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 this paper, it is first shown that a smooth scalar pressure function
p must exist at all times t where the fluid velocity u(x,t) has a smooth spatial profile. From this, it is shown that the solution u must be smooth for as long as it exists. The proof of existence of solution arises from the fact that if a blowup occurs where the fluid velocity reaches infinite values at some point xb in finite time, then xb must be a global maximum point of |u|. At such a point, however, it is only the ∇p forces that can further accelerate the fluid. The only other force acting on it is viscosity, which retards the fluid flow at points of maximum |u|. But it is also shown that the time integral of |∇p| is finite for all points x and all times t > 0. Therefore, the momentum per unit volume of the fluid at each of these points, including xb, can only change by a finite amount over any time interval. Hence, a blowup where the fluid velocity reaches infinite values in finite time is not possible.

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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           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: svetlingeorgiev1@gmail.com, gal.davidi11@gmail.com

   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)                                         
          
                     
          
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           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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