# Vol.4 : Computational Turbulent Incompressible Flow

Dear Claes,

I attach a new paper of mine on the periodic Navier-Stokes equation. Here I show that the regularity problem is equivalent to that of asserting a global H^1 bound on solutions in terms of the H^1 bound on the initial data.

Since you are particularly fond of stability results, let me point out that this result also proves that the regularity problem is equivalent to that of global stability of the Navier-Stokes flow in the H^1 norm.

More precisely, the regularity problem is equivalent to the following assertion: if u, v are two global weak solutions in H^1 to periodic Navier-Stokes, then one has the stability estimate

sup_{0 < T < infty} || u(T) - v(T) ||_{H^1} <~ || u(0) - v(0)

 _{H^1} F( u(0) _{H^1} + v(0) _{H^1} ) (*)

for all times T, and some function F: (0,\infty) -> (0,\infty). One can deduce this result from those in my paper by standard energy arguments based on Gronwall's inequality; I am sure you are capable of filling in the details.

In particular, your stability conjecture in the case of periodic H^1 solutions is in fact _equivalent_ to the regularity conjecture.

Best,

Terry