On solutions of Vlasov-Poisson-Landau equations for slowly varying in space initial data

Abstract

<p style='text-indent:20px;'>The paper is devoted to analytical and numerical study of solutions to the Vlasov-Poisson-Landau kinetic equations (VPLE) for distribution functions with typical length <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon = r_D/L &lt;&lt; 1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ r_D $\end{document}</tex-math></inline-formula> stands for the Debye radius. It is also assumed that the Knudsen number <inline-formula><tex-math id="M4">\begin{document}$ \mathrm{K\!n} = l/L = O(1) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ l $\end{document}</tex-math></inline-formula> denotes the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon &gt; 0 $\end{document}</tex-math></inline-formula>. It is known that the formal limit of VPLE at <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon = 0 $\end{document}</tex-math></inline-formula> does not describe a rapidly oscillating part of the electrical field. Our aim is to fill this gap and to study the behavior of the "true" electrical field near this limit. We show that, in the problem with standard isotropic in velocities Maxwellian initial conditions, there is almost no damping of these oscillations in the collisionless case. An approximate formula for the electrical field is derived and then confirmed numerically by using a simplified BGK-type model of VPLE. Another class of initial conditions that leads to strong oscillations having the amplitude of order <inline-formula><tex-math id="M8">\begin{document}$ O(1/\varepsilon ) $\end{document}</tex-math></inline-formula> is considered. A formal asymptotic expansion of solution in powers of <inline-formula><tex-math id="M9">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> is constructed. Numerical solutions of that class are studied for different values of parameters <inline-formula><tex-math id="M10">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \mathrm{K\!n} $\end{document}</tex-math></inline-formula>.</p>