# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
\begin{split}&\bar{\rho}\big{(}\mathcal{D}_{t}\mathbf{u}+2\omega(\mathbf{e}_{ 3}\times\mathbf{u})\big{)}-\bar{\mu}\
abla^{2}\mathbf{u}+\
abla q=\bar{\rho}\hat{f}+I_ {1}(\mathbf{u},q,r)\equiv\mathbf{f}_{1},\\ &\
abla\cdot\mathbf{u}=l_{2}(\mathbf{u},r)\qquad\text{in}\cup\mathcal{F}^ {\pm},\ \ t>0,\\ &\mathbf{u}\big{|}_{t=0}=\mathbf{u}_{0}\quad\text{in}\cup\mathcal{F}^{\pm },\quad r\big{|}_{t=0}=r_{0}\quad\text{on}\ \mathcal{G},\\ &\mu^{-}\Pi_{\mathcal{G}}\mathcal{S}(\mathbf{n})\mathbf{N}=I_{3}^{-}(\bm {u},r)\quad\text{on}\ \mathcal{G}^{-},\qquad[\bar{\mu}\Pi_{\mathcal{G}}\mathcal{S}(\mathbf{n})\mathbf{N}] \big{|}_{\mathcal{G}^{+}}=I_{3}^{+}(\mathbf{u},r)\quad\text{on}\ \mathcal{G}^{+},\\ &-q+\mu^{-}\mathbf{N}\cdot\mathbb{S}(\mathbf{u})\mathbf{N}+\mathcal{B}_{0}^{-} r=l_{4}^{-}(\mathbf{u},r)+l_{5}^{-}(r)\quad\text{on}\ \mathcal{G}^{-},\\ &[\mathbf{u}]\big{|}_{\mathcal{G}^{+}}=0,\quad[-q+\bar{\mu}\mathbf{N}\cdot \mathbb{S}(\mathbf{u})\mathbf{N}]\big{|}_{\mathcal{G}^{+}}+\mathcal{B}_{0}^{+}r=l_{4}^{ +}(\mathbf{u},r)+l_{5}^{+}(r)\quad\text{on}\ \mathcal{G}^{+},\\ &\mathcal{D}_{t}r-\mathbf{u}\cdot\mathbf{N}=l_{6}(\mathbf{u},r)\quad\text{on} \ \mathcal{G},\end{split}  \tag{44}\]where \(\mathbf{u}(z,t)=\vartheta\big{(}e_{r}(z,t),t\big{)}\), \(\mathbf{u}_{0}(z)=\vartheta\big{(}e_{r_{0}}(z,0),0\big{)}\), \(q(z,t)=\vartheta\big{(}e_{r}(z,t),t\big{)}\), \(\widehat{f}(z,t)=\widehat{f}\big{(}e_{r}(z,t),t\big{)}\),  
\[I_{1}(u,q,r)=\vartheta(\widehat{\
abla}^{2}-\
abla^{2})\mathbf{u}+( \
abla-\widehat{\
abla})q+\vartheta\mathcal{D}_{t}r^{*}(\mathbb{L}^{-1}N^{*} \cdot\
abla)\mathbf{u}-\vartheta(\mathbb{L}^{-1}\mathbf{u}\cdot\
abla)\mathbf{u},\] \[I_{2}(u,r)=(\
abla-\widehat{\
abla})\cdot\mathbf{u}\equiv\
abla\cdot \mathbf{L}_{2}(\mathbf{u},r),\] \[I_{3}^{-}(\mathbf{u},r)=\mu^{-}\Pi_{\mathcal{G}}\big{(}\Pi_{ \mathcal{G}}\mathbb{S}(\mathbf{u})\mathbf{N}-\widehat{\Pi\widetilde{\mathbb{S}}}(\mathbf{u })\mathbf{\tilde{n}}(e_{r})\big{)},\] \[I_{4}^{-}(\mathbf{u},r)=\mu^{-}\big{(}\mathbf{N}\cdot\mathbb{S}(\mathbf{u}) \mathbf{N}-\mathbf{\tilde{n}}(e_{r})\cdot\widetilde{\mathbb{S}}(\mathbf{u})\mathbf{\tilde{n} }(e_{r})\big{)}, \tag{45}\] \[I_{4}^{+}(\mathbf{u},r)=\big{[}\bar{\eta}\big{(}\mathbf{N}\cdot\mathbb{S} (\mathbf{u})\mathbf{N}-\mathbf{\tilde{n}}(e_{r})\cdot\widetilde{\mathbb{S}}(\mathbf{u})\mathbf{ \tilde{n}}(e_{r})\big{)}\big{]}\big{|}_{\mathcal{G}^{+}},\] \[I_{5}^{-}(r)=\sigma^{-}\int_{0}^{1}(1-s)\,\frac{\mathrm{d}^{2}}{ \mathrm{d}\mathrm{s}^{2}}\big{(}\mathbb{L}^{-T}(z,sr)\
abla_{\mathcal{G}} \cdot\frac{\widehat{\mathbb{L}}^{T}(z,sr)\mathbf{N}}{|\mathbb{L}^{T}(z,sr)\mathbf{N}|} \big{)}\,\mathrm{d}s+\frac{\omega^{2}}{2}\rho^{-}|\mathbf{N}^{\prime}|^{2}r^{2}\] \[\qquad\qquad+\varkappa\rho^{-}\int_{0}^{1}(1-s)\,\frac{\mathrm{d} ^{2}}{\mathrm{d}\mathrm{s}^{2}}\widetilde{U}(e_{sr}(z),t)\,\mathrm{d}s,\] \[I_{5}^{+}(r)=\sigma^{+}\int_{0}^{1}(1-s)\,\frac{\mathrm{d}^{2}} {\mathrm{d}\mathrm{s}^{2}}\big{(}\mathbb{L}^{-T}(z,sr)\
abla_{\mathcal{G}} \cdot\frac{\widehat{\mathbb{L}}^{T}(z,sr)\mathbf{N}}{|\mathbb{L}^{T}(z,sr)\mathbf{N}|} \big{)}\,\mathrm{d}s+\frac{\omega^{2}}{2}[\vartheta]\big{|}_{\mathcal{G}^{+}} |\mathbf{N}^{\prime}|^{2}r^{2}\] \[\qquad\qquad+\varkappa[\vartheta]\big{|}_{\mathcal{G}^{+}}\int_{0 }^{1}(1-s)\,\frac{\mathrm{d}^{2}}{\mathrm{d}\mathrm{s}^{2}}\widetilde{U}(e_{ sr}(z),t)\,\mathrm{d}s,\] \[I_{6}(\mathbf{u},r)=\Big{(}\frac{\widehat{\mathbb{L}}^{T}\mathbf{N}}{ \mathbf{N}\cdot\widehat{\mathbb{L}}^{T}\mathbf{N}}-\mathbf{N}\Big{)}\cdot\mathbf{u},\]  
\(\mathbb{L}\) is the Jacobi matrix of transformation (15):  
\[I_{2}(u,r)=(\
abla-\widehat{\
abla})\cdot\mathbf{u}\equiv\
abla\cdot \mathbf{L}_{2}(\mathbf{u},r),\]0  
\(\widehat{\mathbb{L}}\equiv L\mathbb{L}^{-1}\), \(L\equiv\mathrm{det}\mathbb{L}\).