  We prove the global-in-time existence of nonnegative weak solutions to a
class of fourth order partial differential equations on a convex bounded domain
in arbitrary spatial dimensions. Our proof relies on the formal gradient flow
structure of the equation with respect to the $L^2$-Wasserstein distance on the
space of probability measures. We construct a weak solution by approximation
via the time-discrete minimizing movement scheme; necessary compactness
estimates are derived by entropy-dissipation methods. Our theory essentially
comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.
