{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:38:08Z","timestamp":1787323088466,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>We derive a kinetic formulation for the parabolic scalar conservation law $\\partial_t u + \\mathrm{div}_y A(y,u) - \\Delta_y u=0$. This allows us to define a weaker notion of solutions in $L^1$, which is enough to recover the $L^1$ contraction principle. We also apply this kinetic formulation to a homogenization problem studied in a previous paper; namely, we prove that the kinetic solution $u^{\\varepsilon}$ of $\\partial_t u^{\\varepsilon} + \\mathrm{div}_x A\\left({x}\/{\\varepsilon}, u^{\\varepsilon} \\right)- \\varepsilon\\Delta_x u^{\\varepsilon}=0$ behaves in $L^1_{\\text{loc}}$ as $v\\left( {x}\/{\\varepsilon}, \\bar{u}(t,x)\\right)$, where v is the solution of a cell problem and $\\bar{u}$ the solution of the homogenized problem.<\/jats:p>","DOI":"10.1137\/060662770","type":"journal-article","created":{"date-parts":[[2007,9,20]],"date-time":"2007-09-20T09:50:05Z","timestamp":1190281805000},"page":"891-915","source":"Crossref","is-referenced-by-count":6,"title":["Kinetic Formulation for a Parabolic Conservation Law. Application to Homogenization"],"prefix":"10.1137","volume":"39","author":[{"given":"Anne-Laure","family":"Dalibard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,9,19]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0523084"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500003863"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2006.04.001"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-007-0063-7"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2005.05.005"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160450304"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0152055"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1992-5402"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"D. Gilbarg and N. Trudinger,\n                      Elliptic Partial Differential Equations of Second Order\n                      , 3rd ed., Springer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"R10","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1007\/BF00971654","volume":"10","author":"Kruzkhov S. N.","year":"1969","journal-title":"Soviet Math. Dokl.","ISSN":"https:\/\/id.crossref.org\/issn\/0197-6788","issn-type":"print"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1070\/SM1970v010n02ABEH002156"},{"key":"R12","unstructured":"P.L. Lions, G. Papanicolaou, and S. R. S. Varadhan,\n                      Homogenization of Hamilton-Jacobi Equations\n                      , unpublished manuscript, 1987."},{"key":"R13","first-page":"97","volume":"312","author":"Lions P.-L.","year":"1991","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1201239-3"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0520043"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-7824(99)80003-8"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"B. Perthame,\n                      Kinetic Formulation of Conservation Laws\n                      , Oxford Lecture Ser. Math. Appl. 21, Oxford University Press, Oxford, UK, 2002.","DOI":"10.1093\/oso\/9780198509134.001.0001"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099071"}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060662770","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:41:20Z","timestamp":1787319680000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060662770"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":18,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/060662770"],"URL":"https:\/\/doi.org\/10.1137\/060662770","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}