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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2512.20899 |
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| _version_ | 1866912787622526976 |
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| author | Gualdani, Maria Pia Sun, Weiran |
| author_facet | Gualdani, Maria Pia Sun, Weiran |
| contents | We prove the uniqueness of $H$-solutions to the homogeneous Landau-Coulomb equation satisfying $\langle v \rangle^{k_0} f \in C([0, T]; L^{3/2}(\mathbb{R}^3))$ and $\langle v \rangle^{-3/2} \nabla_v ((\langle v \rangle^{k_0} f)^{3/4}) \in L^2((0, T) \times \mathbb{R}^3)$ for any $k_0 \geq 5$. In particular, this shows that the solutions constructed in~\cite{GGL25} are unique. The present work thus completes the global well-posedness theory in the critical space $L^{3/2}(\mathbb{R}^3)$. Our proof is part of a broader effort to use the $\mathcal{M}$-operator technique developed in~\cite{AGS2025, AMSY2020} to establish the uniqueness of rough solutions to nonlinear kinetic equations. When applied to the space-homogeneous case, the $\mathbb{M}$-operator can be taken simply as a Bessel potential operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20899 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness for the Homogeneous Landau-Coulomb Equation in $L^{3/2}$ Gualdani, Maria Pia Sun, Weiran Analysis of PDEs We prove the uniqueness of $H$-solutions to the homogeneous Landau-Coulomb equation satisfying $\langle v \rangle^{k_0} f \in C([0, T]; L^{3/2}(\mathbb{R}^3))$ and $\langle v \rangle^{-3/2} \nabla_v ((\langle v \rangle^{k_0} f)^{3/4}) \in L^2((0, T) \times \mathbb{R}^3)$ for any $k_0 \geq 5$. In particular, this shows that the solutions constructed in~\cite{GGL25} are unique. The present work thus completes the global well-posedness theory in the critical space $L^{3/2}(\mathbb{R}^3)$. Our proof is part of a broader effort to use the $\mathcal{M}$-operator technique developed in~\cite{AGS2025, AMSY2020} to establish the uniqueness of rough solutions to nonlinear kinetic equations. When applied to the space-homogeneous case, the $\mathbb{M}$-operator can be taken simply as a Bessel potential operator. |
| title | Uniqueness for the Homogeneous Landau-Coulomb Equation in $L^{3/2}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2512.20899 |