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

Patricia Alonso Ruiz, Gigliola Staffilani

TL;DR

This work analyzes the nonlinear Schrödinger equation on the Sierpinski gasket with a $|u|^{2k}u$ nonlinearity and proves $C^{2k+1}$-ill-posedness for all $0<s<\sigma_\infty$, where $\sigma_\infty=\frac{d_S}{2}$ is the Sobolev embedding threshold and $d_S=\frac{\log 9}{\log 5}$ is the spectral dimension. The key mechanism is the presence of localized eigenfunctions of the SG Laplacian, which drives the sharp, nonuniform behavior of the flow map below the embedding threshold and makes the threshold independent of $k$. The analysis relies on a Taylor-expansion framework for the flow map and a construction of highly localized eigenmodes via spectral decimation, showing that Strichartz-type improvements do not occur on SG in the low-regularity regime. The results highlight fractal geometry as a fundamental obstacle to well-posedness techniques that are effective on smooth compact manifolds, with implications for studying dispersive equations on fractal spaces.

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 $H^s\subseteq L^\infty$. 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.

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

TL;DR

This work analyzes the nonlinear Schrödinger equation on the Sierpinski gasket with a nonlinearity and proves -ill-posedness for all , where is the Sobolev embedding threshold and is the spectral dimension. The key mechanism is the presence of localized eigenfunctions of the SG Laplacian, which drives the sharp, nonuniform behavior of the flow map below the embedding threshold and makes the threshold independent of . The analysis relies on a Taylor-expansion framework for the flow map and a construction of highly localized eigenmodes via spectral decimation, showing that Strichartz-type improvements do not occur on SG in the low-regularity regime. The results highlight fractal geometry as a fundamental obstacle to well-posedness techniques that are effective on smooth compact manifolds, with implications for studying dispersive equations on fractal spaces.

Abstract

We show that the nonlinear Schrödinger equation on the Sierpinski gasket with a power nonlinearity of order is not locally well-posed for initial data just below the regularity threshold for the Sobolev embedding . More precisely, the flow map fails to be -continuous in any Sobolev space 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.
Paper Structure (11 sections, 8 theorems, 62 equations, 3 figures, 1 table)

This paper contains 11 sections, 8 theorems, 62 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. The Sierpinski gasket
  4. Fractional Sobolev spaces
  5. The Schrödinger equation
  6. Sobolev embedding
  7. Eigenfunctions on the Sierpinski gasket
  8. Sobolev embedding
  9. C2k-ill-posedness
  10. Map derivatives
  11. Proof of main result

Key Result

Theorem 1

Let $0<s<\sigma_\infty({\rm SG})$. For $\mu=\pm 1$, for every integer $k\geq 1$, the non-linear Schrödinger equation is $C^{2k+1}$-ill-posed.

Figures (3)

  • Figure 1: The standard Sierpinski gasket and the measure of different cells.
  • Figure 2: Graph approximations of the Sierpinski gasket.
  • Figure 3: A possible function $v\colon V_3\to\mathbb{R}$. In green the cells $F_w(V_0)$ and $F_{w'}(V_0)$.

Theorems & Definitions (21)

  • Definition 1: Strong well-posedness
  • Theorem 1
  • Remark 1
  • Definition 2
  • Remark 2
  • Theorem 2
  • Corollary 3
  • Proposition 4
  • proof
  • Remark 3
  • ...and 11 more