Galerkin-type methods for strictly parabolic equations on compact Riemannian manifolds

TL;DR

This paper establishes the existence of weak solutions for strictly parabolic equations on compact Riemannian manifolds, including stochastic cases, using a Galerkin method that supports finite element construction.

Contribution

It introduces a Galerkin-type approach to prove existence of solutions for parabolic equations on manifolds, extending to stochastic equations with linear diffusion.

Findings

Existence of weak solutions proven for parabolic equations on manifolds.

Method supports construction of convergent finite element schemes.

Applicable to stochastic parabolic equations with linear diffusion.

Abstract

We prove existence of weak solutions to the Cauchy problem corresponding to various strictly parabolic equations on a compact Riemannian manifold $(M,g)$. This also includes strictly parabolic equations with stochastic forcing with linear diffusion. Existence is proved through a variant of the Galerkin method and can be used to construct a convergent finite element method.