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Hydrodynamic limits of the Vlasov equation. (English) Zbl 0787.35070
The authors consider a spatially periodic plasma which, after a rescaling $x \to \varepsilon x$, $t \to \varepsilon t$, is described by the system $$\partial\sb t f\sp \varepsilon+v \cdot \partial\sb xf\sp \varepsilon+F\sp \varepsilon \cdot \partial\sb vf\sp \varepsilon=0,\ x \in[-1,1]\sp d,\ v \in \bbfR\sp d,$$ $$F\sp \varepsilon(x,t)=-\partial\sb x \int\sb{\bbfR\sp d} \varphi\sp \varepsilon (x-y) \rho\sp \varepsilon (y,t) dy,$$ $$\rho\sp \varepsilon (x,t)=\int f\sp \varepsilon (x,v,t) dv,\ \varphi\sp \varepsilon (x)=\varepsilon\sp{-d} \varphi(x/ \varepsilon),\ \varepsilon>0,$$ with a $C\sp \infty$ interaction potential $\varphi$ satisfying certain assumptions. For data of the form $f\sb 0(x,v)=\rho\sb 0(x) \delta(v-u\sb 0(x))$ the solutions are of the form $f\sp \varepsilon (x,v,t)= \rho\sp \varepsilon(x,t)\delta (v-u\sp \varepsilon(x,t))$ where the pair $(\rho\sp \varepsilon,\ u\sp \varepsilon)$ solves the system $$\partial\sb tu\sp \varepsilon (x,t)+(u\sp \varepsilon \cdot \nabla) u\sp \varepsilon(x,t)=- \partial\sb x \int \rho\sp \varepsilon(y,t) \varphi\sp \varepsilon(x-y)dy,$$ $$\partial\sb t \rho\sp \varepsilon+\text{div} (\rho\sp \varepsilon u\sp \varepsilon)=0.$$ It is shown that for $\varepsilon \to 0$ the solutions of the latter system converge weakly to a solution of the system $$\partial\sb tu+(u\cdot \nabla)u=-\partial\sb x \rho,\ \partial\sb t \rho+\text{div} (\rho u)= 0.$$ Provided $\rho>0$ the latter are the compressible Euler equations with $p={1 \over 2} \rho\sp 2$ as equation of state. In the same sense the incompressible limit of the Vlasov equation is established by the scaling $x \to \varepsilon x$, $t \to \varepsilon \lambda\sp{-1}t$, $v \to \lambda v$ with $\lambda=\varepsilon\sp{- \alpha}$, $0<\alpha<1$. Since the assumption on $f\sb 0$ rules out local fluctuations around the mean velocity $u$, one might wonder whether this result should be called a hydrodynamic limit of the {\sl Vlasov} equation, but this is merely a question of terminology.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76P05 Rarefied gas flows, Boltzmann equation 76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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