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

TL;DR

This paper proves the global existence of non-negative weak solutions for a family of 1D fourth order gradient flow equations, generalizing models like the Quantum-Drift-Diffusion and Thin-Film equations, without upper bounds on the density exponent.

Contribution

It establishes the existence of solutions for a broad class of fourth order equations of gradient flow type, extending previous results to cases without upper bounds on the density exponent.

Findings

Proved global-in-time existence of non-negative weak solutions.

Generalized models including Quantum-Drift-Diffusion and Thin-Film equations.

No upper bound required on the exponent of the density in the energy.

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.