Problems in the dynamics of flotation liquids

Abstract

This survey presents new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid. A flotation liquid is a liquid on whose surface heavy particles are floating; the particles may consist of arbitrary materials or may be particles of frozen liquid.The first part of the article considers initial-and boundary-value problems in the theory, their solvability, and the behavior of the solutions over long periods. In the second part of the survey, theorems are proved on the existence of nonlinear standing waves within the framework of an exact physical model, and both internal and free waves are considered. Also, the fundamental equations for shallow flotation waves are derived and examined.

On the coupled time-harmonic motion of a freely floating body and water covered by brash ice

“…(See, for example, Peters 1950, p. 321, who supposed both densities to be constant; more details can be found in Gabov & Sveshnikov 1991, § § 1.2 and 1.3.) The values of σ are less than or equal to one, and we assume that σ is constant throughout F; the value of this constant is defined by the water salinity.…”

“…According to his model, brash ice is considered as an infinitely thin mat whose particles do not interact. It should be noted that instead of the motion of a mat some authors speak of waves at an inertial surface (see Rhodes-Robinson 1984, Mandal & Kundu 1986 and references cited therein) or of the dynamics of flotation liquid (see Gabov & Sveshnikov 1991, where one finds the derivation of the corresponding nonlinear time-dependent boundary condition and its linearization).…”

Nonlocal boundary-value problem for linear partial differential equations unsolved with respect to the higher time derivative

“…Boundary-value problems for equations unsolved with respect to the higher time derivative are encountered in the study of small vibrations of ideal [1,2] and viscous [3] fluids in rotating vessels, in the problems of filtration of fluid in cracked rocks [4], in the analysis of small vibrations of exponentially stratified fluid in gravitational fields [5,6], etc. In the cited papers and in [7 -9], the Cauchy problem and mixed problems are studied for differential and differential-operator equations unsolved with respect to the higher derivative.…”

On the behavior at infinity of solutions to differential equations in Hilbert spaces

“…For instance, the propagation of small oscillations in a visco-elastic compressible medium is defined by an equation of the form (1) with n = 1 [1,2], and the dynamics of a stratified rotating compressible fluid is described by an equation of the form (1) with n = 2 [3,4].…”