måndag 28 januari 2019

Solution of the Clay Navier-Stokes Problem by Computer-Assisted Proof

This is a reminder of the resolution of the Clay Navier-Stokes Millennium Problem which I have presented together with Johan Hoffman and Johan Jansson in different forms over the years:
Hopefully our suggested resolution will now be considered by the Clay Institute.

The Clay problem concerns existence of smooth solutions to Navier-Stokes equations as formulated by Charles Fefferman. No progress towards a solution using techniques of analytical mathematics has been reported in the literature since the problem formulation in 2000.  

Terence Tao has attempted to construct by analytical methods a solution which shows blow-up towards infinite velocities in finite time and thus would give a negative answer to the question of 
existence of smooth solutions for all smooth data. But Tao has not yet (fully) succeeded. 

We suggest to seek an answer instead by a computational method in the form of Direct Finite Element Simulation DFS on a sequence of finite element meshes with mesh size tending to zero. 

DFS is a Galerkin method stabilised by weighted least squares control of the Navier-Stokes residual R(U, P) with U velocity and P pressure. DFS introduces turbulent dissipation as an effect of residual least squares stabilisation and can be seen as a solver of the Euler equations (Navier-Stokes with vanishing viscosity) with an automatic turbulence model.

DFS produces on a given mesh a piecewise linear (U, P) with residual R(U,P) which is small in a weak sense (H-1) by construction (Galerkin orthogonality). The key point is then that the residual R(U, P) shows to be large in a strong sense (L2) as an expression of non-smoothness 
of turbulent solutions.

DFS produces/constructs/computes solutions to Euler/Navier-Stokes which show to be non-smooth/turbulent even if initiated as smooth potential solutions.  

DFS thus puts on the table for inspection a sequence of approximate solutions of Navier-Stokes equations with residuals tending to zero weakly in H-1, while showing blow up in L2 as an expression of non-smoothness of turbulent flow (finite rate of turbulent dissipation). DFS offers simulation/prediction of a very large range of important engineering applications in aero and hydro-mechanics of slightly viscous flow.

DFS shows Navier-Stokes equations to have non-smooth turbulent solutions and thus gives an answer to the Clay Problem. 

Stability analysis in computational form through an associated dual solution gives the further information that mean-value quantities such as lift and drag are computable by DFS with error tending to zero as the square root of the mesh-size.  But point-wise quantities are not computable to arbitrary precision. 

Finally, stability analysis shows that any smooth solution is unstable (as conjectured by Birkhoff) and thus cannot persist over time. Potential solutions are examples of smooth solutions, which thus do not persist over time but turn into non-smooth turbulent solutions. 

Our suggested resolution of the Clay Problem is based on computing approximate solutions to Navier-Stokes/Euler equations, which show to be non-smooth turbulent. 

We thus compute solutions of Navier-Stokes equations and put them on the table for anyone to check that they are non-smooth turbulent. 

Will this convince a jury of mathematicians used to analytical methods? Is it thinkable that for example Tao would give our argument a moment of scrutiny? We argue that we contribute the following basic elements to the scientific discussion of Navier-Stokes/Euler equations in the case of slightly viscous flow:
  1. DFS computes solutions of Navier-Stokes/Euler without user-specied turbulence model. DFS thus solves the basic open problem of designing a mathematical model of turbulence.
  2. Inspection of computed solutions shows them to be non-smooth/turbulent. It is concluded that solutions of Navier-Stokes/Euler for slightly viscous flow are non-smooth turbulent, which gives the Clay problem an answer.
  3. Slightly viscous flow is identified as flow with Reynolds number larger than say $10^6$ associated with a reduction of drag resulting from delayed separation due to an effective slip boundary condition.  
Remark 1 Tao discusses Onsager's conjecture that Navier-Stokes solutions are (less smooth than) Hölder 1/3  resulting in finite rate of turbulent dissipation. DFS solutions typically show to be Hölder 1/3 with gradients $\nabla U$ blowing up like $h^{-0.5}$ with variations $h^{0.25}$ on scales of size $h^{0.75}$ consistent with weigthed least squares stabilisation $\int h\vert\nabla U\vert^2dx\sim 1$.

Here is one DFS Navier-Stokes solution put on the table for inspection showing turbulent flow with finite rate of turbulent dissipation around a jumbojet (with finite drag and lift):

Remark 2 Sabine Hossenfelder reminds us in Quanta Magazine about
  • The End of Theoretical Physics As We Know It:
  • Computer simulations and custom-built quantum analogues are changing what it means to search for the laws of nature.
Yes, computational techniques are changing the way physics is done and so also the mathematical physics of fluid mechanics and the related mathematics. The formulation, meaning and practical utility of a mathematical model for som physical phenomena, typically in the form a (differential) equation like Navier-Stokes equations,  closely connects to techniques for computing solutions and thus it is natural to expect that questions concerning the nature of solutions can be answered by computation with thus the computer offering a powerful new tool for mathematical modeling and analysis. The Clay Navier-Stokes problem can be seen as the outstanding open problem of classical continuum physics. This is the problem of predicting turbulent flow, which can now be viewed to be solved by computation.

Remark 3 The Clay problem is officially presented in the following words:
  • 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. 
We see the connection with turbulence and we see that we can indeed solve the Navier-Stokes equations and thus predict turbulent flow representing world-unique breakthrough of making NASA Vision 2030 Grand Challenge into a reality already today. We are also proud to see that our New Theory of Flight indeed unlocks secrets hidden in the equations. We thus believe that we have something substantial to contribute which is worthy of consideration. But it is a new kind of science with new standards and so reviewers must be open-minded.

Remark 4 Recall that we argue that the problem formulation by Fefferman is unfortunate by not including the aspect of well-posedness, which is very well understood by mathematicians since Hadamard to be a necessary quality for physical relevance. One can thus argue that the Navier-Stokes problem essentially concerns the question of well-posedness and that our resolution is to give this question a positive answer: Computed solutions show to be non-smooth/turbulent and as such show to be well-posed physical solutions with stable mean-value outputs (such as lift and drag persisting over time making flight possible). We also argue that a solution initiated as smooth does not have stable (mean-value) outputs persisting over time and thus is not well-posed.

We thus give a positive answer to the Clay problem formulated as a question of well-posedness.
In short: Computed solutions show to be non-smooth/turbulent and well-posed with stable outputs. A solution initialised as smooth (for example as potential flow) is unstable and develops over time into a non-smooth/turbulent solution.

It is the oscillating nature of turbulent solutions which make them well-posed as expressed by the dual solution which over an oscillating velocity field shows little growth. On the other hand, a smooth solution is not oscillating and thus can give the dual solution consistent growth into non-wellposedness.

Remark 5 With Clay problems in mind, we are led to a counterexample to the  P = NP problem in the form of Turbulent Irreversible Solutions of the Euler Equations. Take a look and see if you buy the argument!

Remark 6 DFS can be seen as an incompressible Euler solver with automatic turbulence model which produces a non-smooth turbulent solution with finite rate of turbulent dissipation. DFS applied to the compressible Euler equations typically produce solutions with dissipative (energy-consuming) shocks.  DFS for Euler thus introduces dissipation from residual stabilisation as automatic turbulence/shock modeling.

Remark 7 Recall that the discussion involves the following elements:
  1. Physics (of fluid particles): (i) Newton's 2nd Law + (ii) Incompressibility. 
  2. Mathematics: Formulation of (i) + (ii) in terms of Calculus = Navier-Stokes/Euler.
  3. Computation: DFS as solver of Navier-Stokes/Euler.
Physical fluid particles move so as to satisfy (i) and (ii), while DFS computes motion of virtual fluid particles from a mathematical principle best possible satisfaction of (i) and (ii) in the form of Navier-Stokes/Euler. 

DFS can be given an interpretation in physical terms through Navier-Stokes/Euler residuals and DFS can thus be viewed to be the physical model to be analysed, rather than the Navier-Stokes/Euler equations in conventional Calculus form which occurs in the problem formulation by Fefferman. 

Concerning the mathematical formulation of an equation/model describing som physics the technique  for solving the equation usually connects to the formulation of the equation, and so solution and formulation are intertwined and often cannot be separated. 

This is the case for Navier-Stokes/Euler where the classical approach of first seeking to formulate a turbulence model and then solve the corresponding equations, has failed. With DFS we instead directly aim at solving Navier-Stokes/Euler in original formulation without turbulence model and where the computational technique of residual stabilisation automatically produces the turbulence model.

DFS expresses best possible satisfaction of (i)+(ii) on a given mesh with piecewise linear velocity-pressure, which is not exact satisfaction. Similarly, a physical fluid can be expected to seek best possible satisfaction of (i)+(ii), which may not mean exact satisfaction of e.g. incompressibility.

We argue that DFS can be a more meaningful object of study than the Navier-Stokes/Euler equations in a conventional strong or weak formulation asking for interpretations. 

Remark 8 Our proposed solution of the Clay problem has the form of an open-source computer program (Unicorn FEniCS/HPC), which upon execution delivers a solution of Navier-Stokes/Euler showing to be non-smooth/turbulent. When executed on a sequence of automatically generated adaptive meshes with decreasing mesh size, mean-value quantities such as lift and drag are seen to converge to specific values within tolerances which can be estimated by duality, and can be made as small as desired. Residuals of computed solutions are seen to tend to zero weakly while becoming large strongly as expression of non-smoothness.

Our solution can be seen as  a computer-assisted proof similar to the celebrated computer-assisted proof of the four-color theorem and the Feit-Thompson theorem on group classification.

The FEniCS/HPC code is open-source and thus available for inspection, evaluation and testing by anyone. It uses the automated modeling of FEniCS and as such can be expected to be correct, or at least possible to be made correct.

It is thus possible to check if our claim of having solved the Clay problem is correct or not. The question is if anyone with connection to Clay is willing to make the check.

Remark 9 The standard following Prandtl is to specify the boundary condition for Navier-Stokes to be a no-slip condition $U=0$ asking both tangential and normal velocity to vanish, while for Euler only asking the normal velocity to vanish (non-penetration) leaving the tangential velocity free as a slip condition.

However, it is more natural from physical point of view to specify slip also for Navier-Stokes as an expression of small friction for slight viscosity, as mixed Dirichlet-Neumann condition. This is what we do, which opens a whole new perspective for computation without requirement of resolving thin boundary layers beyond the capacity of any forseeable  computer.  DFS thus solves Navier-Stokes /Euler with a slip boundary condition and so allows prediction of virtually any slightly viscous flow at affordable cost.

With slip we do not make a distinction between Navier-Stokes with slight viscosity and Euler with formally zero viscosity, since in the numerics it is the residual stabilisation which introduces the main viscosity/dissipation and not a vanishingly small standard viscosity term.

Altogether, we argue that we have given a resolution of the Clay problem by computation offering
  1. Accurate prediction for arbitrary data (geometry and forcing) at affordable computational cost.
  2. Understanding of the nature of solutions of Navier-Stokes/Euler from observations of computed solutions. In particular we observe that computed solutions can be described as non-smooth turbulent dissipative Euler solutions with Hölder continuity 1/3 in accordance with Onsager's conjecture and Kolmogorov's 4/5 law. 
Remark 10 Recall that we consider the official formulation by Fefferman to be incorrect from mathematical point by not including the crucial aspect of well-posednedness (here and here). It may well be that the question posed by Fefferman (existence of smooth solution for all time for all smooth data) will be impossible to answer, since mathematical techniques for proving global smoothness will remain hidden to humans together with techniques for construction of blow-up.

We suggest to reformulate the problem into a question of wellposedness of weak solutions for which a positive answer is offered by DFS.  This is the relevant question from physical point of view and then also from mathematical point of view, since Navier-Stokes is a mathematical equation with physical meaning.

Remark 11 We argue that only a notion of approximate solution of Navier-Stokes/Euler is meaningful, and this is what DFS delivers and which upon inspection shows to be non-smooth/turbulent with undetermined point-values of velocity and pressure, but with mean-value outputs such as lift and drag computable with quantitative error control.  In particular,
we argue that it does not make much sense to ask about exact solutions in a situation where solutions are non-smooth without well determined point-values and thus have the form of distributions which to be defined require the specification of a wealth/infinity of integrals weighted with smooth test functions. In short, does it make any sense to ask for exact specification of mean-values requiring a wealth of information. Isn't it more reasonable to be satisfied with specification of a piecewise linear DFS velocity-pressure, which is an approximate solution with error controled output? What more you could you ask for?

Remark 12 Our resolution includes the following ingredients:
  1. Open-source computer code FEniCS/Dolphin (about 100.000 lines) for automated discretization of the Navier-Stokes/Euler equations in standard analytical form into a system of algebraic equations in piecewise linear DFS velocity-pressure on a given finite element mesh (millions of mesh points) expressing asking residuals to vanish weakly combined with weighted least squares stabilisation.
  2. Open-source computer code PETc (about 100.000 lines) for automated computation of DFS solution.
  3. Open source computer code FEniCS/Unicorn for quality assessment of computed DFS solution as quantitative measure of accuracy of chosen (mean-value) output by computation of a dual solution expressing sensitivity of output with respect to DFS residuals. 
  4. Open-source code FEniCS/Unicorn for automated mesh adaptivity to reach specify output accuracy.  
The codes express a massive volume of analytical mathematics and execution of the codes massive volume of computational work.  Our resolution is the result of a combination of analytical mathematics and brute computational force with the goal/scope of delivering answers to "all that can be asked for".  It is not to be expected that non-linear pde-theory within the frame of Fefferman's problem formulation, can deliver anything near this volume of information.

Remark 13 There is only one notable mathematical result for Navier-Stokes/Euler in the literature and that is the existence proof by Leray from 1934 of weak solutions, however without any information on uniqueness/wellposedness. Leray gives a short mathematically simple argument based on basic energy estimate everyone knows. And after Leray basically no progress! No existence of unique strong solutions and nothing about wellposedness of weak solutions.

What we do is to continue Leray's work by (i) computing weak solutions and (ii) assessing wellposed of weak solutions. From the pictures above of computed solutions it is clear that Fefferman's question about existence of unique smooth solutions has a negative answer, and so the remaining question concerns wellposedness of weak solutions, a question we answer.  

PS For perspective browse this talk on the Clay problem by Titi, where at the end the question of computer-assisted proof is raised, and we learn that Titi believes it will take 5000 years to compute solutions to Navier-Stokes/Euler. We know that the reality today is that it takes hours.

Here is a discussion of the relevance of the problem formulation by Fefferman.