Smooth steady solutions of the planar Vlasov-Poisson system with a magnetic obstacle
Author:
Jack Schaeffer
Journal:
Quart. Appl. Math. 72 (2014), 753-772
MSC (2010):
Primary 35L60, 35Q83, 82C22, 82D10
DOI:
https://doi.org/10.1090/S0033-569X-2014-01358-7
Published electronically:
November 7, 2014
MathSciNet review:
3291827
Full-text PDF Free Access
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Abstract: The solar wind interacting with a magnetized obstacle is modeled with the Vlasov equation. The domain considered is a disk in the plane. Inflowing boundary conditions are given for the particle density. A magnetic field is prescribed, and the electric field is computed self consistently with potential zero on the boundary. Taking the boundary condition for the particle density to be sufficiently small, it is shown that there is a natural smooth steady solution. The speed of the inflowing plasma and the magnetic field are not size restricted.
References
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- Jack Schaeffer, Steady states for a one-dimensional model of the solar wind, Quart. Appl. Math. 59 (2001), no. 3, 507–528. MR 1848532, DOI https://doi.org/10.1090/qam/1848532
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References
- J. Batt, W. Faltenbacher, and E. Horst, Stationary spherically symmetric models in stellar dynamics, Arch. Rational Mech. Anal. 93 (1986), no. 2, 159–183. MR 823117 (87i:85001), DOI https://doi.org/10.1007/BF00279958
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
- Robert T. Glassey, The Cauchy problem in kinetic theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR 1379589 (97i:82070)
- Yan Guo and Gerhard Rein, Isotropic steady states in galactic dynamics, Comm. Math. Phys. 219 (2001), no. 3, 607–629. MR 1838751 (2002g:85001), DOI https://doi.org/10.1007/s002200100434
- Yan Guo, Singular solutions of the Vlasov-Maxwell system on a half line, Arch. Rational Mech. Anal. 131 (1995), no. 3, 241–304. MR 1354697 (96h:35228), DOI https://doi.org/10.1007/BF00382888
- Yan Guo, Regularity for the Vlasov equations in a half-space, Indiana Univ. Math. J. 43 (1994), no. 1, 255–320. MR 1275462 (95d:35178), DOI https://doi.org/10.1512/iumj.1994.43.43013
- Hyung Ju Hwang, Regularity for the Vlasov-Poisson system in a convex domain, SIAM J. Math. Anal. 36 (2004), no. 1, 121–171 (electronic). MR 2083855 (2005f:35036), DOI https://doi.org/10.1137/S0036141003422278
- P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math. 105 (1991), no. 2, 415–430 (English, with French summary). MR 1115549 (92e:35160), DOI https://doi.org/10.1007/BF01232273
- K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), no. 2, 281–303. MR 1165424 (93d:35170), DOI https://doi.org/10.1016/0022-0396%2892%2990033-J
- Gerhard Rein, Collisionless kinetic equations from astrophysics—the Vlasov-Poisson system, Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 383–476. MR 2549372 (2011b:85005), DOI https://doi.org/10.1016/S1874-5717%2807%2980008-9
- Jack Schaeffer, Steady states for a one-dimensional model of the solar wind, Quart. Appl. Math. 59 (2001), no. 3, 507–528. MR 1848532 (2002j:82114)
- Jack Schaeffer, Slow decay for a linearized model of the solar wind, Quart. Appl. Math. 70 (2012), no. 1, 181–198. MR 2920623, DOI https://doi.org/10.1090/S0033-569X-2011-01252-2
- Jack Schaeffer, Steady states of the Vlasov-Maxwell system, Quart. Appl. Math. 63 (2005), no. 4, 619–643. MR 2187923 (2006k:82148)
- D. Tidman and N. Krall, Shock waves in collisionless plasmas, Wiley-Interscience, 1971.
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Additional Information
Jack Schaeffer
Affiliation:
Department of Mathematics Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email:
js5m@andrew.cmu.edu
Received by editor(s):
January 15, 2013
Published electronically:
November 7, 2014
Article copyright:
© Copyright 2014
Brown University