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Uniqueness of the solution to the Vlasov–Poisson system with bounded density. (English) Zbl 1111.35045

The Vlasov-Poisson system describes the evolution of a cloud of electrons or gravitational matter by the equations
\[ \partial_t f+\xi\cdot\nabla_xf-\nabla\Psi\cdot\nabla_\xi f=0,\quad -\Delta\psi=\varepsilon\rho \]
where \(\rho(t,x)=\int f(t,x,\xi)\,d\xi\) and \(\varepsilon>0\) in the electrostatic (repulsive) case, \(\varepsilon<0\) in the gravitational (attractive) case. Here
\[ f(t,x,\xi)\geq 0 \]
denotes the density of electrons (or matter) at time \(t\in \mathbb{R}^+\), position \(x\in\mathbb{R}^3\), velocity \(\xi\in\mathbb{R}^3\). The second equation is to be understood in the following sense
\[ \Psi(t,x)=\varepsilon\int_{\mathbb{R}^3}\rho(t,y)\frac{1}{4\pi| x-y| }dy. \]
There is shown the uniqueness of weak solutions of this system under the only condition that the macroscopic density \(\rho(t,x)\) is bounded in \(L^{\infty}\).

MSC:

35Q35 PDEs in connection with fluid mechanics
82D10 Statistical mechanics of plasmas
82C70 Transport processes in time-dependent statistical mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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References:

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