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Numer. Anal."],"published-print":{"date-parts":[[2026,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Velocity fields with low regularity (below the Lipschitz threshold) naturally arise in many models from mathematical physics, such as the inhomogeneous incompressible Navier\u2013Stokes equations, and play a fundamental role in the analysis of nonlinear PDEs. The DiPerna\u2013Lions theory ensures existence and uniqueness of the flow associated with a divergence-free velocity field with Sobolev regularity. In this paper, we establish a priori error estimates showing a logarithmic rate of convergence of numerical solutions, constructed via the [Formula: see text]-method, towards the exact (analytic) flow for a velocity field with Sobolev regularity. In addition, we derive analogous a priori error estimates for Lagrangian solutions of the associated transport equation, exhibiting the same logarithmic rate of convergence. Our theoretical results are supported by numerical experiments, which confirm the predicted logarithmic behavior.<\/jats:p>","DOI":"10.1137\/25m177432x","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T07:26:06Z","timestamp":1782890766000},"page":"1162-1189","source":"Crossref","is-referenced-by-count":0,"title":["A Priori Error Estimates for the \\({\\theta}\\)-Method for the Flow of Nonsmooth Velocity Fields"],"prefix":"10.1137","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1582-274X","authenticated-orcid":true,"given":"Gennaro","family":"Ciampa","sequence":"first","affiliation":[{"name":"DISIM - Dipartimento di Ingegneria e Scienze dell\u2019Informazione e Matematica, Universit\u00e0 degli Studi dell\u2019Aquila, 67100 L\u2019Aquila, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8183-5987","authenticated-orcid":true,"given":"Tommaso","family":"Cortopassi","sequence":"additional","affiliation":[{"name":"Scuola Normale Superiore, 56126 Pisa, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6825-4275","authenticated-orcid":true,"given":"Gianluca","family":"Crippa","sequence":"additional","affiliation":[{"name":"Departement Mathematik und Informatik, Universit\u00e4t Basel, CH-4051 Basel, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5178-5416","authenticated-orcid":true,"given":"Raffaele","family":"D\u2019Ambrosio","sequence":"additional","affiliation":[{"name":"DISIM - Dipartimento di Ingegneria e Scienze dell\u2019Informazione e Matematica, Universit\u00e0 degli Studi dell\u2019Aquila, 67100 L\u2019Aquila, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8373-4541","authenticated-orcid":true,"given":"Stefano","family":"Spirito","sequence":"additional","affiliation":[{"name":"DISIM - Dipartimento di Ingegneria e Scienze dell\u2019Informazione e Matematica, Universit\u00e0 degli Studi dell\u2019Aquila, 67100 L\u2019Aquila, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,7,1]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1090\/jams\/913"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-004-0367-2"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1081\/PDE-200040210"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9121-0_3"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-019-01396-3"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2022.09.005"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1142\/S0219891613500100"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drr031"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-021-01628-5"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.4171\/jems\/1756"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-021-03993-4"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2021.1883650"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s40818-020-00091-x"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2024.17.2161"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-019-1659-0"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-020-09635-8"},{"key":"ref17","unstructured":"M. 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