Potential Analysis

, Volume 45, Issue 4, pp 755–776 | Cite as

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure



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.


Fourth-order equations Gradient flow Wasserstein distance Weak solution Minimizing movement scheme 

Mathematics Subject Classification (2010)

Primary: 35K30 Secondary: 35A15 35D30 


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Zentrum für MathematikTechnische Universität MünchenGarchingGermany

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