{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T08:00:57Z","timestamp":1777622457926,"version":"3.51.4"},"reference-count":31,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100002850","name":"Fondo Nacional de Desarrollo Cient\u00edfico y Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["1160774"],"award-info":[{"award-number":["1160774"]}],"id":[{"id":"10.13039\/501100002850","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011264","name":"FP7 People: Marie-Curie Actions","doi-asserted-by":"publisher","award":["777778"],"award-info":[{"award-number":["777778"]}],"id":[{"id":"10.13039\/100011264","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"Abstract<\/jats:title>\n While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space \n \n \n \n \n H<\/m:mi>\n 0<\/m:mn>\n 1<\/m:mn>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H_{0}^{1}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the Banach Sobolev space \n \n \n \n \n W<\/m:mi>\n 0<\/m:mn>\n \n 1<\/m:mn>\n ,<\/m:mo>\n q<\/m:mi>\n <\/m:mrow>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W^{1,q}_{0}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n \n \n 1<\/m:mn>\n <<\/m:mo>\n q<\/m:mi>\n <<\/m:mo>\n \u221e<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n {1<q<{\\infty}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, is more general allowing more irregular solutions. In this paper we present a well-posedness theory for the convection-diffusion-reaction equation in the \n \n \n \n \n W<\/m:mi>\n 0<\/m:mn>\n \n 1<\/m:mn>\n ,<\/m:mo>\n q<\/m:mi>\n <\/m:mrow>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W^{1,q}_{0}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-\n \n \n \n \n W<\/m:mi>\n 0<\/m:mn>\n \n 1<\/m:mn>\n ,<\/m:mo>\n \n q<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {W_{0}^{1,q^{\\prime}}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> functional setting, \n \n \n \n \n \n 1<\/m:mn>\n q<\/m:mi>\n <\/m:mfrac>\n +<\/m:mo>\n \n 1<\/m:mn>\n \n q<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n <\/m:mfrac>\n <\/m:mrow>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {\\frac{1}{q}+\\frac{1}{q^{\\prime}}=1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The theory is based on directly establishing the inf-sup conditions. Apart from a standard assumption on the advection and reaction coefficients, the other key assumption pertains to a subtle regularity requirement for the standard Laplacian. An elementary consequence of the well-posedness theory is the stability and convergence of Galerkin\u2019s method in this setting, for a diffusion-dominated case and under the assumption of \n \n \n \n W<\/m:mi>\n \n 1<\/m:mn>\n ,<\/m:mo>\n \n q<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n \n {W^{1,q^{\\prime}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-stability of the \n \n \n \n H<\/m:mi>\n 0<\/m:mn>\n 1<\/m:mn>\n <\/m:msubsup>\n <\/m:math>\n \n {H_{0}^{1}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-projector.<\/jats:p>","DOI":"10.1515\/cmam-2018-0198","type":"journal-article","created":{"date-parts":[[2019,6,27]],"date-time":"2019-06-27T09:02:47Z","timestamp":1561626167000},"page":"503-522","source":"Crossref","is-referenced-by-count":8,"title":["The Convection-Diffusion-Reaction Equation in Non-Hilbert Sobolev Spaces: A Direct Proof of the Inf-Sup Condition and Stability of Galerkin\u2019s Method"],"prefix":"10.1515","volume":"19","author":[{"given":"Paul","family":"Houston","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences , The University of Nottingham , University Park , Nottingham , NG7 2RD , United Kingdom"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4430-5167","authenticated-orcid":false,"given":"Ignacio","family":"Muga","sequence":"additional","affiliation":[{"name":"Instituto de Mathem\u00e1ticas , Pontificia Universidad Cat\u00f3lica de Valpara\u00edso , Casilla 4059 , Valpara\u00edso , Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0436-1340","authenticated-orcid":false,"given":"Sarah","family":"Roggendorf","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , The University of Nottingham , University Park , Nottingham , NG7 2RD , United Kingdom"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6830-8031","authenticated-orcid":false,"given":"Kristoffer G.","family":"van der Zee","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences , The University of Nottingham , University Park , Nottingham , NG7 2RD , United Kingdom"}]}],"member":"374","published-online":{"date-parts":[[2019,6,27]]},"reference":[{"key":"2023033110340708801_j_cmam-2018-0198_ref_001_w2aab3b7e1458b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"V. 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Math. 112 (2009), no. 2, 221\u2013243.","DOI":"10.1007\/s00211-009-0213-y"},{"key":"2023033110340708801_j_cmam-2018-0198_ref_019_w2aab3b7e1458b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"T. Jakab, I. Mitrea and M. Mitrea,\nSobolev estimates for the Green potential associated with the Robin\u2013Laplacian in Lipschitz domains satisfying a uniform exterior ball\ncondition,\nSobolev Spaces in Mathematics. II,\nInt. Math. Ser. (N.\u2009Y.) 9,\nSpringer, New York (2009), 227\u2013260.","DOI":"10.1007\/978-0-387-85650-6_11"},{"key":"2023033110340708801_j_cmam-2018-0198_ref_020_w2aab3b7e1458b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"D. Jerison and C. E. Kenig,\nThe inhomogeneous Dirichlet problem in Lipschitz domains,\nJ. Funct. Anal. 130 (1995), no. 1, 161\u2013219.","DOI":"10.1006\/jfan.1995.1067"},{"key":"2023033110340708801_j_cmam-2018-0198_ref_021_w2aab3b7e1458b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"J. P. 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