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.


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