A Vlasov equation with Dirac potential used in fusion plasmas

Abstract

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.

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

“…4. The next few integrals in (4.8) involve (4.6), and hence these contributions are similarly treated after applying Lemma 4.1.…”

“…Our proof applies as is to any W satisfying 0 ≤ W (k) |k| −1 , in particular, the regularizing effects of the collisions can be used to deal with more singular interactions than are treated in [40,12] (note that more singular interactions can arise in certain limits; see e.g. [4,30] and the references therein). We further remark that if one supplements the hypotheses of Theorem 1 with a Penrose stability criterion such as that used in [40,12,11] one can extend our theorem to cover some non-repulsive kernels as well.…”

“…4; the details are omitted for brevity. Similarly, the contributions of the nonlinear collision term J h are not significantly harder than the analogous estimates in §3.4.…”

About a Variant of the 1d Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"

“…In this case one can describe the evolution of the linearized problem (near this profile) by a distribution group of operators a described in [3]. This means a "almost well posed Cauchy problem" or more precisely an evolution equation well posed with a finite lost of regularity.…”

“…This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass.…”