Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation $${\mathrm{\partial }}_{t}u+\mathrm{d}\mathrm{i}\mathrm{v}{\mathfrak{f}}_{\mathbf{x}}(u)=\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{d}\mathrm{i}\mathrm{v}(A_{\mathbf{x}}(u))),\mathbf{x}\in M,t\ge 0$$ on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in \mathbb{R}$, ${\bf x}\mapsto {\mathfrak f}{\bf x}(u)$isavectorfieldand${\bf x}\mapsto A{\bf x}(u)$isa$(1,1)$-tensorfieldon$M$suchthat$u\mapsto \langle A_{\bf x}(u) {\boldsymbol \xi},{\boldsymbol \xi} \rangle$,${\boldsymbol \xi}\in T_{\bf x} M$,isnon-decreasingwithrespectto$u$. 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.