Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the Fokker-Planck Equations with inflow boundary conditions

Authors: Hyung Ju Hwang and Du Phan
Journal: Quart. Appl. Math. 75 (2017), 287-308
MSC (2010): Primary 35Q84
DOI: https://doi.org/10.1090/qam/1462
Published electronically: January 30, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The results in this paper extend those of a 2014 work of the first author, Jang and Velázquez. Instead of considering absorbing boundary data, we treat the general inflow boundary conditions and obtain the well-posedness, regularity up to the singular set, and asymptotic behavior of solutions to the Fokker-Planck equation in an interval with the inflow boundary conditions.

References [Enhancements On Off] (What's this?)

  • [1] M. Bostan, Existence and uniqueness of the mild solution for the 1D Vlasov-Poisson initial-boundary value problem, SIAM J. Math. Anal. 37 (2005), no. no. 1, 156-188. MR 2176927
  • [2] François Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 122 (1995), no. no. 2, 225-238. MR 1355890
  • [3] Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal. 197 (2010), no. no. 3, 713-809. MR 2679358
  • [4] Frédéric Hérau and Francis Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal. 171 (2004), no. no. 2, 151-218. MR 2034753
  • [5] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474
  • [6] Hyung Ju Hwang, Juhi Jang, and Juan J. L. Velázquez, The Fokker-Planck equation with absorbing boundary conditions, Arch. Ration. Mech. Anal. 214 (2014), no. no. 1, 183-233. MR 3237885
  • [7] A. M. Il'in, On a class of ultraparabolic equations, Dokl. Akad. Nauk SSSR 159 (1964), 1214-1217 (Russian). MR 0171084
  • [8] A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. no. 1, 116-117 (German). MR 1503147
  • [9] Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. no. 950, iv+141. MR 2562709
  • [10] Maria Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. Amer. Math. Soc. 71 (1951), 24-37. MR 0042035

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35Q84

Retrieve articles in all journals with MSC (2010): 35Q84

Additional Information

Hyung Ju Hwang
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea
Address at time of publication: Department of Mathematics, Brown University, 151 Thayer street, Providence, Rhode Island 02912
Email: hjhwang@postech.ac.kr

Du Phan
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea
Email: phandu@postech.ac.kr

DOI: https://doi.org/10.1090/qam/1462
Received by editor(s): December 24, 2016
Published electronically: January 30, 2017
Article copyright: © Copyright 2017 Brown University

American Mathematical Society