Conservative Interaction of <i>N</i> Internal Waves in Three Dimensions

Abstract

The Navier-Stokes system of equations is reduced to a system of the vorticity, continuity, Helmholtz, and Lamb-Helmholtz equations. The periodic Dirichlet problems are formulated for internal waves vanishing at infinity in the upper and lower domains. Stationary kinematic Fourier (SKF) structures, stationary exponential kinematic Fourier (SKEF) structures, stationary dynamic exponential (SDEF) Fourier structures, and SKEF-SDEF structures of three spatial variables and time are constructed in the current paper to treat kinematic and dynamic problems of the three-dimensional theory of the Newtonian flows with harmonic velocity. Two exact solutions for conservative interaction of N internal waves in three dimensions are developed by the method of decomposition in invariant structures and implemented through experimental and theoretical programming in Maple?. Main results are summarized in a global existence theorem for the strong solutions. The SKEF, SDEF, and SKEF-SDEF structures of the cumulative flows are visualized by two-parametric surface plots for six fluid-dynamic variables.