# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem
We cite the result obtained in ([9], SS 5.3) about the solvability of the following linear problem for the Stokes system in an unbounded domain \(\Omega^{-}\cup\Omega^{+}\), \(\overline{\Omega}^{+}\cup\Omega^{-}=\mathbb{R}^{3}\), with the closed interface \(\Gamma=\partial\Omega^{+}\):  
\[\mathcal{D}_{t}\mathbf{v}-v^{\pm}\
abla^{2}\mathbf{v}+\frac{1}{\rho^{\pm}}\
abla p=f, \quad\
abla\cdot\mathbf{v}=g\,\,\,\text{in}\,\,\,D_{T}\equiv\cup Q_{T}^{\pm}= \Omega^{\pm}\times(0,T),\]  
\[\mathbf{v}|_{t=0}=\mathbf{v}_{0}\quad\text{in}\,\,\,\Omega^{-}\cup\Omega^{+},\quad \qquad\mathbf{v}\xrightarrow[|x|\to\infty]{}0,\]  
\[[\mathbf{v}]|_{\Gamma}=0,\quad\quad[\Pi_{0}\mathbb{T}\mathbf{n}]|_{\Gamma}=\mathbf{b} \quad\quad\big{(}\mathbf{b}\cdot\mathbf{n}=0\big{)}, \tag{24}\]  
\[\big{[}\mathbf{n}\cdot\mathbb{T}\mathbf{n}\big{]}\big{|}_{\Gamma}-\sigma\mathbf{n}\cdot \int_{0}^{t}\Lambda_{T}\mathbf{v}\,\mathrm{d}t^{\prime}=b^{\prime}+\sigma\int_{0 }^{t}B\,\mathrm{d}t^{\prime}\quad\text{ on }G_{T}\equiv\Gamma\times(0,T),\]  
where \(\mathbf{f},g,\mathbf{b},b^{\prime},B,\mathbf{v}_{0}\) are given functions; \(\mathbf{n}\) denotes the outward unit normal to \(\Gamma\); \(\Pi_{0}\mathbf{a}=\mathbf{a}-(\mathbf{n}\cdot\mathbf{a})\mathbf{n}\). We assume first that there hold two representations:  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]0  
and second, that compatibility conditions  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]1  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]2  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]3  
are satisfied. The last of relations (26) follows from the tangential part of the necessary condition of the continuity of velocity derivative \(\mathcal{D}_{t}\mathbf{v}\) at \(t=0:\,\,[\mathcal{D}_{t}\mathbf{v}]|_{\Gamma}=0\). The normal part of this equality \([\mathcal{D}_{t}\mathbf{v}\cdot\mathbf{n}]|_{\Gamma,t=0}=0\) holds if the initial pressure \(p_{0}\equiv p(x,0)\) is a solution to the problem  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]4  
\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]5\[\int_{\mathcal{F}}\bar{\rho}\mathbf{w}^{\perp}\cdot\mathbf{\eta}\,\mathrm{d}z=0,\]6  
(Here the first equation is understood in a weak sense.) **Theorem 1**. : _Let assumptions (25)-(27) be fulfilled. In addition, we assume for \(\alpha,\gamma\in(0,1)\) and \(\gamma<\alpha\) when \(\sigma>0\) and \(T<\infty\) that \(\Gamma\in C^{2+\alpha}\), \(\mathbf{f}\in\mathbf{C}^{\alpha,\frac{\pi}{2}}(D_{\Gamma})\), \(g\in C^{1+\alpha,\frac{1+\alpha}{2}}(D_{\Gamma})\), \(\mathbf{R}\in C^{(\gamma,1+\alpha)}(D_{\Gamma})\), \(\mathcal{D}_{t}\mathbf{R}\in C^{\alpha,\frac{\pi}{2}}(D_{\Gamma})\), \(\left[\mathbf{R}\cdot\mathbf{n}\right]\big{|}_{G_{\Gamma}}=0\), \(\mathbf{v}_{0}\in C^{2+\alpha}(\Omega^{-}\cup\Omega^{+})\), \(\mathbf{b}\in\mathbf{C}_{\mathbf{n}}^{1+\alpha,\frac{1+\alpha}{2}}(G_{\Gamma})\), \(b^{\prime}\in C^{(\gamma,1+\alpha)}(G_{\Gamma})\), \(B\in C^{\alpha,\alpha/2}(G_{\Gamma})\), and the elements of the tensor \(\mathbb{M}\) have finite semi-norms \(\left|M_{ik}\right|_{D_{\Gamma}}^{(\gamma,1+\alpha)}\), \(\left.\left\langle M_{ik}\right\rangle_{X,D_{\Gamma}}^{(\gamma)}\right.\), \(i,k=1,2,3.\) We suppose also that all given functions decrease quickly enough for \(|x|\to\infty\) (for example, in a power-law way).