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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

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