The open letter to the Clay Mathematics Institute (CMI) «On the sixth problem of millennium»

1 To: Martin R Bridson FRS President, Clay Mathematics Institute From Solomon I. Khmelnik ORCID: https://orcid.org/0000-0002-1493-6630 The open letter to the Clay Mathematics Institute (CMI) «On the sixth problem of millennium» In 2010, I have sent my method of resolving of this problem to the Clay Mathematics Institute (CMI). I received the following reply: «CMI does not receive nor comment on proposed solutions to these problems. The proposed solution must be refereed and published in a leading mathematics journal for a period of at least two years before CMI will consider it.» I am not professional mathematician, do not work at any university (currently I am 86, my short CV can be found in my book on the Navier-Stokes equation) and know the “wish” of leading mathematics journals to publish papers by authors like me. Therefore, I understood that I cannot find such leading mathematics journal, that can publish and moreover present my work on his own initiative to the CMI. Therefore, I did not try. However, I continued my work in this direction and have published my results but not in leading mathematics journals. But in 2021, Dr. A.A. Zakharenko (the Editor of the Canadian Journal of Pure and Applied Sciences) became interested in my works and he drew Christopher's attention to them (Christopher O’Neill BA Fine Arts Media, University College Dublin, Ireland.) Christopher O'Neill has written his letter to the CMI, see Appendix 1. The response from the CMI was received in three hours, almost instantly, see Appendix 2. This meant that the CMI was familiar with my work and believed that it did not resolve the specified problem. So, I want to respond to the response from the CMI in the following open letter: The absolute advantage of the CMI is the formulation of the Millennium problems and the offer of Millennium prizes for their solution from its not infinite budget. But the very formulation of the problems should not be the prerogative of the CMI: these problems can be adjusted by scientists of our time and the scientists of future generations that will work during this millennium. After such a statement, I can be suspected of pulling out the Millennium prize with my teeth. Therefore, first of all, I say that I did not fulfill the condition for receiving the Millennium prize in the wording that is compiled in the CMI [1], namely: Existence and smoothness of the Navier-Stokes equation. (1) I have no formal right to apply for the Millennium prize. However, I do not agree with form (1) itself and with the wording of the refusal, which is given in the mentioned CMI answer, but I believe that I have actually resolved the sixth problem. But, first of all, it is necessary to say about the formulation of problem (1). I want to cite the opinion of Olga Ladyzhenskaya, who is an indisputable authority on the problems of differential equations, hydrodynamics, and the Navier-Stokes equations, see her work [4] in Appendix 3. Appendix 3 contains a translation of an excerpt from paper [3] of Appendix 3, where Ladyzhenskaya, in particular, writes the following. 2 The researcher does not need any prescriptions in advance concerning infinite or any other smoothness of solutions. One thing must be demanded, namely the uniqueness theorem should take place in the selected class of generalized solutions. Thus, according to Ladyzhenskaya, the following problem is important: Existence and uniqueness of the solution of the Navier-Stokes equation. (2) However, elsewhere on the CMI website, the problem is formulated by following way ([2] in Appendix 3): Navier–Stokes Equation This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding. It almost coincides with formulation (2) but significantly different from original formulation (1). I resolved the problem in formulation (2), not only for incompressible fluid but also for compressible fluid, as well as (with additional assumptions) for turbulent solutions (where it is not necessary to speak about the smoothness of solutions). As for the result, it is obvious. A functional with a single optimal point is found and it is shown that the necessary and sufficient condition for the optimum of this functional coincides with the Navier-Stokes equations. If someone undertakes to prove that the problem is not resolved, he must prove the fallacy of this conclusion. The last edition of my book with my solution is in [5] in Appendix 3. As for the answer from the CMI itself, it surprised me not with a denial but with an explanation of this denial. First, my task was to prove existence and uniqueness, not smoothness. Uniqueness is more important than smoothness, and more difficult to prove. Secondly, the answer assumes that I only have heuristics and numerical methods. Apparently, my method is called a heuristic because the proof does not move along a well-trodden track. However, it was obtained not by numerical but by an analytical method. Numerous programs illustrate not numerical calculation methods, but analytical results formulated in the MATLAB language. In addition, they can be used for possible applications. ------------------------------------------------------------------------------------------------- Appendices Appendix 1 The letter by Christopher O'Neill to the CMI From: Christopher O'Neill <[email protected]> Subject: Millennium Prize: Navier-Stokes Equation problem, Dr. Solomon I. Khmelnik Date: 4 August 2021 at 15:18:26 BST To: [email protected] To Naomi Kraker, I have been commissioned by the editor of the Canadian Journal of Pure and Applied Sciences, Dr. A.A. Zakharenko, to present a solution to the Navier-Stokes Equation problem. The solution 3 was devised by Dr. Solomon Khmelnik and is laid out in his book; ‘Navier-Stokes equations: On the existence and the search method for global solutions’, which accompanies this letter. The book consists of 10 Chapters and a total of 261 pages. It was first published in 2011 and is currently in its Sixth Edition. The publisher in question is Mathematics in Computer Corp. (MiC), which supplies the following contact information: BOX 15302, Bene-Ayish, Israel, 0060860 email: [email protected] This organization is in connection with the DNA Journal [http://dna.izdatelstwo.com/homeng.htm], who has a list of employees; Etkin, Valery A.; Grinshtein, Mark M.; Samokhvalov, Vladimir N.; Vilshansky, Alexander N. Currently and Dr. Khmelnik himself [1]. The journal is available in both Russian and English. As per the ‘Millenium Prize Description and Rules’, it will be left to the discretion of Clay Mathematics Institute to discern whether or not either of these publishing entities constitute a “Qualifying Outlet” [2]. Dr. Khmelnik’s solution has received increasing attention from the global mathematical community in the last number of years. Recently, a question posted on the popular internet forum Research Gate garnered over a 100 page views in a matter of days [3]. Moreover, his book has garnered 8 citations, from over 12 authors working in a wide range of physics fields [4]. Of these 8 citations, 6 have been in the last 3 years, and half in the last year alone. This indicates that Dr. Khmelnik’s work is gathering favour among the global scientific community. The reason behind this sudden uptick in citations may be attributed to the addition of “computer algorithms” to later editions of his book. Indeed, a full quarter of the most recent citations appear in the Journal of Applied and Computational Mechanics [https://jacm.scu.ac.ir/], which is a recognized by and registered with the DOAJ (Directory of Open Access Journals) [5][6]. This suggests that the method is finding favour with physicists and mathematicians who are working in Computer Science first and foremost. One of these papers describes how Dr. Khmelnik’s solution to the Navier-Stokes equation can be used in conjunction with the equation for convective heat transfer to provide “the mathematical model of the processes of heat and mass transfer at PCM phase transformations” [6]. The inclusion of this equation and the computational simulations accompanying it strongly suggest the realworld applications of Khmelnik’s method. According to one source, the Navier-Stokes equations are described as “the extremum conditions of some functional” with Khmelnik’s method for finding a solution to them being described as “the gradient motion to the extremum of this functional”[7]. Sincerely, Christopher O’Neill BA Fine Arts Media, University College Dublin, Ireland References: [1] http://www.hmel.iri-as.org/index_e.html [2] https://www.claymath.org/sites/default/files/millennium_prize_rules_0.pdf [3] https://www.researchgate.net/post/The_method_of_resolving_of_the_NavierStokes_equations_by_Dr_Khmelnik#view=60ff125060b20020db459865 [4] https://www.semanticscholar.org/paper/Navier-Stokes-equations%3A-On-the-existence-andthe-Khmelnik/a1becbda4fce352f19f048354dacd0cc1171bc30?sort=relevance 4 [5] Gorobets, V., Trokhaniak, V., Bohdan, Y., & Antypov, I. (2020). Numerical Modeling of Heat Transfer and Hydrodynamics in Compact Shifted Arrangement Small Diameter Tube Bundles. Applied and Computational Mechanics. [6] Antypov, I., Gorobets, V., & Trokhaniak, V. (2020). Experimental and Numerical Investigation of Heat and Mass Transfer Processes for Determining the Optimal Design of an Accumulator with Phase Transformations. Applied and Computational Mechanics. [7] Fedosin, S. G. (2014). Four-dimensional equation of motion for viscous compressible and charged fluid with regard to the acceleration field, pressure field and dissipation field. International Journal of Thermodynamics, 18, 13-24. ------------------------------------------------------------------------------------------ Appendix 2 The answer-letter from the CMI On 4 Aug 2021, at 18:47, Martin Bridson <[email protected]> wrote: Dear Mr O'Neill I am replying to your letter on behalf of CMI. Nothing that I say should be interpreted in any way as indicating that CMI will deviate from the published rules for the millennium prizes, which are available on our website. The Navier Stokes problem, as described in the official problem description, asks for rigorous mathematical proof of the existence and smoothness of solutions to the Navier Stokes equations on R^3, the 3-torus R^3/Z^3, or else a proof of breakdown of solutions as described in (C) or (D) of the problem statement. *Rigorous mathematical proof* is key. Nothing short of this is relevant in the context of the millennium problem, although heuristics and numerics, for example, might be of interest to many studying fluid flows in science and engineering. The contents of Dr. Khmelnik's book do not alter the fact that the millennium problem concerning existence and smoothness for the Navier Stokes equations remains open. Yours sincerely, Martin Bridson Martin R Bridson FRS President, Clay Mathematics Institute http://www.claymath.org ------------------------------------------------------------------------------------------------- Appendix 3. O.A. Ladyzhenskaya. Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness. [3] At the middle of 2000 the site of the Clay Mathematics Institute (http://claymath.org/) has published one article called “Millennium Prize Problems”. This article identifies seven problems called the problems of the new millennium. Each of the problems has its own subsection. The sixth subsection in [1] was entitled “Navier-Stokes, existence and regularity” (now this title was changed), where it is concerned only on incompressible fluids. The author of the article (Charles L. Fefferman, the Princeton University, USA) The author confines himself to the two simplest problems for the Navier-Stokes equations: the Cauchy problem and the problem with boundary conditions periodic in spatial variables. In these simplest problems, the fluid fills the entire space. There are no obstacles (bodies) on the way of the fluid. These problems reflect some of the 5 difficulties that exist in the initial boundary value problems for fluids that fill only part of the space Rn, n = 2 or 3. Charles Fefferman considers physically meaningful only those solutions that are infinitely smooth functions and, in the case of the Cauchy problem, decreasing faster than any degree of x. For both tasks, the following two questions are posed: 1) has it a solution of the specified type for all the time moments t Є R1+ = [0, ∞) or 2) is there a finite point in time t1, at which the required smoothness is violated? In this case, the given parameters of the problem (the initial velocity field and external forces) are assumed to be infinitely smooth and rapidly decreasing at x or (x, t) approaching an infinity. With my almost fifty years of experience in the study of Navier-Stokes equations and more than half a century of experience in the study of boundary value and initial boundary value problems for linear and nonlinear partial differential equations of various types, I would formulate the main question regarding the Navier-Stokes equations in a completely different way, namely: Problem 1. Can or cannot the Navier–Stokes equations together with initial and boundary conditions give deterministic description of the dynamics of an incompressible fluid? In this article, I want to bring to a wide range of mathematicians what it is enough to do for the Navier-Stokes equations to give an answer to Problem 1. Impatient reader or those who like to read works (books) in reverse order can get acquainted with my conclusions in the "Conclusion" section at the end of Paragraph 4. When solving Problem 1, the choice of the phase space and the class of generalized solutions should be left to the researcher, not prescribe to the researcher in advance an infinite or any other smoothness of solutions. One thing must be demanded: the uniqueness theorem should take place in the selected class of generalized solutions. It is advisable to start the study of any initial boundary value problem (as well as the Cauchy problem) by finding uniqueness classes. References: 1. Charles L. Fefferman. Existence & smoothness of the Navier-Stokes equation, http://www.claymath.org/sites/default/files/navierstokes.pdf 2. Navier-Stokes equation, http://www.claymath.org/millennium-problems 3. O.A. Ladyzhenskaya. Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness. Russian Mathematical Surveys, 2003, volume 58, issue 2(350), http://www.mathnet.ru/links/5f7064e54fb50225e7443f91783364c8/rm610.pdf 4. Olga Ladyzhenskaya, https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya 5. Khmelnik Solomon. Navier-Stokes equations. On the existence and the search method for global solutions. Version 3, pp. 1–129. "MiC" - Mathematics in Computer Corp., Israel, https://doi.org/10.5281/zenodo.5037579