Existence of weak solutions to a class of fourth order partial differential equations with Wasserstein gradient structure

2015/07/20 by Daniel Loibl, Loibl, Daniel, Daniel Matthes +3
Mathematics · #35D30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Primary: 35K30 #Secondary: 35A15

paper · pdf · doi:10.48550/arxiv.1507.05507

openalex publication_date 2015/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 L²-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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