Equilibrium Figures for a Rotating Compressible Capillary Two-Layer Liquid

1. Introduction

Existence of an equilibrium surface for an isolated compressible liquid mass rotating about a fixed axis was first proved in [9]. Our aim is to prove the existence of equilibrium figures for a rotating compressible two-layer fluid.

The problem of the rotation of an isolated incompressible liquid mass about a fixed axis as a rigid body was considered by many famous mathematicians, among them were Newton, Maclaurin, Jacobi, Kovalevskaya, Lyapunov, Poincare and others [1,2,3], who mainly studied the movement without capillarity. The capillary fluids were first investigated by Globa-Mikhailenko [4], Boussinesq and Charrueau in the begining of 20th century. The latter gave a detailed analysis of the problem, calculated the shape of equilibrium figures, including the toroidal case, and considered some aspects of the stability [5,6]. These results were included in a big review on this subject presented in the book of Appell [7]. Stability problem for various ellipsoidal equilibrium figures is analyzed in monograph [8].

Now we state, in a complete setting, the problem on unsteady motion of two compressible barotropic fluids of finite volume separated by a closed unknown interface.

At the initial instant $t=0$, let a fluid with dynamic viscosities ${\mu }^{+}$, ${\mu }_1^{+}$ be in a bounded domain ${\Omega }_0^{+}\subset {\mathbb{R}}^3$, and in the domain ${\Omega }_0^{-}$, surrounding it, there be a fluid with dynamic viscosities ${\mu }^{-}$, ${\mu }_1^{-}$;

$${\mu }^{\pm }>0,2{\mu }^{\pm }+3{\mu }_1^{\pm }\ge 0.$$

The domain ${\Omega }_0\equiv \overline{{\Omega }_0^{+}}\cup {\Omega }_0^{-}$ is bounded by the free surface ${\Gamma }_0^{-}$ and includes the closed interface ${\Gamma }_0^{+}\equiv \partial {\Omega }^{+}$; ${\Gamma }_0^{\pm }$ are given. This two-component cloud rotates about the vertical axis $x_3$ with an angular velocity $\omega$.

For $t>0$, it is necessary to find the surfaces ${\Gamma }_t^{-}$, ${\Gamma }_t^{+}$, as well as velocity vector field $\mathit{v}(x,t)=(v_1,v_2,v_3)$ and the density $\rho (x,t)>0$ of the fluids satisfying diffraction problem for the Navier – Stokes system

(1)

where

$$\mathbb{T}=\left(-p\left(\rho \right)+{\mu }_1^{\pm }\nabla ·\mathit{v}\right)\mathbb{I}+{\mu }^{\pm }\mathbb{S}\left(\mathit{v}\right)$$

is stress tensor, ${\left(\mathbb{S}\left(\mathit{v}\right)\right)}_{ij}=\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}$ is double strain rate tensor, $\mathbb{I}$ is identity matrix; ${\mu }^{\pm }$, ${\mu }_1^{\pm }$ are step functions of dynamic viscosities, equal to ${\mu }^{+}$, ${\mu }_1^{+}$ in ${\Omega }_t^{+}$ and ${\mu }^{-}$, ${\mu }_1^{-}$ in ${\Omega }_t^{-}$ respectively; $p\left(\rho \right)$ is fluid pressure given by a known smooth density function; ${\mathit{v}}_0$ and ${\rho }_0>0$ are initial distributions of velocity and density of the liquids, $\mathit{n}$ is the outward normal vector to the union ${\Gamma }_t$; $H^{\pm }(x,t)$ are twice the mean curvatures of the surfaces ${\Gamma }_t^{\pm }$ (moreover, $H^{+}<0$ at points of convexity ${\Gamma }_t^{+}$ towards ${\Omega }_t^{-}$); ${\sigma }^{-},{\sigma }^{+}>0$ are surface tension coefficients on ${\Gamma }_t^{-}$ and ${\Gamma }_t^{+}$, respectively; $V_{\mathit{n}}$ is the rate of evolution of ${\Gamma }_t$ in the direction $\mathit{n}$. We assume that the Cartesian coordinate system $\left\{x\right\}$ is introduced in the space ${\mathbb{R}}^3$. The central dot denotes the Cartesian scalar product.

We mean summation over repeated indices from 1 to 3 if they are denoted by Latin letters, and from 1 to 2 if they are Greek. We mark vectors and vector spaces in bold. The notation $\nabla ·\mathbb{T}$ denotes the vector with the components ${(\nabla ·\mathbb{T})}_j=\frac{\partial T_{ij}}{\partial x_i}$, $j=1,2,3$.

The kinematic boundary condition $\mathit{v}·\mathit{n}=V_{\mathit{n}}$ excludes mass transfer across fluid boundaries. It follows from our assumption that the fluid particles do not leave the boundaries ${\Gamma }_t^{\pm }$ during the time.

Local (in time) solvability was proved for problem (1) in the whole space ${\mathbb{R}}^3$ with a closed interface between the fluids. The result was obtain both in the Sobolev – Slobodetskiǐ classes of functions [10] and in the Hölder ones [11]. One can get similar results for a two-component domain bounded by a free boundary if one takes into account the estimates for a model problem in a half-space [12,13].

As we have mentioned, we suppose the liquids to be barotropic which implies that the pressure p is a known increasing function of the density: $p^{\prime }\left(\rho \right)>0$. Let, in addition, $\rho =\rho (|x^{\prime }|)$, $x^{\prime }=(x_1,x_2,0)$.

We assume that equilibrium figures ${\mathcal{F}}^{+}$, $\mathcal{F}$ are nearly globular domains with the radiuses $R_0^{\pm }$ ($R_0^{+}<R_0^{-}$), and the motion of fluids is close to the state of rest, i.e., the velocity $\mathit{v}$ is small, and the density $\rho$ differs little from a step function ${\rho }^{\pm }>0$. This picture is schematically presented in Figure 1. We denote the balls $\{x\in {\mathbb{R}}^3:|x|\le R_0^{\pm }\}$ by $B_{R_0^{\pm }}$.

We are going to prove the existence of ${\mathcal{G}}^{+}$ and ${\mathcal{G}}^{-}$, the boundaries of the figures ${\mathcal{F}}^{+}$ and $\mathcal{F}$, respectively. We follow the plan of paper [9].

Figure 1. Equilibrium Figures for a Two-Layer Compressible Fluid.

Figure 1. Equilibrium Figures for a Two-Layer Compressible Fluid.

At rest, the bubble consisted of nested spherical two layers $B_{R_0^{+}}$ and $B_{R_0^{-}}\setminus B_{R_0^{+}}$ with uniform distributions of densities ${\rho }^{\pm }$ has the piecewise constant pressure:

$$\begin{array}{c}p\left({\rho }^{-}\right)=\frac{2{\sigma }^{-}}{R_0^{-}}\mathrm{in}{B}_{R_0^{-}}\setminus B_{R_0^{+}},\\ p\left({\rho }^{+}\right)=\frac{2{\sigma }^{+}}{R_0^{+}}+\frac{2{\sigma }^{-}}{R_0^{-}}\mathrm{in}{B}_{R_0^{+}}.\end{array}$$

(2)

The masses of the layers are

$${\rho }^{+}|B_{R_0^{+}}|=m^{+},{\rho }^{-}\left(|B_{R_0^{-}}|-|B_{R_0^{+}}|\right)=m^{-}.$$

(3)

Steady motion of a two-layer gaseous body $\mathcal{F}\equiv \overline{{\mathcal{F}}^{+}}\cup {\mathcal{F}}^{-}$ uniformly rotating about the axis $x_3$ with a constant angular velocity $\omega$ is governed by the homogeneous stationary Navier–Stokes equations

$$\rho (\mathcal{V}·\nabla )\mathcal{V}-\nabla ·\mathbb{T}=0,\nabla ·\left(\rho \mathcal{V}\right)=0\mathrm{in}\cup {\mathcal{F}}^{\pm }$$

(4)

(here the density $\rho$ and velocity $\mathcal{V}$ depend only on x) and the boundary conditions

$$\begin{array}{cc}\hfill & \mathbb{T}(\mathcal{V},\mathcal{P})\mathit{n}{|}_{{\mathcal{G}}^{-}}={\sigma }^{-}{\mathcal{H}}^{-}\mathit{n}\mathrm{on}{\mathcal{G}}^{-},\hfill \\ \hfill & \left[\mathcal{V}\right]{|}_{{\mathcal{G}}^{+}}=0,\left[\mathbb{T}(\mathcal{V},\mathcal{P})\mathit{n}\right]{|}_{{\mathcal{G}}^{+}}={\sigma }^{+}{\mathcal{H}}^{+}\mathit{n}\mathrm{on}{\mathcal{G}}^{+},\hfill \\ \hfill & \mathcal{V}·\mathit{n}=0\mathrm{on}\mathcal{G}={\mathcal{G}}^{+}\cup {\mathcal{G}}^{-},\hfill \end{array}$$

(5)

where ${\mathcal{H}}^{-}$, ${\mathcal{H}}^{+}$ are twice the mean curvatures of ${\mathcal{G}}^{-}$, ${\mathcal{G}}^{+}$, respectively. The last relation follows from the boundary condition $\mathit{v}·\mathit{n}=V_{\mathit{n}}$. The pressure $\mathcal{P}$ depends on $\rho$.

It is easily seen that velocity vector field

$$\mathcal{V}\left(x\right)=\omega {\mathit{e}}_3\times \mathit{x}\equiv \omega (-x_2,x_1,0)$$

(6)

satisfies (4) together with pressure function gradient

$$\nabla \mathcal{P}\left(\rho \right)=\rho {\omega }^2{x}^{\prime }\equiv \rho \frac{{\omega }^2}{2}\nabla {|x^{\prime }|}^2,$$

(7)

where ${\mathit{e}}_i$ is the ith basis vector, $|x^{\prime }{|}^2=x_1^2+x_2^2$.

First, we consider the simple case when equality (7) coincides with the following one

$$\nabla \mathcal{P}\left(\rho \right)={\mathcal{P}}^{\prime }\left(\rho \right)\nabla \left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2\right),$$

whence ${\mathcal{P}}^{\prime }\left(\rho \right)=\rho \left(x\right)$ and ${\mathcal{P}}^{\pm }\left(\rho \right)\equiv \frac{{\rho }^2\left(x\right)}{2}+p^{\pm }$ in ${\mathcal{F}}^{\pm }$ with constants $p^{\pm }$, because pressure functions can differ each other in different domains by a constant. These constants can be found from relations (2):

$$\begin{array}{c}{p}^{-}=\frac{2{\sigma }^{-}}{R_0^{-}}-\frac{{{\rho }^{-}}^2}{2},\\ p^{+}=\frac{2{\sigma }^{+}}{R_0^{+}}+\frac{2{\sigma }^{-}}{R_0^{-}}-\frac{{{\rho }^{+}}^2}{2}.\end{array}$$

(8)

Let $S_1$ be the unit sphere in ${\mathbb{R}}^3$ with the center in zero, $\xi =\frac{x}{|x|}\in S_1$. We suppose ${\mathcal{G}}^{\pm }$ to be given by functions $R^{\pm }\left(\xi \right)$ on $S_1$. In addition, let $R^{\pm }\left(\xi \right)$ be rotationally symmetric, i.e., they depend only on $|{\xi }^{\prime }|=\sqrt{{\xi }_1^2+{\xi }_2^2}$ and ${\xi }_3$, and be even in ${\xi }_3$.

By substituting $\mathcal{V}$ given by (6) and $\mathcal{P}={\mathcal{P}}^{\pm }$ into boundary conditions (5), we obtain the equations for the surface ${\mathcal{G}}^{-}$ of the domain $\mathcal{F}$ and for the interface ${\mathcal{G}}^{+}$ between the fluids:

$$\begin{array}{c}{\sigma }^{-}{\mathcal{H}}^{-}\left(x\right)+\mathcal{P}\left(\rho \right)=0,x\in {\mathcal{G}}^{-},\\ {\sigma }^{+}{\mathcal{H}}^{+}\left(x\right)+\left[\mathcal{P}\left(\rho \right)\right]{|}_{{\mathcal{G}}^{+}}=0,x\in {\mathcal{G}}^{+}.\end{array}$$

(9)

Rotationally symmetry implies that $R^{\pm }\left(\xi \right)$ do not depend on $arctan\left(\frac{{\xi }_2}{{\xi }_1}\right)$. It is clear that $\mathit{n}=(\frac{{\xi }_1}{c_0},\frac{{\xi }_2}{c_0},n_3)$, since the first two components of $\mathit{n}$ are proportional to ones of the radial vector of the circles to be horizontal sections of ${\mathcal{G}}^{\pm }$. Therefore $\mathcal{V}·\mathit{n}=0$ on ${\mathcal{G}}^{\pm }$.

Obviously, the density is given by the formulas

$$\begin{array}{c}\rho \left(x\right)=\frac{{\omega }^2}{2}|x^{\prime }{|}^2+c^{+}\mathrm{in}{\mathcal{F}}^{+},\\ \rho \left(x\right)=\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{-}\mathrm{in}{\mathcal{F}}^{-}\end{array}$$

(10)

with arbitrary positive constants $c^{+}$ and $c^{-}$, equations (9) taking the form

$$\begin{array}{c}{\sigma }^{-}{\mathcal{H}}^{-}\left(x\right)+{\mathcal{P}}^{-}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{-}\right)=0,x\in {\mathcal{G}}^{-},\\ {\sigma }^{+}{\mathcal{H}}^{+}\left(x\right)+{\mathcal{P}}^{+}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{+}\right)-{\mathcal{P}}^{-}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{-}\right)=0,x\in {\mathcal{G}}^{+}.\end{array}$$

(11)

One can determine the constants $c^{\pm }$ by prescribing the masses of fluids to be the same as that of the nested spherical liquid layers (3):

$${\int }_{{\mathcal{F}}^{\pm }}\rho \left(x\right)\mathrm{d}x\equiv {\int }_{{\mathcal{F}}^{\pm }}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{\pm }\right)\mathrm{d}x=m^{\pm }.$$

(12)

We consider the angular momentum to be one more parameter of the problem. It is given:

$$\beta \equiv {\int }_{{\mathcal{F}}^{+}\cup {\mathcal{F}}^{-}}\rho \left(x\right)\mathcal{V}·{\eta }_3\mathrm{d}x=\omega {\int }_{\cup {\mathcal{F}}^{\pm }}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{\pm }\right){|x^{\prime }|}^2\mathrm{d}x,$$

(13)

where ${\eta }_i={\mathit{e}}_i\times \mathit{x}$ is the ith solid rotation vector, $i=1,2,3$. Then the angular velocity $\omega$ is a function of $\beta$.

We denote by $C^s$, $s>0$, $s\notin \mathbb{N}$, the Hölder space of functions f on the sphere $S_1$ with the norm

$${|f|}_{C^s\left(S_1\right)}\equiv \underset{k=\{1,...,N\}}{max}\left(\sum _{|\mathit{j}|<s}\underset{\xi \in {\zeta }_k}{sup}|{\mathcal{D}}^{\mathit{j}}f\left(\xi \right)|+\sum _{|\mathit{j}|=\left[s\right]}\underset{\xi ,\overline{\xi }\in {\zeta }_k}{sup}|\xi -\overline{\xi }{|}^{-(s-[s\left]\right)}|{\mathcal{D}}^{\mathit{j}}f\left(\xi \right)-{\mathcal{D}}^{\mathit{j}}f\left(\overline{\xi }\right)|\right),$$

where ${\mathcal{D}}^{\mathit{j}}f$ is $|\mathit{j}|$th derivative of f calculated in local coordinates on the subdomain ${\zeta }_k\subset S_1$, ${\bigcup }_{k=1}^N{\zeta }_k=S_1$. Under ${\tilde{C}}^s\left(S_1\right)$, we mean the subspace of $C^s\left(S_1\right)$ consisting of rotationally symmetric functions that are even with respect to ${\xi }_3$.

Theorem 1.

Let $\alpha >0$, $\alpha \notin \mathbb{N}$, and let the data of problem (4), (5) be such that condition (26) holds. Then for an arbitrary β satisfying the estimate

$$|\beta |<\epsilon$$

(14)

with small enough ε, there exists a unique solution $(R^{\pm },\omega ,c^{\pm })\in {\tilde{C}}^{2+\alpha }\left(S_1\right)\times {\tilde{C}}^{2+\alpha }\left(S_1\right)\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ to system (11)–(13). It obeys the inequality

$$\sum _{\pm }\left\{|R^{\pm }-R_0^{\pm }{|}_{C^{2+\alpha }\left(S_1\right)}+|c^{\pm }-{\rho }^{\pm }|\right\}+|\omega |<c|\beta |.$$

(15)

2. Proof of Theorem 1

Proof. 

In order to linearize system (11), we apply the formula for the first variation of a functional ${\delta }_{0}R\left[r\right]=\frac{\mathrm{d}}{\mathrm{d}s}R\left[sr\right]{|}_{s=0}$. According to [14,15], the first variation of twice the mean curvature of ${\mathcal{G}}^{\pm }$ with respect to the double curvature of the sphere $S_{R_0^{\pm }}$ is

$${\delta }_0\left({\mathcal{H}}^{\pm }\left[r\left(\xi \right)\right]+\frac{2}{R_0^{\pm }}\right)=\frac{1}{R_0^{{\pm }^2}}\left({\Delta }_{S_1}{r}^{\pm }+2r^{\pm }\right),$$

where ${\Delta }_{S_1}$ is the Laplace – Beltrami operator on $S_1$ and $r^{\pm }\equiv R^{\pm }-R_0^{\pm }$; ${\mathcal{G}}^{\pm }=\{x=R_0^{\pm }\xi +r^{\pm }\left(\xi \right)\mathit{N},\xi \in S_1\}$, $\mathit{N}$ is the outward normal to $S_1$. This yields

$$\begin{array}{c}\frac{{\sigma }^{-}}{{R_0^{-}}^2}\left({\Delta }_{S_1}{r}^{-}+2r^{-}\right)=k^{-}+f^{-},\xi \in S_1,\\ \frac{{\sigma }^{+}}{{R_0^{+}}^2}\left({\Delta }_{S_1}{r}^{+}+2r^{+}\right)=k^{+}+f^{+},\xi \in S_1,\end{array}$$

(16)

where $k^{-}=\frac{2{\sigma }^{-}}{R_0^{-}}-{\mathcal{P}}^{-}\left(c^{-}\right)\equiv {\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\mathcal{P}}^{-}\left(c^{-}\right)$, $k^{+}=\frac{2{\sigma }^{+}}{R_0^{+}}-{\mathcal{P}}^{+}\left(c^{+}\right)+{\mathcal{P}}^{-}\left(c^{-}\right)\equiv {\mathcal{P}}^{+}\left({\rho }^{+}\right)-{\mathcal{P}}^{+}\left(c^{+}\right)+{\mathcal{P}}^{-}\left(c^{-}\right)-{\mathcal{P}}^{-}\left({\rho }^{-}\right)$ in view of (2);

$$\begin{array}{c}{f}^{-}={\sigma }^{-}\left({\delta }_0\left({\mathcal{H}}^{-}\left(x\right)+\frac{2}{R_0^{-}}\right)-\left({\mathcal{H}}^{-}\left(x\right)+\frac{2}{R_0^{-}}\right)\right)-{\mathcal{P}}^{-}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{-}\right)\\ +{\mathcal{P}}^{-}\left(c^{-}\right),\\ f^{+}={\sigma }^{+}\left({\delta }_0\left({\mathcal{H}}^{+}\left(x\right)+\frac{2}{R_0^{+}}\right)-\left({\mathcal{H}}^{+}\left(x\right)+\frac{2}{R_0^{+}}\right)\right)-{\mathcal{P}}^{+}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{+}\right)\\ +{\mathcal{P}}^{-}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+c^{-}\right)+{\mathcal{P}}^{+}\left(c^{+}\right)-{\mathcal{P}}^{-}\left(c^{-}\right).\end{array}$$

(17)

We integrate equations (16) over $S_1$ by parts, then we get

$$\frac{2{\sigma }^{\pm }}{{R_0^{\pm }}^2}{\int }_{S_1}{r}^{\pm }\mathrm{d}\xi ={\int }_{S_1}(k^{\pm }+f^{\pm })\mathrm{d}\xi .$$

(18)

The integrals ${\int }_{S_1}{r}^{\pm }\mathrm{d}\xi$ can be expressed in terms of the differences of the volumes $|\mathcal{F}|-|B_{R_0^{-}}|$ and $|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|$, $|B_{R_0^{\pm }}|=\frac{4\pi }{3}{R_0^{\pm }}^3$. For example,

$$|\mathcal{F}|-|B_{R_0^{-}}|=\frac{1}{3}{\int }_{S_1}({R^{-}}^3-{R_0^{-}}^3)\mathrm{d}\xi ={R_0^{-}}^2{\int }_{S_1}{r}^{-}\mathrm{d}\xi +R_0^{-}{\int }_{S_1}{r^{-}}^2\mathrm{d}\xi +\frac{1}{3}{\int }_{S_1}{r^{-}}^3\mathrm{d}\xi ,$$

and hence

$$\begin{array}{c}{\int }_{S_1}{r}^{+}\mathrm{d}\xi =\frac{1}{{R_0^{+}}^2}\left(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|\right)+Q\left[r^{+}\right],\\ {\int }_{S_1}{r}^{-}\mathrm{d}\xi =\frac{1}{{R_0^{-}}^2}\left(|\mathcal{F}|-|B_{R_0^{-}}|\right)+Q\left[r^{-}\right],\end{array}$$

(19)

where $Q\left[r^{\pm }\right]\equiv -\frac{1}{R_0^{\pm }}{\int }_{S_1}{r^{\pm }}^2\mathrm{d}\xi -\frac{1}{3{R_0^{\pm }}^2}{\int }_{S_1}{r^{\pm }}^3\mathrm{d}\xi$.

We rewrite (12) as follows

$$\begin{array}{c}{m}^{+}={\int }_{S_1}\mathrm{d}\xi {\int }_0^{R^{+}\left(\xi \right)}\frac{{\omega }^2}{2}|{\xi }^{\prime }{|}^2{s}^4\mathrm{d}s+c^{+}|{\mathcal{F}}^{+}|=\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +c^{+}|{\mathcal{F}}^{+}|,\\ m^{-}={\int }_{S_1}\mathrm{d}\xi {\int }_{R^{+}\left(\xi \right)}^{R^{-}\left(\xi \right)}\frac{{\omega }^2}{2}|{\xi }^{\prime }{|}^2{s}^4\mathrm{d}s+c^{-}|{\mathcal{F}}^{-}|=\frac{{\omega }^2}{10}{\int }_{S_1}({R^{-}}^5-{R^{+}}^5)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +c^{-}|{\mathcal{F}}^{-}|,\end{array}$$

where $|{\xi }^{\prime }{|}^2={\xi }_1^2+{\xi }_2^2$. On the other hand, $m^{+}={\rho }^{+}|B_{R_0^{+}}|$, $m^{-}={\rho }^{-}(|B_{R_0^{-}}|-|B_{R_0^{+}}|)$. That’s why

$$\begin{array}{c}0=\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +{\rho }^{+}\left(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|\right)+|{\mathcal{F}}^{+}|\left(c^{+}-{\rho }^{+}\right),\\ 0=\frac{{\omega }^2}{10}{\int }_{S_1}({R^{-}}^5-{R^{+}}^5)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +{\rho }^{-}\left(|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|+|B_{R_0^{+}}|\right)+|{\mathcal{F}}^{-}|\left(c^{-}-{\rho }^{-}\right).\end{array}$$

(20)

We express $|{\mathcal{F}}^{\pm }|-|B_{R_0^{\pm }}|$ from (20) and substitute in (19). Then equalities (18) imply that

$$\begin{array}{cc}\hfill & \frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{-1}{{\rho }^{+}{R_0^{+}}^2}\left(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +|{\mathcal{F}}^{+}|\left(c^{+}-{\rho }^{+}\right)\right)+Q\left[r^{+}\right]\right\}={\int }_{S_1}(k^{+}+f^{+})\mathrm{d}\xi \hfill \end{array}$$

(21)

and

$$\frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left(\frac{1}{{R_0^{-}}^2}\left(|{\mathcal{F}}^{+}|+|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|\pm |B_{R_0^{+}}|\right)+Q\left[r^{-}\right]\right)={\int }_{S_1}(k^{-}+f^{-})\mathrm{d}\xi .$$

Finally, we have

$$\begin{array}{cc}\hfill & \frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{-1}{{\rho }^{-}{R_0^{-}}^2}\right\{\frac{{\omega }^2}{10}{\int }_{S_1}({R^{-}}^5-{R^{+}}^5)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +|{\mathcal{F}}^{-}|\left(c^{-}-{\rho }^{-}\right)\hfill \\ \hfill & +\frac{{\rho }^{-}}{{\rho }^{+}}\left(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +|{\mathcal{F}}^{+}|\left(c^{+}-{\rho }^{+}\right)\right)\}\hfill \\ \hfill & +Q\left[r^{-}\right]\}={\int }_{S_1}(k^{-}+f^{-})\mathrm{d}\xi .\hfill \end{array}$$

(22)

In order to prove the solvability of system (16), (13), (21), (22), we use implicit function theorem. We represent this system as a nonlinear vector equation:

$$\mathsf{\Phi }\left(\mathit{\phi }\right)=0,$$

(23)

where $\mathsf{\Phi }\left(\mathit{\phi }\right)=({\mathsf{\Phi }}_1^{\pm },{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3^{\pm })$, $\mathit{\phi }\equiv (r^{\pm },\omega ,{\lambda }^{\pm })$, ${\lambda }^{\pm }\equiv c^{\pm }-{\rho }^{\pm }$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1^{\pm }\left(\mathit{\phi }\right)& =\frac{{\sigma }^{\pm }}{R_0^{{\pm }^2}}\left({\Delta }_{S_1}{r}^{\pm }+2r^{\pm }\right)-k^{\pm }-f^{\pm },\hfill \\ \hfill {\mathsf{\Phi }}_2\left(\mathit{\phi }\right)& =\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_0^{R_0^{+}+r^{+}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+c^{+}\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}+r^{+}}^{R_0^{-}+r^{-}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+c^{-}\right)s^4\mathrm{d}s-\beta ,\hfill \\ \hfill {\mathsf{\Phi }}_3^{+}\left(\mathit{\phi }\right)& =\frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{1}{{\rho }^{+}{R_0^{+}}^2}\left(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +|{\mathcal{F}}^{+}|{\lambda }^{+}\right)-Q\left[r^{+}\right]\right\}\hfill \\ \hfill & +{\int }_{S_1}(k^{+}+f^{+})\mathrm{d}\xi ,\hfill \\ \hfill {\mathsf{\Phi }}_3^{-}\left(\mathit{\phi }\right)& =\frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{1}{{\rho }^{-}{R_0^{-}}^2}\right\{{\int }_{S_1}\mathrm{d}\xi {\int }_{R_0^{+}+r^{+}}^{R_0^{-}+r^{-}}\frac{{\omega }^2}{2}|{\xi }^{\prime }{|}^2{s}^4\mathrm{d}s+|{\mathcal{F}}^{-}|{\lambda }^{-}\hfill \\ \hfill & +\frac{{\rho }^{-}}{{\rho }^{+}}\left(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +|{\mathcal{F}}^{+}|{\lambda }^{+}\right)\}\hfill \\ \hfill & -Q\left[r^{-}\right]\}+{\int }_{S_1}(k^{-}+f^{-})\mathrm{d}\xi .\hfill \end{array}$$

We linearize (23) at zero and show that the derivative of ${\mathsf{\Phi }}^{\prime }$ at this point is not equal to zero. Thus,

$${\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }+\mathit{\psi }\left(\mathit{\phi }\right)=0,$$

(24)

where ${\mathsf{\Phi }}^{\prime }\left(0\right)=({{\mathsf{\Phi }}_1^{\pm }}^{\prime },{\mathsf{\Phi }}_2^{\prime },{{\mathsf{\Phi }}_3^{\pm }}^{\prime })\left(0\right)$ is the Fréchet derivative of $\mathsf{\Phi }$, that is,

$$\begin{array}{cc}\hfill {{\mathsf{\Phi }}_1^{+}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{{\sigma }^{+}}{R_0^{{+}^2}}\left({\Delta }_{S_1}{r}^{+}+2r^{+}\right)+{\rho }^{+}{\lambda }^{+}-{\rho }^{-}{\lambda }^{-},\hfill \\ \hfill {{\mathsf{\Phi }}_1^{-}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{{\sigma }^{-}}{R_0^{{-}^2}}\left({\Delta }_{S_1}{r}^{-}+2r^{-}\right)+{\rho }^{-}{\lambda }^{-},\hfill \\ \hfill {\mathsf{\Phi }}_2^{\prime }\left(0\right)\mathit{\phi }& =\omega {\rho }^{+}{\int }_{S_1}|{\xi }^{\prime }{|}^2\mathrm{d}\xi {\int }_0^{R_0^{+}}{s}^4\mathrm{d}s+\omega {\rho }^{-}{\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{-}}{s}^4\mathrm{d}s\hfill \\ \hfill & =\frac{8\pi }{15}\omega \left\{{\rho }^{+}{R_0^{+}}^5+{\rho }^{-}\left({R_0^{-}}^5-{R_0^{+}}^5\right)\right\},\hfill \\ \hfill {{\mathsf{\Phi }}_3^{+}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{2{\sigma }^{+}}{{R_0^{+}}^2}\frac{|B_{R_0^{+}}|{\lambda }^{+}}{{\rho }^{+}{R_0^{+}}^2}-4\pi {\rho }^{+}{\lambda }^{+}+4\pi {\rho }^{-}{\lambda }^{-}\hfill \\ \hfill & =4\pi \left\{\left(\frac{{\mathcal{P}}^{+}\left({\rho }^{+}\right)-{\mathcal{P}}^{-}\left({\rho }^{-}\right)}{3{\rho }^{+}}-{\rho }^{+}\right){\lambda }^{+}+{\rho }^{-}{\lambda }^{-}\right\},\hfill \\ \hfill {{\mathsf{\Phi }}_3^{-}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{2{\sigma }^{-}}{{R_0^{-}}^2}\frac{1}{{\rho }^{-}{R_0^{-}}^2}\left\{\left(|B_{R_0^{-}}|-|B_{R_0^{+}}|\right){\lambda }^{-}+\frac{{\rho }^{-}}{{\rho }^{+}}|B_{R_0^{+}}|{\lambda }^{+}\right\}-4\pi {\rho }^{-}{\lambda }^{-}\hfill \\ \hfill & =4\pi \left\{\frac{{\mathcal{P}}^{-}\left({\rho }^{-}\right){R_0^{+}}^3}{3{\rho }^{+}{R_0^{-}}^3}{\lambda }^{+}+\left(\frac{{R_0^{-}}^3-{R_0^{+}}^3}{3{\rho }^{-}{R_0^{-}}^3}{\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\rho }^{-}\right){\lambda }^{-}\right\},\hfill \end{array}$$

and $\mathit{\psi }\left(\mathit{\phi }\right)=\mathsf{\Phi }\left(\mathit{\phi }\right)-{\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }\equiv ({\mathit{\psi }}_1^{\pm },{\mathit{\psi }}_2,{\mathit{\psi }}_3^{\pm })$ with

$$\begin{array}{cc}\hfill {\mathit{\psi }}_1^{+}\left(\mathit{\phi }\right)=& {\mathcal{P}}^{+}\left(c^{+}\right)-{\mathcal{P}}^{+}\left({\rho }^{+}\right)-{\mathcal{P}}^{-}\left(c^{-}\right)+{\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\rho }^{+}{\lambda }^{+}+{\rho }^{-}{\lambda }^{-}-f^{+},\hfill \\ \hfill {\mathit{\psi }}_1^{-}\left(\mathit{\phi }\right)=& {\mathcal{P}}^{-}\left(c^{-}\right)-{\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\rho }^{-}{\lambda }^{-}-f^{-},\hfill \\ \hfill {\mathit{\psi }}_2\left(\mathit{\phi }\right)=& \omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_0^{R_0^{+}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+{\lambda }^{+}\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{+}+r^{+}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+c^{+}\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{-}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+{\lambda }^{-}\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{-}}^{R_0^{-}+r^{-}}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+c^{-}\right)s^4\mathrm{d}s-\beta ,\hfill \\ \hfill {\mathit{\psi }}_3^{+}\left(\mathit{\phi }\right)=& \frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{1}{{\rho }^{+}{R_0^{+}}^2}\left(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|){\lambda }^{+}\right)-Q\left[r^{+}\right]\right\}\hfill \\ \hfill & +{\int }_{S_1}(k^{+}-{\rho }^{+}{\lambda }^{+}+{\rho }^{-}{\lambda }^{-}+f^{+})\mathrm{d}\xi ,\hfill \\ \hfill {\mathit{\psi }}_3^{-}\left(\mathit{\phi }\right)=& \frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{1}{{\rho }^{-}{R_0^{-}}^2}\right\{\frac{{\omega }^2}{10}{\int }_{S_1}({R^{-}}^5-{R^{+}}^5)|{\xi }^{\prime }{|}^2\mathrm{d}\xi +\frac{{\rho }^{-}}{{\rho }^{+}}(\frac{{\omega }^2}{10}{\int }_{S_1}{R^{+}}^5\left(\xi \right){|{\xi }^{\prime }|}^2\mathrm{d}\xi \hfill \\ \hfill & +(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|){\lambda }^{+})+(|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|+|B_{R_0^{+}}|){\lambda }^{-}\}-Q\left[r^{-}\right]\}\hfill \\ \hfill & +{\int }_{S_1}(k^{-}-{\rho }^{-}{\lambda }^{-}+f^{-})\mathrm{d}\xi .\hfill \end{array}$$

Let $\mathcal{X}={\tilde{C}}^{2+\alpha }\times {\tilde{C}}^{2+\alpha }\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ and $\mathcal{Y}={\tilde{C}}^{\alpha }\times {\tilde{C}}^{\alpha }\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$. Using fixed point theorem, we will show that equation (23) is solvable in $\mathcal{X}.$

In [16] (see the corollary to Theorem 2.1), it was proved that for any $f\in {\tilde{C}}^{\alpha }\left(S_1\right)$ the equation

$${\Delta }_{S_1}r+2r=f$$

(25)

has a unique solution $r\in {\tilde{C}}^{2+\alpha }\left(S_1\right)$ and

$${|r|}_{C^{2+\alpha }\left(S_1\right)}\le c_1{|f|}_{C^{\alpha }\left(S_1\right)}.$$

Moreover, due to the assumptions of the theorem, the determinant of the matrix in the left hand sides of two equations ${{\mathsf{\Phi }}_3^{\pm }}^{\prime }\left(0\right)\mathit{\phi }=-{\mathit{\psi }}_3^{\pm }\left(\mathit{\phi }\right)$ is not equal to zero:

$$\left(\frac{2{\sigma }^{+}}{3{\rho }^{+}{R}_0^{+}}-{\rho }^{+}\right)\left(\frac{2{\sigma }^{-}({R_0^{-}}^3-{R_0^{+}}^3)}{3{\rho }^{-}{R_0^{-}}^4}-{\rho }^{-}\right)-\frac{2{\sigma }^{-}{\rho }^{-}{R_0^{+}}^3}{3{\rho }^{+}{R_0^{-}}^4}\ne 0.$$

(26)

(Here we have taken into account relations (8).) Hence, this matrix is invertible. Thus, the whole operator ${\mathsf{\Phi }}^{\prime }\left(0\right):\mathcal{X}\to \mathcal{Y}$ is also invertible and

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}\le c_2{\parallel {\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }\parallel }_{\mathcal{Y}},$$

where ${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}=|r^{+}{|}_{C^{2+\alpha }\left(S_1\right)}+|r^{-}{|}_{C^{2+\alpha }\left(S_1\right)}+|\omega |+|{\lambda }^{+}|+|{\lambda }^{-}|$ and

$\parallel {\mathsf{\Phi }}^{\prime }{\left(0\right)\mathit{\phi }\parallel }_{\mathcal{Y}}=|{{\mathsf{\Phi }}_1^{+}}^{\prime }{\left(0\right)\mathit{\phi }|}_{C^{\alpha }\left(S_1\right)}+|{{\mathsf{\Phi }}_1^{-}}^{\prime }{\left(0\right)\mathit{\phi }|}_{C^{\alpha }\left(S_1\right)}+|{{\mathsf{\Phi }}_2}^{\prime }\left(0\right)\mathit{\phi }|+|{{\mathsf{\Phi }}_3^{+}}^{\prime }\left(0\right)\mathit{\phi }|+|{{\mathsf{\Phi }}_3^{-}}^{\prime }\left(0\right)\mathit{\phi }|.$

Therefore, equation (24) can be written in the form:

$$\mathit{\phi }+\mathcal{A}\mathit{\psi }\left(\mathit{\phi }\right)=0,\mathcal{A}={\mathsf{\Phi }}^{\prime }{\left(0\right)}^{-1}.$$

(27)

Let us now estimate the non-linear operator $\mathit{\psi }$. The term $f^{-}$ given by (17) can be written as

$$f^{-}=-{\sigma }^{-}{\int }_0^1(1-s)\frac{{\mathrm{d}}^2}{\mathrm{d}{s}^2}{\mathcal{H}}^{-}\left[sr\right]\mathrm{d}s-{{\mathcal{P}}^{-}}^{\prime }\left({\overline{c}}^{-}\right)\frac{{\omega }^2}{2}{R^{+}}^2\left(\xi \right){|{\xi }^{\prime }|}^2,$$

(28)

where ${\mathcal{H}}^{-}\left[sr\right]$ is twice the mean curvature of the surface ${\mathcal{G}}_s^{-}$; ${\mathcal{G}}_s^{\pm }=\{x=R_0^{\pm }\xi +s{r}^{\pm }\left(\xi \right)\mathit{N},\xi \in S_1\}$, $\mathit{N}\equiv \frac{\xi }{|\xi |}$; ${\overline{c}}^{-}$ is some point of the interval $[c^{-},\frac{{\omega }^2}{2}{R^{+}}^2\left(\xi \right)|{\xi }^{\prime }{|}^2+c^{-}]$. One can write a similar formula for $f^{+}$. The terms $f^{\pm }$ have the second order of smallness with respect to $\mathit{\phi }$. After simple calculations for two values of the arguments $\mathit{\phi }=(r^{\pm },\omega ,{\lambda }^{\pm })$ and ${\mathit{\phi }}^{\prime }=({r^{\pm }}^{\prime },{\omega }^{\prime },{{\lambda }^{\pm }}^{\prime })\in \mathcal{X}$ such that

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}},{\parallel {\mathit{\phi }}^{\prime }\parallel }_{\mathcal{X}}\le \eta ,\eta \le min\{1,R_0^{+}/2,(R_0^{-}-R_0^{+})/2\},$$

it can be shown that the estimates

$$\begin{array}{cc}\hfill {\parallel \mathit{\psi }\left(\mathit{\phi }\right)\parallel }_{\mathcal{Y}}& \le {c\parallel \mathit{\phi }\parallel }_{\mathcal{X}}^{1+\alpha }+|\beta |\le c_3{\eta \parallel \mathit{\phi }\parallel }_{\mathcal{X}}^{\alpha }+|\beta |,\hfill \\ \hfill \parallel \mathit{\psi }\left(\mathit{\phi }\right)-\mathit{\psi }\left({\mathit{\phi }}^{\prime }\right){\parallel }_{\mathcal{Y}}& \le c_4{\eta }^{\alpha }{\parallel \mathit{\phi }-{\mathit{\phi }}^{\prime }\parallel }_{\mathcal{X}}\hfill \end{array}$$

hold. This implies that $\mathcal{A}\mathit{\psi }\left(\mathit{\phi }\right)$ maps the ball $\{\mathit{\phi }\in \mathcal{X}:\parallel \mathit{\phi }{\parallel }_{\mathcal{X}}\le \eta \}$ into itself and it is a contraction operator if

$$c_2(c_3{\eta }^{1+\alpha }+|\beta |)\le \eta ,c_2{c}_4{\eta }^{\alpha }<1.$$

These inequalities are satisfied for $\eta$ such that

$$2c_2|\beta |\le \eta \le min\{{\left(2c_2{c}_3\right)}^{-1/\alpha },{\left(c_2{c}_4\right)}^{-1/\alpha }\}.$$

(29)

So if

$$|\beta |\le \frac{1}{2c_2}min\{{\left(2c_2{c}_3\right)}^{-1/\alpha },{\left(c_2{c}_4\right)}^{-1/\alpha }\},$$

(30)

then, by fixed point theorem, equation (27) has a unique solution in the ball $\{\parallel \mathit{\phi }{\parallel }_{\mathcal{X}}\le \eta \}$ with $\eta$ satisfying (29). Thus, equation (23) is also uniquely solvable for $\beta$ obeying inequality (30), which gives the estimate of $\epsilon$ in condition (14).

We now prove estimate (15). Since

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}\le {\parallel \mathcal{A}\parallel \parallel \mathit{\psi }\left(\mathit{\phi }\right)\parallel }_{\mathcal{Y}}\le c_2\left(c_3{\eta }^{\alpha }{\parallel \mathit{\phi }\parallel }_{\mathcal{X}}+|\beta |\right),$$

then for $c_2{c}_3{\eta }^{\alpha }\le \frac{1}{2}$

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}\le 2c_2|\beta |,$$

which coincides with inequality (15). □

Thus, we have shown that, for certain data of the problem, there are equilibrium figures for a two-layer compressible fluid, pressure function being ${\mathcal{P}}^{\pm }=\frac{{\rho }^2\left(x\right)}{2}+p^{\pm }$ and the density $\rho$ being determined by formulas (10).

3. General case

Now, we study the general case of pressure function. We assume only that ${\mathcal{P}}^{\prime }\left(\rho \right)$ is positive and may have a jump across ${\mathcal{G}}^{+}$.

Velocity vector field remains the same as above:

$$\mathcal{V}\left(x\right)=\omega (-x_2,x_1,0).$$

It satisfies (4) together with pressure function gradient $\nabla \mathcal{P}\left(\rho \right)=\rho \frac{{\omega }^2}{2}\nabla {|x^{\prime }|}^2.$ We introduce $\mathcal{Q}\left(\rho \right)$ such that

$$\nabla \mathcal{Q}\left(\rho \right)\equiv \frac{{\mathcal{P}}^{\prime }\left(\rho \right)\nabla \rho }{\rho }=\nabla \left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2\right).$$

(31)

The function $\mathcal{Q}\left(\rho \right)={\int }_{{\rho }_1}^{\rho }\frac{{\mathcal{P}}^{\prime }\left(s\right)}{s}\mathrm{d}s$, ${\rho }_1\ge 0$. Since ${\mathcal{Q}}^{\prime }\left(\rho \right)=\frac{{\mathcal{P}}^{\prime }\left(\rho \right)}{\rho }>0$, there is an inverse function ${\mathcal{Q}}^{-1}$. And (31) implies

$$\rho ={\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{\pm }\right)\mathrm{in}{\mathcal{F}}^{\pm }$$

(32)

with arbitrary constants $C^{\pm }$.

Substituting $\rho$ into equations (9), we have

$$\begin{array}{c}{\sigma }^{-}{\mathcal{H}}^{-}\left(x\right)+{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{-}\right)\right)=0,x\in {\mathcal{G}}^{-},\\ {\sigma }^{+}{\mathcal{H}}^{+}\left(x\right)+{\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{+}\right)\right)-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{-}\right)\right)=0,x\in {\mathcal{G}}^{+}.\end{array}$$

(33)

We prescribe the masses of fluids and apply (32):

$$m^{\pm }={\int }_{{\mathcal{F}}^{\pm }}\rho \left(x\right)\mathrm{d}x\equiv {\int }_{{\mathcal{F}}^{\pm }}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{\pm }\right)\mathrm{d}x.$$

(34)

Then one has the equations for determining the constants $C^{\pm }$.

Similar, a given angular momentum $\beta$ defines the angular velocity $\omega$:

$$\beta \equiv {\int }_{{\mathcal{F}}^{+}\cup {\mathcal{F}}^{-}}\rho \left(x\right)\mathcal{V}·{\eta }_3\mathrm{d}x=\omega {\int }_{\cup {\mathcal{F}}^{\pm }}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{\pm }\right){|x^{\prime }|}^2\mathrm{d}x.$$

(35)

Let us state the main theorem.

Theorem 2.

Let $\alpha >0$, $\alpha \notin \mathbb{N}$, and let ${\mathcal{P}}^{\pm }\left(\rho \right)\in C^{\alpha }\left({\mathbb{R}}_{+}\right)$ be positive increasing functions such that equalities (2) are satisfied for it. Here ${\mathbb{R}}_{+}\equiv \{x\in \mathbb{R}|x>0\}$. We assume also that the data of problem (4), (5) are subjected to condition (47). Then for an arbitrary β satisfying the estimate

$$|\beta |<\epsilon$$

(36)

with ε small enough, there exists a unique solution $(R^{\pm },\omega ,C^{\pm })\in {\tilde{C}}^{2+\alpha }\left(S_1\right)\times {\tilde{C}}^{2+\alpha }\left(S_1\right)\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ to system (33)–(35), and the inequality

$$\sum _{\pm }\left\{|R^{\pm }-R_0^{\pm }{|}_{C^{2+\alpha }\left(S_1\right)}+|{\mathcal{Q}}^{-1}\left(C^{\pm }\right)-{\rho }^{\pm }|\right\}+|\omega |<c|\beta |$$

(37)

holds.

Proof. 

After linearisation, system (33) takes the form of (16) with

$$\begin{array}{cc}\hfill k^{-}& =\frac{2{\sigma }^{-}}{R_0^{-}}-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right)\equiv {\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right),\hfill \\ \hfill k^{+}& =\frac{2{\sigma }^{+}}{R_0^{+}}-{\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(C^{+}\right)\right)+{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right)\hfill \\ \hfill & \equiv {\mathcal{P}}^{+}\left({\rho }^{+}\right)-{\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(C^{+}\right)\right)+{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right)-{\mathcal{P}}^{-}\left({\rho }^{-}\right)\hfill \end{array}$$

(38)

due to (2) and

$$\begin{array}{c}{f}^{-}={\sigma }^{-}\left({\delta }_0\left({\mathcal{H}}^{-}\left(x\right)+\frac{2}{R_0^{-}}\right)-\left({\mathcal{H}}^{-}\left(x\right)+\frac{2}{R_0^{-}}\right)\right)-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{-}\right)\right)\\ +{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right),\\ f^{+}={\sigma }^{+}\left({\delta }_0\left({\mathcal{H}}^{+}\left(x\right)+\frac{2}{R_0^{+}}\right)-\left({\mathcal{H}}^{+}\left(x\right)+\frac{2}{R_0^{+}}\right)\right)-{\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{+}\right)\right)\\ +{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|x^{\prime }|}^2+C^{-}\right)\right)+{\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(C^{+}\right)\right)-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right).\end{array}$$

(39)

Following Sect. 2, we obtain (18) by integrating equations (16) by parts but now with new $k^{\pm }$ (38) and $f^{\pm }$ (39). We substitute (19) in equality (18):

$$\begin{array}{c}\frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left(\frac{1}{{R_0^{+}}^2}\left(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|\right)+\tilde{Q}\left[r^{+}\right]\right)={\int }_{S_1}(k^{+}+f^{+})\mathrm{d}\xi ,\\ \frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left(\frac{1}{{R_0^{-}}^2}\left(|{\mathcal{F}}^{+}|+|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|\pm |B_{R_0^{+}}|\right)+\tilde{Q}\left[r^{-}\right]\right)={\int }_{S_1}(k^{-}+f^{-})\mathrm{d}\xi ,\end{array}$$

(40)

where $\tilde{Q}\left[r^{\pm }\right]\equiv -\frac{1}{R_0^{\pm }}{\int }_{S_1}{r^{\pm }}^2\mathrm{d}\xi -\frac{1}{3{R_0^{\pm }}^2}{\int }_{S_1}{r^{\pm }}^3\mathrm{d}\xi$.

We can write (34) as follows

$$\begin{array}{c}{m}^{+}={\int }_{S_1}\mathrm{d}\xi {\int }_0^{R^{+}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s+{\mathcal{Q}}^{-1}\left(C^{+}\right)|{\mathcal{F}}^{+}|,\\ m^{-}={\int }_{S_1}\mathrm{d}\xi {\int }_{R^{+}\left(\xi \right)}^{R^{-}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)-{\mathcal{Q}}^{-1}\left(C^{-}\right)\right)s^2\mathrm{d}s+{\mathcal{Q}}^{-1}\left(C^{-}\right)|{\mathcal{F}}^{-}|.\end{array}$$

In view of (3), we have

$$\begin{array}{c}0={\int }_{S_1}\mathrm{d}\xi {\int }_0^{R^{+}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\\ +{\rho }^{+}\left(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|\right)+|{\mathcal{F}}^{+}|\left({\mathcal{Q}}^{-1}\left(C^{+}\right)-{\rho }^{+}\right),\\ 0={\int }_{S_1}\mathrm{d}\xi {\int }_{R^{+}\left(\xi \right)}^{R^{-}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)-{\mathcal{Q}}^{-1}\left(C^{-}\right)\right)s^2\mathrm{d}s\\ +{\rho }^{-}\left(|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|+|B_{R_0^{+}}|\right)+|{\mathcal{F}}^{-}|\left({\mathcal{Q}}^{-1}\left(C^{-}\right)-{\rho }^{-}\right).\end{array}$$

(41)

By expressing the differences $|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|$ and $|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|+|B_{R_0^{+}}|$ from equalities (41) and substituting them in (40), we arrive at

$$\begin{array}{cc}\hfill \frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{-1}{{\rho }^{+}{R_0^{+}}^2}\right({\int }_{S_1}\mathrm{d}\xi & {\int }_0^{R^{+}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +|{\mathcal{F}}^{+}|\left({\mathcal{Q}}^{-1}\left(C^{+}\right)-{\rho }^{+}\right))+\tilde{Q}\left[r^{+}\right]\}={\int }_{S_1}(k^{+}+f^{+})\mathrm{d}\xi \hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill & \frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{-1}{{\rho }^{-}{R_0^{-}}^2}\right\{{\int }_{S_1}\mathrm{d}\xi {\int }_{R^{+}\left(\xi \right)}^{R^{-}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)-{\mathcal{Q}}^{-1}\left(C^{-}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +|{\mathcal{F}}^{-}|\left({\mathcal{Q}}^{-1}\left(C^{-}\right)-{\rho }^{-}\right)\hfill \\ \hfill & +\frac{{\rho }^{-}}{{\rho }^{+}}({\int }_{S_1}\mathrm{d}\xi {\int }_0^{R^{+}\left(\xi \right)}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +|{\mathcal{F}}^{+}|\left({\mathcal{Q}}^{-1}\left(C^{+}\right)-{\rho }^{+}\right)\left)\right\}+\tilde{Q}\left[r^{-}\right]\}={\int }_{S_1}(k^{-}+f^{-})\mathrm{d}\xi .\hfill \end{array}$$

(43)

Next, we represent system (16), (35), (42), (43) in the form of vector equation (23) with $\mathsf{\Phi }\left(\mathit{\phi }\right)=({\mathsf{\Phi }}_1^{\pm },{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3^{\pm })$, $\mathit{\phi }\equiv (r^{\pm },\omega ,{\lambda }^{\pm })$, ${\lambda }^{\pm }\equiv {\mathcal{Q}}^{-1}\left(C^{\pm }\right)-{\rho }^{\pm }$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_1^{\pm }\left(\mathit{\phi }\right)& =\frac{{\sigma }^{\pm }}{R_0^{{\pm }^2}}\left({\Delta }_{S_1}{r}^{\pm }+2r^{\pm }\right)-k^{\pm }-f^{\pm },\hfill \\ \hfill {\mathsf{\Phi }}_2\left(\mathit{\phi }\right)& =\omega \underset{S_1}{\int }{|{\xi }^{\prime }|}^2\mathrm{d}\xi \underset{0}{\overset{R_0^{+}+r^{+}}{\int }}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega \underset{S_1}{\int }{|{\xi }^{\prime }|}^2\mathrm{d}\xi \underset{R_0^{+}+r^{+}}{\overset{R_0^{-}+r^{-}}{\int }}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)s^4\mathrm{d}s-\beta ,\hfill \\ \hfill {\mathsf{\Phi }}_3^{+}\left(\mathit{\phi }\right)& =\frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{1}{{\rho }^{+}{R_0^{+}}^2}\right(\underset{S_1}{\int }\mathrm{d}\xi \underset{0}{\overset{R_0^{+}+r^{+}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +|{\mathcal{F}}^{+}|{\lambda }^{+})-\tilde{Q}\left[r^{+}\right]\}+\underset{S_1}{\int }(k^{+}+f^{+})\mathrm{d}\xi ,\hfill \\ \hfill {\mathsf{\Phi }}_3^{-}\left(\mathit{\phi }\right)& =\frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{1}{{\rho }^{-}{R_0^{-}}^2}\right\{\underset{S_1}{\int }\mathrm{d}\xi \underset{R_0^{+}+r^{+}}{\overset{R_0^{-}+r^{-}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)-{\mathcal{Q}}^{-1}\left(C^{-}\right)\right)s^2\mathrm{d}s+|{\mathcal{F}}^{-}|{\lambda }^{-}\hfill \\ \hfill & +\frac{{\rho }^{-}}{{\rho }^{+}}(\underset{S_1}{\int }\mathrm{d}\xi \underset{0}{\overset{R_0^{+}+r^{+}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +|{\mathcal{F}}^{+}|{\lambda }^{+}\left)\right\}-\tilde{Q}\left[r^{-}\right]\}+\underset{S_1}{\int }(k^{-}+f^{-})\mathrm{d}\xi .\hfill \end{array}$$

(44)

Then we apply again implicit function theorem to (23). To this end, we calculate the Fréchet derivative of $\mathsf{\Phi }$ at zero and linearize (23) at this point. As a result, one has

$${\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }+\mathit{\psi }\left(\mathit{\phi }\right)=0$$

(45)

with ${\mathsf{\Phi }}^{\prime }\left(0\right)=({{\mathsf{\Phi }}_1^{\pm }}^{\prime },{\mathsf{\Phi }}_2^{\prime },{{\mathsf{\Phi }}_3^{\pm }}^{\prime })\left(0\right)$,

$$\begin{array}{cc}\hfill {{\mathsf{\Phi }}_1^{+}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{{\sigma }^{+}}{R_0^{{+}^2}}\left({\Delta }_{S_1}{r}^{+}+2r^{+}\right)+{\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right){\lambda }^{+}-{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-},\hfill \\ \hfill {{\mathsf{\Phi }}_1^{-}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{{\sigma }^{-}}{R_0^{{-}^2}}\left({\Delta }_{S_1}{r}^{-}+2r^{-}\right)+{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-},\hfill \\ \hfill {\mathsf{\Phi }}_2^{\prime }\left(0\right)\mathit{\phi }& =\omega {\rho }^{+}{\int }_{S_1}|{\xi }^{\prime }{|}^2\mathrm{d}\xi {\int }_0^{R_0^{+}}{s}^4\mathrm{d}s+\omega {\rho }^{-}{\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{-}}{s}^4\mathrm{d}s\hfill \\ \hfill & =\frac{8\pi }{15}\omega \left\{{\rho }^{+}{R_0^{+}}^5+{\rho }^{-}\left({R_0^{-}}^5-{R_0^{+}}^5\right)\right\},\hfill \\ \hfill {{\mathsf{\Phi }}_3^{+}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{2{\sigma }^{+}}{{R_0^{+}}^2}\frac{|B_{R_0^{+}}|{\lambda }^{+}}{{\rho }^{+}{R_0^{+}}^2}-4\pi {\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right){\lambda }^{+}+4\pi {\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}\hfill \\ \hfill & =4\pi \left\{\left(\frac{2{\sigma }^{+}}{3{\rho }^{+}{R}_0^{+}}-{\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right)\right){\lambda }^{+}+{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}\right\},\hfill \\ \hfill {{\mathsf{\Phi }}_3^{-}}^{\prime }\left(0\right)\mathit{\phi }& =\frac{2{\sigma }^{-}}{{R_0^{-}}^2}\frac{1}{{\rho }^{-}{R_0^{-}}^2}\left\{\left(|B_{R_0^{-}}|-|B_{R_0^{+}}|\right){\lambda }^{-}+\frac{{\rho }^{-}}{{\rho }^{+}}|B_{R_0^{+}}|{\lambda }^{+}\right\}-4\pi {\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}\hfill \\ \hfill & =4\pi \left\{\frac{2{\sigma }^{-}{R_0^{+}}^3}{3{\rho }^{+}{R_0^{-}}^4}{\lambda }^{+}+\left(\frac{2{\sigma }^{-}({R_0^{-}}^3-{R_0^{+}}^3)}{3{\rho }^{-}{R_0^{-}}^4}-{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right)\right){\lambda }^{-}\right\},\hfill \end{array}$$

(46)

and $\mathit{\psi }\left(\mathit{\phi }\right)=\mathsf{\Phi }\left(\mathit{\phi }\right)-{\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }\equiv \left({\mathit{\psi }}_1^{\pm }\left(\mathit{\phi }\right),{\mathit{\psi }}_2\left(\mathit{\phi }\right),{\mathit{\psi }}_3^{\pm }\left(\mathit{\phi }\right)\right)$, where

$$\begin{array}{cc}\hfill {\mathit{\psi }}_1^{+}\left(\mathit{\phi }\right)=& {\mathcal{P}}^{+}\left({\mathcal{Q}}^{-1}\left(C^{+}\right)\right)-{\mathcal{P}}^{+}\left({\rho }^{+}\right)-{\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right)+{\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right){\lambda }^{+}\hfill \\ \hfill & +{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}-f^{+},\hfill \\ \hfill {\mathit{\psi }}_1^{-}\left(\mathit{\phi }\right)=& {\mathcal{P}}^{-}\left({\mathcal{Q}}^{-1}\left(C^{-}\right)\right)-{\mathcal{P}}^{-}\left({\rho }^{-}\right)-{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}-f^{-},\hfill \\ \hfill {\mathit{\psi }}_2\left(\mathit{\phi }\right)=& \omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_0^{R_0^{+}}\left\{{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+\mathcal{Q}({\rho }^{+}+{\lambda }^{+})\right)-{\rho }^{+}\right\}{s}^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{+}+r^{+}}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+\mathcal{Q}({\rho }^{+}+{\lambda }^{+})\right)s^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{+}}^{R_0^{-}}\left\{{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+\mathcal{Q}({\rho }^{-}+{\lambda }^{-})\right)-{\rho }^{-}\right\}{s}^4\mathrm{d}s\hfill \\ \hfill & +\omega {\int }_{S_1}{|{\xi }^{\prime }|}^2\mathrm{d}\xi {\int }_{R_0^{-}}^{R_0^{-}+r^{-}}{\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+\mathcal{Q}({\rho }^{-}+{\lambda }^{-})\right)s^4\mathrm{d}s-\beta ,\hfill \\ \hfill {\mathit{\psi }}_3^{+}\left(\mathit{\phi }\right)=& \frac{2{\sigma }^{+}}{{R_0^{+}}^2}\left\{\frac{1}{{\rho }^{+}{R_0^{+}}^2}\right(\underset{S_1}{\int }\mathrm{d}\xi \underset{0}{\overset{R_0^{+}+r^{+}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|){\lambda }^{+})-\tilde{Q}\left[r^{+}\right]\}\hfill \\ \hfill & +{\int }_{S_1}\left(k^{+}-{\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right){\lambda }^{+}+{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}+f^{+}\right)\mathrm{d}\xi ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathit{\psi }}_3^{-}\left(\mathit{\phi }\right)=& \frac{2{\sigma }^{-}}{{R_0^{-}}^2}\left\{\frac{1}{{\rho }^{-}{R_0^{-}}^2}\right\{\underset{S_1}{\int }\mathrm{d}\xi \underset{R_0^{+}+r^{+}}{\overset{R_0^{-}+r^{-}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{-}\right)-{\mathcal{Q}}^{-1}\left(C^{-}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +\frac{{\rho }^{-}}{{\rho }^{+}}(\underset{S_1}{\int }\mathrm{d}\xi \underset{0}{\overset{R_0^{+}+r^{+}}{\int }}\left({\mathcal{Q}}^{-1}\left(\frac{{\omega }^2}{2}{|{\xi }^{\prime }|}^2{s}^2+C^{+}\right)-{\mathcal{Q}}^{-1}\left(C^{+}\right)\right)s^2\mathrm{d}s\hfill \\ \hfill & +(|{\mathcal{F}}^{+}|-|B_{R_0^{+}}|){\lambda }^{+})+\left(|{\mathcal{F}}^{-}|-|B_{R_0^{-}}|+|B_{R_0^{+}}|\right){\lambda }^{-}\}-\tilde{Q}\left[r^{-}\right]\}\hfill \\ \hfill & +{\int }_{S_1}\left(k^{-}-{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){\lambda }^{-}+f^{-}\right)\mathrm{d}\xi .\hfill \end{array}$$

We recall the notation: $\mathcal{X}={\tilde{C}}^{2+\alpha }\times {\tilde{C}}^{2+\alpha }\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ and $\mathcal{Y}={\tilde{C}}^{\alpha }\times {\tilde{C}}^{\alpha }\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}$. By means of fixed point theorem, we prove the solvability of system (23) with (44) in $\mathcal{X}.$

The existence of a unique solution $r\in {\tilde{C}}^{2+\alpha }\left(S_1\right)$ to two first equations in (25) and the estimate for it are discussed in Sec. 2. The operator ${\mathsf{\Phi }}_2^{\prime }\left(0\right)$ is invertible. In addition, we have assumed that the determinant of two equations ${{\mathsf{\Phi }}_3^{\pm }}^{\prime }\left(0\right)\mathit{\phi }=-{\mathit{\psi }}_3^{\pm }\left(\mathit{\phi }\right)$ with relations (46) is not equal to zero:

$$\left(\frac{2{\sigma }^{+}}{3{\rho }^{+}{R}_0^{+}}-{\mathcal{P}}^{{+}^{\prime }}\left({\rho }^{+}\right)\right)\left(\frac{2{\sigma }^{-}({R_0^{-}}^3-{R_0^{+}}^3)}{3{\rho }^{-}{R_0^{-}}^4}-{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right)\right)-\frac{2{\sigma }^{-}{\mathcal{P}}^{{-}^{\prime }}\left({\rho }^{-}\right){R_0^{+}}^3}{3{\rho }^{+}{R_0^{-}}^4}\ne 0.$$

(46)

Therefore, the vector value operator ${\mathsf{\Phi }}^{\prime }\left(0\right):\mathcal{X}\to \mathcal{Y}$ is invertible too and the solution obeys the inequality

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}\le c_2{\parallel {\mathsf{\Phi }}^{\prime }\left(0\right)\mathit{\phi }\parallel }_{\mathcal{Y}},$$

where ${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}=|r^{+}{|}_{C^{2+\alpha }\left(S_1\right)}+|r^{-}{|}_{C^{2+\alpha }\left(S_1\right)}+|\omega |+|{\lambda }^{+}|+|{\lambda }^{-}|$,

$\parallel {\mathsf{\Phi }}^{\prime }{\left(0\right)\mathit{\phi }\parallel }_{\mathcal{Y}}=|{{\mathsf{\Phi }}_1^{+}}^{\prime }{\left(0\right)\mathit{\phi }|}_{C^{\alpha }\left(S_1\right)}+|{{\mathsf{\Phi }}_1^{-}}^{\prime }{\left(0\right)\mathit{\phi }|}_{C^{\alpha }\left(S_1\right)}+|{{\mathsf{\Phi }}_2}^{\prime }\left(0\right)\mathit{\phi }|+|{{\mathsf{\Phi }}_3^{+}}^{\prime }\left(0\right)\mathit{\phi }|+|{{\mathsf{\Phi }}_3^{-}}^{\prime }\left(0\right)\mathit{\phi }|.$Hence, one can write equation (45) in the form (27).

We estimate the operator $\mathit{\psi }$ in the same way as in Sec. 2. Using formulas similar to (28) for the new functions $f^{\pm }$, we conclude again that

$$\begin{array}{cc}\hfill {\parallel \mathit{\psi }\left(\mathit{\phi }\right)\parallel }_{\mathcal{Y}}& \le {c\parallel \mathit{\phi }\parallel }_{\mathcal{X}}^{1+\alpha }+|\beta |\le c_3{\eta \parallel \mathit{\phi }\parallel }_{\mathcal{X}}^{\alpha }+|\beta |,\hfill \\ \hfill \parallel \mathit{\psi }\left(\mathit{\phi }\right)-\mathit{\psi }\left({\mathit{\phi }}^{\prime }\right){\parallel }_{\mathcal{Y}}& \le c_4{\eta }^{\alpha }{\parallel \mathit{\phi }-{\mathit{\phi }}^{\prime }\parallel }_{\mathcal{X}}\hfill \end{array}$$

for two values of the arguments $\mathit{\phi }=(r^{\pm },\omega ,{\lambda }^{\pm })$ and ${\mathit{\phi }}^{\prime }=({r^{\pm }}^{\prime },{\omega }^{\prime },{{\lambda }^{\pm }}^{\prime })\in \mathcal{X}$ such that

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}},{\parallel {\mathit{\phi }}^{\prime }\parallel }_{\mathcal{X}}\le \eta ,\eta \le min\{1,R_0^{+}/2,(R_0^{-}-R_0^{+})/2\}.$$

By repeating the arguments of Sec. 2, we deduce that $\mathcal{A}\mathit{\psi }\left(\mathit{\phi }\right)$ is a contraction operator if inequality (30) holds. Hence, fixed point theorem guarantees the existence of a unique solution to equation (27) in the ball $\{\parallel \mathit{\phi }{\parallel }_{\mathcal{X}}\le \eta \}$ with $\eta$ satisfying (29). Therefore (23) with $\mathsf{\Phi }\left(\mathit{\phi }\right)$ given by (44) is also uniquely solvable for $\beta$ such that condition (30) holds. This inequality implies an estimate for $\epsilon$ in (36).

Estimate (37) follows from

$${\parallel \mathit{\phi }\parallel }_{\mathcal{X}}\le {\parallel \mathcal{A}\parallel \parallel \mathit{\psi }\left(\mathit{\phi }\right)\parallel }_{\mathcal{Y}}\le c_2\left(c_3{\eta }^{\alpha }{\parallel \mathit{\phi }\parallel }_{\mathcal{X}}+|\beta |\right),$$

if $c_2{c}_3{\eta }^{\alpha }\le \frac{1}{2}$. □