Well-posedness theory for degenerate parabolic equations on Riemannian manifolds
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2016
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| _version_ | 1866913485723533312 |
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| author | Graf, Melanie Kunzinger, Michael Mitrovic, Darko |
| author_facet | Graf, Melanie Kunzinger, Michael Mitrovic, Darko |
| contents | We consider the degenerate parabolic equation $$ \partial_t u +\mathrm{div} {\mathfrak f}_{\bf x}(u)=\mathrm{div}(\mathrm{div} ( A_{\bf x}(u) ) ), \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in {\mathbb R}$, ${\bf x}\mapsto {\mathfrak f}_{\bf x}(u)$ is a vector field and ${\bf x}\mapsto A_{\bf x}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto \langle A_{\bf x}(u) {\boldsymbol ξ},{\boldsymbol ξ} \rangle$, ${\boldsymbol ξ}\in T_{\bf x} M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1612_08195 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Well-posedness theory for degenerate parabolic equations on Riemannian manifolds Graf, Melanie Kunzinger, Michael Mitrovic, Darko Analysis of PDEs 35K65, 42B37, 76S99 We consider the degenerate parabolic equation $$ \partial_t u +\mathrm{div} {\mathfrak f}_{\bf x}(u)=\mathrm{div}(\mathrm{div} ( A_{\bf x}(u) ) ), \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in {\mathbb R}$, ${\bf x}\mapsto {\mathfrak f}_{\bf x}(u)$ is a vector field and ${\bf x}\mapsto A_{\bf x}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto \langle A_{\bf x}(u) {\boldsymbol ξ},{\boldsymbol ξ} \rangle$, ${\boldsymbol ξ}\in T_{\bf x} M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem. |
| title | Well-posedness theory for degenerate parabolic equations on Riemannian manifolds |
| topic | Analysis of PDEs 35K65, 42B37, 76S99 |
| url | https://arxiv.org/abs/1612.08195 |