## 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
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### References:

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