Abstract
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.
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This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”
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Loibl, D., Matthes, D. & Zinsl, J. Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure. Potential Anal 45, 755–776 (2016). https://doi.org/10.1007/s11118-016-9565-y
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DOI: https://doi.org/10.1007/s11118-016-9565-y