Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

We consider the degenerate parabolic equation∂tu+divfx(u)=div(div(Ax(u))),x∈M,t≥0 on a smooth, compact, d-dimensional Riemannian manifold (M,g). Here, for each u∈R, x↦fx(u) is a vector field and x↦Ax(u) is a (1,1)-tensor field on M such that u↦〈Ax(u)ξ,ξ〉, ξ∈TxM, 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.