{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T02:11:04Z","timestamp":1780452664233,"version":"3.54.1"},"reference-count":13,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T00:00:00Z","timestamp":1752624000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the existence and uniqueness of local and global solutions to the incompressible three-dimensional Navier\u2013Stokes equations within the framework of homogeneous Lei\u2013Lin\u2013Gevrey spaces Xa,\u03b3\u03c1(R3), where \u03c1\u2208[\u22121,0),a&gt;0, and \u03b3\u2208(0,1). These function spaces combine the critical scaling structure of the Lei\u2013Lin spaces with the exponential regularity of Gevrey classes, thereby enabling a refined treatment of analytic regularity and frequency localization. The main results are obtained under the assumption of small initial data in the critical Lei\u2013Lin space X\u03c1(R3), extending previous works and improving regularity thresholds. In particular, we establish that for suitable initial data, the Navier\u2013Stokes system admits unique solutions globally in time. The influence of the Gevrey parameter \u03b3 on the high-frequency behavior of solutions is also discussed. This work contributes to a deeper understanding of regularity and decay properties in critical and supercritical regimes.<\/jats:p>","DOI":"10.3390\/sym17071138","type":"journal-article","created":{"date-parts":[[2025,7,16]],"date-time":"2025-07-16T08:24:17Z","timestamp":1752654257000},"page":"1138","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Local and Global Solutions of the 3D-NSE in Homogeneous Lei\u2013Lin\u2013Gevrey Spaces"],"prefix":"10.3390","volume":"17","author":[{"given":"Lotfi","family":"Jlali","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF00276188","article-title":"On the Navier-Stokes initial value problem. I","volume":"16","author":"Fujita","year":"1964","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1137\/0523002","article-title":"Remarques sur l\u2019sexistence globale pour le syst\u00e8me de Navier-Stokes incompressible","volume":"23","author":"Chemin","year":"1992","journal-title":"SIAM J. Math. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/BF01174182","article-title":"Lp- solution of the Navier Stokes in Rm with applications to weak solutions","volume":"187","author":"Kato","year":"1984","journal-title":"Math. Z."},{"key":"ref_4","unstructured":"(1995). Ondelettes, Paraproduits et Navier-Stokes, Arts et Sciences."},{"key":"ref_5","first-page":"A15","article-title":"Symmetry breaking in Navier-Stokes flows: An analytic study","volume":"945","author":"Schorlepp","year":"2022","journal-title":"J. Fluid Mech."},{"key":"ref_6","first-page":"909","article-title":"Analytic and Gevrey regularity for solutions to the Navier-Stokes equations","volume":"365","author":"Cannone","year":"2023","journal-title":"Math. Ann."},{"key":"ref_7","unstructured":"Sawada, O., and Taniuchi, Y. (J. Math. Fluid Mech., 2021). A new class of critical spaces for the Navier-Stokes equations, J. Math. Fluid Mech., to appear."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1297","DOI":"10.1002\/cpa.20361","article-title":"Global mild solutions of Navier Stokes equations","volume":"64","author":"Lei","year":"2011","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1002\/mma.4193","article-title":"Global well posedness of 3D-NSE in Fourier-Lei-Lin spaces","volume":"40","author":"Jlali","year":"2017","journal-title":"Math. Meth. Appl. 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