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Relativistic Vlasov-Poisson equation (RVM)

A collisionfree plasma is modelled by the Vlasov-Maxwell system. In the case of very large velocities relativistic corrections are necessary. Let $E$ be the elctric field and $B$ the magnetic induction then the RVM is given by:
\begin{eqnarray*}\partial _tf_\alpha +\hat{v}\cdot \nabla_xf_\alpha +e_\alpha (......&\rho,\\\partial _t B&=&-c\nabla\times E,\\\nabla\cdot B&&0.\end{eqnarray*}
It denotes $x\in\R^3$ the position,$v\in\R^3$ the momentum, $c$ the velocity of light, $f_\alpha $ the partical density of species $\alpha $ with mass $m_\alpha $ and charge $e_\alpha $. The velocity of these particles is given by
\begin{eqnarray*}\hat{v}_\alpha =(m_\alpha ^2+c^{-2}\vert v\vert^2)^{-2}v.\end{eqnarray*}
Charge density is defined by
\begin{eqnarray*}\rho(t,x):=4\pi\int\limits \sum_\alpha e_\alpha f_\alpha dv\end{eqnarray*}
Similarely is determind the current density by
\begin{eqnarray*}j(t,x):=4\pi\int\limits \sum_\alpha e_\alpha f_\alpha \hat{v}_\alpha dv.\end{eqnarray*}
Furthermore, initial data are given for $f_\alpha $$E$ and $B$. Without loss of generality it can be reduced to the case of only one species, $e_\alpha =m_\alpha =c=1$. Also the normalization factors $4\pi, \rho$ and $j$ can be dropped out. We note that for $c\to \infty$ we get the Vlassov-Maxwell system
\begin{eqnarray*}\partial _tf+v\cdot \nabla_xf+E\cdot \nabla_vf=0\end{eqnarray*}
with$ E=\nabla U$$\Delta U=\rho$.
 

A last reference: R.T.Glassey, J. Schaeffer (2001) On global symmetric solutions to the relativistic Vlasov-Poisson equation in three dimensions, MMAS 24, 143 - 157.


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Nächste Seite:Frictional viscoelastic contact problemAufwärts:Models of Mathematical PhysicsVorherige Seite:One dimensional hydrodynamic model
collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg