# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
The other conservation laws in (5) take the form  
\[\int_{\Omega_{t}}\rho^{\pm}v(x,t)dx=\int_{\mathcal{F}}\bar{\rho}\boldsymbol{ \mathcal{V}}(x)dx=0, \tag{10}\]  
\[\int_{\Omega_{t}}\rho^{\pm}v(x,t)\cdot\boldsymbol{\eta}_{i}(x)dx=\int_{ \mathcal{F}}\bar{\rho}\boldsymbol{\mathcal{V}}(x)\cdot\boldsymbol{\eta}_{i}(x) dx=\delta_{i}^{3}\beta,\quad i=1,2,3.\]Let us consider the perturbations of the velocity and pressure  
\[\mathbf{v}_{r}(x,t)=\mathbf{v}(x,t)-\mathbf{\mathcal{V}}(x),\quad p_{r}(x,t)=p(x,t)-\mathcal{ P}(x).\]  
We introduce the new coordinates \(\{y_{i}\}\) rotating about the \(x_{3}\)-axis with the angular velocity \(\omega\) and the new unknown functions (\(\tilde{\mathbf{v}}\), \(\tilde{p}\)) by the formulas  
\[x=\mathcal{Z}(\omega t)y,\]  
\[\tilde{\mathbf{v}}(y,t)=\mathcal{Z}^{-1}(\omega t)\mathbf{v}_{r}(\mathcal{Z}(\omega t )y,t),\quad\tilde{p}(y,t)=p_{r}(\mathcal{Z}(\omega t)y,t),\]  
where  
\[\mathcal{Z}(\theta)=\left( \begin{array}{ccc}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{array} \right).\]  
We note that \(\
abla_{x}=\mathcal{Z}\
abla_{y}\), \(\mathcal{Z}^{-1}(\omega t)(\mathbf{\mathcal{V}}\cdot\
abla_{x})\mathbf{v}_{r}=\omega \big{(}\mathbf{\eta}_{3}(x)\cdot\
abla_{x}\big{)}\tilde{\mathbf{v}}(y,t)=\omega\big{(} \mathcal{Z}\eta_{3}(y)\cdot\mathcal{Z}\
abla_{y}\big{)}\tilde{\mathbf{v}}=\omega \big{(}\mathbf{\eta}_{3}(y)\cdot\
abla_{y}\big{)}\tilde{\mathbf{v}}(y,t)=\omega\big{(} y_{2}\frac{\partial\tilde{\mathbf{v}}}{\partial y_{1}}-y_{1}\frac{\partial \tilde{\mathbf{v}}}{\partial y_{2}}\big{)}\) and \(\mathcal{D}_{t}\mathbf{v}_{r}|_{x=\mathcal{Z}y}=\mathcal{D}_{t}\mathbf{v}_{r}( \mathcal{Z}y,t)-(\mathbf{\mathcal{V}}\cdot\
abla)\mathbf{v}_{r}\). By substituting this in (6) and (7), acting by \(\mathcal{Z}^{-1}\) and taking (8) into account in the boundary conditions, we obtain the following interface problem for the perturbations \(\tilde{\mathbf{v}}\) and \(\tilde{\mathbf{v}}\):  
\[ \begin{split}\rho^{\pm}\big{(}\mathcal{D}_{t}\tilde{\mathbf{v}}+( \tilde{\mathbf{v}}\cdot\
abla)\tilde{\mathbf{v}}+2\omega(\varepsilon_{3}\times\tilde{ \mathbf{v}})\big{)}-\mu^{\pm}\
abla^{2}\tilde{\mathbf{v}}+\
abla\tilde{\mathbf{v}}=\rho^{ \pm}\tilde{\mathbf{J}},\\ \
abla\cdot\tilde{\mathbf{v}}=0\qquad\text{ in }\cup\tilde{\Omega}_{t}^{\pm} \equiv\tilde{\Omega}_{t}^{-}\cup\tilde{\Omega}_{t}^{+},\qquad t>0,\\ \tilde{\mathbf{v}}(y,0)=\mathbf{v}_{0}(y)-\mathbf{\mathcal{V}}(y)\equiv\tilde{ \mathbf{v}}_{0}(y),\quad y\in\cup\tilde{\Omega}_{0}^{\pm},\\ -\tilde{p}\tilde{\mathbf{n}}+\mu^{-}\mathbb{S}(\tilde{\mathbf{v}})\tilde{ \mathbf{n}}\big{|}_{\Gamma_{t}^{-}}=\big{\{}\sigma^{-}\big{(}\tilde{H}^{-}(y)- \mathcal{H}^{-}(z)\big{)}+\rho^{-}\omega^{2}\big{(}|y|^{2}-|z^{\prime}|^{2} \big{)}\,/2\\ +\varkappa\rho^{-}\big{(}\tilde{U}(y,t)-\mathcal{U}(z)\big{)} \big{\}}\tilde{\mathbf{n}},\\ [\tilde{\mathbf{v}}]\big{|}_{\Gamma_{t}^{+}}=0,\\ [-\tilde{p}\tilde{\mathbf{n}}+\mu^{\pm}\mathbb{S}(\tilde{\mathbf{v}}) \tilde{\mathbf{n}}]|_{\Gamma_{t}^{+}}=\big{\{}\sigma^{+}\big{(}\tilde{H}^{+}(y)- \mathcal{H}^{+}(z)\big{)}+[\rho^{\pm}]\omega^{2}\big{(}|y|^{2}-|z^{\prime}|^{2 }\big{)}\,/2\\ +\varkappa[\rho^{\pm}]|_{\Gamma_{t}^{+}}\big{(}\tilde{U}(y,t)- \mathcal{U}(z)\big{)}\big{\}}\tilde{\mathbf{n}},\\ \tilde{\mathbf{V}}_{\tilde{\mathbf{n}}}=\tilde{\mathbf{v}}\cdot\tilde{\mathbf{n}} \quad\text{on }\tilde{\Gamma}_{t}\equiv\tilde{\Gamma}_{t}^{-}\cup\tilde{\Gamma}_{t}^{+}, \end{split}  \tag{11}\]  
where \(\tilde{\Omega}_{t}^{\pm}=\mathcal{Z}^{-1}(\omega t)\Omega_{t}^{\pm}\), \(\tilde{\Gamma}_{t}^{\pm}=\mathcal{Z}^{-1}(\omega t)\Gamma_{t}^{\pm}\), \(\tilde{\mathbf{f}}(y,t)=\mathcal{Z}^{-1}(\omega t)\mathbf{f}(\mathcal{Z}(\omega t)y,t)\), \(\tilde{\mathbf{n}}\) is the outward normal to \(\Gamma_{t}\), \(\mathbf{n}=\mathcal{Z}\tilde{\mathbf{n}}\), \(y^{\prime}=(y_{1},y_{2},0)\), etc. We note that the kinematic boundary condition \(V_{\mathbf{n}}=\mathbf{v}\cdot\mathbf{n}\), conserves its form (see [10]).