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On the initial value problem for the Vlasov‐Poisson‐Fokker‐Planck system with initial data in L p spaces
Author(s) -
Carrillo José A.,
Soler Juan
Publication year - 1995
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670181006
Subject(s) - mathematics , initial value problem , sequence (biology) , fokker–planck equation , mathematical analysis , linearization , a priori estimate , uniqueness , a priori and a posteriori , convergence (economics) , weak convergence , vlasov equation , term (time) , differential equation , nonlinear system , physics , quantum mechanics , computer security , computer science , asset (computer security) , electron , biology , philosophy , genetics , epistemology , economics , economic growth
In this paper the global existence of weak solutions for the Vlasov‐Poisson‐Fokker‐Planck equations in three dimensions is proved with an L 1 ∩ L p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker‐Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non‐linear regulation of the problem together with a linearization via a time retarded mollification of the non‐linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.