torsdag 15 maj 2014

Towards Computational Solution of Clay Navier-Stokes Problem 1

The Clay Navier-Stokes problem as formulated by Fefferman asks for a mathematical proof of (i) existence for smooth initial data of smooth solutions for all time to the incompressible Navier-Stokes equations, or (ii) blow-up of a solution in finite time. No progress towards an answer has been made since the problem was announced in 2000. It appears that the available tools of mathematical analysis by pen and paper are too crude to give an answer.

Let me here sketch (see also earlier posts) an approach based on digital computation which may give an answer. We then consider the incompressible Navier-Stokes equations in velocity $u=u_\epsilon (t,x)$ and pressure $p=p_\epsilon (t,x)$:
  • $\frac{\partial u}{\partial t}+u\cdot\nabla u +\nabla p =\epsilon\Delta u$ 
  • $\nabla\cdot u =0$
for  time $t > 0$ and $x\in\Omega$ with $\Omega$ a three-dimensional domain, subject to smooth initial data $u_\epsilon (0,x)=u^0(x)$ and and slip or no-slip boundary conditions. Here $\epsilon > 0$ is a constant viscosity, which we assume to be small compared to data ($\Omega$ and $u^0$).

Computed solutions show the following dependence on $\epsilon$ under constant data:
  1. $\Vert\epsilon^{\frac{1}{2}}\nabla u_\epsilon\Vert_{L_2(L_2)} \sim 1$
  2. $\Vert\epsilon\Delta u_\epsilon\Vert_{L_2(H^{-1})}\sim \epsilon^{\frac{1}{2}}$
  3. $\Vert\epsilon\Delta u_\epsilon\Vert_{L_2(L_2)}\sim \epsilon^{-\frac{1}{2}}$.
Here 1 reaches the upper bound of the standard energy estimate, which can be proved analytically,
which shows that $\nabla u_\epsilon$ becomes large with decreasing $\epsilon$ as a quantitative expression of non-smoothness, with 2 a variant thereof.  Also 3 expresses non-smoothness in quantitative form with $\epsilon\Delta u_\epsilon$ being small in a weak norm but large in a strong norm.

Computation thus is observed to produce solutions to the Navier-Stokes equations with increasing degree of non-smoothness as $\epsilon$ tends to zero, which can be seen as an answer to the Clay question in direction of  (ii) but not quite since the solution does not cease to exist by "blow up" and continues as a non-smooth weak solution.

Computed solutions satisfying 3 are turbulent. Mean-value outputs of turbulent solutions show small variation as the viscosity becomes small, in particular with slip. This can be seen to express weak well-posedness under variation of small viscosity, which may allow to carry the conclusion from computationally resolvable small viscosity to vanishingly small viscosity beyond computation.

We may compare with the attempt by Terence Tao to construct a non-smooth solution by pen and paper in a thought experiment, where the computation is left to the reader of a 70 page dense "computer program" expressed in analytical mathematics. We let instead the computer compute the solution following a standard (freely accessible) computer program, which allows the reader to do the same and then inspect the solution and verify 3 and thus get an answer to the Clay problem.     

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