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The new Vavilov-Cherenkov radiation theory which is based on the relativistic generalization of the Landau theory for superfluid threshold velocity and Abraham theory of the electromagnetic field (EMF) in medium is represented. The new exact solution of the Cauchy problem in unbounded space is obtained for the n-dimensional Euler-Helmholtz (EH) equation in the case of a nonzero-divergence velocity field for an ideal compressible medium. The solution obtained describes the inertial vortex motion and coincides with the exact solution to the n-dimensional Hopf equation which simulates turbulence without pressure. Due to the introduction of a fairly large external friction or by introducing an arbitrary small effective volume viscosity, a new analytic solution of the Cauchy problem for the threedimensional Navier-Stokes (NS) equation is obtained for compressible flows. This gives the positive solution to the Clay problem (www.clamath.org) generalization on the compressible NS equation. This solution also gives the possibility to obtain a new class of regular solutions to the n-dimensional modification of the KuramotoSivashinsky equation, which is ordinarily used for the description of the nonlinear propagation of fronts in active media. The example for potential application of the new exact solution to the Hopf equation is considered in the connection of nonlinear geometrical optics with weak nonlinear medium at the nonlocality of the small action radii.

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory in Medium