# Vol.4 : Computational Turbulent Incompressible Flow

Dear Terry;

> 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.

What is the function F? I see that you normalize to viscosity = 1, which effectively means that the H1 norm of initial data will blow up as the viscosity tends to zero. Is F exponential?

A main point is that a Gronwall stability estimate is meaningless if the exponential is very large, which typically is the case for small viscosity or large Reynolds number, which is the interesting case.

Since the solution goes turbulent from laminar initial data, the estimate you present can only be true with a very large F, and then it has no meaning, as far as I can see at least.

I hope I can stimulate you to look into the books Computational Turbulent Incompressible Flow and Computational Thermodynamics. I am convinced that we have something interesting to say even to analysts: The concept of weak uniqueness seems to capture the essence of turbulence, in mathematical terms. Isn't that potentially of interest?

I also hope you can take a look at our new resolution of d'Alemberts paradox. We have rewritten the Wikipedia article, to include the new resolution, and nobody is protesting, so it represents the new truth.

Best regards,

Claes