Abstract
We establish the convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations in this paper. The convergence is rigorously proved on the time interval where the smooth solution to the incompressible Euler equations exists. The proof relies on the compactness argument and the so-called relative-entropy method.
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References
Chandrasekhar, S., Stochastic problems in physics and astronomy, Rev. Modern Physics, 1943, 15: 1–89.
Chandrasekhar, S., Brownian motion, dynamical friction, and stellar dynamics, Rev. Modern Physics, 1949, 21: 383–388.
Bouchut, F., Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal., 1993, 111(1): 239–258.
Degond, P., Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions, Ann. Sci. 'Ecole Norm. Sup. (4), 1986, 19(4): 519–542.
Victory, Jr. H. D., O'Dwyer, B. P., On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana Univ. Math. J., 1990, 39(1): 105–156.
Havlak, K. J., Victory, Jr. H. D., On deterministic particle methods for solving Vlasov-Poisson-Fokker-Planck systems, SIAM J. Numer. Anal., 1998, 35(4): 1473–1519.
Schaeffer, J., Convergence of a difference scheme for the Vlasov-Poisson-Fokker-Planck system in one dimension, SIAM J. Numer. Anal., 3 1998, 5(3): 1149–1175.
Poupaud, F., Soler J., Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Models Methods Appl. Sci., 2000, 10(7): 1027–1045.
Goudon, T., Nieto, J., Poupaud, F., Soler, J., Multidimessional high-field limit of the electrostatic vlasov-poisson-fokker-planck system, J. Differential Equations, 2005, 213(2): 418–442.
Nieto, J., Poupaud, F., Soler, J., High-field limit for the Vlasov-Poisson-Fokker-Planck system, Arch. Ration. Mech. Anal., 2001, 158(1): 29–59.
Arnold, A., Carrillo, J. A., Gamba, I., Shu, C.-W., Low and high field scaling limits for the Vlasovand Wigner-Poisson-Fokker-Planck systems, Transport Theory Statist. Phys., 2001, 30(2-3): 121–153.
Beale, J. T., Kato, T., Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., 1984, 94(1): 61–66.
Lions, P. L., Mathematical Topics in Fluid Mechanics. Vol. 1, Incompressible models, volume 3 of Oxford Lecture Series in Mathematics and its Applications, New York: The Clarendon Press Oxford University Press, 1996.
McGrath, F. J., Nonstationary plane flow of viscous and ideal fluids, Arch. Rational Mech. Anal., 1967, 27: 329–348.
Dafermos, C. M., Hyperbolic Conservation Laws in Continuum Physics, Vol. 325 of Grundlehren der Mathematischen Wissenschaften, Berlin: Springer-Verlag, 2000.
Brenier, Y., Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 2000, 25(3-4): 737–754.
Jüngel, A., Wang, S., Convergence of nonlinear Schrödinger-Poisson systems to the compressible Euler equations, Comm. Partial Differential Equations, 2003, 28(5-6): 1005–1022.
Puel, M., Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field, M2AN Math. Model. Numer. Anal., 2002, 36(6): 1071–1090.
Puel, M., Convergence of the Schrödinger-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 2002, 27(11-12): 2311–2331.
Masmoudi, N., From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 2001, 26(9-10): 1913–1928.
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Xiao, L., Li, F. & Wang, S. Convergence of the Vlasov-Poisson-Fokker-Planck system to the incompressible Euler equations. SCI CHINA SER A 49, 255–266 (2006). https://doi.org/10.1007/s11425-005-0062-9
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DOI: https://doi.org/10.1007/s11425-005-0062-9