Asymptotic growth bounds for the Vlasov-Poisson system with radiation damping

Estimate of largest velocities for Vlasov–Poisson plasma with radiation damping

“…And we notice that R(s) ≲ (s + 1) 67∕84 was proved in Ma and Zhang, 24 , Theorem 1.1 combining it with (1.4) and (2.1), we have…”

“…Here, we choose $$$$ L=\sqrt{\frac{P\varphi (t)}{\ln^2\left(t+2\right)}},P=\frac{\ln^{6/11}\left(t+2\right)}{{\left(t+1\right)}^{4/11}{\varphi}^{3/11}(t)}. $$$$ Since $$$ R(t)\lesssim {\left(t+1\right)}^{67/84} $$$ was proved in Ma and Zhang, 24 , Theorem 1.1 combining it with () and (), we have $$$ S(t)\lesssim {\left(t+1\right)}^{151/84} $$$, and then, we have $$$ \varphi (t)\le C{\left(t+1\right)}^{67/84} $$$; hence, we can obtain that $$$ P $$$ defined above satisfies $$$ \frac{\ln^{6/1...$…”

“…We need to find some new techniques to overcome the difficulty induced by radiation damping. Recently, Ma and Zhang 24 studied the large time behavior of velocity supports and obtained a sublinear estimate with exponent 67/84. Obviously, from a kinetic point of view, this result is not optimal.…”