Elliptic and Parabolic Equations - Abstract

Matthes, Daniel

The Wasserstein gradient flow of energies

Just as the gradient flow of entropy functionals in the $L2$-Wasserstein metric generate second order diffusion equations, the gradient flow of the energies $E(u) = int nabla u^p ^2 dx$ produce non-linear denerate parabolic PDEs of fourth order. These generalize the Thin Film (p = 1) and DLSS (p = 1/2) equations. We sketch the proof for existence of solutions to these equations on the whole space $R^d$, and we estimate the rate of convergence to the steady state in the case of quadratic confinement potential, and in the unconfined situation.
Joint work with: Giuseppe Savare, Robert J McCann.