I am looking for a reference which rigorously explores the heat equation as a gradient flow of the Dirichlet energy (say in $L^2$ ? or some other inner-product space). I don't know this literature well at all so struggled to find actual research papers on it.
2026-03-27 12:20:09.1774614009
Heat equation as gradient flow of Dirichlet energy
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Although I am not an expert on this subject (I do not know of specific books on the subject), I know that the paper "Repulsive Curves" by Yu, Schumacher, and Crane introduces the relationship between heat equation and the gradient flow of the Dirichlet energy in $L^2$; the paper also discusses the gradient flow of the Dirichlet energy in Sobolev spaces ($H^1$, $H^2$, etc.).
With regard to the derivation of the heat equation from the Dirichlet energy (in $L^2$), we can directly obtain the heat equation from the energy by using the limit definition provided in the paper; Green's first identity is needed to complete the derivation.