Motion of a rigid body in a compressible fluid with Navier-slip boundary condition

Abstract

In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.

Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded smooth domain occupied by a fluid and a rigid body. Let the rigid body $\mathcal{S}(t)$ be a regular, bounded domain and moving inside Ω. The motion of the rigid body is governed by the balance equations for linear and angular momentum. We assume that the fluid domain $\mathcal{F}(t)=\mathrm{\Omega }\setminus \overline{\mathcal{S}(t)}$ is filled with a viscous isentropic compressible fluid. We also assume the Navier-slip boundary conditions at the interface of the interaction of the fluid and the rigid body as well as at ∂Ω. The fluid occupies, at $t=0$, the domain ${\mathcal{F}}_0=\mathrm{\Omega }\setminus {\mathcal{S}}_0$, where the initial position of the rigid body is ${\mathcal{S}}_0$. In equations (1.4)–(1.9), $\nu (t,x)$ is the unit normal to $\partial \mathcal{S}(t)$ at the point $x\in \partial \mathcal{S}(t)$, directed to the interior of the body. In (1.3) and (1.4)–(1.5), $g_{\mathcal{F}}$ and $g_{\mathcal{S}}$ are the specific body forces. Moreover, $\alpha >0$ is a coefficient of friction. Here, the notation $u\otimes v$ is the tensor product of two vectors $u,v\in {\mathbb{R}}^3$ and it is defined as $u\otimes v={(u_i{v}_j)}_{1⩽i,j⩽3}$. In the equations, ${\rho }_{\mathcal{F}}$ and $u_{\mathcal{F}}$ represent respectively the mass density and the velocity of the fluid, and the pressure of the fluid is denoted by $p_{\mathcal{F}}$.

We assume that the flow is in the barotropic regime and we focus on the isentropic case where the relation between $p_{\mathcal{F}}$ and ${\rho }_{\mathcal{F}}$ is given by the constitutive law:

$$p_{\mathcal{F}}=a_{\mathcal{F}}{\rho }_{\mathcal{F}}^{\gamma },$$

with $a_{\mathcal{F}}>0$ and the adiabatic constant $\gamma >\frac{3}{2}$, which is a necessary assumption for the existence of a weak solution of compressible fluids (see for example [9]).

As it is common, we set

$$\mathbb{T}(u_{\mathcal{F}})=2{\mu }_{\mathcal{F}}\mathbb{D}(u_{\mathcal{F}})+{\lambda }_{\mathcal{F}}\mathrm{div}{u}_{\mathcal{F}}\mathbb{I},$$

where $\mathbb{D}(u_{\mathcal{F}})=\frac{1}{2}(\mathrm{\nabla }{u}_{\mathcal{F}}+\mathrm{\nabla }{u}_{\mathcal{F}}^{\top })$ denotes the symmetric part of the velocity gradient, $\mathrm{\nabla }{u}_{\mathcal{F}}^{\top }$ is the transpose of $\mathrm{\nabla }{u}_{\mathcal{F}}$, ${\lambda }_{\mathcal{F}}$ and ${\mu }_{\mathcal{F}}$ are the viscosity coefficients satisfying

$${\mu }_{\mathcal{F}}>0,3{\lambda }_{\mathcal{F}}+2{\mu }_{\mathcal{F}}⩾0.$$

The evolution of this fluid-structure system can be described by the following equations

$$\frac{\partial {\rho }_{\mathcal{F}}}{\partial t}+\mathrm{div}({\rho }_{\mathcal{F}}{u}_{\mathcal{F}})=0,t\in (0,T),x\in \mathcal{F}(t),$$

$$\frac{\partial ({\rho }_{\mathcal{F}}{u}_{\mathcal{F}})}{\partial t}+\mathrm{div}({\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\otimes u_{\mathcal{F}})-\mathrm{div}\mathbb{T}(u_{\mathcal{F}})+\mathrm{\nabla }{p}_{\mathcal{F}}={\rho }_{\mathcal{F}}{g}_{\mathcal{F}},t\in (0,T),x\in \mathcal{F}(t),$$

$$m{h}^{″}(t)=-\underset{\partial \mathcal{S}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu d\mathrm{\Gamma }+\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{g}_{\mathcal{S}}dx,\text{in}(0,T),$$

$$\begin{gathered} {(J\omega )}^{\prime }(t)=-\underset{\partial \mathcal{S}(t)}{\int }(x-h(t))\times (\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu d\mathrm{\Gamma } \\ +\underset{\mathcal{S}(t)}{\int }(x-h(t))\times {\rho }_{\mathcal{S}}{g}_{\mathcal{S}}dx,\text{in}(0,T), \end{gathered}$$

the boundary conditions

$$u_{\mathcal{F}}\cdot \nu =u_{\mathcal{S}}\cdot \nu ,\text{for}t\in (0,T),x\in \partial \mathcal{S}(t),$$

$$(\mathbb{T}(u_{\mathcal{F}})\nu )\times \nu =-\alpha (u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu ,\text{for}t\in (0,T),x\in \partial \mathcal{S}(t),$$

$$u_{\mathcal{F}}\cdot \nu =0,\text{on}(t,x)\in (0,T)\times \partial \mathrm{\Omega },$$

$$(\mathbb{T}(u_{\mathcal{F}})\nu )\times \nu =-\alpha (u_{\mathcal{F}}\times \nu ),\text{on}(t,x)\in (0,T)\times \partial \mathrm{\Omega },$$

and the initial conditions

$${\rho }_{\mathcal{F}}(0,x)={\rho }_{{\mathcal{F}}_0}(x),({\rho }_{\mathcal{F}}{u}_{\mathcal{F}})(0,x)=q_{{\mathcal{F}}_0}(x),\forall x\in {\mathcal{F}}_0,$$

$$h(0)=0,h^{\prime }(0)={\ell }_0,\omega (0)={\omega }_0.$$

The Eulerian velocity $u_{\mathcal{S}}(t,x)$ at each point $x\in \mathcal{S}(t)$ of the rigid body is given by

$$u_{\mathcal{S}}(t,x)=h^{\prime }(t)+\omega (t)\times (x-h(t)),$$

where $h(t)$ is the position of the centre of mass and $h^{\prime }(t)$, $\omega (t)$ are the translational and angular velocities of the rigid body.

The solid domain at time t is given by

$$\mathcal{S}(t)=\{h(t)+\mathbb{O}(t)x|x\in {\mathcal{S}}_0\},$$

where $\mathbb{O}(t)\in SO(3)$ is associated to the rotation of the rigid body:

$${\mathbb{O}}^{\prime }(t){\mathbb{O}}^{\top }(t)x=\omega (t)\times x\forall x\in {\mathbb{R}}^3,\mathbb{O}(0)=\mathbb{I}.$$

Observe that ${\mathbb{O}}^{\prime }{\mathbb{O}}^{\top }$ is skew-symmetric as $\mathbb{O}{\mathbb{O}}^{\top }=\mathbb{I}$. The initial velocity of the rigid body is given by

$$u_{\mathcal{S}}(0,x)=u_{{\mathcal{S}}_0}:={\ell }_0+{\omega }_0\times x,x\in {\mathcal{S}}_0.$$

Here the mass density ${\rho }_{\mathcal{S}}$ of the body satisfies the following transport equation

$$\frac{\partial {\rho }_{\mathcal{S}}}{\partial t}+u_{\mathcal{S}}\cdot \mathrm{\nabla }{\rho }_{\mathcal{S}}=0,t\in (0,T),x\in \mathcal{S}(t),{\rho }_{\mathcal{S}}(0,x)={\rho }_{{\mathcal{S}}_0}(x),\forall x\in {\mathcal{S}}_0.$$

Moreover, m is the mass of the solid and $J(t)$ is the moment of inertia tensor of the solid calculated with respect to $h(t)$. We express $h(t)$, m and $J(t)$ in the following way:

$$m=\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}dx,$$

$$h(t)=\frac{1}{m}\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}xdx,$$

$$J(t)=\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}[|x-h(t){|}^2\mathbb{I}-(x-h(t))\otimes (x-h(t))]dx.$$

In the remainder of this introduction, we present the weak formulation of the system, discuss our main result regarding the existence of weak solutions and put it in a larger perspective.

We derive a weak formulation with the help of multiplication by appropriate test functions and integration by parts along with the application of the boundary conditions. Due to the presence of the Navier-slip boundary condition, the test functions will be discontinuous across the fluid-solid interface. We introduce the set of rigid velocity fields:

$$\mathcal{R}(\mathrm{\Omega })=\{\zeta :\mathrm{\Omega }\to {\mathbb{R}}^3|\text{There exist }V,r,a\in {\mathbb{R}}^3\text{ such that }\zeta (x)=V+r\times (x-a)\text{ for any }x\in \mathrm{\Omega }\}.$$

For any $T>0$, we define the test function space $V_T$ as follows:

$$V_T=\left\{\begin{array}{cc}\hfill & \varphi \in C([0,T];L^2(\mathrm{\Omega }))\text{ such that there exist }{\varphi }_{\mathcal{F}}\in \mathcal{D}([0,T);\mathcal{D}(\bar{\mathrm{\Omega }})),{\varphi }_{\mathcal{S}}\in \mathcal{D}([0,T);\mathcal{R}(\mathrm{\Omega }))\hfill \\ \hfill & \text{satisfying }\varphi (t,\cdot )={\varphi }_{\mathcal{F}}(t,\cdot )\text{ on }\mathcal{F}(t),\varphi (t,\cdot )={\varphi }_{\mathcal{S}}(t,\cdot )\text{ on }\mathcal{S}(t)\text{ with }\hfill \\ \hfill & {\varphi }_{\mathcal{F}}(t,\cdot )\cdot \nu ={\varphi }_{\mathcal{S}}(t,\cdot )\cdot \nu \text{ on }\partial \mathcal{S}(t),{\varphi }_{\mathcal{F}}(t,\cdot )\cdot \nu =0\text{ on }\partial \mathrm{\Omega }\text{ for all }t\in [0,T]\hfill \end{array}\right\},$$

where $\mathcal{D}$ denotes the set of all infinitely differentiable functions that have compact support. We multiply equation (1.3) by a test function $\varphi \in V_T$ and integrate over $\mathcal{F}(t)$ to obtain

$$\frac{d}{dt}\underset{\mathcal{F}(t)}{\int }{\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\cdot {\varphi }_{\mathcal{F}}-\underset{\mathcal{F}(t)}{\int }{\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\cdot \frac{\partial }{\partial t}{\varphi }_{\mathcal{F}}-\underset{\mathcal{F}(t)}{\int }({\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\otimes u_{\mathcal{F}}):\mathrm{\nabla }{\varphi }_{\mathcal{F}}+\underset{\mathcal{F}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I}):\mathbb{D}({\varphi }_{\mathcal{F}})=\underset{\partial \mathrm{\Omega }}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{F}}+\underset{\partial \mathcal{S}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{F}}+\underset{\mathcal{F}(t)}{\int }{\rho }_{\mathcal{F}}{g}_{\mathcal{F}}\cdot {\varphi }_{\mathcal{F}}.$$

We use the identity $(A\times B)\cdot (C\times D)=(A\cdot C)(B\cdot D)-(B\cdot C)(A\cdot D)$ to have

$$\mathbb{T}(u_{\mathcal{F}})\nu \cdot {\varphi }_{\mathcal{F}}=[\mathbb{T}(u_{\mathcal{F}})\nu \cdot \nu ]({\varphi }_{\mathcal{F}}\cdot \nu )+[\mathbb{T}(u_{\mathcal{F}})\nu \times \nu ]\cdot ({\varphi }_{\mathcal{F}}\times \nu ),$$

$$\mathbb{T}(u_{\mathcal{F}})\nu \cdot {\varphi }_{\mathcal{S}}=[\mathbb{T}(u_{\mathcal{F}})\nu \cdot \nu ]({\varphi }_{\mathcal{S}}\cdot \nu )+[\mathbb{T}(u_{\mathcal{F}})\nu \times \nu ]\cdot ({\varphi }_{\mathcal{S}}\times \nu ).$$

Now by using the definition of $V_T$ and the boundary conditions (1.6)–(1.9), we get

$$\underset{\partial \mathrm{\Omega }}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{F}}=-\alpha \underset{\partial \mathrm{\Omega }}{\int }(u_{\mathcal{F}}\times \nu )\cdot ({\varphi }_{\mathcal{F}}\times \nu ),$$

$$\underset{\partial \mathcal{S}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{F}}=-\alpha \underset{\partial \mathcal{S}(t)}{\int }[(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu ]\cdot [({\varphi }_{\mathcal{F}}-{\varphi }_{\mathcal{S}})\times \nu ]+\underset{\partial \mathcal{S}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{S}}.$$

Using the rigid body equations (1.4)–(1.5), equation (1.14) and Reynolds' transport theorem, we obtain

$$\begin{gathered} \underset{\partial \mathcal{S}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I})\nu \cdot {\varphi }_{\mathcal{S}} \\ =-\frac{d}{dt}\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{u}_{\mathcal{S}}\cdot {\varphi }_{\mathcal{S}}+\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{u}_{\mathcal{S}}\cdot \frac{\partial }{\partial t}{\varphi }_{\mathcal{S}}+\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{g}_{\mathcal{S}}\cdot {\varphi }_{\mathcal{S}}. \end{gathered}$$

Thus by combining the above relations (1.20)–(1.23) and then integrating from 0 to T, we have

$$-\underset{0}{\overset{T}{\int }}\underset{\mathcal{F}(t)}{\int }{\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\cdot \frac{\partial }{\partial t}{\varphi }_{\mathcal{F}}-\underset{0}{\overset{T}{\int }}\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{u}_{\mathcal{S}}\cdot \frac{\partial }{\partial t}{\varphi }_{\mathcal{S}}-\underset{0}{\overset{T}{\int }}\underset{\mathcal{F}(t)}{\int }({\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\otimes u_{\mathcal{F}}):\mathrm{\nabla }{\varphi }_{\mathcal{F}}+\underset{0}{\overset{T}{\int }}\underset{\mathcal{F}(t)}{\int }(\mathbb{T}(u_{\mathcal{F}})-p_{\mathcal{F}}\mathbb{I}):\mathbb{D}({\varphi }_{\mathcal{F}})+\alpha \underset{0}{\overset{T}{\int }}\underset{\partial \mathrm{\Omega }}{\int }(u_{\mathcal{F}}\times \nu )\cdot ({\varphi }_{\mathcal{F}}\times \nu )+\alpha \underset{0}{\overset{T}{\int }}\underset{\partial \mathcal{S}(t)}{\int }[(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu ]\cdot [({\varphi }_{\mathcal{F}}-{\varphi }_{\mathcal{S}})\times \nu ]=\underset{0}{\overset{T}{\int }}\underset{\mathcal{F}(t)}{\int }{\rho }_{\mathcal{F}}{g}_{\mathcal{F}}\cdot {\varphi }_{\mathcal{F}}+\underset{0}{\overset{T}{\int }}\underset{\mathcal{S}(t)}{\int }{\rho }_{\mathcal{S}}{g}_{\mathcal{S}}\cdot {\varphi }_{\mathcal{S}}+\underset{\mathcal{F}(0)}{\int }({\rho }_{\mathcal{F}}{u}_{\mathcal{F}}\cdot {\varphi }_{\mathcal{F}})(0)+\underset{\mathcal{S}(0)}{\int }({\rho }_{\mathcal{S}}{u}_{\mathcal{S}}\cdot {\varphi }_{\mathcal{S}})(0).$$

Definition 1.1

Let $T>0$, and let Ω and ${\mathcal{S}}_0⋐\mathrm{\Omega }$ be two regular bounded domains of ${\mathbb{R}}^3$. A triplet $(\mathcal{S},\rho ,u)$ is a bounded energy weak solution to system (1.2)–(1.11) if the following holds:

• $\mathcal{S}(t)⋐\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^3$ for all $t\in [0,T)$ such that

  $${\chi }_{\mathcal{S}}(t,x):={\mathbb{1}}_{\mathcal{S}(t)}(x)\in L^{\infty }((0,T)\times \mathrm{\Omega }).$$

• u belongs to the following space

  $$U_T=\left\{\begin{array}{cc}\hfill & u\in L^2(0,T;L^2(\mathrm{\Omega }))\text{ such that there exist }{u}_{\mathcal{F}}\in L^2(0,T;H^1(\mathrm{\Omega })),u_{\mathcal{S}}\in L^2(0,T;\mathcal{R})\hfill \\ \hfill & \text{satisfying }u(t,\cdot )=u_{\mathcal{F}}(t,\cdot )\text{ on }\mathcal{F}(t),u(t,\cdot )=u_{\mathcal{S}}(t,\cdot )\text{ on }\mathcal{S}(t)\text{ with }\hfill \\ \hfill & u_{\mathcal{F}}(t,\cdot )\cdot \nu =u_{\mathcal{S}}(t,\cdot )\cdot \nu \text{ on }\partial \mathcal{S}(t),u_{\mathcal{F}}\cdot \nu =0\text{ on }\partial \mathrm{\Omega }\text{ for a.e }t\in [0,T]\hfill \end{array}\right\}.$$

• $\rho ⩾0$, $\rho \in L^{\infty }(0,T;L^{\gamma }(\mathrm{\Omega }))$ with $\gamma >3/2$, $\rho |u{|}^2\in L^{\infty }(0,T;L^1(\mathrm{\Omega }))$, where

  $$\rho =(1-{\mathbb{1}}_{\mathcal{S}}){\rho }_{\mathcal{F}}+{\mathbb{1}}_{\mathcal{S}}{\rho }_{\mathcal{S}},u=(1-{\mathbb{1}}_{\mathcal{S}})u_{\mathcal{F}}+{\mathbb{1}}_{\mathcal{S}}{u}_{\mathcal{S}}.$$

• The continuity equation is satisfied in the weak sense, i.e.

  $$\frac{\partial {\rho }_{\mathcal{F}}}{\partial t}+\mathrm{div}({\rho }_{\mathcal{F}}{u}_{\mathcal{F}})=0\text{ in }{\mathcal{D}}^{\prime }([0,T)\times \mathrm{\Omega }),{\rho }_{\mathcal{F}}(0,x)={\rho }_{{\mathcal{F}}_0}(x),x\in \mathrm{\Omega }.$$

  Also, a renormalized continuity equation holds in a weak sense, i.e.

  $${\partial }_{t}b({\rho }_{\mathcal{F}})+\mathrm{div}(b({\rho }_{\mathcal{F}})u_{\mathcal{F}})+(b^{\prime }({\rho }_{\mathcal{F}})-b({\rho }_{\mathcal{F}}))\mathrm{div}{u}_{\mathcal{F}}=0\text{ in }{\mathcal{D}}^{\prime }([0,T)\times \mathrm{\Omega }),$$

  for any $b\in C([0,\infty ))\cap C^1((0,\infty ))$ satisfying

  $$|b^{\prime }(z)|⩽c{z}^{-{\kappa }_0},z\in (0,1],{\kappa }_0<1,|b^{\prime }(z)|⩽c{z}^{{\kappa }_1},z⩾1,-1<{\kappa }_1<\infty .$$

• The transport of $\mathcal{S}$ by the rigid vector field $u_{\mathcal{S}}$ holds (in the weak sense)

  $$\frac{\partial {\chi }_{\mathcal{S}}}{\partial t}+\mathrm{div}(u_{\mathcal{S}}{\chi }_{\mathcal{S}})=0\text{ in }(0,T)\times \mathrm{\Omega },{\chi }_{\mathcal{S}}(0,x)={\mathbb{1}}_{{\mathcal{S}}_0}(x),x\in \mathrm{\Omega }.$$

• The density ${\rho }_{\mathcal{S}}$ of the rigid body $\mathcal{S}$ satisfies (in the weak sense)

  $$\frac{\partial {\rho }_{\mathcal{S}}}{\partial t}+\mathrm{div}(u_{\mathcal{S}}{\rho }_{\mathcal{S}})=0\text{ in }(0,T)\times \mathrm{\Omega },{\rho }_{\mathcal{S}}(0,x)={\rho }_{{\mathcal{S}}_0}(x),x\in \mathrm{\Omega }.$$

• Balance of linear momentum holds in a weak sense, i.e. for all $\varphi \in V_T$ the relation (1.24) holds.

• The following energy inequality holds for almost every $t\in (0,T)$:

  $$E(t)+\underset{0}{\overset{t}{\int }}\underset{\mathcal{F}(\tau )}{\int }(2{\mu }_{\mathcal{F}}|\mathbb{D}(u_{\mathcal{F}}){|}^2+{\lambda }_{\mathcal{F}}|\mathrm{div}{u}_{\mathcal{F}}{|}^2)+\alpha \underset{0}{\overset{t}{\int }}\underset{\partial \mathrm{\Omega }}{\int }|u_{\mathcal{F}}\times \nu {|}^2+\alpha \underset{0}{\overset{t}{\int }}\underset{\partial \mathcal{S}(\tau )}{\int }|(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu {|}^2⩽\underset{0}{\overset{t}{\int }}\underset{\mathcal{F}(\tau )}{\int }{\rho }_{\mathcal{F}}{g}_{\mathcal{F}}\cdot u_{\mathcal{F}}+\underset{0}{\overset{t}{\int }}\underset{\mathcal{S}(\tau )}{\int }{\rho }_{\mathcal{S}}{g}_{\mathcal{S}}\cdot u_{\mathcal{S}}+E_0,$$

  where $E(t)$ and $E_0$ are given by

  $$E(t)=\underset{\mathcal{F}(t)}{\int }\frac{1}{2}{\rho }_{\mathcal{F}}|u_{\mathcal{F}}{|}^2+\underset{\mathcal{S}(t)}{\int }\frac{1}{2}{\rho }_{\mathcal{S}}|u_{\mathcal{S}}{|}^2+\underset{\mathcal{F}(t)}{\int }\frac{a_{\mathcal{F}}}{\gamma -1}{\rho }_{\mathcal{F}}^{\gamma },E_0=\underset{{\mathcal{F}}_0}{\int }\frac{1}{2}\frac{|q_{{\mathcal{F}}_0}{|}^2}{{\rho }_{{\mathcal{F}}_0}}+\underset{{\mathcal{S}}_0}{\int }\frac{1}{2}{\rho }_{{\mathcal{S}}_0}|u_{{\mathcal{S}}_0}{|}^2+\underset{{\mathcal{F}}_0}{\int }\frac{a_{\mathcal{F}}}{\gamma -1}{\rho }_{{\mathcal{F}}_0}^{\gamma }.$$

Remark 1.2

We stress that in the definition of the set $U_T$ (in Definition 1.1) the function $u_{\mathcal{F}}$ on Ω is a regular extension of the velocity field $u_{\mathcal{F}}$ from $\mathcal{F}(t)$ to Ω, see (5.10)–(5.11). Correspondingly, $u_{\mathcal{S}}\in \mathcal{R}$ denotes a rigid extension from $\mathcal{S}(t)$ to Ω as in (1.12). Moreover, by the density ${\rho }_{\mathcal{F}}$ in (1.26), we mean an extended fluid density ${\rho }_{\mathcal{F}}$ from $\mathcal{F}(t)$ to Ω by zero, see (5.16)–(5.17). Correspondingly, ${\rho }_{\mathcal{S}}$ refers to an extended solid density from $\mathcal{S}(t)$ to Ω by zero.

Remark 1.3

In (1.26), the initial fluid density ${\rho }_{{\mathcal{F}}_0}$ on Ω represents a zero extension of ${\rho }_{{\mathcal{F}}_0}$ (defined in (1.10)) from ${\mathcal{F}}_0$ to Ω. Correspondingly, ${\rho }_{{\mathcal{S}}_0}$ in equation (1.30) stands for an extended initial solid density (defined in (1.14)) from ${\mathcal{S}}_0$ to Ω by zero. Obviously, $q_{{\mathcal{F}}_0}$ refers to an extended initial momentum from ${\mathcal{F}}_0$ to Ω by zero and $u_{{\mathcal{S}}_0}\in \mathcal{R}$ denotes a rigid extension from ${\mathcal{S}}_0$ to Ω as in (1.13).

Remark 1.4

We note that our continuity equation (1.26) is different from the corresponding one in [7]. We have to work with $u_{\mathcal{F}}$ instead of u because of the Navier boundary condition. The reason is that we need the $H^1(\mathrm{\Omega })$ regularity of the velocity in order to achieve the validity of the continuity equation in Ω. Observe that $u\in L^2(0,T;L^2(\mathrm{\Omega }))$ but the extended fluid velocity has better regularity, in particular, $u_{\mathcal{F}}\in L^2(0,T;H^1(\mathrm{\Omega }))$, see (5.10)–(5.11).

Remark 1.5

In the weak formulation (1.24), we need to distinguish between the fluid velocity $u_{\mathcal{F}}$ and the solid velocity $u_{\mathcal{S}}$. Due to the presence of the discontinuities in the tangential components of u and ϕ, neither ${\partial }_t\varphi$ nor $\mathbb{D}(u)$, $\mathbb{D}(\varphi )$ belong to $L^2(\mathrm{\Omega })$. That's why it is not possible to write (1.24) in a global and condensed form (i.e. integrals over Ω).

Remark 1.6

Let us mention that in the whole paper we assume the regularity of domains Ω and ${\mathcal{S}}_0$ as $C^{2+\kappa }$, $\kappa >0$. However, we expect that our assumption on the regularity of the domain can be relaxed to a less regular domain like in the work of Kukučka [26].

The mathematical analysis of systems describing the motion of a rigid body in a viscous incompressible fluid is nowadays well developed. The proof of existence of weak solutions until a first collision can be found in several papers, see [3], [4], [18], [24], [33]. Later, the possibility of collisions in the case of a weak solution was included, see [8], [32]. Moreover, it was shown that under Dirichlet boundary conditions collisions cannot occur, which is paradoxical with respect to real situations; for details see [20], [22], [23]. Neustupa and Penel showed that under a prescribed motion of the rigid body and under Navier-type of boundary conditions collision can occur [29]. After that Gérard-Varet and Hillairet showed that to construct collisions one needs to assume less regularity of the domain or different boundary conditions, see e.g. [15], [16], [17]. In the case of very high viscosity, under the assumption that rigid bodies are not touching each other or not touching the boundary at the initial time, it was shown that collisions cannot occur in finite time, see [10]. For an introduction we refer to the problem of a fluid coupled with a rigid body in the work by Galdi, see [13]. Let us also mention results on strong solutions, see e.g. [14], [34], [35].

A few results are available on the motion of a rigid structure in a compressible fluid with Dirichlet boundary conditions. The existence of strong solutions in the $L^2$-framework for small data up to a collision was shown in [1], [31]. The existence of strong solutions in the $L^p$ setting based on $\mathcal{R}$-bounded operators was applied in the barotropic case [21] and in the full system [19].

The existence of a weak solution, also up to a collision but without smallness assumptions, was shown in [5]. Generalization of this result allowing collisions was given in [7]. The weak-strong uniqueness of a compressible fluid with a rigid body can be found in [25]. Existence of weak solutions in the case of Navier boundary conditions is not available yet; it is the topic of this article.

For many years, the no-slip boundary condition has been the most widely used given its success in reproducing the standard velocity profiles for incompressible/compressible viscous fluids. Although the no-slip hypothesis seems to be in good agreement with experiments, it leads to certain rather surprising conclusions. As we have already mentioned, the most striking one being the absence of collisions of rigid objects immersed in a linearly viscous fluid [20], [22].

The so-called Navier boundary conditions, which allow for slip, offer more freedom and are likely to provide a physically acceptable solution at least to some of the paradoxical phenomena resulting from the no-slip boundary condition, see, e.g. Moffat [28]. Mathematically, the behavior of the tangential component ${[u]}_{tan}$ is a delicate issue.

The main result of our paper (Theorem 1.7) asserts the local-in-time existence of a weak solution for the system involving the motion of a rigid body in a compressible fluid in the case of Navier boundary conditions at the interface with the solid and at the outer boundary. It is the first result in the context of rigid body-compressible fluid interaction in the case of Navier type of boundary conditions. Let us mention that the main difficulty which arises in our problem is the jump in the velocity through the interface boundary between the rigid body and the compressible fluid. This difficulty cannot be resolved by the approach introduced in the work of Desjardins, Esteban [5], or Feireisl [7] since they consider the velocity field continuous through the interface. Moreover, the techniques in the works by Gérard-Varet, Hillairet [16] and Chemetov, Nečasová [2] cannot be used directly as they are in the incompressible framework. Our weak solutions have to satisfy the jump of the velocity field through the interface boundary.

Our idea is to introduce a novel approximate scheme which combines the theory of compressible fluids introduced by P. L. Lions [27] and then developed by Feireisl [9] to get the strong convergence of the density (renormalized continuity equations, effective viscous flux, artificial pressure) together with ideas from Gérard-Varet, Hillairet [16] and Chemetov, Nečasová [2] concerning a penalization of the jump. We remark that such type of difficulties do not arise for the existence of weak solutions of compressible fluids without a rigid body neither for Dirichlet nor for Navier type of boundary conditions.

We emphasize the main issues that arise in the analysis of our system and the novel methodology that we adapt to deal with it:

• It is not possible to define a uniform velocity field as in [5], [7] due to the presence of a discontinuity through the interface of interaction. This is the reason why we introduce the regularized fluid velocity $u_{\mathcal{F}}$ and the solid velocity $u_{\mathcal{S}}$ and why we treat them separately.

• We introduce approximate problems and recover the original problem as a limit of the approximate ones. In fact, we consider several levels of approximations; in each level we ensure that our solution and the test function do not show a jump across the interface so that we can use several available techniques of compressible fluids (without body). In the limit, however, the discontinuity at the interface is recovered. The particular construction of the test functions is a delicate and crucial issue in our proof of Proposition 5.1.

• Recovering the velocity fields for the solid and fluid parts separately is also a challenging issue. We introduce a penalization in such a way that, in the last stage of the limiting process, this term allows us to recover the rigid velocity of the solid, see (5.8)–(5.9). The introduction of an appropriate extension operator helps us to recover the fluid velocity, see (5.12)–(5.13).

• Since we consider the compressible case, our penalization with parameter $\delta >0$, see (2.3), is different from the penalization for the incompressible fluid in [16].

• Due to the Navier-slip boundary condition, no $H^1$ bound on the velocity on the whole domain is possible. We can only obtain the $H^1$ regularity of the extended velocities of the fluid and solid parts separately. We have introduced an artificial viscosity that vanishes asymptotically on the solid part so that we can capture the $H^1$ regularity for the fluid part (see step 1 of the proof of Theorem 1.7 in Section 5).

• We have already mentioned that the main difference with [16] is that we consider compressible fluid whereas they have considered an incompressible fluid. We have encountered several issues that are present due to the compressible nature of the fluid (vanishing viscosity in the continuity equation, recovering the renormalized continuity equation, identification of the pressure). One important point is to see that passing to the limit as δ tends to zero in the transport for the rigid body is not obvious because our velocity field does not have regularity $L^{\infty }(0,T,L^2(\mathrm{\Omega }))$ as in the incompressible case see e.g. [16] but $L^2(0,T,L^2(\mathrm{\Omega }))$ (here we have $\sqrt{\rho }u\in L^{\infty }(0,T,L^2(\mathrm{\Omega }))$ only). To handle this problem, we apply Proposition 3.5 in the δ-level, see Section 5.

Next we present the main result of our paper.

Theorem 1.7

Let Ω and ${\mathcal{S}}_0⋐\mathit{\Omega }$ be two regular bounded domains of ${\mathbb{R}}^3$. Assume that for some $\sigma >0$

$$\mathrm{dist}({\mathcal{S}}_0,\partial \mathit{\Omega })>2\sigma .$$

Let $g_{\mathcal{F}}$, $g_{\mathcal{S}}\in L^{\infty }((0,T)\times \mathit{\Omega })$ and the pressure $p_{\mathcal{F}}$ be determined by (1.1) with $\gamma >3/2$. Assume that the initial data (defined in the sense of Remark 1.3) satisfy

$${\rho }_{{\mathcal{F}}_0}\in L^{\gamma }(\mathit{\Omega }),{\rho }_{{\mathcal{F}}_0}⩾0\mathit{\text{a.e. in}}\mathit{\Omega },{\rho }_{{\mathcal{S}}_0}\in L^{\infty }(\mathit{\Omega }),{\rho }_{{\mathcal{S}}_0}>0\mathit{\text{a.e. in}}{\mathcal{S}}_0,$$

$$q_{{\mathcal{F}}_0}\in L^{\frac{2\gamma }{\gamma +1}}(\mathit{\Omega }),q_{{\mathcal{F}}_0}{\mathbb{1}}_{\{{\rho }_{{\mathcal{F}}_0}=0\}}=0\mathit{\text{a.e. in}}\mathit{\Omega },\frac{|q_{{\mathcal{F}}_0}{|}^2}{{\rho }_{{\mathcal{F}}_0}}{\mathbb{1}}_{\{{\rho }_{{\mathcal{F}}_0}>0\}}\in L^1(\mathit{\Omega }),$$

$$u_{{\mathcal{S}}_0}={\ell }_0+{\omega }_0\times x\forall x\in \mathit{\Omega }\mathit{\text{with}}{\ell }_0,{\omega }_0\in {\mathbb{R}}^3.$$

Then there exists $T>0$ (depending only on ${\rho }_{{\mathcal{F}}_0}$, ${\rho }_{{\mathcal{S}}_0}$, $q_{{\mathcal{F}}_0}$, $u_{{\mathcal{S}}_0}$, $g_{\mathcal{F}}$, $g_{\mathcal{S}}$, $\mathrm{dist}({\mathcal{S}}_0,\partial \mathit{\Omega })$) such that a finite energy weak solution to (1.2)–(1.11) exists on $[0,T)$. Moreover,

$$\mathcal{S}(t)⋐\mathit{\Omega },\mathrm{dist}(\mathcal{S}(t),\partial \mathit{\Omega })⩾\frac{3\sigma }{2},\forall t\in [0,T].$$

Remark 1.8

We want to mention that in the absence of rigid body, the existence of at least one bounded energy weak solution for compressible fluid with Navier-slip on the outer boundary has been stated in [30, Theorem 7.69]. This is the same class of regularity as the fluid part in our main result.

Remark 1.9

We can establish the existence result Theorem 1.7 in the case when the frictional coefficients for the outer boundary and the moving solid are not the same. Precisely, we can replace (1.7) and (1.9) by the following boundary conditions:

$$(\mathbb{T}(u_{\mathcal{F}})\nu )\times \nu =-{\alpha }_1(u_{\mathcal{F}}-u_{\mathcal{S}})\times \nu ,\text{for}t\in (0,T),x\in \partial \mathcal{S}(t),(\mathbb{T}(u_{\mathcal{F}})\nu )\times \nu =-{\alpha }_2(u_{\mathcal{F}}\times \nu ),\text{on}(t,x)\in (0,T)\times \partial \mathrm{\Omega },$$

where ${\alpha }_1>0$, ${\alpha }_2>0$ are the frictional coefficients for the rigid body and the outer boundary respectively.

The outline of the paper is as follows. We introduce three levels of approximation schemes in Section 2. In Section 3, we describe some results on the transport equation, which are needed in all the levels of approximation. The existence results of approximate solutions have been proved in Section 4. Section 4.1 and Section 4.2 are dedicated to the construction and convergence analysis of the Faedo-Galerkin scheme associated to the finite dimensional approximation level. We discuss the limiting system associated to the vanishing viscosity in Section 4.3. Section 5 is devoted to the main part: we derive the limit as the parameter δ tends to zero.