Bardos, Claude; Nouri, Anne A Vlasov equation with Dirac potential used in fusion plasmas. (English) Zbl 1457.82405 J. Math. Phys. 53, No. 11, 115621, 16 p. (2012). 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} Cited in 1 ReviewCited in 15 Documents 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 Keywords:Vlasov equation; fusion plasmas PDFBibTeX XMLCite \textit{C. Bardos} and \textit{A. Nouri}, J. Math. Phys. 53, No. 11, 115621, 16 p. (2012; Zbl 1457.82405) Full Text: DOI References: [1] DOI: 10.1007/s00205-010-0392-9 · Zbl 1229.35297 · doi:10.1007/s00205-010-0392-9 [2] DOI: 10.3934/krm.2009.2.39 · Zbl 1185.35292 · doi:10.3934/krm.2009.2.39 [3] DOI: 10.1016/0022-1236(71)90027-9 · Zbl 0211.12902 · doi:10.1016/0022-1236(71)90027-9 [4] Da Prato G., Ann. Scuola Norm. Sup. Pisa 19 (3) pp 367– (1965) [5] DOI: 10.1090/S0002-9947-1986-0825714-8 · doi:10.1090/S0002-9947-1986-0825714-8 [6] DOI: 10.1080/00207216208937448 · doi:10.1080/00207216208937448 [7] Feix, M. R. , Hohl, F., and Staton, L. D. Nonlinear Effects in Plasmas, edited by G. Kalman and M. Feix (Gordon and Breach, 1969), pp. 3–21. [8] DOI: 10.1090/S0894-0347-09-00652-3 · Zbl 1197.35204 · doi:10.1090/S0894-0347-09-00652-3 [9] Gerard-Varet D., Asymptotic Anal. 77 (1) pp 71– (2012) [10] DOI: 10.3934/krm.2009.2.707 · Zbl 1195.82087 · doi:10.3934/krm.2009.2.707 [11] Grenier E., C.R. Acad. Sci. Paris Sér. I Math. 320 (6) pp 691– (1995) [12] DOI: 10.1002/cpa.20377 · Zbl 1232.35126 · doi:10.1002/cpa.20377 [13] DOI: 10.1512/iumj.2011.60.4193 · Zbl 1248.35153 · doi:10.1512/iumj.2011.60.4193 [14] Hauray M., Ann. Inst. Henri Poincare, Anal. Non Lineaire 4 (1) pp 109– (2011) [15] DOI: 10.1016/j.crma.2011.03.024 · Zbl 1219.35046 · doi:10.1016/j.crma.2011.03.024 [16] DOI: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L · Zbl 0935.35148 · doi:10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L [17] Krall N. A., Principles of Plasma Physics (1973) [18] Lions J.-L., Port. Math. 19 pp 141– (1960) [19] DOI: 10.1063/1.1706024 · Zbl 0090.22801 · doi:10.1063/1.1706024 [20] Vlasov A., Zh. Eksp. Teor. Fiz. 8 pp 291– (1938) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.