Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation

RESUMO

We apply the variational method of Weighted Energy Dissipation (WED) to obtain a global-in-time approximation of solutions to fourth order degenerate parabolic equations of Cahn-Hilliard type. Differently from the standard approach to the existence theory, WED induces an elliptic regularization in time, not in space. The confinement on the phase field is guaranteed already for the approximation, a priori estimates follow without modification of Lyapunov functionals, and the approximations are weakly differentiable in time --- even twice in the case of linear mobility. While WED has been widely used for the construction of gradient flows in Hilbert spaces, this appears to be the first application of WED to a metric gradient flow beyond second order PDEs of Wasserstein type.