Last week I found a way to solve the Riccati form of Navier-stokes equations using Mathematica. So what do you think? Your comments are welcome.
Article An Exact Solution of Riccati Form of Navier-Stokes Equations...
This is a good question.
To get started, see
M. Badra, Some contributions to the control and stabilization of parabolic systems in fluid mechanics (my translation of the French title), Habilitation, Universite de Pau, 2013:
http://mbadra.perso.univ-pau.fr/HDR-BADRA.pdf
Infinite dimensional Riccati equations are considered, startlng with (1.1.3) on page 12. Navuer-Stokes equations are considered in 3D and 2D, staring on page 14.
M.K. Stoyanov, Reduced Order Methods for Large Scale Ricatti Equations, Ph.D. thesis, Virgina Polytechnic Institute and State University, 2009:
http://scholar.lib.vt.edu/theses/available/etd-05212009-155950/unrestricted/MiroslavStoyanovDissertation.pdf
Navier-Stokes equations and the 2D Burgers equation are considered in Section 1.2, starting on page 2 (see, especially, Section 5.2, starting on page 80).
A good overview of linearized Navier-Stokes is given in
E. Bansch, P. Benner, J. Saak, Numerical solution of descriptor Riccati equations for linearized Navier-Stokes control problems, 2010:
http://www.birs.ca/workshops/2010/10w5029/files/Benner.pdf
Dear Dr. James F. Peters, thank you for the references. best wishes
Wolfram Mathworld says "As used in physics, the term "exact" generally refers to a solution that captures the entire physics and mathematics of a problem as opposed to one that is approximate, perturbative, etc."
One can not find an exact solution starting on equations that do not capture all the terms. I do not think NS captures all the terms. It won't be hard to find opposition to this but I'm positive about it. There are missing terms associated to the geometry that are not portrayed within NS. I got an upcoming publication on this topic.
But we tend to quote things as exact and seem to love to sit and let the equations do the job for us. This is a poor approach as equations do not supersede imagination. If one lets this approach take over there is no more truth to be uncovered and we cease to move forward.
@Yuri: thank you for your answer. I only refer to exact solution to a given problem, not exact representation of turbulence. If you have a paper, you can send me. Thanks. Best wishes
Thank you for your polite clarification. I don't think I'm able to help. I just meant to highlight some views regarding the equations but of course, can not obscure the powerful tool that NS equations are, with the limitations they may have.
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