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November  2020, 19(11): 5197-5217. doi: 10.3934/cpaa.2020233

On the inelastic Boltzmann equation for soft potentials with diffusion

1. 

School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

2. 

Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

*Corresponding author

Received  January 2020 Revised  May 2020 Published  September 2020

Fund Project: The work is supported by NSF grant of China (No.11501292) and NUPTSF (Grant No. NY218091)

We are concerned with the Cauchy problem of the inelastic Boltzmann equation for soft potentials, with a Laplace term representing the random background forcing. The inelastic interaction here is characterized by the non-constant restitution coefficient. We prove that under the assumption that the initial datum has bounded mass, energy and entropy, there exists a weak solution to this equation. The smoothing effect of weak solutions is also studied. In addition, it is shown the solution is unique and stable with respect to the initial datum provided that the initial datum belongs to $ L^{2}(R^{3}) $.

Citation: Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233
References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J, 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar

[2]

R. J. AlonsoE. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Commun. Math. Phys., 298 (2010), 293-322.  doi: 10.1007/s00220-010-1065-0.  Google Scholar

[3]

R. J. Alonso and B. Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal., 42 (2010), 2499-2538.  doi: 10.1137/100793979.  Google Scholar

[4]

R. J. Alonso and B. Lods, Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.  doi: 10.1016/j.jmaa.2012.07.045.  Google Scholar

[5]

R. J. Alonso and B. Lods, Boltzmann model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity, Commun. Math. Phys., 331 (2014), 545-591.  doi: 10.1007/s00220-014-2089-7.  Google Scholar

[6]

M. BisiJ. A. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equillibria, J. Stat. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5.  Google Scholar

[7]

M. Bisi, J. A. Carrillo and G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model, J. Stat. Phys., 124 (2006), 625–653. doi: 10.1007/s10955-006-9035-9.  Google Scholar

[8]

A. V. BobylevJ. A. Carrillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800.  Google Scholar

[9]

A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys., 106 (2002), 547-567.  doi: 10.1023/A:1013754205008.  Google Scholar

[10]

A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375.  doi: 10.1023/A:1021031031038.  Google Scholar

[11]

C. CercignaniR. Illner and C. Stoica, On diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352.  doi: 10.1023/A:1012246513712.  Google Scholar

[12]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.  Google Scholar

[13]

S. H. Choi and S.Y. Ha, Global existence of classical solutions to the inelastic Vlasov-Poisson-Boltzmann system, J. Stat. Phys., 156 (2014), 948-974.  doi: 10.1007/s10955-014-1041-8.  Google Scholar

[14]

M. H. ErnstE. Trizac and A. Barrat, The Boltzmann equation for driven systems of inelastic soft spheres, J. Stat. Phys., 124 (2006), 549-586.  doi: 10.1007/s10955-006-9062-6.  Google Scholar

[15]

N. Fournier and H. Guérin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, J. Funct. Anal., 256 (2008), 2542-2560.  doi: 10.1016/j.jfa.2008.11.008.  Google Scholar

[16]

I. M. GambaV. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5.  Google Scholar

[17]

T. Gustafsson, Global $L^{p}$ properties for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 103 (1988), 1-38.  doi: 10.1007/BF00292919.  Google Scholar

[18]

F. Meng and X. P. Yang, The existence and convergence of the Maxwell model for granular materials, Applicable Anal., 95 (2016), 2383-2396.  doi: 10.1080/00036811.2015.1091068.  Google Scholar

[19]

S. MischlerC. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres. Part I: The Cauchy problem, J. Stat. Phys., 124 (2006), 605-702.  doi: 10.1007/s10955-006-9096-9.  Google Scholar

[20]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. Part II: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.  Google Scholar

[21]

S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Commun. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9.  Google Scholar

[22]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[23]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann equation and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[24]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

[25]

K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal., 266 (2014), 3134-3155.  doi: 10.1016/j.jfa.2013.11.005.  Google Scholar

[26]

J. Wei and X. Zhang, On the Cauchy problem for the inelastic Boltzmann equation with external force, J. Stat. Phys., 146 (2012), 592-609.  doi: 10.1007/s10955-011-0410-9.  Google Scholar

show all references

References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J, 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar

[2]

R. J. AlonsoE. Carneiro and I. M. Gamba, Convolution inequalities for the Boltzmann collision operator, Commun. Math. Phys., 298 (2010), 293-322.  doi: 10.1007/s00220-010-1065-0.  Google Scholar

[3]

R. J. Alonso and B. Lods, Free cooling and high-energy tails of granular gases with variable restitution coefficient, SIAM J. Math. Anal., 42 (2010), 2499-2538.  doi: 10.1137/100793979.  Google Scholar

[4]

R. J. Alonso and B. Lods, Two proofs of Haff's law for dissipative gases: the use of entropy and the weakly inelastic regime, J. Math. Anal. Appl., 397 (2013), 260-275.  doi: 10.1016/j.jmaa.2012.07.045.  Google Scholar

[5]

R. J. Alonso and B. Lods, Boltzmann model for viscoelastic particles: asymptotic behavior, pointwise lower bounds and regularity, Commun. Math. Phys., 331 (2014), 545-591.  doi: 10.1007/s00220-014-2089-7.  Google Scholar

[6]

M. BisiJ. A. Carrillo and G. Toscani, Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equillibria, J. Stat. Phys., 118 (2005), 301-331.  doi: 10.1007/s10955-004-8785-5.  Google Scholar

[7]

M. Bisi, J. A. Carrillo and G. Toscani, Decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model, J. Stat. Phys., 124 (2006), 625–653. doi: 10.1007/s10955-006-9035-9.  Google Scholar

[8]

A. V. BobylevJ. A. Carrillo and I. M. Gamba, On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Stat. Phys., 98 (2000), 743-773.  doi: 10.1023/A:1018627625800.  Google Scholar

[9]

A. V. Bobylev and C. Cercignani, Moment equations for a granular material in a thermal bath, J. Stat. Phys., 106 (2002), 547-567.  doi: 10.1023/A:1013754205008.  Google Scholar

[10]

A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys., 110 (2003), 333-375.  doi: 10.1023/A:1021031031038.  Google Scholar

[11]

C. CercignaniR. Illner and C. Stoica, On diffusive equilibria in generalized kinetic theory, J. Stat. Phys., 105 (2001), 337-352.  doi: 10.1023/A:1012246513712.  Google Scholar

[12]

Y. Chen and L. He, Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.  Google Scholar

[13]

S. H. Choi and S.Y. Ha, Global existence of classical solutions to the inelastic Vlasov-Poisson-Boltzmann system, J. Stat. Phys., 156 (2014), 948-974.  doi: 10.1007/s10955-014-1041-8.  Google Scholar

[14]

M. H. ErnstE. Trizac and A. Barrat, The Boltzmann equation for driven systems of inelastic soft spheres, J. Stat. Phys., 124 (2006), 549-586.  doi: 10.1007/s10955-006-9062-6.  Google Scholar

[15]

N. Fournier and H. Guérin, Well-posedness of the spatially homogeneous Landau equation for soft potentials, J. Funct. Anal., 256 (2008), 2542-2560.  doi: 10.1016/j.jfa.2008.11.008.  Google Scholar

[16]

I. M. GambaV. Panferov and C. Villani, On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., 246 (2004), 503-541.  doi: 10.1007/s00220-004-1051-5.  Google Scholar

[17]

T. Gustafsson, Global $L^{p}$ properties for the spatially homogeneous Boltzmann equation, Arch. Rat. Mech. Anal., 103 (1988), 1-38.  doi: 10.1007/BF00292919.  Google Scholar

[18]

F. Meng and X. P. Yang, The existence and convergence of the Maxwell model for granular materials, Applicable Anal., 95 (2016), 2383-2396.  doi: 10.1080/00036811.2015.1091068.  Google Scholar

[19]

S. MischlerC. Mouhot and M. Rodriguez Ricard, Cooling process for inelastic Boltzmann equations for hard spheres. Part I: The Cauchy problem, J. Stat. Phys., 124 (2006), 605-702.  doi: 10.1007/s10955-006-9096-9.  Google Scholar

[20]

S. Mischler and C. Mouhot, Cooling process for inelastic Boltzmann equations for hard spheres. Part II: Self-similar solutions and tail behavior, J. Stat. Phys., 124 (2006), 703-746.  doi: 10.1007/s10955-006-9097-8.  Google Scholar

[21]

S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Commun. Math. Phys., 288 (2009), 431-502.  doi: 10.1007/s00220-009-0773-9.  Google Scholar

[22]

C. Mouhot and C. Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal., 173 (2004), 169-212.  doi: 10.1007/s00205-004-0316-7.  Google Scholar

[23]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann equation and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.  Google Scholar

[24]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

[25]

K. C. Wu, Global in time estimates for the spatially homogeneous Landau equation with soft potentials, J. Funct. Anal., 266 (2014), 3134-3155.  doi: 10.1016/j.jfa.2013.11.005.  Google Scholar

[26]

J. Wei and X. Zhang, On the Cauchy problem for the inelastic Boltzmann equation with external force, J. Stat. Phys., 146 (2012), 592-609.  doi: 10.1007/s10955-011-0410-9.  Google Scholar

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