On the existence of non-negative weak solutions for 1D fourth order equations of gradient flow type

2025/11/11 by Georgiadis, Stefanos, Spirito, Stefano · 1 citation
#35D30 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 35G20 #Secondary: 35K65

paper · doi:10.48550/arxiv.2511.08776

Abstract

In this paper, we consider a family of one-dimensional fourth order evolution equations arising as gradient flows of the Korteweg energy, i.e. the L²-norm of the first derivative of some power of the density. This family of equations generalizes the Quantum-Drift-Diffusion equation and the Thin-Film equation. We prove the global-in-time existence of \em non-negative weak solutions without requiring any upper bound on the exponent of the power of the density in the energy.

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