9

I know that existence and uniqueness for incompressible viscous flow in the 2-D case has already been established$^1$, and that doing the same for the 3-D case has yet to be shown. Not only that, but it's one of the hardest problems out there in mathematics today.

Clearly then, the 2-D case is considerably easier than the 3-D case, but why is this so? As quoted from the Navier-Stokes Millenium Problem's problem statement,

"This gives no hint about the three-dimensional case, since the main difficulties are absent in two dimensions."

What are these main difficulties? I'm looking for any answers, going as deep into any subjects as necessary. I'm ready to read long and deeply for this.

P.S: I've only started self-studying basic partial-differential equations, but I'm really interested in numerical (and analytic too) methods for solving PDE's.

[1] Ladyzhenskaya, Olga A. Mathematics and Its Applications : The Mathematical Theory of Viscous Incompressible Flow. 2nd ed. 2. Camberwell, Australia: Gordon and Breach Science Publishers, 1969. 224. Print.

Arturo don Juan
  • 4,098

2 Answers2

1

Not really a direct answer to your question, but Terry Tao has a splendid article on the Navier-Stokes Millenium problem, accessible via arXiv:

"Localisation and compactness properties of the Navier-Stokes global regularity problem"

Jose Arnaldo Bebita Dris
  • 8,471
1

In two dimensions it is possible to take advantage of the "enstrophy miracle". This is not possible in three dimensions.

user190649
  • 26