# Holder Space Theory for the Rotation Problem of a Two-Phase Drop
## 2 Setting of a problem  
We give a mathematical statement of the problem. We assume two immiscible viscous incompressible fluids of densities \(\rho^{\pm}\) and viscosities \(\mu^{\pm}\) to be situated in a domain \(\Omega_{t}\subset\mathbb{R}^{3}\) which is separated by a variable closed interface \(\Gamma_{t}^{+}\) and bounded by a free surface \(\Gamma_{t}^{-}\), \(\Gamma_{t}^{+}\) being the boundary of the domain \(\Omega_{t}^{+}\) filled with a fluid of the density \(\rho^{+}\). It is surrounded by another fluid of the density \(\rho^{-}\) contained in the domain \(\Omega_{t}^{-}=\Omega_{t}\setminus\overline{\Omega_{t}^{+}}\) (see Figure 1). At the initial moment \(t=0\), the boundaries \(\Gamma_{0}^{\pm}\) are given. This two-layer fluid mass rotates about the vertical axis \(x_{3}\). One should find the surfaces \(\Gamma_{t}^{\pm}\), velocity vector field \(\mathbf{v}(x,t)\) and pressure function \(p(x,t)\) which satisfy the interface problem for the Navier-Stokes equations:  
\[\rho^{\pm}\big{(}\mathcal{D}_{t}\mathbf{v}+(\mathbf{v}\cdot\
abla)\mathbf{v} \big{)}-\mu^{\pm}\
abla^{2}\mathbf{v}+\
abla p=\rho^{\pm}(\mathbf{f}+\varkappa\
abla U ),\] \[\
abla\cdot\mathbf{v}=0\quad\text{in}\ \cup\Omega_{t}^{\pm}=\Omega_{t}^{+}\cup\Omega_{t}^{-},\ \ t>0,\] \[\mathbf{v}(x,0)=\mathbf{v}_{0}(x),\ x\in\cup\Omega_{0}^{\pm},\] \[\big{[}\mathbf{v}\big{]}\big{|}_{\Gamma_{t}^{+}}\equiv\lim_{ \begin{subarray}{c}x\to x_{0}\in\Gamma_{t}^{+},\\ x\in\Omega_{t}^{+}\end{subarray} }\mathbf{v}(x,t)-\lim_{ \begin{subarray}{c}x\to x _{0}\in\Gamma_{t}^{+},\\ x\in\Omega_{t}^{-}.\end{subarray} }\mathbf{v}(x,t)=0, \tag{1}\] \[\big{[}\mathbb{T}(\mathbf{v},p)\mathbf{n}\big{]}\big{|}_{\Gamma_{t}^{+}}= \sigma^{+}H^{+}\mathbf{n}\quad\text{on}\ \Gamma_{t}^{+},\qquad\mathbb{T}(\mathbf{v},p)\mathbf{n}\big{|}_{\Gamma_{t}^{-}}= \sigma^{-}H^{-}\mathbf{n}\quad\text{on}\ \Gamma_{t}^{-},\] \[V_{\mathbf{n}}=\mathbf{v}\cdot\mathbf{n}\quad\text{on}\ \Gamma_{t}=\Gamma_{t}^{+}\cup\Gamma_{t}^{-},\]  
where \(\mathcal{D}_{t}=\partial/\partial t\), \(\
abla=(\partial/\partial x_{1},\partial/\partial x_{2},\partial/\partial x _{3})\), \(\
abla^{2}\equiv\
abla\cdot\
abla\); \(\mathbf{f}\) is the vector of mass forces,  
\[U(x,t)=\int_{\Omega_{t}}\frac{\rho^{\pm}\,\text{d}z}{|x-z|};\]  
\(\varkappa\geqslant 0\) is the gravitational constant; \(\mathbf{v}_{0}\) is initial velocity vector field; the stress tensor is  
\[\mathbb{T}(\mathbf{v},p)=-p\mathbb{I}+\mu^{\pm}\mathbb{S}(\mathbf{v}),\]  
where \(\mathbb{I}\) is identity matrix, \(\mathbb{S}(\mathbf{v})=(\
abla\mathbf{v})+(\
abla\mathbf{v})^{\text{T}}\) is the twice rate-of-strain tensor, the superscript \(T\) means the transposition, the step-functions of density and dynamical viscosity \(\rho^{\pm},\mu^{\pm}>0\) are equal to \(\rho^{-}\), \(\mu^{-}\) in \(\Omega_{t}^{-}\) and \(\rho^{+}\), \(\mu^{+}\) in \(\Omega_{t}^{+}\), respectively; \(\sigma^{\pm}>0\) are surface tension coefficients on \(\Gamma_{t}^{\pm}\), \(H^{\pm}\) are the doubled mean curvatures of the surfaces \(\Gamma_{t}^{\pm}\) and \((H^{+}<0\) at the points where \(\Gamma_{t}^{+}\) is convex toward \(\Omega_{t}^{-}\)), the vector \(\mathbf{n}\) is the outward normal to \(\Gamma_{t}^{-}\cup\Gamma_{t}^{+}\) and \(V_{\mathbf{n}}\) is the speed of evolution of the surfaces \(\Gamma_{t}^{\pm}\) in the direction of \(\mathbf{n}\). A Cartesian coordinate system \(\{x\}\) is assumed to be introduced in \(\mathbb{R}^{3}\). The centered dot \(\cdot\) denotes the Cartesian scalar product. The vectors and vector spaces are marked by boldface letters. We imply the summation from 1 to 3 by repeated indices, indicated in Latin letters.