{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:16Z","timestamp":1760147356286,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,26]],"date-time":"2023-01-26T00:00:00Z","timestamp":1674691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["2248"],"award-info":[{"award-number":["2248"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Navier\u2013Stokes equations (NS-equations) are applied extensively for the study of various waves phenomena where the symmetries are involved. In this paper, we discuss the NS-equations with the time-fractional derivative of order \u03b2\u2208(0,1). In fractional media, these equations can be utilized to recreate anomalous diffusion equations which can be used to construct symmetries. We examine the initial value problem involving the symmetric Stokes operator and gravitational force utilizing the Caputo fractional derivative. Additionally, we demonstrate the global and local mild solutions in H\u03b1,p. We also demonstrate the regularity of classical solutions in such circumstances. An example is presented to demonstrate the reliability of our findings.<\/jats:p>","DOI":"10.3390\/sym15020343","type":"journal-article","created":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T01:58:59Z","timestamp":1674784739000},"page":"343","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Existence of Global and Local Mild Solution for the Fractional Navier\u2013Stokes Equations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6447-6361","authenticated-orcid":false,"given":"Muath","family":"Awadalla","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4501-9269","authenticated-orcid":false,"given":"Azhar","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department Mathematics, University of Chakwal, Chakwal 48800, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Farva","family":"Hafeez","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Lahore, Sargodha 40100, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2744-6320","authenticated-orcid":false,"given":"Kinda","family":"Abuasbeh","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,26]]},"reference":[{"key":"ref_1","unstructured":"Varnhorn, W. (1994). The Stokes Equations, Akademie Verlag."},{"key":"ref_2","first-page":"29","article-title":"Nombres de Reynolds, stabilit\u00e9 et Navier\u2013Stokes","volume":"52","author":"Cannone","year":"2000","journal-title":"Banach Cent. Publ."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lemari-Rieusset, P.G. (2002). Recent Developments in the Navier\u2013Stokes Problem, CRC Press.","DOI":"10.1201\/9780367801656"},{"key":"ref_4","unstructured":"Wojciech, S.O., and Benjamin, C.P. (2018). Partial Differential Equations in Fluid Mechanics, Cambridge University Press."},{"key":"ref_5","unstructured":"Bermudez, B., Huerta, A.R., Guerrero-Sanchez, W.F., and Alans, J.D. (2018). Computational Fluid Dynamics: Basic Instruments & Applications in Science, BoD."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1090\/S0002-9947-10-04744-6","article-title":"Large, global solutions to the Navier\u2013Stokes equations, slowly varying in one direction","volume":"362","author":"Chemin","year":"2010","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/0022-1236(91)90136-S","article-title":"Abstract Lp estimates for the Cauchy problem with applications to the Navier\u2013Stokes equations in exterior domains","volume":"102","author":"Giga","year":"1991","journal-title":"J. Funct. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1515","DOI":"10.1016\/j.jde.2013.11.005","article-title":"Lp-theory for Stokes and Navier\u2013Stokes equations with Navier boundary condition","volume":"256","author":"Amrouche","year":"2014","journal-title":"J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/PL00004644","article-title":"On a larger class of stable solutions to the Navier\u2013Stokes equations in exterior domains","volume":"228","author":"Kozono","year":"1998","journal-title":"Math. Z."},{"key":"ref_10","first-page":"503","article-title":"Navier\u2013Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions","volume":"6","author":"Raugel","year":"1993","journal-title":"J. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2171","DOI":"10.1016\/j.jfa.2014.12.016","article-title":"Boundary regularity of suitable weak solution for the Navier\u2013Stokes equations","volume":"268","author":"Choe","year":"2015","journal-title":"J. Funct. Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00211-014-0685-2","article-title":"An analysis of the Rayleigh\u2013Stokes problem for a generalized second-grade fluid","volume":"131","author":"Bazhlekova","year":"2015","journal-title":"Numer. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/3779"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Herrmann, R. (2011). Fractional Calculus: An Introduction for Physicists, World Scientific.","DOI":"10.1142\/8072"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1216\/JIE-2013-25-4-557","article-title":"Existence of mild solutions for fractional evolution equations","volume":"25","author":"Zhou","year":"2013","journal-title":"J. Integral Equ. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"507","DOI":"10.3934\/eect.2015.4.507","article-title":"Controllability for fractional evolution inclusions without compactness","volume":"4","author":"Zhou","year":"2015","journal-title":"Evol. Equ. Control Theory"},{"key":"ref_17","first-page":"119","article-title":"Abstract Cauchy problem for fractional functional differential equations","volume":"42","author":"Zhou","year":"2013","journal-title":"Topol. Methods Nonlinear Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1016\/j.amc.2005.11.025","article-title":"Analytical solution of a time-fractional Navier\u2013Stokes equation by Adomian decomposition method","volume":"177","author":"Momani","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1002\/num.20420","article-title":"Analytical solution of time-fractional Navier\u2013Stokes equation in polar coordinate by homotopy perturbation method","volume":"26","author":"Ganji","year":"2010","journal-title":"Numer. Methods Partial Differ. Equ. Int. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/j.amc.2003.07.022","article-title":"On the generalized Navier\u2013Stokes equations","volume":"156","author":"Salem","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_21","unstructured":"Wahl, W.V. (2013). The Equations of Navier\u2013Stokes and Abstract Parabolic Equations, Springer."},{"key":"ref_22","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_23","unstructured":"Galdi, G.P. (1998). An Introduction to the Mathematical Theory of the Navier\u2013Stokes Equations: Nonlinear Steady Problems, Springer. Springer Tracts in Natural Philosophy."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Shafqat, R., Niazi, A.U.K., Yavuz, M., Jeelani, M.B., and Saleem, K. (2022). Mild Solution for the Time-Fractional Navier\u2013Stokes Equation Incorporating MHD Effects. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6100580"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/343\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:16:12Z","timestamp":1760120172000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/343"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,26]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020343"],"URL":"https:\/\/doi.org\/10.3390\/sym15020343","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,26]]}}}