Usual tricks are: - reducing one degrees of liberty per symmetry (this can involve creating abstract symmetries); - making 2 coupled dimensions uncoupled by introducing perturbations that tends to 0; - making weired assumption on the form of the solution (definite, continuous, and derivable in all points) because if you are not in quantum mechanics, your solution rarely accepts discrete change of values);
Thus I am intuitively thinking the navier stockes solution cannot be found by a mathematician. I was pretty much expecting an approximation of the solution by a physicist because it requires the kind of free spirit accepting to make hypothesis that normally makes a mathematician have an heart attack.
The physical solutions tends to be a subset of the mathematical equations. Physicists tends to discard solutions that would involve the world exploding every time a fluid is flowing (I don't know why).
Am I the only one that noticed that Maxwell Equations are looking like a weired simplified sets of Navier Stokes equation in a case of a very weired perfect gaz? (viscosity = 0, compressibility = 0, and a weired twist about vortex (rotational seems related to vortex), and at the opposite of NS a time dependency on two fluids (E and B) that are kind of independent but dual (rot and div meaning seem to be swapped in their role for B and E), and a ... weired kind of coupling on B and E over time?)
DivE = is expressing the compressibility RotE is expressing the vorticity of E ...
And the funniest of Maxwell law is that at the end they seem to be relativistic (c is appearing in the equation of propagation and is the limit)
Man, I so would like to see a geometrical interpretation of the maxwell equation as a dual space coupling, so we can use intuition instead of math to solve it.
I so wish I had time to study this... I would really try to solve NS by first solving Maxwell. :)
I am betting the solution as for relativity is involving at least one non euclidean geometry. In fact since it involves "time" I could call it not a geometry (wich is time agnostic) but a dynametry :)
The equivalents of formal axiom of geometry describing formal solutions of canonical "dynamic space" that would be attractors of stable solutions and rules to transform them.