TY - JOUR

T1 - Steady states of the Vlasov-Poisson-Fokker-Planck system

AU - Glassey, Robert

AU - Schaeffer, Jack

AU - Zheng, Yuxi

N1 - Funding Information:
* Research supported in part by NSF DMS 9321383, NSF MCS 91-01517, and NSF DMS 9303414. ²E-mail address: glassey@indiana.edu. ³E-mail address: js5mq@andrew.cmu.edu. ¶E-mail address: yzheng@zu.math.indiana.edu.

PY - 1996/9/15

Y1 - 1996/9/15

N2 - The form of steady state solutions to the Vlasov-Poisson-Fokker-Planck system is known from the works of Dressler and others. In these papers an external potential is present which tends to infinity as |x| → ∞. It is shown here that this assumption is needed to obtain nontrivial steady states. This is achieved by showing that for a given nonnegative background density satisfying certain integrability conditions, only the trivial solution is possible. This result is sharp and exactly matches the known existence criteria of F. Bouchut and J. Dolbeault (Differential Integral Equations 8, 1995, 487-514) and others. These steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity. For a given background density which is asymptotically constant, it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely determined global solution.

AB - The form of steady state solutions to the Vlasov-Poisson-Fokker-Planck system is known from the works of Dressler and others. In these papers an external potential is present which tends to infinity as |x| → ∞. It is shown here that this assumption is needed to obtain nontrivial steady states. This is achieved by showing that for a given nonnegative background density satisfying certain integrability conditions, only the trivial solution is possible. This result is sharp and exactly matches the known existence criteria of F. Bouchut and J. Dolbeault (Differential Integral Equations 8, 1995, 487-514) and others. These steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity. For a given background density which is asymptotically constant, it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely determined global solution.

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U2 - 10.1006/jmaa.1996.0360

DO - 10.1006/jmaa.1996.0360

M3 - Article

AN - SCOPUS:0030587377

SN - 0022-247X

VL - 202

SP - 1058

EP - 1075

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 3

ER -