×

Convergence of the Vlasov-Poisson system to the incompressible Euler equations. (English) Zbl 0970.35110

The paper deals with the displacement of an electronic cloud generated by the local difference of charge with a uniform neutralizing background of non-moving ions. The equations are given by the Vlasov-Poisson system, with a coupling constant \(\varepsilon =(\frac{\tau}{2\pi})^2\) where \(\tau\) is the constant oscillation period of the electrons. As \(\varepsilon\to 0\), the convergence to a solution of the incompressible Euler equations is proved under some restrictions. A link between Lions’ dissipative solutions and Diperna-Majda’s measure-valued solutions of the Euler equations is established.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q05 Euler-Poisson-Darboux equations
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
Full Text: DOI

References:

[1] Arnold V.I., Topological methods in hydrodynamics (1998) · Zbl 0902.76001
[2] Batt J., C.R. Acad. Sci. Paris 313 pp 411– (1991)
[3] Brenier Y., Journées Equations aux Dérivées Partielles (1995)
[4] Brenier Y., C.R. Acad. Sci. Paris 318 pp 121– (1994)
[5] Chemin J.-Y., Fluides parfaits incompressibles 230 (1995) · Zbl 0829.76003
[6] DOI: 10.1090/S0894-0347-1991-1102579-6 · doi:10.1090/S0894-0347-1991-1102579-6
[7] DOI: 10.1007/BF01214424 · Zbl 0626.35059 · doi:10.1007/BF01214424
[8] Golse F., personal communication (1998)
[9] DOI: 10.1080/03605309608821189 · Zbl 0849.35107 · doi:10.1080/03605309608821189
[10] DOI: 10.1080/03605309508821129 · Zbl 0828.35106 · doi:10.1080/03605309508821129
[11] DOI: 10.1002/(SICI)1097-0312(199709)50:9<821::AID-CPA2>3.0.CO;2-7 · Zbl 0884.35183 · doi:10.1002/(SICI)1097-0312(199709)50:9<821::AID-CPA2>3.0.CO;2-7
[12] Grenier E., On the stability of boundary layers for Euler equations (1997) · Zbl 0914.35032
[13] Lions P.-L., Oxford Lecture Series in Mathematics and its Applications (1996)
[14] DOI: 10.1007/BF01232273 · Zbl 0741.35061 · doi:10.1007/BF01232273
[15] Marchioro C., Mathematical theory of incompressible nonviscous fluids (1994) · Zbl 0789.76002
[16] DOI: 10.1016/0022-0396(92)90033-J · Zbl 0810.35089 · doi:10.1016/0022-0396(92)90033-J
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.