Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

Abstract

We consider the degenerate parabolic equation $${\partial }_{t}u+\mathrm{div}{\mathfrak{f}}_{\mathbf{x}}(u)=\mathrm{div}(\mathrm{div}(A_{\mathbf{x}}(u))),\mathbf{x}\in M,t\ge 0$$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in \mathbb{R}$, $\mathbf{x}\mapsto {\mathfrak{f}}_{\mathbf{x}}(u)$ is a vector field and $\mathbf{x}\mapsto A_{\mathbf{x}}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto ⟨A_{\mathbf{x}}(u)\mathbf{\xi },\mathbf{\xi }⟩$, $\mathbf{\xi }\in T_{\mathbf{x}}M$, 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.

arXiv comment32 pages, proof of Th. 8 corrected