onsdag 20 april 2016

Turbulent Euler Solutions and the Clay Navier-Stokes Problem 1

Turbulent solutions of the incompressible Navier-Stokes equations with viscosity $\nu >0$ can be characterised as having substantial turbulent dissipation, that is, satisfying for all sufficiently small positive $\nu$ with normalisation of the velocity $u$ (for $t>0$ say):
  • $\int\nu\vert \nabla u(x,t)\vert^2\, dx > C$ 
where $C$ is a positive constant.

Dimension analysis suggests that turbulent solutions are non-smooth Hölder continuous with exponent 1/3 on a smallest scale in space of size $\nu^{\frac{3}{4}}$ with $\vert\nabla u\vert\sim \nu^{-1/2}$. 

We view such solutions as approximate weak solutions of the Euler equations (formally corresponding to $\nu =0$), or turbulent Euler solutions, thus characterised by substantial turbulent dissipation. Stability analysis and computation strongly suggest that all smooth solutions to the Navier-Stokes with small $\nu$ and the Euler equations, become turbulent over time, see Computational Turbulent Incompressible Flow.

Terence Tao struggles to analytically construct solutions to the incompressible Euler equations with blow up in finite time, which could possibly show blow-up also for Navier-Sokes,  but does "not fully achieve" the goal, which is to answer the Clay Navier-Stokes problem. 

Let us compare our approach based on stability analysis/computation with that of Tao based on analytical construction of solution with blow-up. We thus give evidence that (i) turbulent solutions can be computed over global time, (ii) all smooth solutions become turbulent because of inherent instability, while Tao seeks to (iii) construct a very specific solution with blow-up for Euler and Navier-Stokes.

We see that our approach is complementary to that of Tao, or the other way around: (i)-(ii) concerns the general problem and gives life to solutions after blow-up as turbulent solutions, while (iii) concerns a very specific problem without life after blow-up.  

The evidence of (i)-(ii) consists of stability analysis + high performance computation, while (iii) is 
based on analytical computation by hand.  It may be that (i)-(iii) together capture the core aspects of the Clay Navier-Stokes problem using different forms of mathematics and "proofs".   

Inga kommentarer:

Skicka en kommentar