C0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules

ABSTRACT

In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation . We then study the asymptotic behaviour of the mild solution as the coefficients , and an estimate on is derived.

In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwellian molecules. We first show that the global existence in time of the mild solution of the viscosity equation . We then study the asymptotic behaviour of the mild solution as the coefficients , and an estimate on is derived.

Cite this paper

nullM. Zheng, "C0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 504-509. doi: 10.4236/am.2010.16066.

nullM. Zheng, "C0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules,"

References

[1] H. L. Li and A. Matsumura, “Behaviour of the Fokker- Planck-Boltzmann Equation near a Maxwellian,” Archive for Rational Mechanics and Analysis, Vol. 189, No. 1, 2008, pp. 1-44.

[2] M. Zheng and X. P. Yang, “Viscosity Analysis on the Spatially Homogeneous Botzmann Equation,” Asymptotic Analysis, Vol. 53, 2007, pp. 13-28.

[3] L. Arkeryd, “On the Boltzmann Equation,” Archive for Rational Mechanics and Analysis, Vol. 45, 1972, pp. 1-34.

[4] L. Arkeryd, “Stability in L1 for the Spatially Homogeneous Boltzmann Equation,” Archive for Rational Mechanics and Analysis, Vol. 103, 1988, pp. 151-167.

[5] T. Elmroth, “Global Boundedness of Moments of Solution of the Boltzmann Equation for Forces of Infinite Range,” Archive for Rational Mechanics and Analysis, Vol. 82, 1983, pp. 1-12.

[6] L. Desvillettes, “Some Applications of the Method of Moments for the Homogeneous Boltzmann Equation and Kac Equation,” Archive for Rational Mechanics and Analysis, Vol. 123, 1993, pp. 387-395.

[7] S. Mischler and B. Wennberg, “On the Spatially Homogeneous Boltzmann Equaion,” Annales de l’Institut Henri Poincar′e - Analyse non lin′eaire, Vol. 16, No. 4, 1999, pp. 467-501.

[8] B. Wennberg, “Entropy Dissipation and Moment Production for the Boltzmann Equation,” Journal of Statistical Physics, Vol. 86, 1997, pp. 1053-1066.

[9] B. Wennberg, “Stability and Exponential Convergence in Lp for the Spatially Homogeneous Boltzmann Equation,” Nonlinear Analysis, Theory, Methods & Applications, Vol. 20, 1993, pp. 935-964.

[10] C. Mouhot and C. Villani, “Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut- off,” Archive for Rational Mechanics and Analysis, Vol. 173, No. 2, 2004, pp. 169-212.

[11] C. Mouhot, “Rate of Convergence to Equalibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials,” Communications in Mathematical Physics, Vol. 261, 2006, pp. 629-672.

[12] D. Morgenstern, “General Existence and Uniqueness Proof for Spatial Homogeneous Solution of the Maxwell-Boltzm-ann Equation in the Case of Maxwellian Molecules,” Proceedings of the National Academy of Sciences, Vol. 40, 1954, pp. 719-721.

[13] R. J. DiPerna and P. L. Lions, “On the Fokker-Planck- Boltzmann Equation,” Communications in Mathematical Physics, Vol. 120, No. 1, 1988, pp. 1-23.

[14] I. M. Gamba, V. Panferno and C. Villani, “On the Boltzmann Equation for Diffusively Excited Granular Media,” Communications in Mathematical Physics, Vol. 246, No. 3, 2004, pp. 503-541

[1] H. L. Li and A. Matsumura, “Behaviour of the Fokker- Planck-Boltzmann Equation near a Maxwellian,” Archive for Rational Mechanics and Analysis, Vol. 189, No. 1, 2008, pp. 1-44.

[2] M. Zheng and X. P. Yang, “Viscosity Analysis on the Spatially Homogeneous Botzmann Equation,” Asymptotic Analysis, Vol. 53, 2007, pp. 13-28.

[3] L. Arkeryd, “On the Boltzmann Equation,” Archive for Rational Mechanics and Analysis, Vol. 45, 1972, pp. 1-34.

[4] L. Arkeryd, “Stability in L1 for the Spatially Homogeneous Boltzmann Equation,” Archive for Rational Mechanics and Analysis, Vol. 103, 1988, pp. 151-167.

[5] T. Elmroth, “Global Boundedness of Moments of Solution of the Boltzmann Equation for Forces of Infinite Range,” Archive for Rational Mechanics and Analysis, Vol. 82, 1983, pp. 1-12.

[6] L. Desvillettes, “Some Applications of the Method of Moments for the Homogeneous Boltzmann Equation and Kac Equation,” Archive for Rational Mechanics and Analysis, Vol. 123, 1993, pp. 387-395.

[7] S. Mischler and B. Wennberg, “On the Spatially Homogeneous Boltzmann Equaion,” Annales de l’Institut Henri Poincar′e - Analyse non lin′eaire, Vol. 16, No. 4, 1999, pp. 467-501.

[8] B. Wennberg, “Entropy Dissipation and Moment Production for the Boltzmann Equation,” Journal of Statistical Physics, Vol. 86, 1997, pp. 1053-1066.

[9] B. Wennberg, “Stability and Exponential Convergence in Lp for the Spatially Homogeneous Boltzmann Equation,” Nonlinear Analysis, Theory, Methods & Applications, Vol. 20, 1993, pp. 935-964.

[10] C. Mouhot and C. Villani, “Regularity Theory for the Spatially Homogeneous Boltzmann Equation with Cut- off,” Archive for Rational Mechanics and Analysis, Vol. 173, No. 2, 2004, pp. 169-212.

[11] C. Mouhot, “Rate of Convergence to Equalibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials,” Communications in Mathematical Physics, Vol. 261, 2006, pp. 629-672.

[12] D. Morgenstern, “General Existence and Uniqueness Proof for Spatial Homogeneous Solution of the Maxwell-Boltzm-ann Equation in the Case of Maxwellian Molecules,” Proceedings of the National Academy of Sciences, Vol. 40, 1954, pp. 719-721.

[13] R. J. DiPerna and P. L. Lions, “On the Fokker-Planck- Boltzmann Equation,” Communications in Mathematical Physics, Vol. 120, No. 1, 1988, pp. 1-23.

[14] I. M. Gamba, V. Panferno and C. Villani, “On the Boltzmann Equation for Diffusively Excited Granular Media,” Communications in Mathematical Physics, Vol. 246, No. 3, 2004, pp. 503-541