Small motions of an ideal stratified fluid in a basin covered with ice

We study the problem on small motions of an ideal stratified fluid with a free surface partially covered with crushed ice. The crushed ice is supposed to be ponderable particles of some matter floating on the free surface. These particles do not interact with each other during oscillations of the free boundary (or this interaction is neglible) and stay on the surface during these oscillations. Using the method of orthogonal projecting of boundary-value conditions on the free surface and introducing auxiliary problems, we reduce the original initial-boundary value problem to the equivalent Cauchy problem for a second-order differential equation in some Hilbert space. We obtain conditions under which there exists a strong with respect to time solution of the initial-boundary value problem describing the evolution of this hydraulic system.