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
MathSciNet review:
3614499
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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
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- François Bouchut, Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system, J. Differential Equations 122 (1995), no. 2, 225–238. MR 1355890, DOI https://doi.org/10.1006/jdeq.1995.1146
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- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI https://doi.org/10.1007/BF02392081
- 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. 1, 183–233. MR 3237885, DOI https://doi.org/10.1007/s00205-014-0758-5
- A. M. Il′in, On a class of ultraparabolic equations, Dokl. Akad. Nauk SSSR 159 (1964), 1214–1217 (Russian). MR 0171084
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- Maria Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. Amer. Math. Soc. 71 (1951), 24–37. MR 42035, DOI https://doi.org/10.1090/S0002-9947-1951-0042035-0
References
- 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
- 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
- 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
- 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
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 0222474
- 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
- A. M. Il’in, On a class of ultraparabolic equations, Dokl. Akad. Nauk SSSR 159 (1964), 1214–1217 (Russian). MR 0171084
- A. Kolmogoroff, Zufällige Bewegungen (zur Theorie der Brownschen Bewegung), Ann. of Math. (2) 35 (1934), no. no. 1, 116–117 (German). MR 1503147
- Cédric Villani, Hypocoercivity, Mem. Amer. Math. Soc. 202 (2009), no. no. 950, iv+141. MR 2562709
- Maria Weber, The fundamental solution of a degenerate partial differential equation of parabolic type, Trans. Amer. Math. Soc. 71 (1951), 24–37. MR 0042035
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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
MR Author ID:
672369
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
Received by editor(s):
December 24, 2016
Published electronically:
January 30, 2017
Article copyright:
© Copyright 2017
Brown University
