{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T01:46:09Z","timestamp":1787881569431,"version":"build-2784847793"},"reference-count":28,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,26]],"date-time":"2022-02-26T00:00:00Z","timestamp":1645833600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (\u22072u(x, y, z) = Fx (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where \u22072u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.<\/jats:p>","DOI":"10.3390\/e24030339","type":"journal-article","created":{"date-parts":[[2022,2,27]],"date-time":"2022-02-27T20:46:17Z","timestamp":1645994777000},"page":"339","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["No Existence and Smoothness of Solution of the Navier-Stokes Equation"],"prefix":"10.3390","volume":"24","author":[{"given":"Hua-Shu","family":"Dou","sequence":"first","affiliation":[{"name":"Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,26]]},"reference":[{"key":"ref_1","unstructured":"Foias, C., Manley, O., Rosa, R., and Temam, R. (2004). Navier-Stokes Equations and Turbulence, Cambridge University Press."},{"key":"ref_2","unstructured":"Fefferman, C.L. (2022, February 23). Existence and Smoothness of the Navier-Stokes Equation, Available online: http:\/\/www.claymath.org\/sites\/default\/files\/navierstokes.pdf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF02547354","article-title":"Sur le mouvement d\u2019un liquide visquex emplissent l\u2019espace","volume":"63","author":"Leray","year":"1934","journal-title":"Acta Math. J."},{"key":"ref_4","unstructured":"Dou, H.-S. (July, January 30). Energy gradient theory of hydrodynamic instability. Proceedings of the Third International Conference on Nonlinear Science, Singapore. Available online: https:\/\/www.researchgate.net\/publication\/2147222."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.ijnonlinmec.2005.12.002","article-title":"Mechanism of flow instability and transition to turbulence","volume":"41","author":"Dou","year":"2006","journal-title":"Inter.J. Non-Linear Mech."},{"key":"ref_6","first-page":"1411","article-title":"Physics of flow instability and turbulent transition in shear flows","volume":"6","author":"Dou","year":"2011","journal-title":"Inter. J. Phys. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"165","DOI":"10.4208\/aamm.10-m1017","article-title":"Investigation of turbulent transition in plane Couette flows using energy gradient method","volume":"3","author":"Dou","year":"2011","journal-title":"Advances in Appl.Math. and Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1422","DOI":"10.1016\/j.ijthermalsci.2007.12.012","article-title":"Instability of Taylor-Couette Flow between Concentric Rotating Cylinders","volume":"47","author":"Dou","year":"2008","journal-title":"Inter. J. Thermal.Sci."},{"key":"ref_9","first-page":"527","article-title":"Singularity of Navier-Stokes equations leading to turbulence, Advances in Applied Mathematics and Mechanics","volume":"13","author":"Dou","year":"2021","journal-title":"arXiv Prepr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/BF01212349","article-title":"Remarks on the breakdown of smooth solutions for the 3D Euler equations","volume":"94","author":"Beale","year":"1984","journal-title":"Commun. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"R2","DOI":"10.1017\/jfm.2020.58","article-title":"On singularity formation via viscous vortex reconnection","volume":"888","author":"Yao","year":"2020","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1017\/jfm.2019.263","article-title":"Towards a finite-time singularity of the Navier-Stokes equations. Part 2. Vortex reconnection and singularity evasion","volume":"870","author":"Moffatt","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"110502","DOI":"10.1103\/PhysRevFluids.4.110502","article-title":"Singularities in fluid mechanics","volume":"4","author":"Moffatt","year":"2019","journal-title":"Phys. Rev. Fluids"},{"key":"ref_14","unstructured":"Schlichting, H. (1979). Boundary Layer Theory, Springer. [7th Ed.]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1017\/S0022112075003254","article-title":"An experimental investigation of the stability of plane Poiseuille flow","volume":"72","author":"Nishioka","year":"1975","journal-title":"J. Fluid Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1017\/S002211208400241X","article-title":"Final stages of transition to turbulence in plane channel flow","volume":"148","author":"Biringen","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1017\/S002211209200048X","article-title":"The late stages of transition to turbulence in channel flow","volume":"245","author":"Sandham","year":"1992","journal-title":"J. Fluid Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1360\/04yw0168","article-title":"Inherent mechanism of breakdown in laminar-turbulent transition of plane channel flows, Science in China Ser. G Physics","volume":"48","author":"Luo","year":"2005","journal-title":"Mech. Astron."},{"key":"ref_19","first-page":"524","article-title":"Timescale and structure of ejections and bursts in turbulent channel flows","volume":"174","author":"Luchikt","year":"1987","journal-title":"J. Fluid Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1017\/S0022112084000380","article-title":"On the detection of turbulence-generating events","volume":"139","author":"Alfredsson","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"N33","DOI":"10.1080\/14685240600602929","article-title":"Large-eddy simulation of spatial transition in plane channel flow","volume":"7","author":"Schlatter","year":"2006","journal-title":"J. Turbul."},{"key":"ref_22","unstructured":"Meyer, R.E. (1981). Wall phenomena in the final stage of transition to turbulence. Transition and Turbulence, Academic Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1017\/S0022112087000892","article-title":"Turbulence statistics in fully developed channel flow at low Reynolds number","volume":"177","author":"Kim","year":"1987","journal-title":"J. Fluid Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1017\/S0022112087000326","article-title":"On the generation of high-amplitude wall-pressure peaks in turbulent boundary layers and spots","volume":"175","author":"Johansson","year":"1987","journal-title":"J. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1017\/S0022112093001843","article-title":"Vortex dynamics and the production of Reynolds stress","volume":"253","author":"Bernard","year":"1993","journal-title":"J. Fluid Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1017\/jfm.2013.501","article-title":"Turbulent structure of high-amplitude pressure peaks within the turbulent boundary layer","volume":"735","author":"Ghaemi","year":"2013","journal-title":"J. Fluid Mech."},{"key":"ref_27","unstructured":"Swearingen, J.D., Blackwelder, R.F., and Spalart, P.R. Inflectional instabilities in the wall region of bounded turbulent shear flows. Proceedings of the Summer Program."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Fletcher, C.A.J. (1991). Computational Techniques for Fluid Dynamics, Springer.","DOI":"10.1007\/978-3-642-58239-4"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/339\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:28:28Z","timestamp":1760135308000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/339"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,26]]},"references-count":28,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030339"],"URL":"https:\/\/doi.org\/10.3390\/e24030339","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,26]]}}}