# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
\begin{split}&\bar{p}(\mathcal{D}_{t}\mathbf{w}+2\omega(\mathbf{e}_{3} \times\mathbf{w}))-\bar{\mu}\
abla^{2}\mathbf{w}+\
abla p_{1}=0,\qquad\
abla\cdot\mathbf{ w}=0\quad\text{in}\cup\mathcal{F}^{\pm},\;\;t>0,\\ &\mathbf{w}\big{|}_{t=0}=\mathbf{w}_{0}\quad\text{in}\cup\mathcal{F}^{\pm },\\ &\big{[}\mathbf{w}\big{]}_{\mathcal{G}^{+}}=0,\quad\big{[}\mathbb{T}( \mathbf{w},p_{1})\mathbf{N}\big{]}_{\mathcal{G}^{+}}+\mathbf{N}\mathcal{B}_{0}^{+}(r)=0 \quad\text{on}\;\mathcal{G}^{+},\\ &\mathbb{T}(\mathbf{w},p_{1})\mathbf{N}\big{|}_{\mathcal{G}^{-}}+\mathbf{N} \mathcal{B}_{0}^{-}(r)=0\quad\text{on}\;\mathcal{G}^{-},\\ &\mathcal{D}_{t}r-\mathbf{w}\cdot\mathbf{N}=0\quad\text{on}\;\mathcal{G}, \qquad r\big{|}_{t=0}=r_{0}\quad\text{on}\;\mathcal{G}.\end{split}  \tag{20}\]  
We assume that the domains \(\mathcal{F}^{\pm}\) are symmetric with respect to \(z_{1},z_{2},z_{3}\) and that the initial data satisfy, due to assumptions (12), (13), orthogonality conditions  
\[ \begin{split}&\int_{\mathcal{G}^{\pm}}r_{0}(z)\,\mathrm{d}\mathcal{ G}=0,\\ &\rho^{-}\int_{\mathcal{G}^{-}}r_{0}(z)z_{j}\,\mathrm{d}\mathcal{ G}+[\bar{p}]\big{|}_{\mathcal{G}^{+}}\int_{\mathcal{G}^{+}}r_{0}(z)z_{j}\, \mathrm{d}\mathcal{G}=0,\quad j=1,2,3,\end{split}  \tag{21}\]  
\[ \begin{split}&\int_{\mathcal{F}}\bar{\rho}\mathbf{w}_{0}(z)\,\mathrm{d}z=0,\\ &\int_{\mathcal{F}}\bar{\rho}\mathbf{w}_{0}(z)\cdot\mathbf{\eta}_{j}(z)\, \mathrm{d}z+\omega\Big{(}\rho^{-}\int_{\mathcal{G}^{-}}r_{0}(z)\mathbf{\eta}_{3}(z )\cdot\mathbf{\eta}_{j}(z)\,\mathrm{d}\mathcal{G}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+[\bar{p}] \big{|}_{\mathcal{G}^{+}}\int_{\mathcal{G}^{+}}r_{0}(z)\mathbf{\eta}_{3}(z)\cdot \mathbf{\eta}_{j}(z)\,\mathrm{d}\mathcal{G}\Big{)}=0.\end{split}  \tag{22}\]  
We put \(\mathcal{Q}_{T}^{\pm}=\mathcal{F}^{\pm}\times(0,T)\), \(G_{T}^{\pm}=\mathcal{G}^{\pm}\times(0,T)\), \(D_{T}=\mathcal{Q}_{T}^{\pm}\cup\mathcal{Q}_{T}^{-}\), \(Q_{T}=\mathcal{Q}_{T}^{+}\cup\mathcal{Q}_{T}^{-}\), \(G_{T}=\mathcal{G}_{T}^{+}\cup\mathcal{G}_{T}^{-}\), \(T\in(0,\infty]\). **Proposition 2**. : _A solution of problem (20) under conditions (21), (22) at the initial instant satisfies (21) and (22) for all \(t>0\)._  
This proposition is proved in the same way as Proposition 2.1 in [10] by virtue of Proposition 3. In view of impulse conservation, the following statement is valid. **Corollary 1**. : _There holds the decomposition_  
\[\mathbf{w}=\mathbf{w}^{\perp}+\sum_{i=1}^{3}d_{i}(r)\mathbf{\eta}_{i}, \tag{23}\]_where \(\mathbf{w}^{\perp}\) means a vector field orthogonal to all the vectors of rigid motion \(\mathbf{\eta}\); i.e.,_  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]  
\[\mathbf{\eta}=\mathbf{e}_{i}\quad\text{or}\quad\mathbf{\eta}(z)=\mathbf{\eta}_{i}(z),\quad i=1,2,3,\quad\text{and}\]  
\[d_{i}(r)=-\frac{\omega}{S_{i}}\big{(}\rho^{-}\int_{\mathcal{G}^{-}}r\mathbf{\eta}_ {3}\cdot\mathbf{\eta}_{i}\,\mathrm{d}\mathcal{G}+[\bar{\rho}]|_{\mathcal{G}^{+}} \int_{\mathcal{G}^{+}}r\mathbf{\eta}_{3}\cdot\mathbf{\eta}_{i}\,\mathrm{d}\mathcal{G} \big{)},\quad S_{i}=\int_{\mathcal{F}}\bar{\rho}|\mathbf{\eta}_{i}|^{2}\,\mathrm{d}z.\]  
**Proposition 3**. : _The following relations hold:_  
\[\mathcal{B}_{0}^{-}\left(\mathbf{\eta}\cdot\mathbf{N}\right) =-\omega^{2}\rho^{-}\mathbf{\eta}\cdot\mathbf{z}^{\prime},\quad z\in \mathcal{G}^{-},\] \[\mathcal{B}_{0}^{+}\left(\mathbf{\eta}\cdot\mathbf{N}\right) =-\omega^{2}[\rho]|_{\mathcal{G}^{+}_{t}}\mathbf{\eta}\cdot\mathbf{z}^{ \prime},\quad z\in\mathcal{G}^{+},\]  
_where \(\mathbf{\eta}\) is an arbitrary vector of rigid motion._  
This statement was proved in [11].