Stochastic mathematical model of internal waves

The paper studies a mathematical model of internal gravitational waves with additive “white noise”, which models the fluctuations and random heterogeneity of the medium. The mathematical model is based on the Sobolev stochastic equation, Dirichlet boundary conditions and the initial Cauchy condition. The Sobolev equation is obtained from the assumption of the propagation of waves in a uniform incompressible rotation with a constant angular velocity of the fluid. The solution to this problem is called the inertial (gyroscopic) wave, since it arises due to the Archimedes's law and under the influence of inertia forces. By “white noise” we mean the Nelson–Gliklikh derivative of the Wiener process. The study was conducted in the framework of the theory of relatively bounded operators, the theory of stochastic equations of Sobolev type and the theory of (semi) groups of operators. It is shown that the relative spectrum of the operator is bounded, and the solution of the Cauchy–Dirichlet problem for the Sobolev stochastic equation is constructed in the operator form.