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Volumn 58, Issue 3, 2009, Pages 999-1022

Existence of global solutions to the cauchy problem for the inelastic Boltzmann equation with near-vacuum data

Author keywords

Granular materials; Inelastic boltzmann equation; Kinetic equations; Viscoelastic spheres

Indexed keywords


EID: 69549101986     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2009.58.3506     Document Type: Article
Times cited : (42)

References (29)
  • 1
    • 10044233885 scopus 로고    scopus 로고
    • On blowup for gain-term-only classical and relativistic Boltzmann equations
    • MR 2102321 (2005h:82093)
    • H. ANDRÉASSON, S. CALOGERO, and R. ILLNER, On blowup for gain-term-only classical and relativistic Boltzmann equations. Math. Methods Appl. Sci. 27 (2004), 2231-2240, http://dx.doi.org/10.1002/mma.555. MR 2102321 (2005h:82093)
    • (2004) Math. Methods Appl. Sci. , vol.27 , pp. 2231-2240
    • Andréasson, H.1    Calogero, S.2    Illner, R.3
  • 2
    • 0007219031 scopus 로고
    • On the Cauchy problem for the nonlinear Boltzmann equation: Global existence, uniqueness and asymptotic stability
    • MR 776503 (86h:82038)
    • N. BELLOMO and G. TOSCANI, On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic stability, J. Math. Phys. 26 (1985), 334-338, http://dx.doi.org/10.1063/L526664. MR 776503 (86h:82038)
    • (1985) J. Math. Phys. , vol.26 , pp. 334-338
    • Bellomo, N.1    Toscani, G.2
  • 3
    • 0037764158 scopus 로고    scopus 로고
    • The collapse phenomenon in one-dimensional inelastic point particle systems
    • MR 1704833 (2001j:74017)
    • D. BENEDETTO and E. Caglioti, The collapse phenomenon in one-dimensional inelastic point particle systems, Phys. D 132 (1999), 457-475, http://dx.doi.org/10.1016/S0167-2789(99)00056-l. MR 1704833 (2001j:74017)
    • (1999) Phys. D , vol.132 , pp. 457-475
    • Benedetto, D.1    Caglioti, E.2
  • 4
    • 0035643773 scopus 로고    scopus 로고
    • On the one-dimensional Boltzmann equation for granular flows
    • DOI 10.1051/m2an:2001141
    • D. BENEDETTO and M. PULVIRENTI, On the one-dimensional Boltzmann equation for granular flows, M2AN Math. Model. Numer. Anal. 35 (2001), 899-905, http://dx.doi.Org/10.1051/m2an:2001l4l. MR 1866273 (2002h:82066) (Pubitemid 33573400)
    • (2001) Mathematical Modelling and Numerical Analysis , vol.35 , Issue.5 , pp. 899-905
    • Benedetto, D.1    Pulvirenti, M.2
  • 5
    • 0037210439 scopus 로고    scopus 로고
    • Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions
    • MR 1966332 (2004g:82110)
    • A.V. BOBYLEV and C. CERCIGNANI, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys. 110 (2003), 333-375, http://dx.doi.Org/10.1023/A:1021031031038. MR 1966332 (2004g:82110)
    • (2003) J. Statist. Phys. , vol.110 , pp. 333-375
    • Bobylev, A.V.1    Cercignani, C.2
  • 6
    • 0037276633 scopus 로고    scopus 로고
    • Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials
    • MR 1964277 (2004d:82032)
    • A.V. BOBYLEV, C. CERCIGNANI, and G. TOSCANI, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Statist. Phys. Ill (2003), 403-417, http://dx.doi.Org/10.1023/A: 1022273528296. MR 1964277 (2004d:82032)
    • (2003) J. Statist. Phys. , vol.111 , pp. 403-417
    • Bobylev, A.V.1    Cercignani, C.2    Toscani, G.3
  • 7
    • 0034343060 scopus 로고    scopus 로고
    • On some properties of kinetic and hydrodynamic equations for inelastic interactions
    • MR 1749231 (2001c:82063)
    • A.V. BOBYLEV, J.A. CARRILLO, and I.M. GAMBA, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys. 98 (2000), 743-773, http://dx.doi.Org/10.1023/A:1018627625800. MR 1749231 (2001c:82063)
    • (2000) J. Statist. Phys. , vol.98 , pp. 743-773
    • Bobylev, A.V.1    Carrillo, J.A.2    Gamba, I.M.3
  • 8
    • 7244221021 scopus 로고    scopus 로고
    • Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions
    • MR2096050 (2005g:82111)
    • A.V. BOBYLEV, I.M. GAMBA, and V.A. PANFEROV, Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions, J. Statist. Phys. 116 (2004), 1651-1682, http://dx.doi.org/10.1023/BJOSS. 0000041751.11664.ea. MR2096050 (2005g:82111)
    • (2004) J. Statist. Phys. , vol.116 , pp. 1651-1682
    • Bobylev, A.V.1    Gamba, I.M.2    Panferov, V.A.3
  • 9
    • 0003840403 scopus 로고    scopus 로고
    • On the singularities of the global small solutions of the full Boltzmann equation
    • MR 1798557(2001m:350l4)
    • L. BOUDIN and L. DESVILLETTES, On the singularities of the global small solutions of the full Boltzmann equation, Monatsh. Math. 131 (2000), 91-108, http://dx.doi.org/10.1007/s006050070015. MR 1798557(2001m:350l4)
    • (2000) Monatsh. Math. , vol.131 , pp. 91-108
    • Boudin, L.1    Desvillettes, L.2
  • 10
    • 10444237637 scopus 로고    scopus 로고
    • Oxford Graduate Texts, Oxford University Press, Oxford, ISBN 0-19-853038-2. MR 2101911 (2006a:76087)
    • N.V. BRILLIANTOV and T. PÖSCHEL, Kinetic Theory of Granular Gases, Oxford Graduate Texts, Oxford University Press, Oxford, 2004, ISBN 0-19-853038-2. MR 2101911 (2006a:76087)
    • (2004) Kinetic Theory of Granular Gases
    • Brilliantov, N.V.1    Pöschel, T.2
  • 11
    • 0003200990 scopus 로고
    • The mathematical theory of dilute gases
    • Springer-Verlag, New York, ISBN 0-387-94294-7. MR 1307620 (96g:82046)
    • C. CERCIGNANI, R. ILLNER, and M. PULVIRENTI, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol.106, Springer-Verlag, New York, 1994, ISBN 0-387-94294-7. MR 1307620 (96g:82046)
    • (1994) Applied Mathematical Sciences , vol.106
    • Cercignani, C.1    Illner, R.2    Pulvirenti, M.3
  • 12
    • 0035540897 scopus 로고    scopus 로고
    • On diffusive equilibria in generalized kinetic theory
    • MR 1861207 (2002j:82093)
    • C. CERCIGNANI, R. ILLNER, and C. STOICA, On diffusive equilibria in generalized kinetic theory, J. Statist. Phys. 105 (2001), 337-352, http://dx.doi.org/10.1007/s00220-004-1051-5. MR 1861207 (2002j:82093)
    • (2001) J. Statist. Phys. , vol.105 , pp. 337-352
    • Cercignani, C.1    Illner, R.2    Stoica, C.3
  • 13
    • 0001089576 scopus 로고
    • On the Cauchy problem for Boltzmann equations: Global existence and weak stability
    • MR 1014927 (90k:82045)
    • R.J. DlPERNA and P.-L LIONS, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. (2) 130 (1989), 321-366, http://dx.doi.org/10.2307/1971423. MR 1014927 (90k:82045)
    • (1989) Ann. of Math. , Issue.130 , pp. 321-366
    • Dlperna, R.J.1    Lions, P.-L.2
  • 14
    • 0001485362 scopus 로고
    • Global existence proof for relativistic Boltzmann equation
    • MR 1151987 (93b:82064)
    • M. DUDYŃSKI and M.L. EKIEL-JEŻEWSKA, Global existence proof for relativistic Boltzmann equation,}. Statist. Phys. 66 (1992), 991-1001, http://dx.doi.org/10.1007/BF01055712. MR 1151987 (93b:82064)
    • (1992) J. Statist. Phys. , vol.66 , pp. 991-1001
    • Dudyński, M.1    Ekiel-Jezewska, M.L.2
  • 15
    • 84864161175 scopus 로고
    • On the modified Enskog equation for elastic and inelastic collisions. Models with spin
    • MR 1127928 (92i:82094) (English, with French summary)
    • MJ. ESTEBAN and B. PERTHAME, On the modified Enskog equation for elastic and inelastic collisions. Models with spin, Ann. Inst. H. Poincaré Anal. Non Lineaire 8 (1991), 289-308. MR 1127928 (92i:82094) (English, with French summary)
    • (1991) Ann. Inst. H. Poincaré Anal. Non Lineaire , vol.8 , pp. 289-308
    • Esteban, M.J.1    Perthame, B.2
  • 16
    • 0003448545 scopus 로고    scopus 로고
    • Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, ISBN 0-8218-0772-2. MR 1625845 (99e:35001)
    • L.C. EVANS, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, American Mathematical Society, Providence, RI, 1998, ISBN 0-8218-0772-2. MR 1625845 (99e:35001)
    • (1998) Partial Differential Equations , vol.19
    • Evans, L.C.1
  • 17
    • 2442489990 scopus 로고    scopus 로고
    • On the Boltzmann equation for diffusively excited granular media
    • MR 2053942 (2005b:82076)
    • I.M. GAMBA, V. PANFEROV, and C. VlLLANI, On the Boltzmann equation for diffusively excited granular media, Comm. Math. Phys. 246 (2004), 503-541, http://dx.doi.org/10.1007/s00220-004-1051-5. MR 2053942 (2005b:82076)
    • (2004) Comm. Math. Phys. , vol.246 , pp. 503-541
    • Gamba, I.M.1    Panferov, V.2    Vlllani, C.3
  • 18
    • 33646509890 scopus 로고    scopus 로고
    • Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data
    • MR 2217287 (2007a:82062)
    • R.T. GLASSEY, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys. 264 (2006), 705-724, http://dx.doi.org/10.1007/s00220-006-1522-y. MR 2217287 (2007a:82062)
    • (2006) Comm. Math. Phys. , vol.264 , pp. 705-724
    • Glassey, R.T.1
  • 19
    • 18144375596 scopus 로고    scopus 로고
    • Nonlinear functionab of the Boltzmann equation and uniform stability estimates
    • MR 2146347 (2006b:35033)
    • S.-Y. HA, Nonlinear functionab of the Boltzmann equation and uniform stability estimates, J. Differential Equations 215 (2005), 178-205, http://dx.doi.Org/10.10l6/j.jde.2004.07.022. MR 2146347 (2006b:35033)
    • (2005) J. Differential Equations , vol.215 , pp. 178-205
    • Ha, S.-Y.1
  • 20
    • 0007222794 scopus 로고
    • Existence in the large and asymptotic behaviour for the boltzmann equation
    • MR 839316 (87j:35322)
    • K. H AM DACH E, Existence in the large and asymptotic behaviour for the Boltzmann equation, Japan J. Appl. Math. 2 (1985), 1-15. MR 839316 (87j:35322)
    • (1985) Japan J. Appl. Math. , vol.2 , pp. 1-15
    • Hamdache, K.1
  • 21
    • 21144478445 scopus 로고
    • Initial-boundary value problems for the Boltzmann equation: Global existence of weak solutions
    • MR 1179690 (93g:35132)
    • -, Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions, Arch. Rational Mech. Anal. 119 (1992), 309-353, http://dx.doi.org/10.1007/BF01837113. MR 1179690 (93g:35132)
    • (1992) Arch. Rational Mech. Anal. , vol.119 , pp. 309-353
  • 22
    • 34250139505 scopus 로고
    • The Boltzmann equation: Global existence for a rare gas in an infinite vacuum
    • MR 760333 (86a:82019)
    • R. ILLNER and M. SHINBROT, The Boltzmann equation: global existence for a rare gas in an infinite vacuum, Comm. Math. Phys. 95 (1984), 217-226, http://dx.doi.org/10.1007/BF01468142. MR 760333 (86a:82019)
    • (1984) Comm. Math. Phys. , vol.95 , pp. 217-226
    • Illner, R.1    Shinbrot, M.2
  • 23
    • 0040943239 scopus 로고
    • The Boltzmann equation. I. Uniqueness and local existence
    • MR 0475532 (57 #15133)
    • S. KANIEL and M. SHINBROT, The Boltzmann equation. I. Uniqueness and local existence, Comm. Math. Phys. 58 (1978), 65-84, http://dx.doi.org/10.1007/ BF01624788. MR 0475532 (57 #15133)
    • (1978) Comm. Math. Phys. , vol.58 , pp. 65-84
    • Kaniel, S.1    Shinbrot, M.2
  • 24
    • 0034348887 scopus 로고    scopus 로고
    • On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system
    • MR 1776840 (200 lf:45013)
    • S. MlSCHLER, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys. 210 (2000), 447-466, http://dx.doi.org/10.1007/s002200050787. MR 1776840 (200 lf:45013)
    • (2000) Comm. Math. Phys. , vol.210 , pp. 447-466
    • Mlschler, S.1
  • 25
    • 33751051996 scopus 로고    scopus 로고
    • Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem
    • MR 2264622 (2007g:82053)
    • S. MlSCHLER, C. MOUHOT, and M. RODRIGUEZ RlCARD, Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem, J. Stat. Phys. 124 (2006), 655-702, http://dx.doi.org/10.1007/sl0955-006-9096-9. MR 2264622 (2007g:82053)
    • (2006) J. Stat. Phys. , vol.124 , pp. 655-702
    • Mlschler, S.1    Mouhot, C.2    Ricard, M.R.3
  • 26
    • 0031488084 scopus 로고    scopus 로고
    • Boltzmann equation with infinite energy: renormalizedsolutions and distributional solutions for small initial data and initial data close to a Maxwellian
    • MR 1466666 (98g:35197)
    • S. MlSCHLER and B. PERTHAME, Boltzmann equation with infinite energy: renormalizedsolutions and distributional solutions for small initial data and initial data close to a Maxwellian, SIAM J. Math. Anal. 28 (1997), 1015-1027, http://dx.doi.org/10.1137/S0036l4l096298102. MR 1466666 (98g:35197)
    • (1997) SIAM J. Math. Anal. , vol.28 , pp. 1015-1027
    • Mischler, S.1    Perthame, B.2
  • 27
    • 18844442701 scopus 로고
    • Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian
    • MR 1016317 (91c:82054)
    • A. PALCZEWSKI and G. TOSCANI, Global solution of the Boltzmann equation for rigid spheres and initial data close to a local Maxwellian, J. Math. Phys. 30 (1989), 2445-2450, http://dx.doi.org/10.1063/L528522. MR 1016317 (91c:82054)
    • (1989) J. Math. Phys. , vol.30 , pp. 2445-2450
    • Palczewski, A.1    Toscani, G.2
  • 28
    • 0007365295 scopus 로고
    • H-theorem and asymptotic trendof the solution for a rarefied gas in the vacuum
    • MR 906131 (89b:82053)
    • G. TOSCANI, H-theorem and asymptotic trendof the solution for a rarefied gas in the vacuum, Arch. Rational Mech. Anal. 100 (1987), 1-12, http://dx.doi.org/10.1007/BF00281245. MR 906131 (89b:82053)
    • (1987) Arch. Rational Mech. Anal. , vol.100 , pp. 1-12
    • Toscani, G.1
  • 29
    • 85008266605 scopus 로고
    • On the Cauchy problem of the Boltzmann equation with a soft potential
    • MR 677262 (84h:82048)
    • S. UKAI and K. ASANO, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci. 18 (1982), 477-519 (57-99), http://dx.doi.org/10.2977/prims/1195183569. MR 677262 (84h:82048)
    • (1982) Publ. Res. Inst. Math. Sci. , vol.18 , Issue.57-99 , pp. 477-519
    • Ukai, S.1    Asano, K.2


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