{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:33:30Z","timestamp":1787322810141,"version":"3.56.0"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>We consider the Dirichlet problem for second-order linear elliptic equations with singular drift terms given by a vector field $u$. $W^{1,p}$-estimates for weak solutions are quite well known, provided that $u$ is sufficiently regular, e.g., $u \\in L^\\infty$. In this paper, we establish the existence and uniqueness of weak solutions satisfying $W^{1,p}$- or $W^{1,2}$-estimates for less regular $u$. First, some $W^{1,p}$-estimates are shown for $u \\in L_\\sigma^n +L^r$, where $n \\le r&lt; \\infty$ if $n \\ge 3$ and $2 &lt; r&lt;\\infty$ if $n=2$. Here $n$ denotes the dimension and $L_\\sigma^n = \\{ v \\in L^n : {\\rm div}\\, v =0 \\}$. The case of more singular $u$ is then studied. Assuming that $n \\ge 3$, $u \\in L^2$, and ${\\rm div}\\, u \\in L^{n\/2} $, we prove the existence, uniqueness and $W^{1,2}$-estimate of weak solutions. Our $W^{1,p}$- and $W^{1,2}$-results are optimal in some sense, as shown by counterexamples due to Moscariello [Adv. Calc. Var., 4 (2011), pp. 421--444].<\/jats:p>","DOI":"10.1137\/14096270x","type":"journal-article","created":{"date-parts":[[2015,4,2]],"date-time":"2015-04-02T11:58:24Z","timestamp":1427975904000},"page":"1271-1290","source":"Crossref","is-referenced-by-count":19,"title":["On Weak Solutions of Elliptic Equations with Singular Drifts"],"prefix":"10.1137","volume":"47","author":[{"given":"Hyunseok","family":"Kim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Young-Heon","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,4,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-010-0340-8"},{"key":"atypb2","first-page":"1037","volume":"248","author":"Bogovski\u012d M.E.","year":"1979","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2011.613079"},{"key":"atypb4","first-page":"247","volume":"72","author":"Coifman R.","year":"1993","journal-title":"J. Math. Pures Appl. (9)"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-009-0228-7"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1015709329011"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1081\/PDE-120019377"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1998.3316"},{"key":"atypb9","first-page":"98","volume":"195","author":"Filonov N.","year":"2013","journal-title":"Y.)"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-09620-9"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01182469"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61798-0"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2008.7.163"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2014.v12.n4.a4"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-008-0168-7"},{"key":"atypb16","unstructured":"M. Kontovourkis,\n                      On Elliptic Equations with Low-Regularity Divergence-Free Drift Terms and the Steady-State Navier-Stokes Equations in Higher Dimensions\n                      , Ph.D. thesis, University of Minnesota, Minneapolis, MN, 2007."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1515\/acv.2011.007"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1090\/S1061-0022-2011-01188-4"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2011.08.039"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.5802\/aif.204"},{"key":"atypb21","volume-title":"Harmonic Analysis","author":"Stein E.M.","year":"1993"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2011.09.022"}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/14096270X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:26:20Z","timestamp":1787318780000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/14096270X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/14096270X"],"URL":"https:\/\/doi.org\/10.1137\/14096270x","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}