Shi-type and Hamilton-type gradient estimates for a general parabolic equation under compact Finsler $CD(-K,N)$ geometric flows
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911010424619008 |
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| author | Miao, Yijie Shen, Bin |
| author_facet | Miao, Yijie Shen, Bin |
| contents | Recently, the Li-Yau-type gradient estimates for positive solutions to parabolic equations
\begin{equation}
\partial_t u=Δu+\mathcal{R}_1u+\mathcal{R}_2u^α+\mathcal{R}_3u(\log u)^β,\notag
\end{equation}
under the general compact Finsler $CD(-K,N)$ geometric flow are studied. Here $\mathcal{R}_1$,$\mathcal{R}_2$,$\mathcal{R}_3$ $\in$ $ C^{1}(M,[0,T])$, $α$ and $β$ are both positive constants, $T$ is the maximal existence time for the flow.
However, compared with the Riemannian case, the curvature conditions impose stricter derivative bounds on the development term in the geometric flow, as well as on the derivative bounds of the distortion of the manifold. In this manuscript, we present Shi-type and Hamilton-type gradient estimates to demonstrate the possibility of removing such conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shi-type and Hamilton-type gradient estimates for a general parabolic equation under compact Finsler $CD(-K,N)$ geometric flows Miao, Yijie Shen, Bin Differential Geometry Analysis of PDEs Recently, the Li-Yau-type gradient estimates for positive solutions to parabolic equations \begin{equation} \partial_t u=Δu+\mathcal{R}_1u+\mathcal{R}_2u^α+\mathcal{R}_3u(\log u)^β,\notag \end{equation} under the general compact Finsler $CD(-K,N)$ geometric flow are studied. Here $\mathcal{R}_1$,$\mathcal{R}_2$,$\mathcal{R}_3$ $\in$ $ C^{1}(M,[0,T])$, $α$ and $β$ are both positive constants, $T$ is the maximal existence time for the flow. However, compared with the Riemannian case, the curvature conditions impose stricter derivative bounds on the development term in the geometric flow, as well as on the derivative bounds of the distortion of the manifold. In this manuscript, we present Shi-type and Hamilton-type gradient estimates to demonstrate the possibility of removing such conditions. |
| title | Shi-type and Hamilton-type gradient estimates for a general parabolic equation under compact Finsler $CD(-K,N)$ geometric flows |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2506.14776 |