Clay Navier-Stokes Millennium Problem Solved

· fluid mechanics, mathematics

One of the 7 Clay Mathematics Institute $1 million Millennium Problems concerns existence and regularity of solutions to the Navier-Stokes equations. The problem was formulated in 2000 and I discussed certain very problematic aspects of the formulation in the following articles from 2008:

also addressed in a blog post in 2009. I suggested a reformulation of the Clay Problem into 1 million €1 problems, each one solved by computing turbulent solutions to the Navier-Stokes equations. Clay did not respond to my suggestion.

There is no progress towards a solution of the problem as formulated by Clay, but the technique of computing turbulent solutions to the Navier-Stokes has advanced substantially: It is now possible to compute virtually any Navier-Stokes problem in the most difficult case of small viscosity with solutions always partially turbulent. It is further possible to do this in an fully automated way using the free software Unicorn as a FEniCS App. One of the problems solved is presented in Computation Can Now Replace Wind Tunnel.

It is thus now possible to solve the reformulated Clay Navier-Stokes problem and harvest $1 million.

As concerns the original formulation all solutions to Navier-Stokes equations with small viscosity are turbulent, and a turbulent solution is non-smooth. The Clay NS problem thus has a negative solution: smooth solutions do not exist, in the case of small viscosity.


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  1. Amman Mahendraraja

    In what way would that show that there is a non-smooth solution with smooth initial data? I thought your work only uses floating point arithmetic.

  2. claesjohnson

    Turbulent solutions are non-smooth and Navier-Stokes solutions for small viscosity are turbulent.

  3. Amman Mahendraraja

    The property “is smooth” for realvalued functions is not recursive, and hence cannot be checked by a Turing machine. Is the Fenics system something which goes beyond the capabilities of Turing machines?

  4. claesjohnson

    To win the prize in it’s original formulation it is sufficient to exhibit one solution for smooth data which is non-smooth and for this task FEniCS is more than sufficient.

    • Matt

      Im sorry but it seems clear to me that your work is not mathematically rigorous, as Clay would expect (as well as the entire math community). Why is a discrete solver sufficient to prove RIGROROUSLY a regularity condition?

      • claesjohnson

        I don’t think the NS problem as stated is solvable, and thus some reformulation is required.
        Our idea is to give evidence both analytical and computational that smooth solutions are unstable and thus both physically and mathematically meaningless, while physically meaningful weak solutions are non-smooth.

  5. Jorma Jormakka

    Dear Claes, I wrote to you an email 2010 but you did not answer. Hope you answer now. The Clay Navier-Stokes problem was solved 2010 in a paper in the journal EJDE, search for the paper by Jorma Jormakka, Solutions to three-dimensional Navier-Stokes… July 10, 2010.
    The solution is to my knowledge correct and has not been broken. It gives a solution that cannot be continued to the whole space. The external force is a feedback force, which is allowed since it is not forbidden, since the only example gravitation is a feedback force (fluid has mass, fluid distribution influences gravitation) and in most practical applications the external force is a feedback force. The Clay problem cannot be fixed since a similar solution as i gave for the periodic case can be made for the nonperiodic.

    • claesjohnson

      Hello Jorma: I looked at your proposed solution, but I was not convinced that it solves the Cay problem.
      But I have not heard of any other solution either. It seems to me that the solution mathematicians are looking for
      does not exist.

      Best regards, Claes

  6. Jorma Jormakka

    Thanks for answering,

    Please, send me your email address. You have my email either from the registration to this web-site or from a paper in arxiv. I will send you full explanations why the paper does solve the Clay problem that I sent to Clay when two years from publishing the paper passed. They have not answered yet, except for noting that they are aware of my paper. Then I hope you answer why you think it does not solve the Clay problem. So far nobody has given any reason why my paper would not solve the Clay problem, and several mathematicians have checked the paper, so it is correct. There is a theorem that has all te assumptions and the claim as in the official problem statement, and it has a correct proof. So how in the world it would not solve the problem?

  7. Jorma Jormakka

    Thanks Claes,

    I sent to both of your emails the letter I sent to CMI.

    The simple question is this: Theorem 2.4 in my paper has the exact content of Statement D in the official problem statement (only external force is stated to be of a particular type). Theorem 2.4 is given a correct proof. Thus, Statement D is given a correct proof. This is what the official problem statement required: prove any of the Statements A, B, C or D.

    So, I proved D. What do you mean by not believing it is a solution? Give a mathematical argument, mathematics is not a question of belief.

  8. Jorma Jormakka

    Dear Claes,I am still waiting for your explanation why you are not convinced that my solution solves the Clay problem. You have stated that you are not convinved, and you have stated that there are no solutions to the Clay problem. I have published in a journal a solution. What you then directly are claiming is that my solution is wrong. You cannot state that somebosy’s solution is wrong unless you give mathematicval reasons. It is not enough to say that you are not convinced. You can check my paper in two hours. If you can point out a place that you are not convinced of, please do so. You should do so, if you consider yourself as a mathematician. I ask you to answer and give your reasons, or to withdraw your statements. The wall of silence is not an accepted strategy.

  9. Jorma Jormakka

    You have so far referred to theorems stating that strong solutions are unique. There are such valid theorems, but they have conditions that must be filled. They are not filled in the formulation of the CMI NSE problem. In the case os Statement D, as the pressure is not required to be periodic and the initial data is defined as it is, strong solutions are not unique. Indeed, the set of solutions admists a free function. See Lemma 2.1 and Theorem 2.2 in my paper.

    Your argument is thus invalid. Do you have other arguments, or are you now convinced that Theorem 2.4 in my article is correct and the CMI NSE problem is solved, i.e., Statement D is proven?

  10. Jorma Jormakka

    Dear Claes. You have not given any new arguments, so we both conclude that you have accepted that the proof of Theorem 2.4 in my paper is correct, that theorem 2.4 is Statement D, and that therefore Statement D is proven. Theorem 2.4 does not give a counterexample of the type that the PDE-community expected, but it gives a valid counterexample of unexpected type. The proof is correct since Statement D does not demand periodicity, thus strong solutions are not unique. The CMI problem statement does not exclude a force of the type in theorem 2.4, thus this force is allowed. There can be no implicit assumptions in the CMI problem statement, as it is intended for a wide public. Thank you that we now agree.

  11. Rokuhachi (@68taileddragon)

    Greetings Dr. Jormakka. I’m trying to solve the NSE problem myself, and I found your solution to be very interesting. I’m studying it right now, and so far I think you might have solved it!

    • Jorma Jormakka

      Thanks for your comment. I think it is a solution. The external force, defined as a difference between a blow-up solution and the velocity vector u, gets small when time grows only if u approaches the blow-up solution. The external force is required to become small when time grows, so when the external force fills the required conditions, u has a blow-up in finite time. No way around it. B.T.W. The Clay math. Institute refused to check my proof. It shows something of the mathematical community.

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