• Previous Article
    Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis
  • KRM Home
  • This Issue
  • Next Article
    Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain
September  2012, 5(3): 551-561. doi: 10.3934/krm.2012.5.551

A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules

1. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan

Received  January 2012 Revised  March 2012 Published  August 2012

The purpose of this paper is to extend the result concerning the existence and the uniqueness of infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules without Grad's angular cutoff assumption in the mild singularity case, to the strong singularity case. This extension follows from a simple observation of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform of the Boltzmann collision term.
Citation: Yoshinori Morimoto. A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules. Kinetic & Related Models, 2012, 5 (3) : 551-561. doi: 10.3934/krm.2012.5.551
References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.   Google Scholar

[2]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff and Maxwellian molecules,, Math. Models Methods Appl. Sci., 15 (2005), 907.   Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff,, Kyoto J. Math., 52 (2012), 433.   Google Scholar

[4]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  doi: 10.1002/cpa.20298.  Google Scholar

[5]

Z. H. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,, Kinetic and Related Models, 1 (2008), 453.   Google Scholar

[6]

N. Jacob, "Pseudo-Differential Operators and Markov Process. Vol. 1. Fourier Analysis and Semigroups,", Imperial College Press, (2001).   Google Scholar

[7]

Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete and Continuous Dynamical Systems, 24 (2009), 187.   Google Scholar

[8]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Wahrsch. Verw. Geb., 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[9]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas,, J. Statist. Phys., 94 (1999), 619.   Google Scholar

[10]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.   Google Scholar

[11]

C. Villani, "A Review of Mathematical Topics in Collisional Kinetic Theory,", in, (2002), 71.   Google Scholar

[12]

C. Villani, private communication,, Kyoto, (2008).   Google Scholar

[13]

Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum,, preprint., ().   Google Scholar

show all references

References:
[1]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions,, Arch. Rational Mech. Anal., 152 (2000), 327.   Google Scholar

[2]

R. Alexandre and M. El Safadi, Littlewood-Paley theory and regularity issues in Boltzmann homogeneous equations. I. Non-cutoff and Maxwellian molecules,, Math. Models Methods Appl. Sci., 15 (2005), 907.   Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Smoothing effect of weak solutions for the spatially homogeneous Boltzmann equation without angular cutoff,, Kyoto J. Math., 52 (2012), 433.   Google Scholar

[4]

M. Cannone and G. Karch, Infinite energy solutions to the homogeneous Boltzmann equation,, Comm. Pure Appl. Math., 63 (2010), 747.  doi: 10.1002/cpa.20298.  Google Scholar

[5]

Z. H. Huo, Y. Morimoto, S. Ukai and T. Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff,, Kinetic and Related Models, 1 (2008), 453.   Google Scholar

[6]

N. Jacob, "Pseudo-Differential Operators and Markov Process. Vol. 1. Fourier Analysis and Semigroups,", Imperial College Press, (2001).   Google Scholar

[7]

Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff,, Discrete and Continuous Dynamical Systems, 24 (2009), 187.   Google Scholar

[8]

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules,, Wahrsch. Verw. Geb., 46 (): 67.  doi: 10.1007/BF00535689.  Google Scholar

[9]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equations for Maxwell gas,, J. Statist. Phys., 94 (1999), 619.   Google Scholar

[10]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations,, Arch. Rational Mech. Anal., 143 (1998), 273.   Google Scholar

[11]

C. Villani, "A Review of Mathematical Topics in Collisional Kinetic Theory,", in, (2002), 71.   Google Scholar

[12]

C. Villani, private communication,, Kyoto, (2008).   Google Scholar

[13]

Y. Morimoto and T. Yang, Villani conjecture on smoothing effect of the homogeneous Boltzmann equation with measure initial datum,, preprint., ().   Google Scholar

[1]

Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289

[2]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[3]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[4]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[5]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495

[6]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[7]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405

[8]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[9]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032

[10]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

[11]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[12]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[13]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[14]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003

[15]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[16]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[17]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[18]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[19]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

[20]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]