Gradient flow of the periodic screened Poisson energy
= Gradient flow of the periodic screened Poisson energy
For smooth $2\pi$-periodic $f$, the equation $u_t-u_{xx}+u=f$ is the negative $L^2$ <gradient flow> of $I_f[u]=\frac12\int(u_x^2+u^2-2fu)\,dx$. Periodic <integration by parts> gives $dI_f/dt=-\int u_t^2\,dx$. Its unique equilibrium has <Fourier coefficients> $f_n/(1+n^2)$, and every transient coefficient decays as $e^{-(1+n^2)t}$. Thus the flow converges to the unique minimizer of the energy.