# On local well-posedness of the thin-film equation via the Wasserstein gradient flow

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## Abstract

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

### Mathematics Subject Classification

49Q20 35A01 76A20 49J52### References

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