The Dirichlet energy $E\colon L^2(\mathbb{R}^n) \to \mathbb{R} $ is defined by (for $u \in L^2(\mathbb{R}^n)$) $$ E(u)= \frac{1}{2} \int_{\mathbb{R}^n} ||\nabla u(x)||^2 dx $$ It is a classical fact that the heat equation $$ \partial_t u = \Delta u $$ can be regarded as the gradient flow of the Dirichlet energy in $L^2$. However I cannot understand
- How to derive this, at least formally (in the sense that I admit manipulations which are not really rigorous)
- What this means rigorously (as I do not know how to compute the gradient of $E$).
Any help?
