Well-posedness theory for degenerate parabolic equations on Riemannian manifolds
| Autor: | Michael Kunzinger, Darko Mitrović, Melanie Graf |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Cauchy problem
Pure mathematics Standard molar entropy Applied Mathematics 010102 general mathematics Degenerate energy levels Mathematics::Analysis of PDEs Riemannian manifold 16. Peace & justice 01 natural sciences Parabolic partial differential equation Manifold Tensor field 010101 applied mathematics 35K65 42B37 76S99 Mathematics - Analysis of PDEs FOS: Mathematics Vector field 0101 mathematics Analysis Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | University of Vienna-u:cris |
| ISSN: | 0022-0396 |
| DOI: | 10.1016/j.jde.2017.06.001 |
| Popis: | 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 $u\mapsto \langle A_{\bf x}(u) {\boldsymbol \xi},{\boldsymbol \xi} \rangle$, ${\boldsymbol \xi}\in T_{\bf x} M$, 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. Comment: 32 pages, proof of Th. 8 corrected |
| Databáze: | OpenAIRE |
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