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December 2017, 10(4): 901-924. doi: 10.3934/krm.2017036

Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction

1. 

Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

2. 

Institute for Analysis, Karlsruhe Institute of Technology (KIT), Englerstraße 2,76131 Karlsruhe, Germany

Received  December 2015 Revised  November 2016 Published  March 2017

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

Citation: Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic & Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036
References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Archive for Rational Mechanics and Analysis, 152 (2000), 327-355. doi: 10.1007/s002050000083.

[2]

L. Arkeryd, On the Boltzmann equation. Ⅰ: Existence, Archive for Rational Mechanics and Analysis, 45 (1972), 1-16. doi: 10.1007/BF00253392.

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Archive for Rational Mechanics and Analysis, 77 (1981), 11-21. doi: 10.1007/BF00280403.

[4]

J. -M. Barbaroux, D. Hundertmark, T. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules to appear in Archive for Rational Mechanics and Analysis preprint 1509. 01444. doi: 10.1007/s00205-017-1101-8.

[5]

C. Cercignani, The Boltzmann Equation and Its Applications Applied Mathematical Sciences, 67 Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[6]

P. J. Cohen, A simple proof of the Denjoy-Carleman Theorem, The American Mathematical Monthly, 75 (1968), 26-31. doi: 10.2307/2315100.

[7]

L. Desvillettes, Boltzmann’s kernel and the spatially homogeneous Boltzmann equation, Rivista di Matematica della Università di Parma (6), 4* (2001), 1–22. Available at http://www.rivmat.unipr.it/vols/2001-4s/indice.html.

[8]

L. DesvillettesG. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Transactions of the American Mathematical Society, 361 (2009), 1731-1747. doi: 10.1090/S0002-9947-08-04574-1.

[9]

S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions 2nd edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston Inc. , Boston, MA, 2002. doi: 10.1007/978-0-8176-8134-0.

[10]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 16 (1999), 467-501. doi: 10.1016/S0294-1449(99)80025-0.

[11]

Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka Journal of Mathematics, 24 (1987), 13-35.

[12]

Y. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete and Continuous Dynamical Systems, 24 (2009), 187-212. doi: 10.3934/dcds.2009.24.187.

[13]

W. Rudin, Real and Complex Analysis 3rd edition, McGraw-Hill Book Co. , New York, 1987.

[14]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, Journal of Statistical Physics, 94 (1999), 619-637. doi: 10.1023/A:1004589506756.

[15]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106.

show all references

References:
[1]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Archive for Rational Mechanics and Analysis, 152 (2000), 327-355. doi: 10.1007/s002050000083.

[2]

L. Arkeryd, On the Boltzmann equation. Ⅰ: Existence, Archive for Rational Mechanics and Analysis, 45 (1972), 1-16. doi: 10.1007/BF00253392.

[3]

L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation, Archive for Rational Mechanics and Analysis, 77 (1981), 11-21. doi: 10.1007/BF00280403.

[4]

J. -M. Barbaroux, D. Hundertmark, T. Ried and S. Vugalter, Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules to appear in Archive for Rational Mechanics and Analysis preprint 1509. 01444. doi: 10.1007/s00205-017-1101-8.

[5]

C. Cercignani, The Boltzmann Equation and Its Applications Applied Mathematical Sciences, 67 Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1039-9.

[6]

P. J. Cohen, A simple proof of the Denjoy-Carleman Theorem, The American Mathematical Monthly, 75 (1968), 26-31. doi: 10.2307/2315100.

[7]

L. Desvillettes, Boltzmann’s kernel and the spatially homogeneous Boltzmann equation, Rivista di Matematica della Università di Parma (6), 4* (2001), 1–22. Available at http://www.rivmat.unipr.it/vols/2001-4s/indice.html.

[8]

L. DesvillettesG. Furioli and E. Terraneo, Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules, Transactions of the American Mathematical Society, 361 (2009), 1731-1747. doi: 10.1090/S0002-9947-08-04574-1.

[9]

S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions 2nd edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston Inc. , Boston, MA, 2002. doi: 10.1007/978-0-8176-8134-0.

[10]

S. Mischler and B. Wennberg, On the spatially homogeneous Boltzmann equation, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 16 (1999), 467-501. doi: 10.1016/S0294-1449(99)80025-0.

[11]

Y. Morimoto, Hypoellipticity for infinitely degenerate elliptic operators, Osaka Journal of Mathematics, 24 (1987), 13-35.

[12]

Y. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete and Continuous Dynamical Systems, 24 (2009), 187-212. doi: 10.3934/dcds.2009.24.187.

[13]

W. Rudin, Real and Complex Analysis 3rd edition, McGraw-Hill Book Co. , New York, 1987.

[14]

G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, Journal of Statistical Physics, 94 (1999), 619-637. doi: 10.1023/A:1004589506756.

[15]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106.

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