{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:06:45Z","timestamp":1760148405598,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of Xinjiang Province","award":["2022TSYCTD0019","2022D01D32","12071406"],"award-info":[{"award-number":["2022TSYCTD0019","2022D01D32","12071406"]}]},{"name":"National Natural Science Foundation of China","award":["2022TSYCTD0019","2022D01D32","12071406"],"award-info":[{"award-number":["2022TSYCTD0019","2022D01D32","12071406"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we study the finite element method of the Navier\u2013Stokes equations with the initial data belonging to the L2 space for all time t&gt;0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t\u2208[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.<\/jats:p>","DOI":"10.3390\/e25050726","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T02:27:56Z","timestamp":1682648876000},"page":"726","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Uniform Error Estimates of the Finite Element Method for the Navier\u2013Stokes Equations in R2 with L2 Initial Data"],"prefix":"10.3390","volume":"25","author":[{"given":"Shuyan","family":"Ren","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4180-6150","authenticated-orcid":false,"given":"Kun","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Girault, V., and Raviart, P. 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