Estimates of inverse operator kernels in problems on small vibrations of stratified viscous fluid (Q2719456)

The paper deals with the initial boundary value problem NEWLINE\[NEWLINE \begin{gathered} \begin{gathered} A\bigg(d_x, \frac{\partial}{\partial t}\bigg)U=0;\qquad U=\{U_1,U_2,U_3,U_4\}^T,\\ x\in\mathbb{R}^2_+ = \{(x_1,x_2)\: x_1\in\mathbb{R}^1,\;x_2>0\},\quad t>0; \end{gathered}\tag{1}\\ \begin{gathered} U_k(x,+0) = 0,\quad x\in\mathbb{R}^2_+,\quad k=1, 2, 3;\quad U_2(x_1,+0,t) = V(x_1,t);\\ U_3(x_1,+0,t) = p_0(x_1,t), \end{gathered}\tag{2} \end{gathered} NEWLINE\]NEWLINE where the operator \(A\) is of the form NEWLINE\[NEWLINE A = \begin{pmatrix} \frac{\partial}{\partial t} - \nu\Delta_x & 0 & 0 & \frac{\partial}{\partial x_1}\\ 0 & \frac{\partial}{\partial t} - \nu\Delta_x & g & \frac{\partial}{\partial x_2}\\ 0 & -\omega_0^2g^{-1} & \frac{\partial}{\partial t} & 0\\ \frac{\partial}{\partial x_1} & \frac{\partial}{\partial x_2} & 0 & 0 \end{pmatrix}. NEWLINE\]NEWLINE The author constructs the Green matrix for the problem (1)--(2) and studies the behaviour of the solutions as \( t\to+\infty\).