Iterative methods fail to solve NLS below the Sobolev embedding threshold on the Sierpinski gasket

TL;DR

This paper demonstrates that iterative methods for solving the nonlinear Schrödinger equation on the Sierpinski gasket fail below a certain Sobolev regularity threshold, due to localized eigenfunctions affecting well-posedness.

Contribution

It reveals a new failure of local well-posedness for NLS on fractals, linked to eigenfunction localization, differing from classical compact spaces.

Findings

Flow map not $C^{2k+1}$-continuous below threshold

Threshold independent of nonlinearity power

Localized eigenfunctions cause ill-posedness

Abstract

We show that the nonlinear Schr\"odinger equation on the Sierpinski gasket with a power nonlinearity of order $2k{+}1$ is not locally well-posed for initial data just below the regularity threshold for the Sobolev embedding $H^s\subseteq L^\infty$. More precisely, the flow map fails to be $C^{2k+1}$-continuous in any Sobolev space $H^s$ below that threshold, and the threshold is independent of the power nonlinearity. This novel behavior significantly differs from other compact spaces such as the torus or the sphere, and it is directly connected to the existence of localized eigenfunctions.