Proposed Proof of Existence and Smoothness of Solutions to the
3-Dimensional Driving-Force Free Navier-Stokes Equation


One of the seven Millennium Problems sponsored by the Clay Mathematics Institute (CMI) involves proving existence and smoothness of solutions of the Navier-Stokes (NS) equation; or showing a breakdown thereof. In sponsoring this problem, the CMI hopes to make substanial progress toward a mathematical theory that will aid in understanding fluid motion and possibly explain both laminar and turbulent flow. Currently, these two types of flow must be handled separately when analyzing fluid motion.

In late January 2020, I completed a paper that I believe shows that finite energy, zero driving-force solutions of the 3-dimensional NS equation must have smooth solutions (ie. finite and possessing spatial derivatives to all orders at all points), provided the initial conditions are also smooth. This paper, if correct, would prove statement (A) of the Official Problem Description from CMI for the Navier-Stokes Millennium Problem. After several unsuccessful attempts to get this work published in a refereed journal or even a moderated archive, however, I believe that the only way others may see this paper is to archive it on this website. Please click on the link below to download the most recent version of this paper.

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                                                     Copyright Notice:  This paper may be downloaded and copied, in part or in whole, provided that the author’s name and email address is included with each such copy or excerpt. This information is already contained in the PDF files for the paper available for download.                                                     

           Below are direct links to a few references used in the paper. These were taken from various online archives and may be difficult to find or only available for a limited time.          

                A. Tsionskiy, M. Tsionskiy Existence, Uniqueness, and Smoothness of Solution for 3D Navier-Stokes Equations with Any Smooth Initial Velocity, arXiv:1201.1609v8 [math.AP]    1 September 2013.

    Download (287993 bytes)                                         
              Mihir Kumar Jha, The complete Solution for existence and smoothness of Navier-Stokes equation, Global Academy of Technology, Karnataka, India ‐ 560098

   Download (294912 bytes)                                         
              Svetlin G. Georgiev and Gal Davidi, Existence and Smoothness of Navier-Stokes Equations, emails:,

   Download (620721 bytes)                                         
              Asset A. Durmagambetov, Leyla S. Fazilova, Navier-Stokes Equations ‐ Millennium Prize Problems, System Research “Factor” Company, Astana, Kazakhstan.

   Download (868446 bytes)                                         
              A. Prástaro, Geometry of PDE's. IV: Navier-Stokes equation and integral bordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140-1151. DOI: 10.1016/j.jmaa.2007.06.009. MR2386488(2009j:58028); Zbl 1135.35064]   Download (242521 bytes)                                         

           The proof arises from the fact that the scalar pressure gradient ∇p is a solution of Poisson’s equation where the inhomogeneous term (a function called Q in the paper) depends primarily on the first spatial derivatives of the fluid velocity solution u. From elementary potential theory, the solution of this equation which vanishes in the limit of large values of |x| is given by the Poisson integral. In this paper, we show that the time integral of |Q| and therefore the time integral of |∇p| remain continuous and finite for finite values of t. Now, whenever a new (increased) global maximum of the kinetic energy density K = (1/2) u • u  forms, the point x* at which it forms becomes a maximum point of the smooth function K. Therefore, we must have ∇K(x*) = 0 and the second derivatives of K with respect to each of the xi must be zero or negative at this point. With these conditions necessary for a maximum point of a smooth function, the viscosity forces acting on the fluid can only oppose an increase in K at this point. The scalar pressure gradient ∇p is the only force that can act in the correct direction to cause K to increase at it s global maximum. But since the time integral of |∇p| is finite for all points x, it follows that the momentum per unit volume of the fluid at each of these points can only change by a finite amount for finite values of t. Then, since the total energy of fluid motion approaches zero as t approaches infinity, the solution u must be bounded and smooth for all t.

Finally, I have a suggestion for those interested in reviewing this paper which I believe would expedite the process. Although this is a new paper, two sections were taken from previous publication attempts that were not part of the reviewers concerns in declining the articles for publication. These are the sections titled
Existence and Spatial Dependence of Scalar Pressure Function and Spatial Dependence of Solution. Since these sections have already undergone extensive review by myself and the reviewers, they can probably be taken at face value for an initial review. Their primary purpose is to verify that the solutions ui are of finite energy and comply with the boundary conditions “at infinity”. (Not only that, but it is in these sections where the most grueling equations and inequalities are located. One could easily get bogged down for a long, long time.) Also, the uniqueness proof was found to be generally correct.

It is the following two sections titled
Existence of Pressure Gradient Integral over Time and Existence and Smoothness of Solution over Time that contain my original thoughts that are unique to this paper and probably should be the first to get the “fine tooth comb” review. If my proof does contain an error or fallacy, it would most likely be found in one of these sections.

Download Previous Versions
           Since I first posted this website and my paper in March 2020, I have on occasions taken a fresh look at this work and discovered corrections that needed to be made and explanations that should be clarified. For completion, I decided that these previous versions should also be posted along with the most recent version. Therefore, if you have downloaded this paper and find it difficult to follow in places, you may want to check to see if you have the latest version.

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ns_smooth_solutions_ver_019.pdfJune 15, 2021
ns_smooth_solutions_ver_018.pdfJune 14, 2021
ns_smooth_solutions_ver_017.pdfJune 12, 2021
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ns_smooth_solutions_ver_012a.pdfJanuary 7, 2021
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ns_smooth_solutions_ver_002.pdfApril 14, 2020
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Last Site Update: June 15, 2021