| Abstract: |
| In this talk, we are concerned with the fluid limit to KdV equations for the one-dimensional Vlasov-Poisson-Landau system which describes the dynamics of ions in plasma with the electron density determined by the self-consistent electric potential through the so-called Boltzmann relation. Formally, it is well known that as the Knudsen number $\varepsilon\to 0$ the Vlasov-Poisson-Landau system in the compressible scaling converges to the Euler-Poisson equations which further under the Gardner-Morikawa transformation
$$
(t,x)\to (\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t))
$$
converge to the KdV equations as the parameter $\delta\to 0$. Our goal of this paper is to construct smooth solutions of the correspondingly rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the KdV equations as $\varepsilon\to 0$ and $\delta\to 0$ simultaneously under an extra condition $ \varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is established by an appropriately chosen scaling and an intricate weighted energy method through the macro-micro decomposition around local Maxwellians. We design a $\varepsilon$-$\delta$-dependent high order energy functional to capture the singularity of such fluid limit problem. |
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