Abstract
A local existence theorem is proved for classical solutions of theVlasov-Poisswell system, a set of collisionless equations used in plasmaphysics.Although the method employed is standard,there are several technical difficulties in the treatment of this systemthatarise mainly from the, compared to related systems, special form of theelectric-field term.Furthermore, uniquenessof classical solutions is proved and a continuation criterion forsolutions well knownfor other collisionless kinetic equations is established. Finally, aglobal existence result for a regularized versionof the system is derived and comments are given on the problem ofobtaining global weak solutions.
Highlights
In the present paper the initial value problem (IVP) is studied for a nonlinear system of partial differential equations originating in plasma physics
Our main concern is the proof of a local existence result for classical solutions including a continuation criterion
This result is not surprising at all and in principle the methods developed for Vlasov-Poisson system (VP) and relativistic Vlasov-Maxwell system (RVM) are applicable, there were some traps resulting mainly from the term that had to be circumvented
Summary
In the present paper the initial value problem (IVP) is studied for a nonlinear system of partial differential equations originating in plasma physics. 2000 Mathematics Subject ClassificationPrimary: 35F20; Secondary: 82D10 Key word and phrases Vlasov equation, collisionless plasma potential respectively. With these quantities RVM may be written as. Our main concern is the proof of a local existence result for classical solutions including a continuation criterion. This result is not surprising at all and in principle the methods developed for VP and RVM are applicable, there were some traps resulting mainly from the term 1 c. In this final section we discuss a regularisation of the system for which we can prove global existence
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