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arXiv · 2511.08776

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

Stefanos Georgiadis·Stefano Spirito
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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 $L^2$-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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Stefanos Georgiadis, Stefano Spirito. 2025-11-11. On the existence of non-negative weak solutions for $1D$ fourth order equations of gradient flow type. https://arxiv.org/abs/2511.08776

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