We consider the Vlasov–Poisson equation in R3 with initial data which are not L1 in space and have unbounded support in the velocities. Assuming for the density a slight decay in space and a strong decay in velocities, we prove the existence and uniqueness of the solution, thus generalizing the analogous result by the same authors for data compactly supported in the velocities.
The aim of the paper is to test on a simple model how impenetrable may be a magnetic shield. We study the time evolution of a single species positive plasma, confined in the half-space x_1 > 0. The confinement is the result of a balance between a magnetic field and an external field, both singular at x_1 = 0; the magnetic field forbids the entrance of plasma particles in the region x_1 ≤ 0, whereas the external field attracts the plasma particles towards x_1 = 0. The plasma has finite total charge and velocities distributed with a Maxwell–Boltzmann law.
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