{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T19:46:36Z","timestamp":1768679196962,"version":"3.49.0"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2009,5,1]],"date-time":"2009-05-01T00:00:00Z","timestamp":1241136000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Stat Phys"],"published-print":{"date-parts":[[2009,5]]},"DOI":"10.1007\/s10955-009-9741-1","type":"journal-article","created":{"date-parts":[[2009,5,13]],"date-time":"2009-05-13T13:10:50Z","timestamp":1242220250000},"page":"681-736","source":"Crossref","is-referenced-by-count":18,"title":["On Strong Convergence to Equilibrium for\u00a0the\u00a0Boltzmann Equation with Soft Potentials"],"prefix":"10.1007","volume":"135","author":[{"given":"Eric A.","family":"Carlen","sequence":"first","affiliation":[]},{"given":"Maria C.","family":"Carvalho","sequence":"additional","affiliation":[]},{"given":"Xuguang","family":"Lu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2009,5,14]]},"reference":[{"issue":"1","key":"9741_CR1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s002050000083","volume":"152","author":"R. Alexandre","year":"2000","unstructured":"Alexandre, R., Desvillettes, L., Villani, C., Wennberg, W.: Entropy dissipation and long-range interaction. Arch. Ration. Mech. Anal. 152(1), 327\u2013355 (2000)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"9741_CR2","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/BF00280403","volume":"77","author":"L. Arkeryd","year":"1981","unstructured":"Arkeryd, L.: Intermolecular forces of infinite range and the Boltzmann equation, entropy dissipation and long-range interaction. Arch. Ration. Mech. Anal. 77(1), 11\u201321 (1981)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"9741_CR3","doi-asserted-by":"crossref","first-page":"47","DOI":"10.4171\/RMI\/233","volume":"14","author":"F. Bouchut","year":"1998","unstructured":"Bouchut, F., Desvillettes, L.: A proof of the smoothing properties of the positive part of Boltzmann\u2019s kernel. Rev. Mat. Iberoam. 14(1), 47\u201361 (1998)","journal-title":"Rev. Mat. Iberoam."},{"key":"9741_CR4","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/BF01197579","volume":"74","author":"R. Caflisch","year":"1980","unstructured":"Caflisch, R.: The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Commun. Math. Phys. 74, 71\u201395 (1980)","journal-title":"Commun. Math. Phys."},{"key":"9741_CR5","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF01197752","volume":"74","author":"R. Caflisch","year":"1980","unstructured":"Caflisch, R.: The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Commun. Math. Phys. 74, 97\u2013109 (1980)","journal-title":"Commun. Math. Phys."},{"issue":"3\u20134","key":"9741_CR6","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/BF02188578","volume":"74","author":"E.A. Carlen","year":"1994","unstructured":"Carlen, E.A., Carvalho, M.C.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74(3\u20134), 743\u2013782 (1994)","journal-title":"J. Stat. Phys."},{"issue":"1\u20132","key":"9741_CR7","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1023\/A:1023623503092","volume":"112","author":"E.A. Carlen","year":"2003","unstructured":"Carlen, E.A., Lu, X.G.: Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Stat. Phys. 112(1\u20132), 59\u2013134 (2003)","journal-title":"J. Stat. Phys."},{"issue":"3","key":"9741_CR8","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s002200050511","volume":"199","author":"E.A. Carlen","year":"1999","unstructured":"Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199(3), 521\u2013546 (1999)","journal-title":"Commun. Math. Phys."},{"key":"9741_CR9","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1039-9","volume-title":"The Boltzmann Equation and Its Applications","author":"C. Cercignani","year":"1988","unstructured":"Cercignani, C.: The Boltzmann Equation and Its Applications. Springer, New York (1988)"},{"key":"9741_CR10","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-8524-8","volume-title":"The Mathematical Theory of Dilute Gases","author":"C. Cercignani","year":"1994","unstructured":"Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994)"},{"key":"9741_CR11","volume-title":"The Mathematical Theory of Non-Uniform Gases","author":"S. Chapman","year":"1970","unstructured":"Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1970)","edition":"3"},{"issue":"4","key":"9741_CR12","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/BF00375586","volume":"123","author":"L. Desvillettes","year":"1993","unstructured":"Desvillettes, L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Ration. Mech. Anal. 123(4), 387\u2013404 (1993)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3\u20134","key":"9741_CR13","first-page":"235","volume":"54","author":"L. Desvillettes","year":"2007","unstructured":"Desvillettes, L., Mouhot, C.: Large time behavior of the a priori bounds for the solutions to the spatially homogeneous Boltzmann equations with soft potentials. Asymptot. Anal. 54(3\u20134), 235\u2013245 (2007)","journal-title":"Asymptot. Anal."},{"issue":"1","key":"9741_CR14","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/BF00375684","volume":"114","author":"R.J. DiPerna","year":"1991","unstructured":"DiPerna, R.J., Lions, P.-L.: Global solutions of Boltzmann\u2019s equation and the entropy inequality. Arch. Ration. Mech. Anal. 114(1), 47\u201355 (1991)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3\u20134","key":"9741_CR15","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1007\/BF02765543","volume":"89","author":"T. Goudon","year":"1997","unstructured":"Goudon, T.: On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions. J.\u00a0Stat. Phys. 89(3\u20134), 751\u2013776 (1997)","journal-title":"J.\u00a0Stat. Phys."},{"key":"9741_CR16","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s00205-007-0067-3","volume":"187","author":"Y. Guo","year":"2008","unstructured":"Guo, Y., Strain, R.: Decay for soft potentials near Maxwellian. Arch. Ration. Mech. Anal. 187, 287\u2013339 (2008)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"9741_CR17","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1215\/kjm\/1250519017","volume":"34","author":"P.L. Lions","year":"1994","unstructured":"Lions, P.L.: Compactness in Boltzmann\u2019s equation via Fourier integral operators and applications, I. J.\u00a0Math. Kyoto Univ. 34(2), 391\u2013427 (1994)","journal-title":"J.\u00a0Math. Kyoto Univ."},{"issue":"1","key":"9741_CR18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/S0764-4442(97)82709-7","volume":"326","author":"P.L. Lions","year":"1998","unstructured":"Lions, P.L.: Regularity and compactness for Boltzmann collision operators without angular cut-off. C.R. Acad. Sci. Paris, S\u00e9r. I 326(1), 37\u201341 (1998)","journal-title":"C.R. Acad. Sci. Paris, S\u00e9r. I"},{"issue":"2","key":"9741_CR19","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1006\/jmaa.1998.6141","volume":"228","author":"X.G. Lu","year":"1998","unstructured":"Lu, X.G.: A direct method for the regularity of the gain term in the Boltzmann equation. J. Math. Anal. Appl. 228(2), 409\u2013435 (1998)","journal-title":"J. Math. Anal. Appl."},{"issue":"3\u20134","key":"9741_CR20","first-page":"765","volume":"96","author":"X.G. Lu","year":"1999","unstructured":"Lu, X.G.: Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation. J. Stat. Phys. 96(3\u20134), 765\u2013796 (1999)","journal-title":"J. Stat. Phys."},{"issue":"3","key":"9741_CR21","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1007\/s00220-005-1455-x","volume":"261","author":"C. Mouhot","year":"2006","unstructured":"Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261(3), 629\u2013672 (2006)","journal-title":"Commun. Math. Phys."},{"key":"9741_CR22","volume-title":"The Particle Kinetics of Plasmas","author":"I.P. Shkarofsky","year":"1966","unstructured":"Shkarofsky, I.P., Johnston, T.W., Bachynski, M.P.: The Particle Kinetics of Plasmas. Addison-Wesley, Reading (1966)"},{"issue":"3","key":"9741_CR23","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1007\/s002200050631","volume":"203","author":"G. Toscani","year":"1999","unstructured":"Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commun. Math. Phys. 203(3), 667\u2013706 (1999)","journal-title":"Commun. Math. Phys."},{"issue":"5\u20136","key":"9741_CR24","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1023\/A:1018623930325","volume":"98","author":"G. Toscani","year":"2000","unstructured":"Toscani, G., Villani, C.: On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. J. Stat. Phys. 98(5\u20136), 1279\u20131309 (2000)","journal-title":"J. Stat. Phys."},{"issue":"3","key":"9741_CR25","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s002050050106","volume":"143","author":"C. Villani","year":"1998","unstructured":"Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273\u2013307 (1998)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"9741_CR26","doi-asserted-by":"crossref","first-page":"335","DOI":"10.4171\/RMI\/259","volume":"15","author":"C. Villani","year":"1999","unstructured":"Villani, C.: Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation. Rev. Mat. Iberoam. 15(2), 335\u2013352 (1999)","journal-title":"Rev. Mat. Iberoam."},{"key":"9741_CR27","series-title":"Handbook of Mathematical Fluid Dynamics","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/S1874-5792(02)80004-0","volume-title":"A Review of Mathematical Topics in Collisional Kinetic Theory","author":"C. Villani","year":"2002","unstructured":"Villani, C.: In: A Review of Mathematical Topics in Collisional Kinetic Theory. Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71\u2013305. North-Holland, Amsterdam (2002)"},{"issue":"3","key":"9741_CR28","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1007\/s00220-002-0777-1","volume":"234","author":"C. Villani","year":"2003","unstructured":"Villani, C.: Cercignani\u2019s conjecture is sometimes true and always almost true. Commun. Math. Phys. 234(3), 455\u2013490 (2003)","journal-title":"Commun. Math. Phys."},{"issue":"11\u201312","key":"9741_CR29","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1080\/03605309408821082","volume":"19","author":"B. Wennberg","year":"1994","unstructured":"Wennberg, B.: Regularity in the Boltzmann equation and the Radon transform. Commun. Partial Differ. Eqs. 19(11\u201312), 2057\u20132074 (1994)","journal-title":"Commun. Partial Differ. Eqs."},{"issue":"5\u20136","key":"9741_CR30","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1007\/BF02183613","volume":"86","author":"B. Wennberg","year":"1997","unstructured":"Wennberg, B.: Entropy dissipation and moment production for the Boltzmann equation. J. Stat. Phys. 86(5\u20136), 1053\u20131066 (1997)","journal-title":"J. Stat. Phys."}],"container-title":["Journal of Statistical Physics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10955-009-9741-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10955-009-9741-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10955-009-9741-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,2]],"date-time":"2019-06-02T09:51:10Z","timestamp":1559469070000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10955-009-9741-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5]]},"references-count":30,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,5]]}},"alternative-id":["9741"],"URL":"https:\/\/doi.org\/10.1007\/s10955-009-9741-1","relation":{},"ISSN":["0022-4715","1572-9613"],"issn-type":[{"value":"0022-4715","type":"print"},{"value":"1572-9613","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,5]]}}}