onsdag 6 februari 2019

Wellposedness of Navier-Stokes/Euler: Clay Problem

This is a continuation of the previous post proposing a resolution of the Clay Navier-Stokes Millennium Problem with further remarks on the aspect of wellposedness identified by
Hadamard in 1902 as being necessary in order for a mathematical model to have physical meaning and relevance. The Navier-Stokes equations serve as the basic mathematical model of fluid mechanics and the Clay Problem can be viewed to reduce to the question of wellposedness, since the existence of (weak) solutions was established by Leray in 1932.

And this is the question we give an answer: We show that weak solutions are computable (exist) and are non-smooth/turbulent with wellposed mean value outputs. We do this by solving a (dual) linearized problem with certain data and show a bound of the dual solution (here for lift of a jumbojet) in terms of the data, which we refer to as assessment of stability, and which translates to an error bound on output of a computed solution in terms of its Navier-Stokes residual, showing that the output is well determined under the presence of small disturbance.

The dual linearized problem has a reaction term with coefficient $\nabla u$ with $u$ a computed velocity. The reaction term drives both exponential growth and decay with its trace being zero by incompressibility. The wellposedness of  computed turbulent solutions is reflected by cancellation effects from the reaction term with exponential growth balanced by exponential deacy from  oscillations of turbulent solutions.

We thus argue that we have resolved the Clay Problem by showing that weak solutions are computable/exist and show to be non-smooth/turbulent with wellposed mean-value outputs. In particular we show that lift and drag are wellposed and thus reveal the secret of flight.

It remains to be seen if our resolution will be accepted by the group of pure mathematicians owning the problem including Charles Fefferman responsible for the official problem formulation, Peter Constantin and Terence Tao. One thing is notable: Fefferman’s formulation does not involve the aspect of wellposedness and so missses the heart of the problem, if Navier-Stokes is viewed as a mathematical model of fluid mechanics, which is clearly emphasized in the official problem presentation:
  • Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
All of this is presented in detail in this book supplied as evidence to the Clay problem committee with complementing material listed in the previous post. In particular the book contains a study of the (dual) linearized Navier-Stokes/Euler equations, a topic which for some reason has not attracted the attention of mathematicians despite its fundamental importance from mathematical point of view. In short, we feel that we have made substantial progress toward a mathematical theory which unlocks the secrets vidden in the Navier-Stokes equations, including the Secret of Flight.

Concerning the view of the problem committee recall the opening statement in the opening article Euler Equations, Navier-Stokes Equations and Turbulence by Peter Constantin
(in this book):
  • In 2004 the mathematical world will mark 120 years since the advent of turbulence theory. In his 1884 paper Reynolds introduced the decomposition of turbulent flow into mean and fluctuation and derived the equations that describe the interaction between them. The Reynolds equations are still a riddle. They are based on the Navier-Stokes equations, which are a still a mystery. The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence is a riddle wrapped in a mystery inside an enigma.
In other words, total confusion in the committee in charge of problem formulation and evaluation of proposed resolutions. In particular, Fefferman formulates the problem as the questions of existence and smoothness, forgetting wellposedness, and claims that his problem was solved by standard pde-theory long ago in the case of two space dimensions and evidently has in mind a similar resolution in three dimensions by som ingenious new estimate derived by a clever pure mathematician. But wellposedness is essential also in two space dimensions and so Fefferman exposes the gulf between pure mathematics and mathematics of fluid mechanics, which is not helpful to science.

Fefferman would probably say that wellposedness is a consequence of smoothness, but this is not necessarily so since assessment of smoothness may involve stability factors of arbitrary size and so may say nothing about wellposedness.  But of course questions like this have to remain in the mist since the problem committee is not open to any form of discussion.

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