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.		          
          
533854 bytes

                                                     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: 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)                                         
          
           
          
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.

           
          
Version
    
Filename
     
Date Posted
      
Size (bytes)
2024-07-11
ns_smooth_solutions_ver_2024-07-11.pdfJuly 12, 2024
533854
  Download
2024-06-22
ns_smooth_solutions_ver_2024-06-29.pdfJune 29, 2024
532042
  Download
2024-05-23
ns_smooth_solutions_ver_2024-05-23.pdfMay 23, 2024
456761
  Download
2024-01-10
ns_smooth_solutions_ver_2024-01-10.pdfJanuary 11, 2024
458430
  Download
040E
ns_smooth_solutions_ver_040e.pdfAugust 31, 2023
457184
  Download
040D
ns_smooth_solutions_ver_040d.pdfJune 23, 2023
457187
  Download
040C
ns_smooth_solutions_ver_040c.pdfApril 8, 2023
518710
  Download
040B
ns_smooth_solutions_ver_040b.pdfJanuary 26, 2023
517567
  Removed
040A
ns_smooth_solutions_ver_040a.pdfNovember 18, 2022
519065
  Download
040
ns_smooth_solutions_ver_040.pdfFebruary 15, 2022
489321
  Download
039
ns_smooth_solutions_ver_039.pdfFebruary 4, 2022
489438
  Download
038
ns_smooth_solutions_ver_038.pdfJanuary 20, 2022
489534
  Download
037
ns_smooth_solutions_ver_037.pdfJanuary 19, 2022
488469
  Download
036
ns_smooth_solutions_ver_036.pdfNovember 15, 2021
460662
  Download
035
ns_smooth_solutions_ver_035.pdfNovember 3, 2021
458609
  Download
034
ns_smooth_solutions_ver_034.pdfOctober 16, 2021
457083
  Download
033
ns_smooth_solutions_ver_033.pdfOctober 12, 2021
456728
  Download
032
ns_smooth_solutions_ver_032.pdfOctober 5, 2021
456216
  Download
031
ns_smooth_solutions_ver_031.pdfOctober 4, 2021
454917
  Download
030
ns_smooth_solutions_ver_030.pdfOctober 2, 2021
454917
  Download
029
ns_smooth_solutions_ver_029.pdfSeptember 29, 2021
454466
  Download
028
ns_smooth_solutions_ver_028.pdfSeptember 25, 2021
452627
  Download
027
ns_smooth_solutions_ver_027.pdfSeptember 23, 2021
459447
  Removed
026
ns_smooth_solutions_ver_026.pdfSeptember 18, 2021
453955
  Removed
025
ns_smooth_solutions_ver_025.pdfSeptember 14, 2021
458564
  Download
024
ns_smooth_solutions_ver_024.pdfJuly 24, 2021
453226
  Download
023
ns_smooth_solutions_ver_023.pdfJuly 15, 2021
453003
  Download
022
ns_smooth_solutions_ver_022.pdfJuly 7, 2021
452447
  Download
021
ns_smooth_solutions_ver_021.pdfJuly 5, 2021
454277
  Download
020
ns_smooth_solutions_ver_020.pdfJuly 3, 2021
454561
  Download
019
ns_smooth_solutions_ver_019.pdfJune 21, 2021
484161
  Download
018
ns_smooth_solutions_ver_018.pdfJune 14, 2021
495222
  Download
017
ns_smooth_solutions_ver_017.pdfJune 12, 2021
494972
  Download
016
ns_smooth_solutions_ver_016.pdfJune 6, 2021
495442
  Download
015
ns_smooth_solutions_ver_015.pdfApril 6, 2021
495385
  Download
014
ns_smooth_solutions_ver_014.pdfFebruary 10, 2021
496402
  Download
013
ns_smooth_solutions_ver_013.pdfJanuary 28, 2021
496265
  Download
012a
ns_smooth_solutions_ver_012a.pdfJanuary 7, 2021
494907
  Download
012
ns_smooth_solutions_ver_012.pdfJanuary 4, 2021
497062
  Download
011
ns_smooth_solutions_ver_011.pdfNovember 18, 2020
479069
  Download
010
ns_smooth_solutions_ver_010.pdfNovember 12, 2020
477940
  Download
009
ns_smooth_solutions_ver_009.pdfOctober 21, 2020
477405
  Download
008
ns_smooth_solutions_ver_008.pdfOctober 12, 2020
476241
  Download
007
ns_smooth_solutions_ver_007.pdfSeptember 27, 2020
477431
  Download
006
ns_smooth_solutions_ver_006.pdfSeptember 26, 2020
478705
  Download
005
ns_smooth_solutions_ver_005.pdfAugust 22, 2020
478661
  Download
004
ns_smooth_solutions_ver_004.pdfAugust 20, 2020
478466
  Download
003
ns_smooth_solutions_ver_003.pdfJuly 16, 2020
477704
  Download
002
ns_smooth_solutions_ver_002.pdfApril 14, 2020
473968
  Download
001
ns_smooth_solutions_ver_001.pdfMarch 11, 2020
458604
  Download

Please contact mathguy@navierstokessmoothsolutions.com if you have questions or comments about this paper.

 For those who have the Gnu Privacy Guard (GnuPG) on their systems and would like to secure their email communications with me, please download my GnuPG public key and send me a copy of yours.

Last Site Update: July 12, 2024