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    \title{\bf Free boundary problem of magnetohydrodynamics for two liquids}
    \author{E.V. Frolova
    \footnote{The work is partially supported by the Russian Foundation of Basic Research, grant 14-01-00534.}
\footnote{Communicated by A.I. Nazarov}}
   % \setcounter{equation}{0}
   \maketitle 
   
   \section{Introduction}
\par
We consider the free boundary problem of magnetohydrodynamics in the
bounded domain $\Omega\subset R^3$. It describes the motion of a
finite isolated mass of viscous incompressible electrically
conducting capillary liquid inside the other viscous incompressible
liquid under the action of magnetic field. The interface between the
liquids is unknown. Let the bounded variable domain
 $\Omega_{1t}$
is filled by the liquid of density $d_1$ and viscosity $\nu_1$. The
domain $\Omega_{1t}$ is surrouded by the bounded variable domain
  $\Omega_{2t}=\Omega\setminus\overline{\Omega}_{1t}$, filled by the liquid of density $d_2$ and  viscosity $\nu_2$. The boundary of  $\Omega_{2t}$
  consists of two disjoint components: the free boundary
  $\Gamma_t$ and the fixed boundary  $S=\partial\Omega$.  We assume that both  $\Gamma_0$ and  $S$ are homeomorphic to a sphere, $dist\{\Gamma_0,S\}\geq\delta>0.$
\par The problem consists of determination  for  $t>0$ the variable domains  $\Omega_{it}$, $i=1,2$ together with the velocity vector field $\mathbf{v^{(i)}}$, the pressure $p^{(i)}$, and the magnetic field $\mathbf{H^{(i)}}$.
 Equations in   $\Omega_{it}$ have the form
    \begin{eqnarray}
    &\mathbf{v^{(i)}}_t+(\mathbf{v^{(i)}}\cdot\nabla)
    \mathbf{v^{(i)}}-\nabla\cdot T(\mathbf{v^{(i)}},p^{(i)})-\nabla\cdot T_M(\mathbf{H^{(i)}})=0,\nonumber \\
    &\mu_i\mathbf{H^{(i)}}_t+\alpha_i^{-1}rotrot \mathbf{H^{(i)}}-\mu_irot(\mathbf{v^{(i)}}\times \mathbf{H^{(i)}})=0,\\
    & \nabla\cdot  \mathbf{v^{(i)}}=0,\quad\quad \nabla\cdot \mathbf{H^{(i)}}=0, \quad \quad
    x\in\Omega_{it},
    \nonumber
    \end{eqnarray}
where $\mu_i$,  - magnetic permeability, $\nu_i$ - kinematic
viscosity, $\alpha_i$ - conductivity , $d_i$ - density.  We assume
that  $\nu_i, \alpha_i, d_i, \mu_i$ are positive constants.
$T_M(\mathbf{H})=\mu(\mathbf{H}\otimes\mathbf{H}-\frac{1}{2}I|\mathbf{H}|^2)$
- magnetic stress tensor.

$$T(\mathbf{v},p)=-\frac{1}{d_i}pI+\nu S(\mathbf{v})$$ is the viscous stress tensor,

 $$S(\mathbf{v})=\nabla\mathbf{v}+(\nabla\mathbf{v})^T=\Big(\frac{\partial
    v_i}{\partial x_j}+\frac{\partial v_j}{\partial
    x_i}\Big)_{i,j=1,2,3}$$  is the doubled rate-of-strain tensor.
   \par  On the free surface $\Gamma_t$, which is subject  to capillary forces, we have the following boundary conditions
    \begin{eqnarray}
    &\big ([T( \mathbf{v},p)]+[T_M( \mathbf{H})]\big) \mathbf{n}=\sigma \mathbf{n}
    \mathcal{H}, \nonumber \\
    &\mathbf{V}_n= \mathbf{v}\cdot \mathbf{n}, \quad \quad [\mathbf v]=0,\nonumber \\
    &[\frac{1}{\alpha} (rot\mathbf{H})_{\tau} ]=[\mu
    (\mathbf{v}\times\mathbf{H})_{\tau}], \\
    & [\mu \mathbf{H}\cdot \mathbf{n}]=0,\quad [\mathbf{H}_\tau]=0, \quad
    x\in\Gamma_t,\nonumber
    \end{eqnarray}
    where  $\sigma$ -  coefficient of the surface tension, $ \mathcal{H}$ - is the doubled mean curvature of $\Gamma_t$,
    $\mathbf{V}_n$ is the velocity of evolution of the surface $\Gamma_t$ in
    the direction of the normal $\mathbf {n}$ to $\Gamma_t$,
    which is exterior with respect to the domain  $\Omega_{1t}$.
    By $(rot\mathbf{H})_{\tau}$ we mean the tangential part of the
    rotor. By $[f]$ we denote the jump on $\Gamma_t$:
    $[f]=f^{(1)}-f^{(2)}$.
     Condition on the jump of the tangential part of $rot \mathbf{H}$
     follows from the fact that on the interface tangential part of
     electric field is continuous and Maxwell equations.
\par  We assume that  the fixed boundary $S$ is a perfectly  conducting bounded closed
surface. Boundary conditions on  $S$ have the form
\begin{equation}
\mathbf{H}\cdot\mathbf{n}=0,\quad (rot\mathbf{H})_{\tau}=0, \quad
\mathbf{v}=0,\quad x\in S.
\end{equation}
Finally, we add  the initial conditions
\begin{equation}
\mathbf{v}(x,0)= \mathbf{v}_0(x),\quad \quad \mathbf{H}(x,0)=
\mathbf{H}_0(x),\quad x\in\Omega_{10}\cup \Omega_{20}.
\end{equation}
Free boundary problem governing the motion of a finite isolated mass
of electrically conducting capillary liquid in vacuum has been
studied in [1-3]. In particular, local in time solvability is proved
in \cite {PS1}. The solution is obtained in Sobolev-Slobodetskii
spaces $W_2^{2+l,1+l/2},$ $1/2<l<1$. We obtain the similar result
for the problem $(1.1)-(1.4).$
  \section{Coordinate transform}
  In order to reduce the problem $(1.1)$-$(1.4)$ to a problem set in  a domain with a
  fixed boundary, we use a modification of Hanzawa coordinate
  transform.
\par
  We assume that the initial position of the free boundary
  $\Gamma_0$ can be regarded as a small normal perturbation of the
  given smooth closed surface $G$
  \begin{equation}
  \Gamma_0=\{x=y+\mathbf{N}(y)\rho_0(y),\quad y\in G\}, \nonumber
  \end{equation}
  where $N(y)$ is the external normal to the surface $G$,
  $\rho_0\in W_2^{2+l}(G)$ is a given function, and $|\rho_0|\leq
  \frac{\delta}{4}$. Moreover, we are looking for the free boundary
  in the similar form
  \begin{equation}
  \Gamma_t=\{x=y+\mathbf{N}(y)\rho(y,t),\quad y\in G\}, \nonumber
  \end{equation}
  where the function $\rho(y,t)$ is unknown.
  \par
  We denote by $\mathcal{F}_1$ the domain bounded by $G$,
  $\mathcal{F}_2=\Omega\setminus\overline{\mathcal{F}}_1$.
  We construct the mapping which transforms $\Omega=\mathcal{F}_1\cup
  G\cup\mathcal{F}_2$ to $\Omega=\Omega_{1t}\cup
  \Gamma_t\cup\Omega_{2t}.$ To this end, we extend $N$ and $\rho$
  into $\Omega$. By $N^{*}$ we mean a smooth non-vanishing vector
  field in $\Omega$ which coincides with $N$ on $G$. By
  $\rho^{*}(y,t)$  we denote an extension of unknown function $\rho(y,t)$
  from $G$ into $\Omega$ with preservation of the class, which
  vanishes in a $\frac{\delta_0}{4}$ neighborhood of the surface $S$
  and satisfies the condition $\frac{\partial\rho^{*}(y,t)}{\partial N}\Big
  |_{G}=0.$ We introduce this mapping by the relation
  \begin{equation}
  x=y+\mathbf{N}^{*}(y)\rho^{*}(y,t)=e_{\rho}(y).
  \end{equation}
   When $\rho$ is sufficiently small
(which is certainly the case for small $t$), transform (2.1)
establishes one-to-one correspondence between ${\cal F}_i$ and
$\Omega_{it}$, $i=1,2.$
 We denote by $\mathcal{L}(y,\rho^{*})$ the
Jacobi matrix of the transformation
  $(2.1),$ $L=det\mathcal{L}$,
  $\widehat{\mathcal{L}}=L\mathcal{L}^{-1}$ is the cofactor matrix.
  The normal $\mathbf{n}$ to the free boundary corresponds to
  \begin{equation}
  \mathbf{n}(e_{\rho}(y))=\frac{\widehat{\mathcal{L}}\mathbf{N}(y)}{|\widehat{\mathcal{L}}\mathbf{N}(y)|}.
  \end{equation}
  Let
  \begin{equation}
  \mathbf{v}(e_{\rho},t)=\mathbf{u}(y,t),\quad\quad
  p(e_{\rho},t)=q(y,t). \nonumber
  \end{equation}
  To simplify the calculations, we introduce the new unknown function
  \begin{equation}
  \mathbf{h}=\widehat{\mathcal{L}}\mathbf{H}(e_{\rho},t). \nonumber
  \end{equation}
  As it is demonstrated in \cite{PS1},  $\mathbf{h}$ is a solenoidal vector
  field and satisfies the homogeneous condition $[\mu
  \mathbf{h}\cdot\mathbf{N}]=0,$ $y\in G.$
 Transformation $(2.1)$ converts the problem   $(1.1)-(1.4)$ to a
nonlinear problem in the fixed domain $\Omega={\cal F}_1\cup
S_{R_0}\cup{\cal F}_2,$ for the unknown functions $\mathbf{u}(y,t)$,
 $q(y,t)$,
 $\mathbf{h}(y,t).$  We separate linear and nonlinear parts in this problem and write the boundary condition $(1.2)_1$
 for the tangential and normal parts separately, then it
can be written in the following form:
\begin{eqnarray}\begin{aligned}
&\mathbf{u}^{(i)}_t-\nu_i\nabla^2\mathbf{u}^{(i)}+\frac{1}{d_i}\nabla q^{(i)}=\mathbf{l}^{(i)}_1(\mathbf{u}^{(i)},
q^{(i)},\mathbf{h}^{(i)},\rho),\quad y\in {\cal{F}}_i\\
&\nabla\cdot\mathbf{u}^{(i)}=l^{(i)}_2(\mathbf{u}^{(i)},\rho),\quad y\in{\cal F}_i, \\
&[\nu\Pi_{0}S(\mathbf{u})\mathbf{N}]=\mathbf{l}^{(i)}_3(\mathbf{u},\rho), \quad y\in G,\\
&-[\frac{1}{d}q]+[\nu\mathbf{N}\cdot
S(\mathbf{u})\mathbf{N}(y)]+\sigma B\rho
=l_4(\mathbf{u},\mathbf{h},\rho), \quad y\in G,\\
&\rho_t-\mathbf{u}\cdot{\mathbf{N}}
=l_5(\mathbf{u},\rho),\quad [\mathbf{u}]=0,\quad y\in G,\label{3}\\
&\mu_i\mathbf{h}^{(i)}_t+\alpha_i^{-1}rotrot\mathbf{h}^{(i)}=\mathbf{l}^{(i)}_6(\mathbf{h}^{(i)},\mathbf{u}^{(i)},\rho),
\quad y\in {\cal F}_i, \\
&\nabla\cdot\mathbf{h}^{(i)}=0,\quad y\in{\cal F}_{i}, \\
&[\mu\mathbf{h}\cdot\mathbf{N}]=0,\quad
[\mathbf{h}_\tau]=\mathbf{l}_7(\mathbf{h},\rho), \quad
[\frac{1}{\alpha}(rot\mathbf{h})_{\tau}]=\mathbf{l}_8(\mathbf{h},\mathbf{u},\rho)
\quad y\in G, \\
&\mathbf{h}^{(2)}\cdot\mathbf{n}=0,\quad
(rot\mathbf{h}^{(2)})_{\tau}=0, \quad \mathbf{u}^{(2)}=0\quad
y\in S, \\
&\mathbf{u}^{(i)}(y,0)=\mathbf{u}^{(i)}_0(y),\quad
\mathbf{h}^{(i)}(y,0)=\mathbf{h}^{(i)}_0(y),\quad y\in{\cal
F}_i,\quad \rho(y,0)=\rho_0(y),\quad y\in G.
\end{aligned}
\end{eqnarray}
Here
$\Pi_0\mathbf{u}=\mathbf{u}-\mathbf{N}(\mathbf{u}\cdot\mathbf{N})$
is the tangential part of the vector field $\mathbf{u}$, $-B\rho$ is
the first variation of $\mathcal{H}$ with respect to $\rho$ and has
the form
      $B\rho=-\Delta_{G}\rho+b\rho,$ where
      $\Delta_{G}$ is the Laplace-Beltrami operator on $G$.
 The nonlinear terms $\mathbf{l}_1^{(i)}-\mathbf{l}_7$ are similar to the nonlinear terms calculated in \cite{PS1}.
 The nonlinear term $\mathbf{l}_8$ has the form
 \begin{eqnarray}
 &\mathbf{l}_8=[\frac{1}{\alpha}(rot\mathbf{h})_{\tau}]=[\frac{1}{\alpha}\left (rot\mathbf{h}-(rot\mathbf{h}\cdot\mathbf{N})\mathbf{N}\right
 )] \nonumber \\
 &=[\frac{1}{\alpha}\left (rot\mathbf{h}-\frac{1}{L}\mathcal{L}rot\mathcal{L}^T\frac{1}{L}\mathcal{L}\mathbf{h}\right
 )] \nonumber \\
 &+[\frac{1}{\alpha}\left ((\frac{1}{L}\mathcal{L}rot\mathcal{L}^T\frac{1}{L}\mathcal{L}\cdot\mathbf{n}(e_{\rho})\mathbf{n})(e_{\rho})-
 (rot\mathbf{h}\cdot\mathbf{N})\mathbf{N}\right )]\nonumber \\
 &+[\mu\left (\mathcal{L}^{-1}\mathbf{u}\times\mathbf{h}-
 ((\mathcal{L}^{-1}\mathbf{u}\times\mathbf{h})\cdot\mathbf{n}(e_{\rho})\mathbf{n}(e_{\rho}))\right
 )],\nonumber
 \end{eqnarray}
where $\mathbf{n}(e_{\rho})$ is given in $(2.2)$.
  \section{Main result}
   {\bf Theorem 1.}  Let $\mathbf{u}_{0i}\in W_2^{1+l}({\cal F}_i)$,
 $\mathbf{H_{0i}}\in
 W_2^{1+l}({\cal F}_i),$ $i=1,2,$  $\rho_0\in W_2^{2+l}(G)$ with a certain $l\in (1/2,1)$ and the following compatibility conditions
\begin{eqnarray}
&\nabla\cdot\mathbf{u}_0^{(i)}=l^{(i)}_2(\mathbf{u}_0^{(i)},\rho_0),\quad y\in{\cal F}_i, \nonumber \\
&[\nu\Pi_{0}S(\mathbf{u}_0)\mathbf{N}]=\mathbf{l}_3(\mathbf{u}_0,\rho_0), \quad y\in G, \nonumber\\
&\nabla\cdot\mathbf{h}^{(i)}_0=0,\quad y\in{\cal F}_{i}, \\
&[\mu\mathbf{h}_0\cdot\mathbf{N}]=0,\quad
[(\mathbf{h}_0)\tau]=\mathbf{l}_7(\mathbf{h}_0,\rho_0), \quad
[\frac{1}{\alpha}(rot\mathbf{h}_0)_{\tau}]=\mathbf{l}_8(\mathbf{h}_0,\mathbf{u}_0,\rho_0),\quad
[\mathbf{u}_0]=0
\quad y\in G, \nonumber \\
&\mathbf{h}_0^{(2)}\cdot\mathbf{n}=0,\quad
(rot\mathbf{h}_0^{(2)})_{\tau}=0, \quad \mathbf{u}_0^{(2)}=0\quad
y\in S \nonumber
\end{eqnarray}
are hold.
 We assume that the smallness conditions
  \begin{equation}
 \|\rho_0\|_{W_2^{2+l}(G)}\leq\varepsilon\quad\quad
 \|\mathbf{U}_0-\mathbf{u}_0\|_{W_2^{l+1/2}(G)\leq \varepsilon},
 \end{equation}
 where $\mathbf{U}_0\in W_2^{l+3/2}(G)$ is a given vector field, be satisfied.
Then problem  $(2.3)$ has a unique solution on a certain small time
interval $(0,T)$ with the following regularity properties
 $$\rho\in W_2^{5/2+l,0}(G_T)\cap W_2^{l/2}((0,T),W_2^{5/2}(G)),\quad \rho_t\in W_2^{3/2+l,3/4+l/2}(G_T),$$
 $$\mathbf{u}^{(i)}\in W_2^{2+l,1+l/2}({\cal F}_{i}\times (0,T)),\quad\quad
 \mathbf{h}^{(i)}\in W_2^{2+l,1+l/2}({\cal F}_{i}\times (0,T)),$$
$$ q\in W_2^{1/2+l,0}(G_T)\cap W_2^{l/2}((0,T);W_2^{1/2}(G)),
\quad\quad \nabla q\in W_2^{l,l/2}({\cal F}_{i}\times (0,T)).$$

  {\bf Scheme of the proof.}
 It is clear that problem $(2.3)$ can be decomposed in two parts:
 the hydrodynamical part with linear terms depending on $\mathbf{u}$, $q$, and
 $\rho$ and the magnetic part with linear terms depending on
 $\mathbf{h}$. Linearized hydrodynamical problem is as follows
 \begin{eqnarray}
 \begin{aligned}
  &\mathbf
 {u}^{(i)}_t-\nu^{(i)}\nabla^2\bf{ u}^{(i)}+\frac{1}{d^{(i)}}\nabla
 p^{(i)}=\ \mathbf{f}^{(i)},\quad \nabla\cdot\mathbf{u}
 ^{(i)}=\nabla\cdot \mathbf{F}^{(i)},
 \quad y\in{\cal F}_i, \\
 &[\nu\Pi_0S(\mathbf{u})]\mathbf{N}=\Pi_0\mathbf{A}, \\
 &-[\frac{1}{d}p]+[\nu\mathbf{N}\cdot S(\mathbf{u})\mathbf{N}]+
 \sigma B\rho=\mathbf{A}\cdot\mathbf{N}, \label{4} \\
 &\rho_t-\mathbf{u}\cdot\mathbf{N}=g(y,t),\quad [\mathbf{u}]=0,\quad y\in G, \\
 &\mathbf{u}^{(2)}=0,\quad y\in S,\\
 &\mathbf{u}^{(i)}(y,0)=\mathbf{u}^{(i)}_0(y),\quad y\in{\cal
 F}_i,\quad \rho(x,0)=\rho_0(y),\quad y\in G.
 \end{aligned}
 \end{eqnarray}
 Problem $(3.3)$ similar to the linearized problem  in two phase free boundary problem describes the motion of two liquids
 without action of magnetic field. This linear problem has been studied in \cite{D}, \cite {DS}.
 In particular unique solvability in Sobolev-Slobodetskii spaces is
 obtained.

  The part with linear terms depending on $\mathbf{h}$ is as follows
\begin{eqnarray}
\begin{aligned} &\mu_i\mathbf{H}^{(i)}_t+\frac{1}{\alpha_i}
rotrot\mathbf{H}^{(i)}=\mathbf{f}^{(i)},\quad
\nabla\cdot\mathbf{H}^{(i)}=0,\quad y\in{\cal F}_i, \label{5}\\
&[\mu\mathbf{H}\cdot\mathbf{N}]\big|_{G}=0,\quad
[\mathbf{H}_{\tau}]\big |_{G}=\mathbf{a},\\
&[\frac{1}{\alpha}(rot\mathbf{H})_{\tau}]\Big |_{G}=\mathbf{g},
\\
&\mathbf{H}^{(2)}\cdot\mathbf{n}=0,\quad \quad(rot\mathbf{H}^{(2)})_{\tau}=0,\quad  y\in S    , \\
& \mathbf{H}^{(i)}(y,0)=\mathbf{H}^{(i)}_0(y),\quad y\in {\cal
    F}_i.
    \end{aligned}
\end{eqnarray}
 Problem $(3.4)$ can be reduced to the similar problem with   $\mathbf{g}=0,$, $\mathbf{a}=0$ in
 the same way as in \cite{PS1}, where the solution to the
 auxilliary problem
 \begin{eqnarray}
    \begin{aligned}
 &rot\mathbf{h}(y)=\mathbf{j}(y),\quad\quad\nabla\cdot\mathbf{h}(y)=0,\quad\quad
 y\in {\cal F}_i,\\
 &[\mu\mathbf{h}\cdot\mathbf{n}]\big |_{G}=0,\quad
 [\mathbf{h}_{\tau}]\big |_{G}=\mathbf{a},\\
 &\mathbf{h}\cdot \mathbf{N}(y)=0,\quad\quad y\in S
    \end{aligned}
 \end{eqnarray}
 has been constructed.

{\bf Theorem 2.}\cite{FR} Let in $(3.4)$ $\mathbf{a}=0,$
$\mathbf{g}=0$, $\mathbf{f}^{(i)}\in W_2^{l,l/2}(Q_T^{(i)})$,
$\mathbf{H}^{(i)}_0\in W_2^{l+1}(\mathcal{F}_i)$,  $l\in [0,1)$ and
the following compatibility conditions be satisfied
\begin{eqnarray}
&\nabla\cdot\mathbf{f}^{(i)}=0,\quad\quad
\nabla\cdot\mathbf{H}_0^{(i)}=0,
\quad\quad y\in\mathcal{F}_i,\nonumber \\
&[\mu\mathbf{H}_0\cdot\mathbf{N}]\Big |_{G}=0,\quad\quad
[\mathbf{H}_{0\tau}]\Big |_{G}=0,\quad\quad
[\frac{1}{\alpha}rot_{\tau}\mathbf{H}_0]\Big |_{G}=0,\quad\quad
[\mathbf{f}\cdot\mathbf{N}]\Big
|_{G}=0\nonumber \\
& \mathbf{H}_0\cdot\mathbf{n}\Big |_{S}=0,\quad\quad
(rot\mathbf{H}_0^{(2)})_{\tau}\Big |_{S}=0, \quad\quad
\mathbf{f}^{(2)}\cdot\mathbf{n}\Big |_{S}=0.\nonumber
\end{eqnarray}
(Condition  $\nabla\cdot\mathbf{f}^{(i)}=0$ holds in a week sense.
Compatibility conditions on the tangential part of rotor at the
boundary and for $\mathbf{f}$ on the boundary are set only when
$l\geq1/2$.)

 Then problem $(3.4)$ has a unique solution
$\mathbf{H}^{(i)}\in W_2^{l+2,l/2+1}(Q_T^{(i)}),$
$Q_T^{(i)}=\mathcal{F}_i\times(0,T),$ $i=1,2$. For this solution the
following estimate
\begin{equation}
\sum\limits_{i=1}^2\parallel\mathbf{H}^{(i)}\parallel_{
W_2^{l+2,l/2+1}(Q_T^{(i)})}\leq c\sum\limits_{i=1}^2\left
(\parallel\mathbf{f}^{(i)}\parallel_{
W_2^{l,l/2}(Q_T^{(i)})}+\parallel\mathbf{H}_0^{(i)}\parallel_{
W_2^{l+1}(\mathcal{F}_i)} \right )
\end{equation}
holds. \par Solvability of the nonlinear problem is proved by the
successive approximations method, based on solvability results for
linear problems $(3.3)$, $(3.4)$ and estimates of nonlinear terms.
Assumption              $(3.2)_1$ is stronger as the corresponding
assumption in \cite{PS1}
 $(\|\rho_0\|_{W_2^{3/2+l}(G)}\leq\varepsilon).$ It gives us the
opportunity to obtain for the magnetic field the same regularity
properties as for the velocity vector field. Detailed proof will be
given in subsequent publications.
 \begin{thebibliography}{99}
 \bibitem{PS1}
 M. Padula and   V.~A.~Solonnikov, "On the free boundary problem of magnethydrodynamics",
 Zap. Nauchn. Sem. POMI {\bf385},
 (2010), 135-186.
 \bibitem{SF} V.~A.~Solonnikov and E.V. Frolova,
  "Solvability of a free boundary problem of magnetohydrodynamics in an infinite
 time interval",
 Zap. Nauchn. Sem. POMI {\bf410}(2013), 131-167.
 \bibitem{F}
 E. V. Frolova, "Free boundary problem of magnethydrodynamics"  Zap. Nauchn. Sem. POMI 425(2014), 149-178.
  \bibitem{DS} I.~V.~ Denisova, V.~A.~Solonnikov,  "On the solvability of the linearized problem
  of a drop motion in the fluid stream",  Zap. Nauchn. Sem. POMI {\bf 171}, (1989), 53-65.
   \bibitem{FR}
 E. V. Frolova, "Linear problem arising in the study of free boundary problem of magnethydrodynamics for two
 liquids",
 preprint POMI, (2015) preprint N 10 in russian.
  \bibitem{D} I.~V.~ Denisova, "Apriori estimates for a solution to linear nonstationary problem connected with a
  drop motion in fluid flow."
 \end{thebibliography}



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