Complex Spaces In Hydrodynamics: Complex Navier-Stokes Equations
  The study is devoted to the development of new effective tools and methods of ana-lytical hydrodynamics, including problems of existence, smoothness and structure of laminar and turbulent flows. The main problem is complex Navier-Stokes equations and turbulence in complex spaces. The necessity of introducing complex spaces in hydrodynamics is deter-mined by the mechanism of transition of a laminar flow into a turbulent flow. The author pro-poses a non-traditional scenario of the transition: the cause of turbulence is in destruction (cessation of existence) of a laminar flow. The article contains the mathematical rationale for the necessity of development of the theory of turbulence in the complex configurational space: the complex configurational space is the natural area of existence of turbulence. Hydrody-namic flows are regarded as flows on entropy manifolds that [flows] are supported by the two symmetries: the symmetry of conservation of general entropy and the symmetry of duality of impulse representation. The new symmetry has been introduced and studied: the forminvari-ance of Helmholtz matrix of impulse density. The strict foundation has been provided for the known fact of chaotic mechanics: appearance of the new structure (a turbulent flow) is a result of interaction of two entities: dissipation and vorticity. On the deep level the phenomenology of turbulence in complex spaces is based on the transition from the mechanics of a material point to the mechanics of an oriented material point, that [transition] takes place in a current period of time. 
Anatoly N. Panchenkov