On Vector Fields Describing the 2d Motion of a Rigid Body in a Viscous Fluid and Applications

Abstract

We present some properties of functions in suitable Sobolev spaces which arise naturally in the study of the motion of a rigid body in compressible and incompressible fluid. We relax the regularity assumption of the rigid body by allowing its boundary to be Lipschitz. In the case of a smooth rigid body we obtain a new estimate on the angular velocity. Our results extend and complement related results by V. Starovoitov and moreover we show that they are optimal. As an application we present an example where the rigid body collides with the boundary with non zero speed. Finally, we present a new non collision result concerning a smooth rotating body approaching the boundary, without assuming any special geometry on either the body or the container.

Appendix

We recall that

$$\begin{aligned} \beta = \frac{1+ 2 \alpha }{p(1+ \alpha )} \left( p - \frac{2+ \alpha }{1+ 2 \alpha } \right) . \end{aligned}$$

We then have

Lemma 7.1

Let $p \ge 1$ and $0 \le \alpha \le 1$. Then, for $\varvec{u}$ as defined in (5.2) there holds

$$\begin{aligned} \int _{ I \! \! R_{+}^2} |\varvec{u}|^p dx_1 dx_2 \le \left\{ \begin{array}{ll} C |\dot{h}|^p , &{} \quad \alpha p < 2+\alpha \ , \\ C |\dot{h}|^{p} \, |\ln h|, &{}\quad \alpha p = 2+\alpha , \\ C |\dot{h}|^p h^{ \frac{ 2+\alpha - \alpha p}{1+ \alpha }} , ~~~~ &{}\quad \alpha p > 2+\alpha , \\ \end{array} \right. \end{aligned}$$

(7.1)

$$\begin{aligned} \int _{ I \! \! R_{+}^2} |\nabla \varvec{u}|^p dx_1 dx_2 \le \left\{ \begin{array}{ll} C |\dot{h}|^{p} , ~~~~ &{}\quad p <\frac{2+\alpha }{1+ 2\alpha }, \\ C |\dot{h}|^{\frac{2+\alpha }{1+ 2\alpha }} \, |\ln h|, &{}\quad p = \frac{2+\alpha }{1+ 2\alpha } ,\ \\ C |\dot{h}|^{p} h^{-\beta p }, &{} \quad p>\frac{2+\alpha }{1+ 2\alpha } \ .\\ \end{array} \right. \end{aligned}$$

(7.2)

In addition, the following local estimates hold true for $p \ge 1$

$$\begin{aligned} \int _{\mathcal {G}_{h, \sigma _0}} |\nabla {{\varvec{u}}} |^{p} dx_1 dx_2\le&{} C |\dot{h}|^{p} h^{-\beta p }, ~~~~0<\alpha \le 1, \end{aligned}$$

(7.3)

$$\begin{aligned} \int _{\Pi _{h}} |\nabla {{\varvec{u}}} |^{p} dx_1 dx_2\le&{} C |\dot{h}|^{p} h^{2- p },~~~~\alpha =0, \end{aligned}$$

(7.4)

where $\mathcal {G}_{h, \sigma _0}$ is defined in (4.12) and $\Pi _{h}$ in the proof of Theorem 4.3.

Proof

For

$$\begin{aligned} \phi =\phi \left( \frac{x_2}{k x_1^{1+\alpha }+ h(t)}\right) , \end{aligned}$$

we have that

$$\begin{aligned} \nabla \varvec{u}= & {} \Phi \nabla \nabla ^{\perp } \phi + \nabla \Phi \nabla ^{\perp } \phi + \nabla \phi \nabla ^{\perp } \Phi \ + \cdots \nonumber \\=: & {} I_1 + I_2 + I_3 + \cdots \end{aligned}$$

(7.5)

A straightforward calculation shows that for $i,j=1,2$

$$\begin{aligned} \frac{\partial \phi }{\partial x_i}= & {} 0, ~~~~~~~~~~~~~~~~ x_2 \ge k x_1^{1+\alpha }+ h , \\ \Big | \frac{\partial \phi }{\partial x_i} \Big |\le & {} \frac{C}{k x_1^{1+\alpha } +h}, ~~~~~~0 \le x_2 \le k x_1^{1+\alpha }+ h , \\ \Big | \frac{\partial ^2 \phi }{\partial x_ix_j} \Big |\le & {} \frac{C(1+ \alpha h x_1^{\alpha -1})}{(k x_1^{1+\alpha } + h)^2} ,~~~~0 \le x_2 \le k x_1^{1+\alpha }+ h. \end{aligned}$$

We then compute, for $\gamma >0$ small enough but fixed

$$\begin{aligned} I_1= & {} \int _{ I \! \! R_{+}^2} |\Phi |^p | \nabla \nabla ^{\perp } \phi |^p dx_1 dx_2 \\\le & {} 2 \int _{0}^{\gamma } \int _{0}^{k x_1^{1+\alpha } +h} |\Phi |^p | \nabla \nabla ^{\perp } \phi |^p dx_2 dx_1 + 2 \int _{\gamma }^{2 \rho } \int _{0}^{k x_1^{1+\alpha } +h} |\Phi |^p | \nabla \nabla ^{\perp } \phi |^p dx_2 dx_1 \\\le & {} C |\dot{h}|^p \left( \int _{0}^{\gamma } \int _{0}^{k x_1^{1+\alpha } +h} \frac{x_1^p \, dx_2 dx_1}{(k x_1^{1+\alpha } + h)^{2p} } + \int _{0}^{\gamma } \int _{0}^{k x_1^{1+\alpha } +h} \frac{ \alpha ^p x_1^{\alpha p} \, dx_2 dx_1}{(k x_1^{1+\alpha } + h)^{p} } + O_h(1) \right) \\\le & {} C |\dot{h}|^p \left( \int _{0}^{\gamma } \frac{x_1^p \, dx_1}{(k x_1^{1+\alpha } + h)^{2p-1} } + \alpha ^p \int _{0}^{\gamma } \frac{ x_1^{\alpha p} \, dx_1}{(k x_1^{1+\alpha } + h)^{p-1} } + O_h(1) \right) \end{aligned}$$

To estimate the integrals above we use the fact that for $0 \le \alpha \le 1$ and $b, q \in I \! \! R$ we have

$$\begin{aligned} \int _{0}^{\gamma } \frac{x_1^b \, dx_1}{(k x_1^{1+\alpha } + h)^{q} } = \frac{ h^{\frac{b-\alpha }{1+\alpha }-q+1}}{1+\alpha } \int ^{\frac{\gamma ^{1+\alpha }}{h}}_0 \frac{z^{\frac{b-\alpha }{1+\alpha }} \, dz}{(kz+1)^{q}}, ~~~~~\left( z= \frac{x^{1+\alpha }}{h} \right) . \end{aligned}$$

We then get after elementary manipulations

$$\begin{aligned} I_1 \le \left\{ \begin{array}{ll} C |\dot{h}|^p , ~~~~ &{}\quad p <\frac{2+\alpha }{1+ 2\alpha }, \\ C |\dot{h}|^{\frac{2+\alpha }{1+ 2\alpha }} \, |\ln h|, &{} \quad p = \frac{2+\alpha }{1+ 2\alpha } ,\ \\ C |\dot{h}|^p |h|^{-\frac{1+ 2\alpha }{1+ \alpha } \left( p- \frac{2+\alpha }{1+ 2\alpha }\right) }, &{}\quad p>\frac{2+\alpha }{1+ 2\alpha } \ .\\ \end{array} \right. \end{aligned}$$

We similarly calculate for $i=2,3$

$$\begin{aligned} I_i \le \left\{ \begin{array}{ll} C |\dot{h}|^p , ~~~~ &{}\quad p <\frac{2+\alpha }{1+ \alpha }, \\ C |\dot{h}|^{\frac{2+\alpha }{1+ \alpha }} \, |\ln h|, &{}\quad p = \frac{2+\alpha }{1+ \alpha } ,\ \\ C |\dot{h}|^p |h|^{- \left( p- \frac{2+\alpha }{1+ \alpha }\right) }, &{}\quad p>\frac{2+\alpha }{1+ \alpha } \ .\\ \end{array} \right. \end{aligned}$$

Combining the estimates of $I_i$, $i=1,2,3$ and noticing that the omitted terms in (7.5) are not as important we conclude (7.2). Estimate (7.1) is similar and simpler. $\Box$