Gradient flows of energy functionals on the space of probability measures with Wasserstein metric has proved to be a strong tool in studying certain mass conserving evolution equations. Such gradient flows provide an alternate formulation for the solutions of the corresponding evolution equations. An important condition, which is known to guarantees existence, uniqueness, and continuous dependence on initial data is that the corresponding energy functional be displacement convex. We introduce a relaxed notion of displacement convexity and we show that it still guarantees short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals which are not displacement convex in the standard sense. This extends the applicability of the gradient flow approach to larger family of energies. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.

Gradient flows of energy functionals on the space of probability measures with Wasserstein metric has proved to be a strong tool in studying certain mass conserving evolution equations. Such gradient flows provide an alternate formulation for the solutions of the corresponding evolution equations. An important condition, which is known to guarantees existence, uniqueness, and continuous dependence on initial data is that the corresponding energy functional be displacement convex. We introduce a relaxed notion of displacement convexity and we show that it still guarantees short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals which are not displacement convex in the standard sense. This extends the applicability of the gradient flow approach to larger family of energies. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable.