Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917372403646464 |
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| author | Zhao, Weiren Zi, Ruizhao |
| author_facet | Zhao, Weiren Zi, Ruizhao |
| contents | In this paper, we study the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision. We prove that if the initial perturbation is of size $ν^{\frac{1}{2}}$ in the critical weighted space $H_x^{\log}L^2_{v}(\langle v\rangle^m)$, then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, $\|E\|_{L^2_tL^2_x}\lesssim ν^{\frac{1}{2}}$; and a point-wise type Landau damping holds, namely, $\|E(t)\|_{L^2}\lesssim ν^{1/2}\langle t\rangle^{-N}$ for any $N>0$ for $t\geq ν^{-1}$. We also prove that there exists initial perturbation in $H^{1}_xL^2_v(\langle v\rangle^m)$ with size $ν^{\frac12-\frac32ε_0}$ with any ${ε_0>0}$, such that the enhanced dissipation fails to hold in the following sense: there is $0<T\ll ν^{-\frac13}$ such that
\begin{align*}
\|\langle v\rangle^m f_{\neq}(T)\|_{L^2_xL^2_v}\gtrsim \frac{1}{ν^{δ_1}}\|\langle v\rangle^m f_{\neq}(0)\|_{ H^1_xL^2_v}
\end{align*}
with some $δ_1>0$.
The paper solves the open problem raised in [Bedrossian; arXiv: 2211.13707] about the sharp stability threshold in lower regularity spaces. The main idea is to construct a wave operator $\mathbf{D}$ with a very precise expression to absorb the nonlocal term, namely, \begin{align*}
\mathbf{D}[\partial_tg+v\cdot \nabla_x g+E\cdot\nabla_v μ]=(\partial_t +v\cdot \nabla_x)\mathbf{D}[g]. \end{align*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations Zhao, Weiren Zi, Ruizhao Analysis of PDEs In this paper, we study the optimal stability threshold for the Vlasov-Poisson equation with weak Fokker-Planck collision. We prove that if the initial perturbation is of size $ν^{\frac{1}{2}}$ in the critical weighted space $H_x^{\log}L^2_{v}(\langle v\rangle^m)$, then the solution remains the same size in the same space. Moreover, a space-time type Landau damping holds, namely, $\|E\|_{L^2_tL^2_x}\lesssim ν^{\frac{1}{2}}$; and a point-wise type Landau damping holds, namely, $\|E(t)\|_{L^2}\lesssim ν^{1/2}\langle t\rangle^{-N}$ for any $N>0$ for $t\geq ν^{-1}$. We also prove that there exists initial perturbation in $H^{1}_xL^2_v(\langle v\rangle^m)$ with size $ν^{\frac12-\frac32ε_0}$ with any ${ε_0>0}$, such that the enhanced dissipation fails to hold in the following sense: there is $0<T\ll ν^{-\frac13}$ such that \begin{align*} \|\langle v\rangle^m f_{\neq}(T)\|_{L^2_xL^2_v}\gtrsim \frac{1}{ν^{δ_1}}\|\langle v\rangle^m f_{\neq}(0)\|_{ H^1_xL^2_v} \end{align*} with some $δ_1>0$. The paper solves the open problem raised in [Bedrossian; arXiv: 2211.13707] about the sharp stability threshold in lower regularity spaces. The main idea is to construct a wave operator $\mathbf{D}$ with a very precise expression to absorb the nonlocal term, namely, \begin{align*} \mathbf{D}[\partial_tg+v\cdot \nabla_x g+E\cdot\nabla_v μ]=(\partial_t +v\cdot \nabla_x)\mathbf{D}[g]. \end{align*} |
| title | Optimal stability threshold in lower regularity spaces for the Vlasov-Poisson-Fokker-Planck equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.29204 |