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A Vlasov equation with Dirac potential used in fusion plasmas. (English) Zbl 1457.82405

Summary: Well-posedness of the Cauchy problem is analyzed for a singular Vlasov equation governing the evolution of the ionic distribution function of a quasineutral fusion plasma. The Penrose criterium is adapted to the linearized problem around a time and space homogeneous distribution function showing (due to the singularity) more drastic differences between stable and unstable situations. This pathology appears on the full nonlinear problem, well-posed locally in time with analytic initial data, but generally ill-posed in the Hadamard sense. Eventually with a very different class of solutions, mono-kinetic, which constrains the structure of the density distribution, the problem becomes locally in time well-posed.(Dedicated to Peter Constantin on the occasion of his 60th birthday.){
©2012 American Institute of Physics}

MSC:

82D10 Statistical mechanics of plasmas
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
82D75 Nuclear reactor theory; neutron transport
35R09 Integro-partial differential equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35Q83 Vlasov equations
78A35 Motion of charged particles
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