Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

TL;DR

This paper establishes the well-posedness of degenerate parabolic equations on Riemannian manifolds by developing a geometric entropy framework and kinetic formulation, extending classical PDE theory to curved spaces.

Contribution

It introduces a geometry-compatible entropy admissibility concept and a kinetic formulation for degenerate parabolic equations on manifolds, proving well-posedness of the Cauchy problem.

Findings

Established well-posedness of the PDE on manifolds.

Developed a geometric entropy admissibility framework.

Introduced a kinetic formulation for the equation.

Abstract

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.