On the plasma-charge problem (Q2837180)

This short report is a review on recent results of the author and his collaborators, and mainly concerned with the initial value problem for the following modified Vlasov-Poisson (V-P) equation associated to the evolution of a continuous distribution of charges (plasma) in presence of a finite number of point charges: NEWLINE\[NEWLINE\partial_t f+v\cdot\nabla_x f +(E+F)\cdot\nabla_v f=0,NEWLINE\]NEWLINE where \(f=f(x,v,t)\) is the distribution function of the plasma particle, \((x,v)\in\mathbb R^{2n}\) denotes its position and velocity, \(E(x,t)=\int dy \rho (y,t)K(x-y)\) is the electric field genrated by the plasma, \(\rho (x,t)=\int dv f(x,v,t)\) is the spatial density, \(F(x,t)=\sigma\sum^N_{\alpha =1} K(x-\xi_\alpha (t))\) is the electric field generated by the point charges whose positions and velocities are denoted by \(\xi_\alpha (t)\), \(\eta_\alpha (t)\), \(\alpha =1,\cdots ,N\). The above modified V-P equation is complemented by NEWLINE\[NEWLINE\dot{\xi}_\alpha =\eta_\alpha ,\quad \dot{\eta}_\alpha =\sum^N_{\beta=1;\beta\not =\alpha} K(\xi_\alpha -\xi_\beta )+ \sigma E(\xi_\alpha (t),t).NEWLINE\]NEWLINE Here \(\sigma =\pm 1\) (repulsive or attractive), \(K(x)=-\nabla g(x)\) with \(g(x)=-\log |x|\) for \(n=2\) and \(g(x)=1/|x|\) for \(n=3\).NEWLINENEWLINENEWLINEIn this report, the author gives a short survey on recent mathematical results, which are established for the physical interesting cases, on the existence and uniqueness of solutions to the initial value problem mentioned above. Moreover, the main proof ideas and estimates are described.