Problem on oscillation of a body partially filled with an ideal fluid under the action of an elastic and damping forces

In this paper, we study a problem of small motions of a body partially filled with an ideal homogenous fluid under the action of an elastic damping device. The mathematical statement of the problem consists of the equation of motion of a body with an ideal fluid in a stationary coordinate system
\begin{equation*} m\ddot{x}e_1 + \rho\int\limits_{\Omega}\frac{\partial u}{\partial t} \, d\Omega + (k_0^2+ k_1^2) x e_1 + \alpha \dot{x}e_1 = k_0^2x_0 e_1+ k_1^2 x_1 e_1+ f_{\text{т}} + N(t)e_2 - g m e_2, \end{equation*}
the equations of motion of a fluid in a moving coordinate system rigidly connected with the body
\begin{equation*} \rho\left(\frac{\partial u}{\partial t} + \ddot{x} e_1^{\, (1)}\right) + \nabla p = \rho f_{\text{ж}}, \quad\quad div\, u =0 \; ({\rm in} \,\, \Omega), \end{equation*}
the boundary conditions
\begin{equation*} \begin{split} u_n = u \cdot n= 0 \; ({\rm on} \,\, S),\quad p=\rho g\zeta \; ({\rm on} \,\, \Gamma), \quad \frac{\partial\zeta}{\partial t} = u \cdot e_2^{\, (1)} \; ({\rm on} \,\, \Gamma), \quad \int\limits_{\Gamma} \zeta\, d\Gamma = 0, \end{split} \end{equation*}
and the initial conditions
\begin{equation*} x(0)=x^0, \quad \dot{x}(0) = x^1,\quad u(0,x^{\, (1)}) = u^{\,0} (x^{\, (1)}), \quad \zeta (0,x_1^{\, (1)})=\zeta^0. \end{equation*}
For given initial-boundary value problem we obtain the law of full energy balance in the differential form.
Theorem. If the problem has a classical solution, i.e. all functions in the equations, the boundary and initial conditions are continuous. Then the following equation is valid
\begin{align*} \nonumber\frac{1}{2}\frac{d}{dt}\bigg\{ m_{T} (\dot{x})^{\, 2} + \rho \int\limits_{\Omega} \big| u + \dot{x} e_1^{\, (1)}\big|^2 \, d\Omega + (k_0^2+k_1^2) x^2 + \rho g \int\limits_{\Gamma}|\zeta|^2 d\Gamma\bigg\} =\\ = -\alpha \dot{x}^{\, 2} + (f_{\text{т}} \cdot e_1)\dot{x} + \rho \int\limits_{\Omega} f_{\text{ж}}\cdot u \, d\Omega + k_0^2 x_0 \dot{x} + k_1^2 x_1 \dot{x}. \end{align*}
It is the law of full energy balance of the considered hydromechanical system.
Using necessary Hilbert spaces and subspaces we apply the method of orthogonal projection to the equation of fluid motion. Then problem can be formulated as a Cauchy problem for a first-order differential-operator equation in the orthogonal sum of Hilbert spaces. We study the properties of operator matrices and prove the theorem on existence of a unique strong solution of the final operator equation and initial boundary value problem. The main result of this work is the following theorem.
Theorem. Initial boundary value problem has a unique strong solution whenever
\begin{align*} P_{h, S} u^0 \in \mathcal{D}(\gamma_n), \quad \zeta^0 &\in \mathcal{D}(Q)=H^{1/2}_\Gamma, \\ f_{T} \in C^1 \big(\mathbb{R}_+; \mathbb{C}^2\big),\quad & f_{\text{ж}} \in C^1 \big(\mathbb{R}_+; L_2(\Omega)\big). \end{align*}