HYDRODYNAMIC LIMIT OF THE BOLTZMANN-MONGE-AMPERE SYSTEM

Year 2019, Volume: 9 Issue: 3, 493 - 499, 01.09.2019

F. Ben Belgacem

Abstract

This paper investigates the hydrodynamic limit of the Boltzmann-Monge-Ampere system in the so-called quasineutral regime and more precisely the convergence of the Boltzmann-Monge-Ampere system to the Euler equation by using the relative entropy method.

Keywords

Boltzman equation, Monge-Ampere equation, Euler equations of the incompressible uid.

References

• Brenier, Y. (1991),Polar factorization and monotone rearrangement of vector-valued functions, Pure Appl. Math., 44, (4), pp. 375-417.
• Brenier, Y. and Gr´egoire, L. (2004) , A geometric approximation to the Euler equations: the Vlasov- Monge-Amp`ere system, Geom. Funct. Anal. 14, pp. 1182-1218.
• Caffarelli, L. (1990), Interior W2,pestimates for solutions of Monge-Amp`ere equation, Ann. Math. 131 (1) pp. 135-150.
• Cercignani, C. (1988), The Boltzmann equation and its applications, Applied Mathematical Sciences, 67. Springer, New York.
• Cercignani, C., Illner, R., and Pulvirenti, M. (1994), The mathematical theory of dilute gases, Applied Mathematical Sciences, 106, Springer, New York.
• DiPerna, R. J., Lions, P. L. (1989), On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math., 130, pp. 321-366.
• DiPerna, R. J., Lions, P. L. (1989), Global weak solution of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42, pp. 729-757.
• Desvillettes, L., Dolbeault, J. (1991), On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differential Equations, 16, pp. 451-489.
• Glassey, R. (1996), The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathe- matics (SIAM), Philadelphia.
• Guo, Y. (2002), The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math., 55, pp. 1104-1135.
• Hsiao, L., Cai LI, F. and Wang, S. (2007), Convergence of the Vlasov-Poisson-Boltzmann System to the Incompressible Euler Equations, Acta Mathematica Sinica, English Series, Vol. 23 (4), pp. 761-768.
• Liu, T. P., Yang, T., Yu, S. H., (2004), Energy method for Boltzmann equation. Physica D, 188, pp. 178-192.
• Yang, T., Yu, H. J., Zhao, H. J. (2006), Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch Rational Mech Anal, 182 (3), pp. 415-470.