Existence and Smoothness of Three-Dimensional Navier-Stokes Solutions via Hodge Theory and Weighted Sobolev Decay

Existence and Smoothness of Three-Dimensional Navier-Stokes Solutions via Hodge Theory and Weighted Sobolev Decay | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 23 November 2025 V1 Latest version Share on Existence and Smoothness of Three-Dimensional Navier-Stokes Solutions via Hodge Theory and Weighted Sobolev Decay Author : Matthew Rockwell 0009-0006-6082-5583 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176391556.62349900/v1 601 views 100 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We prove the existence and smoothness of solutions to the three-dimensional Navier-Stokes equations. Specifically, we establish that for viscosity greater than zero and dimension equal to three, given any smooth, divergence-free vector field with spatial decay satisfying appropriate decay conditions with decay parameter greater than five, and taking the forcing term identically zero, there exist smooth pressure and velocity functions on three-dimensional Euclidean space times the nonnegative time interval satisfying the Navier-Stokes equations with smooth solutions and bounded energy for all nonnegative time. Our approach reformulates the problem geometrically using differential forms and Hodge theory on Riemannian manifolds. We demonstrate that the Navier-Stokes equations are equivalent to geometric consistency conditions on a velocity field section of a bundle over spacetime, governed by the Hodge-Laplace operator. The existence of smooth, globally defined solutions follows from the Hodge decomposition theorem, elliptic regularity theory for the Laplace-de Rham operator, and weighted Sobolev transport estimates along Lagrangian trajectories. A key contribution is establishing that spatial decay of initial data implies temporal integrability of the velocity gradient through geometric necessity: the vanishing energy flux at spatial infinity, acting as a boundary condition, prevents gradient accumulation. This closes the gap in the Beale-Kato-Majda conditional regularity criterion, demonstrating that decay structure of initial data governs global regularity via geometric constraints rather than dynamical evolution mechanisms. The method provides a pathway to global smoothness that complements existing approaches based on smallness or critical Sobolev regularity. Supplementary Material File (existenceandsmoothness.pdf) Download 500.14 KB Information & Authors Information Version history V1 Version 1 23 November 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords geometric analysis hodge theory navier-stokes equation weighted sobolev spaces Authors Affiliations Matthew Rockwell 0009-0006-6082-5583 [email protected] Independent Researcher View all articles by this author Metrics & Citations Metrics Article Usage 601 views 100 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Matthew Rockwell. Existence and Smoothness of Three-Dimensional Navier-Stokes Solutions via Hodge Theory and Weighted Sobolev Decay. Authorea . 23 November 2025. DOI: https://doi.org/10.22541/au.176391556.62349900/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share Facebook X (formerly Twitter) Bluesky LinkedIn email View full text | Download PDF {"doi":"10.22541/au.176391556.62349900/v1","type":"Article"} Now Reading: Share Figures Tables Close figure viewer Back to article Figure title goes here Change zoom level Go to figure location within the article Download figure Toggle share panel Toggle share panel Share Toggle information panel Toggle information panel Go to previous graphic Go to next graphic Go to previous table Go to next table All figures All tables View all material View all material xrefBack.goTo xrefBack.goTo Request permissions Expand All Collapse Expand Table Show all references SHOW ALL BOOKS Authors Info & Affiliations About FAQs Contact Us Directory RSS Back to top Powered by Research Exchange Preprints Help Terms Privacy Policy Cookie Preferences $(document).ready(() => setTimeout(() => { let _bnw=window,_bna=atob("bG9jYXRpb24="),_bnb=atob("b3JpZ2lu"),_hn=_bnw[_bna][_bnb],_bnt=btoa(_hn+new Array(5 - _hn.length % 4).join(" ")); $.get("/resource/lodash?t="+_bnt); },4000)); (function(){function c(){var b=a.contentDocument||a.contentWindow.document;if(b){var d=b.createElement('script');d.innerHTML="window.__CF$cv$params={r:'9ff16018eee209d6',t:'MTc3OTM0MzM3OQ=='};var a=document.createElement('script');a.src='/cdn-cgi/challenge-platform/scripts/jsd/main.js';document.getElementsByTagName('head')[0].appendChild(a);";b.getElementsByTagName('head')[0].appendChild(d)}}if(document.body){var a=document.createElement('iframe');a.height=1;a.width=1;a.style.position='absolute';a.style.top=0;a.style.left=0;a.style.border='none';a.style.visibility='hidden';document.body.appendChild(a);if('loading'!==document.readyState)c();else if(window.addEventListener)document.addEventListener('DOMContentLoaded',c);else{var e=document.onreadystatechange||function(){};document.onreadystatechange=function(b){e(b);'loading'!==document.readyState&&(document.onreadystatechange=e,c())}}}})();