{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T19:08:24Z","timestamp":1772478504098,"version":"3.50.1"},"reference-count":28,"publisher":"American Mathematical Society (AMS)","issue":"2","license":[{"start":{"date-parts":[[2007,9,19]],"date-time":"2007-09-19T00:00:00Z","timestamp":1190160000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    We consider a class of kinetic equations, equipped with a single conservation law, which generate\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -contractions. We discuss the hydrodynamic limit to a scalar conservation law and the diffusive limit to a (possibly) degenerate parabolic equation. The limits are obtained in the \u201cdissipative\u201d sense, equivalent to the notion of entropy solutions for conservation laws, which permits the use of the perturbed test function method and allows for simple proofs. A general compactness framework is obtained for the diffusive scaling in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^{1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The radiative transport equations, satisfied by the Wigner function for random acoustic waves, present such a kinetic model that is endowed with conservation of energy. The general theory is used to validate the diffusive approximation of the radiative transport equation.\n                  <\/p>","DOI":"10.1090\/s0002-9947-06-04268-1","type":"journal-article","created":{"date-parts":[[2006,10,17]],"date-time":"2006-10-17T02:57:40Z","timestamp":1161053860000},"page":"529-565","source":"Crossref","is-referenced-by-count":6,"special_numbering":"861","title":["Hydrodynamic limits for kinetic equations and the diffusive approximation of radiative transport for acoustic waves"],"prefix":"10.1090","volume":"359","author":[{"given":"Manuel","family":"Portilheiro","sequence":"first","affiliation":[]},{"given":"Athanasios","family":"Tzavaras","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,9,19]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1002\/cpa.3160400603","article-title":"The Rosseland approximation for the radiative transfer equations","volume":"40","author":"Bardos, C.","year":"1987","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1002\/cpa.3160420609","article-title":"Erratum to the article: \u201cThe Rosseland approximation for the radiative transfer equations\u201d [Comm. Pure Appl. Math. 40 (1987), no. 6, 691\u2013721; MR0910950 (88j:35134)]","volume":"42","author":"Bardos, Claude","year":"1989","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1016\/0022-1236(88)90096-1","article-title":"The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation","volume":"77","author":"Bardos, C.","year":"1988","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"issue":"1-2","key":"4","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1023\/A:1004525427365","article-title":"Construction of BGK models with a family of kinetic entropies for a given system of conservation laws","volume":"95","author":"Bouchut, F.","year":"1999","journal-title":"J. Statist. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4715","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1512\/iumj.2000.49.1811","article-title":"Diffusive BGK approximations for nonlinear multidimensional parabolic equations","volume":"49","author":"Bouchut, F.","year":"2000","journal-title":"Indiana Univ. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2518","issn-type":"print"},{"key":"6","series-title":"Collection Math\\'{e}matiques Appliqu\\'{e}es pour la Ma\\^{i}trise. [Collection of Applied Mathematics for the Master's Degree]","isbn-type":"print","volume-title":"Analyse fonctionnelle","author":"Brezis, Ha\u00efm","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/2225771987"},{"key":"7","volume-title":"Radiative transfer","author":"Chandrasekhar, S.","year":"1960"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"645","DOI":"10.1016\/S0294-1449(02)00014-8","article-title":"Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations","volume":"20","author":"Chen, Gui-Qiang","year":"2003","journal-title":"Ann. Inst. H. Poincar\\'{e} C Anal. Non Lin\\'{e}aire","ISSN":"https:\/\/id.crossref.org\/issn\/0294-1449","issn-type":"print"},{"issue":"3-4","key":"9","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1017\/S0308210500018631","article-title":"The perturbed test function method for viscosity solutions of nonlinear PDE","volume":"111","author":"Evans, Lawrence C.","year":"1989","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1137\/S0036142901399392","article-title":"Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes","volume":"41","author":"Gosse, Laurent","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5-6","key":"11","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1081\/PDE-120004900","article-title":"Kinetic decomposition of approximate solutions to conservation laws: application to relaxation and diffusion-dispersion approximations","volume":"27","author":"Hwang, Seok","year":"2002","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"913","DOI":"10.1137\/S0036142998347978","article-title":"Uniformly accurate diffusive relaxation schemes for multiscale transport equations","volume":"38","author":"Jin, Shi","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1-2","key":"13","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1080\/03605309708821261","article-title":"Contractive relaxation systems and the scalar multidimensional conservation law","volume":"22","author":"Katsoulakis, Markos A.","year":"1997","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"issue":"3-4","key":"14","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1023\/A:1004670308361","article-title":"Multiscale analysis for interacting particles: relaxation systems and scalar conservation laws","volume":"96","author":"Katsoulakis, Markos A.","year":"1999","journal-title":"J. Statist. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4715","issn-type":"print"},{"key":"15","doi-asserted-by":"crossref","unstructured":"S.N. Kruzhkov, First order quasilinear equations with several independent variables, Math. USSR Sbornik 10 (1970), pp. 217\u2013243.","DOI":"10.1070\/SM1970v010n02ABEH002156"},{"key":"16","doi-asserted-by":"publisher","first-page":"259","DOI":"10.2307\/1996565","article-title":"Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics","volume":"186","author":"Kurtz, Thomas G.","year":"1973","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"17","unstructured":"C. Lattanzio and R. Natalini, Convergence of diffusive BGK approximations for nonlinear strongly parabolic systems, Proc. Royal Soc. Edinburgh (to appear)."},{"key":"18","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","volume-title":"Functional analysis","author":"Lax, Peter D.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0471556041"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"473","DOI":"10.4171\/RMI\/228","article-title":"Diffusive limit for finite velocity Boltzmann kinetic models","volume":"13","author":"Lions, Pierre Louis","year":"1997","journal-title":"Rev. Mat. Iberoamericana","ISSN":"https:\/\/id.crossref.org\/issn\/0213-2230","issn-type":"print"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1006\/jdeq.1999.3676","article-title":"Hyperbolic to parabolic relaxation theory for quasilinear first order systems","volume":"162","author":"Marcati, Pierangelo","year":"2000","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1006\/jdeq.1998.3460","article-title":"A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws","volume":"148","author":"Natalini, Roberto","year":"1998","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"22","isbn-type":"print","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1016\/s0165-2125(99)00018-9","article-title":"Waves and transport","author":"Papanicolaou, George","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0821805924"},{"issue":"4","key":"23","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s00205-003-0282-5","article-title":"Dissipative and entropy solutions to non-isotropic degenerate parabolic balance laws","volume":"170","author":"Perthame, Beno\u00eet","year":"2003","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"3","key":"24","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1007\/BF02099071","article-title":"A kinetic equation with kinetic entropy functions for scalar conservation laws","volume":"136","author":"Perthame, Beno\u00eet","year":"1991","journal-title":"Comm. Math. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3616","issn-type":"print"},{"issue":"5","key":"25","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1017\/S0308210500002870","article-title":"Weak solutions for equations defined by accretive operators. I","volume":"133","author":"Portilheiro, Manuel","year":"2003","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1016\/S0022-0396(03)00213-4","article-title":"Weak solutions for equations defined by accretive operators. II. Relaxation limits","volume":"195","author":"Portilheiro, Manuel","year":"2003","journal-title":"J. Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0022-0396","issn-type":"print"},{"key":"27","unstructured":"A.E. Tzavaras, Derivation of fluid equations for kinetic models with one conserved quantity, In \u201cProceedings of International Conf. on Mathematical Analysis\u201d, National Techn. Univ. of Athens, Greece, 2003."},{"key":"28","isbn-type":"print","first-page":"192","article-title":"On the mathematical theory of fluid dynamic limits to conservation laws","author":"Tzavaras, Athanasios E.","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/3540677860"}],"container-title":["Transactions of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/tran\/2007-359-02\/S0002-9947-06-04268-1\/S0002-9947-06-04268-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/tran\/2007-359-02\/S0002-9947-06-04268-1\/S0002-9947-06-04268-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T18:11:55Z","timestamp":1772475115000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/tran\/2007-359-02\/S0002-9947-06-04268-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,9,19]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,2]]}},"alternative-id":["S0002-9947-06-04268-1"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9947-06-04268-1","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6850","0002-9947"],"issn-type":[{"value":"1088-6850","type":"electronic"},{"value":"0002-9947","type":"print"}],"subject":[],"published":{"date-parts":[[2006,9,19]]}}}