad = -\int _{\partial \Omega } P^s_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}} - \int _{\partial \mathcal {B}_1(t)} P^s_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}} + \int _{\mathcal {F}_1(t)} \mathbf {F}\cdot \pmb {\varphi }_{\mathcal {F}}, \end{aligned}$$

(4.6)

where we combine the weak formulation of the continuity equation and the boundary condition. As $\pmb {\varphi }\in V_T$, we have $\pmb {\varphi }_{\mathcal {F}} \cdot \mathbf{n} =0$ on $\partial \Omega$. We use the Reynolds transport theorem 4.2 together with the mass transport of the body to obtain

$$\begin{aligned}&\frac{d}{dt}\int _{{{\mathcal {B}}}_1(t)}\varrho _{{{\mathcal {B}}}} \mathbf{U}^s_{{{\mathcal {B}}}}\cdot \pmb {\varphi }_{\mathcal {B}} = \int _{\mathcal {B}_1(t)} \frac{\partial }{\partial t} (\varrho _{{{\mathcal {B}}}} \mathbf{U} ^s_{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}}) + \int _{\mathcal {B}_1(t)} \mathrm{div}\left( \varrho _{{{\mathcal {B}}}}\mathbf{u} _{1\mathcal {B}} (\mathbf{U} ^s_{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}})\right) \\&\quad = \int _{\mathcal {B}_1(t)} \varrho _{{{\mathcal {B}}}}\mathbf{U} ^s_{\mathcal {B}}\cdot \frac{\partial }{\partial t} \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}_1(t)} \varrho _{{{\mathcal {B}}}} \Big (\frac{\partial }{\partial t} \mathbf{U} ^s_{\mathcal {B}}\\&\qquad + \mathbf{U} ^s_{\mathcal {B}} \cdot \nabla \mathbf{U} ^s_{\mathcal {B}}\Big )\cdot \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}_1(t)} \rho _{{{\mathcal {B}}}}\Big ( (\mathbf {u}_{1\mathcal {B}} \otimes \mathbf{U} ^s_{\mathcal {B}}) : \nabla \pmb {\varphi }_{\mathcal {B}} - (\mathbf {U}^s_{\mathcal {B}} -\mathbf {u}_{1\mathcal {B}})\cdot \nabla \mathbf {U}^s_{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}} \Big ). \end{aligned}$$

Using the rigid body equations (3.22)–(3.23) and the boundary condition $\pmb {\varphi }_{{{\mathcal {F}}}} \cdot \mathbf{n} = \pmb {\varphi }_{{{\mathcal {B}}}} \cdot \mathbf{n}$ on $\partial {\mathcal {B}}_1(t)$, we obtain

$$\begin{aligned}&- \int _{\partial \mathcal {B}_1(t)} P^s_{\mathcal {F}}{\mathbb {I}} \mathbf{n} \cdot \pmb {\varphi }_{\mathcal {F}} = -\frac{d}{dt}\int _{\mathcal {B}_1(t)}\varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}}\nonumber \\&\quad + \int _{\mathcal {B}_1(t)} \varrho _{{\mathcal {B}}}\mathbf{u} _{\mathcal {B}}\cdot \frac{\partial }{\partial t} \pmb {\varphi }_{\mathcal {B}} + \int _{\mathcal {B}_1(t)} \rho _{{{\mathcal {B}}}}\Big ( (\mathbf {u}_{1\mathcal {B}} \otimes \mathbf{U} ^s_{\mathcal {B}}) : \nabla \pmb {\varphi }_{\mathcal {B}} - (\mathbf {U}^s_\mathcal {B} -\mathbf {u}_{1\mathcal {B}})\cdot \nabla \mathbf {U}^s_\mathcal {B}\cdot \pmb {\varphi }_{\mathcal {B}} \Big )\nonumber \\&\quad - ((\mathbf{w} ^s-\mathbf{w} _1)\times (\mathbb {J}_1 \mathbf{w} ^s)\cdot \pmb {\varphi }_{{\mathcal {B}},\mathbf {w}}+m(\mathbf{w} ^s-\mathbf{w} _1)\times \mathbf {V}^s\cdot \pmb {\varphi }_{{\mathcal {B}},\mathbf {V}}). \end{aligned}$$

(4.7)

Thus by combining the above relations (4.6)–(4.7) and then integrating in time, we have for $\tau \in [0,T]$:

$$\begin{aligned}&- \int _0^\tau \int _{\mathcal {F}_1(t)} \mathbf{U} ^s_{\mathcal {F}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {F}} - \int _0^\tau \int _{\mathcal {B}_1(t)}\varrho _{{\mathcal {B}}}\mathbf{U} ^s_{\mathcal {B}}\cdot \frac{\partial }{\partial t}\pmb {\varphi }_{\mathcal {B}}\nonumber \\&\quad - \int _0^\tau \int _{\mathcal {F}_1(t)}\Big ( ( \langle Y_{t,{{\mathbf {x}}}},\mathbf{u} _{1\mathcal {F}}\rangle \otimes \mathbf{U} ^s_{\mathcal {F}}) : \nabla \pmb {\varphi }_{\mathcal {F}} - (\mathbf {U}^s_\mathcal {F} - \langle Y_{t,{{\mathbf {x}}}},\mathbf{u} _{1\mathcal {F}}\rangle )\cdot \nabla \mathbf {U}^s_\mathcal {F}\cdot \pmb {\varphi }_{\mathcal {F}} \Big )\\&\quad - \int _0^\tau \int _{\mathcal {B}_1(t)} \rho _{{{\mathcal {B}}}}\Big ( (\mathbf {u}_{1\mathcal {B}} \otimes \mathbf{U} ^s_{\mathcal {B}}) : \nabla \pmb {\varphi }_{\mathcal {B}} - (\mathbf {U}^s_\mathcal {B} -\mathbf {u}_{1\mathcal {B}})\cdot \nabla \mathbf {U}^s_\mathcal {B}\cdot \pmb {\varphi }_{\mathcal {B}} \Big ) = \int _{0}^{\tau }\int _{\mathcal {F}_1(t)}\big ( \mathbf {F}\cdot \pmb {\varphi }_{\mathcal {F}}\big )\\&\quad - \int _{0}^{\tau }((\mathbf{w} ^s-\mathbf{w} _1)\times (\mathbb {J}_1 \mathbf{w} ^s)\cdot \pmb {\varphi }_{{\mathcal {B}},\mathbf {w}}+m(\mathbf{w} ^s-\mathbf{w} _1)\times \mathbf {V}^s\cdot \pmb {\varphi }_{{\mathcal {B}},\mathbf {V}}) + \int _{\mathcal {F}_0} (\mathbf{U} ^s_{\mathcal {F}}\cdot \pmb {\varphi }_{\mathcal {F}})(0)\\&\quad - \int _{\mathcal {F}_1(\tau )} (\mathbf{U} ^s_{\mathcal {F}}\cdot \pmb {\varphi }_{\mathcal {F}})(\tau ) + \int _{\mathcal {B}_0} (\varrho _{{\mathcal {B}}}\mathbf{U} ^s_{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}})(0) - \int _{\mathcal {B}_1(\tau )} (\varrho _{{\mathcal {B}}}\mathbf{U} ^s_{\mathcal {B}}\cdot \pmb {\varphi }_{\mathcal {B}})(\tau ). \end{aligned}$$

1.5 Derivation of (3.40)

In this section, we follow the calculations of [36, Appendix] to derive our desired identity. We know that

$$\begin{aligned} \mathbf{u} _{1 {{\mathcal {B}}}}(t,x)&= \mathbf{V} _1(t) + \mathbf{w} _1(t) \times ({{\mathbf {x}}}- {{\mathbf {X}}}_1(t)) \qquad \text { in } Q_{{{\mathcal {B}}}_1} \\ \mathbf{U}^s_{{{\mathcal {B}}}}(t,x)&= \mathbf{V} ^s(t) + \mathbf{w} ^s(t) \times ({{\mathbf {x}}}- {{\mathbf {X}}}_1(t)) \qquad \text { in } Q_{{{\mathcal {B}}}_1}. \end{aligned}$$

A simple calculation gives

$$\begin{aligned} \frac{\partial \mathbf{U}^s_{{{\mathcal {B}}}}}{\partial t} + (\mathbf{U}^s_{{{\mathcal {B}}}}\cdot \nabla )\mathbf{U}^s_{{{\mathcal {B}}}} = \frac{\mathrm{d} \mathbf{V} ^s}{\mathrm{d} t} + \frac{\mathrm{d} \mathbf{w} ^s}{\mathrm{d} t}\times ({{\mathbf {x}}}-{{\mathbf {X}}}_1) + \mathbf{w} ^s \times (\mathbf{w} ^s \times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)) \quad \text { in } Q_{{{\mathcal {B}}}_1}. \end{aligned}$$

(4.8)

We use (3.22) to deduce

$$\begin{aligned} \int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}\frac{\mathrm{d} \mathbf{V} ^s}{\mathrm{d} t}\cdot \mathbf{V} _1 = \int _{\partial {{\mathcal {B}}}_1(t)} P^s_{{{\mathcal {F}}}}\mathbf{n} \cdot \mathbf{V} _1 - (m(\mathbf{w} ^s-\mathbf{w} _1)\times \mathbf {V}^s)\cdot \mathbf{V} _1. \end{aligned}$$

(4.9)

Next, by (1.6) we get

$$\begin{aligned}&\int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}\frac{\mathrm{d} \mathbf{V} ^s}{\mathrm{d} t}\cdot (\mathbf{w} _1\times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)) = \left( \frac{\mathrm{d} \mathbf{V} ^s}{\mathrm{d} t} \times \mathbf{w} _1\right) \cdot \int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}({{\mathbf {x}}}-{{\mathbf {X}}}_1) = 0, \end{aligned}$$

(4.10)

$$\begin{aligned}&\int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}\left( \frac{\mathrm{d} \mathbf{w} ^s}{\mathrm{d} t} \times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)\right) \cdot \mathbf{V} _1 = \mathbf{V} _1\cdot \left( \frac{\mathrm{d} \mathbf{w} ^s}{\mathrm{d} t} \times \int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}({{\mathbf {x}}}-{{\mathbf {X}}}_1) \right) = 0, \end{aligned}$$

(4.11)

$$\begin{aligned}&\int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}(\mathbf{w} ^s\times (\mathbf{w} ^s\times ({{\mathbf {x}}}-{{\mathbf {X}}}_1))\cdot \mathbf{V} _1 = \mathbf{V} _1\cdot \left( \mathbf{w} ^s\times \left( \mathbf{w} ^s\times \int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}({{\mathbf {x}}}-{{\mathbf {X}}}_1) \right) \right) = 0. \end{aligned}$$

(4.12)

Using (1.7) and (3.23) we obtain

$$\begin{aligned}&\int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}\left( \frac{\mathrm{d} \mathbf{w} ^s}{\mathrm{d} t} \times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)\right) \cdot (\mathbf{w} _1\times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)) = \mathbb {J}_1\frac{\mathrm{d} \mathbf{w} ^s}{\mathrm{d} t}\cdot \mathbf{w} _1 \nonumber \\&\quad = (\mathbb {J}_1\mathbf{w} ^s \times \mathbf{w} ^s)\cdot \mathbf{w} _1 - ((\mathbf{w} ^s-\mathbf{w} _1)\times (\mathbb {J}_1 \mathbf{w} ^s))\cdot \mathbf{w} _1 + \mathbf{w} _1\cdot \int _{\partial {{\mathcal {B}}}_1(t)} ({{\mathbf {x}}}-{{\mathbf {X}}}_1)\times P^s_{{{\mathcal {F}}}}\mathbf{n} \nonumber \\&\quad = \mathbb {J}_1\mathbf{w} ^s \cdot (\mathbf{w} ^s\times \mathbf{w} _1) - ((\mathbf{w} ^s-\mathbf{w} _1)\times (\mathbb {J}_1 \mathbf{w} ^s))\cdot \mathbf{w} _1 + \int _{\partial {{\mathcal {B}}}_1(t)} P^s_{{{\mathcal {F}}}}\mathbf{n} \cdot (\mathbf{w} _1 \times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)). \end{aligned}$$

(4.13)

We also have

$$\begin{aligned} \int _{{{\mathcal {B}}}_1(t)} \varrho _{{\mathcal {B}}}\left( \mathbf{w} ^s\times \left( \mathbf{w} ^s \times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)\right) \right) \cdot (\mathbf{w} _1\times ({{\mathbf {x}}}-{{\mathbf {X}}}_1)) = - \mathbb {J}_1\mathbf{w} ^s\cdot (\mathbf{w} ^s\times \mathbf{w} _1). \end{aligned}$$

(4.14)

Summing (4.9)–(4.14) and using (4.8) we end up with

$$\begin{aligned} \int _0^\tau \int _{\mathcal {B}_1(t)}\Big (\varrho _{{\mathcal {B}}}\mathbf{U} ^s_{\mathcal {B}}\cdot \nabla \mathbf{U} ^s_{\mathcal {B}} \cdot \mathbf {u}_{1\mathcal {B}}+ \varrho _{{\mathcal {B}}}\mathbf{u} _{1\mathcal {B}}\cdot {\partial _ t}\mathbf{U} ^s_{\mathcal {B}} \Big )= & {} \int _0^\tau \int _{\partial \mathcal {B}_1(t)} (\mathbf{u} _{1\mathcal {B}}\cdot \mathbf{n} ) P^s_{\mathcal {F}} - \int _0^{\tau } (m(\mathbf{w} ^s-\mathbf{w} _1)\times \mathbf {V}^s)\cdot \mathbf{V} _1\\&- \int _0^{\tau }((\mathbf{w} ^s-\mathbf{w} _1)\times (\mathbb {J}_1 \mathbf{w} ^s))\cdot \mathbf{w} _1. \end{aligned}$$