tisdag 10 december 2013

A Variant of the Ill-Posed Clay Navier-Stokes Millennium Problem

I have long argued that the Clay Navier-Stokes Millennium Problem is ill-posed (here and here and previous post) and as such does not have a good answer. The consequence is that nobody will ever get the prize for solving the Navier-Stokes problem, which was not the intention of Clay.

To illustrate the ill-posed nature of the Navier-Stokes problem, let me consider the following related problem which could have been a Clay problem similar to the Navier-Stokes problem: Prove (A) or (B) with

(A) Existence and Uniqueness of Smooth Solutions to the Backward Heat Equation: The backward heat equation with smooth initial data admits a unique smooth solution over finite time.

(B = not A) Breakdown of Backward Heat Equation Solution:  The backward heat equation with smooth initial data does not admit a unique smooth solution over finite time.

To attempt a solution consider the backward heat equation in one space dimension on the interval $(0,\pi )$: Find a smooth function $u(x,t)$ satisfying
  • $\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=$ for $0< x < \pi$ and $0 < t < 1$,
  • $u(x,0) = u^0(x)$ for $0 < x < \pi$,
where the initial data $u^0(x)$ is given by
  • $u^0(x)=\sum_{n=1}^{\infty}c_n\sin(nx)$,
with certain coefficients $c_n$ rapidly decaying to zero as $n$ tends to infinity. Formally the solution $u(x,t)$ is given by the solution formula
  • $u(x,t)=\sum_{n=1}^{\infty}c_n\exp(tn^2)\sin(nx)$. 
where the series converges along with a certain number of derivatives if $c_n$ tends to zero with $n$ sufficiently fast, for example, $c_n\le \exp(n^2)n^{- N}$ for some natural number $N$. 

We could view the formula as a proof of (A), that is, existence of a unique smooth solution for sufficiently smooth initial data. On the other hand, we know (with Hadamard) that the backward heat equation is ill-posed in the sense that an infinitesimal perturbation of the initial data $u^0$ will make the solution blow up (or break down) and thus (B = not A) must hold. We thus have evidence of both (A) and (not A), which shows that the problem formulation is such that no answer can be given.

Unfortunately for mr Clay, the formulation of the Navier-Stokes problem suffers from the same defect.  In other words, the problem formulation is ill-posed and should be reformulated to make sense, to mr Clay and the mathematical world.  It is difficult to understand why the insight of Hadamard was completely neglected by Charles Fefferman in his problem formulation, very difficult.

PS Of course my article was rejected and Fefferman did not want to discuss the issue.

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