TY - JOUR

T1 - Concentrations in the one-dimensional Vlasov-Poisson equations I

T2 - Temporal development and non-unique weak solutions in the single component case

AU - Majda, Andrew J.

AU - Majda, George

AU - Zheng, Yuxi

N1 - Funding Information:
1 Research partially supported by grants NSF DMS-9001805, ARO DAAL03-92-G-0010, ONR N00014-89-J1044.P00003, DARPA N000014-92-J-1796. 2 Research partially supported by grant AFOSR 91-0309, and also as a visitor at the Institute for Advanced Study and the Program in Applied and Computational Mathematics at Princeton University through grants NSF DMS-9001805, ONR N00014-89-J1044.P00003. The numerical computations were supported by the Ohio State Supercomputer Center under Grant PAS330. 3 Research partially supported by grant NSF-DMS 9100383 while a visiting member at the Institute for Advanced Study and grant NSF-DMS-9114456 while a visiting member at the Courant Institute of Mathematical Sciences, New York University.

PY - 1994/7/15

Y1 - 1994/7/15

N2 - Weak solutions of the one-component Vlasov-Poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two-dimensional incompressible Euler equations with non-negative vorticity. The physical, structural, and functional analytic analogies between these two problems are developed in detail here. With this background, explicit solutions for electron sheet initial data, the analogue of vortex sheet initial data, are presented, which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations. Other rigorous explicit examples with charge concentration are developed where there are non-unique weak solutions with the same initial data. In one of these non-unique weak solutions, an electron sheet completely collapses to a point charge in finite time. The detailed limiting behavior of regularizations such as the diffusive Fokker-Planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems. A striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data. In particular, two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the Fokker-Planck equation converges to the other weak solution.

AB - Weak solutions of the one-component Vlasov-Poisson equation in a single space dimension are proposed and studied here as a simpler analogue problem for the behavior of weak solutions of the two-dimensional incompressible Euler equations with non-negative vorticity. The physical, structural, and functional analytic analogies between these two problems are developed in detail here. With this background, explicit solutions for electron sheet initial data, the analogue of vortex sheet initial data, are presented, which display the phenomena of singularity formation in finite time as well as the explicit temporal development of charge concentrations. Other rigorous explicit examples with charge concentration are developed where there are non-unique weak solutions with the same initial data. In one of these non-unique weak solutions, an electron sheet completely collapses to a point charge in finite time. The detailed limiting behavior of regularizations such as the diffusive Fokker-Planck equation are developed through a very efficient numerical method which yields extremely high resolution for these simpler analogue problems. A striking consequences of the numerical results reported here is the fact that there is not a selection principle for a unique weak solution in some situations where there are several weak solutions with charge concentration for the same initial data. In particular, two explicit weak solutions with the same initial data are constructed here where it is demonstrated that the zero smoothing limit of time reversible particle methods converges to one of these solutions while the zero diffusion limit of the Fokker-Planck equation converges to the other weak solution.

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U2 - 10.1016/0167-2789(94)90198-8

DO - 10.1016/0167-2789(94)90198-8

M3 - Article

AN - SCOPUS:0001599323

SN - 0167-2789

VL - 74

SP - 268

EP - 300

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 3-4

ER -