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

2025/05/07 by Ruiz, Patricia Alonso, Staffilani, Gigliola
#28A80 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.04515

Abstract

We show that the nonlinear Schrödinger 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⊆ L^∞. 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.

Citations

• The Second Picard iteration of NLS on the 2d sphere does not regularize Gaussian random initial data
• Minimal gap in the spectrum of the Sierpinski gasket
• Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions
• Szego limit theorems on the Sierpinski gasket
• On ill-posedness for the one-dimensional periodic cubic Schrodinger equation
• Instability of the periodic nonlinear Schrodinger equation
• Bilinear Eigenfunction Estimates and the Nonlinear Schroedinger Equation on Surfaces
• Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

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