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Regularization for the Schrödinger equation with rough potential: high-dimensional case

Ruobing Bai, Yajie Lian, Yifei Wu

Abstract

In this work, we investigate the regularization mechanisms of the Schrödinger equation with a spatial potential $$ i\partial_t u+Δu+ηu =0, $$ where $η$ denotes a given spatial potential. The regularity of solutions constitutes one of the central problems in the theory of dispersive equations. Recent works \cite{Bai-Lian-Wu-2024, M-Wu-Z24} have established the sharp regularization mechanisms for this model in the whole space $\mathbb{R}$ and on the torus $\mathbb{T}$, with $η$ being a rough potential. The present paper extends the line of research to the high-dimensional setting with rough potentials $η\in L_x^r+L_x^{\infty}$. More precisely, we first show that when $1\leq r <\frac d2$, there exists some $η\in L_x^r+L_x^{\infty}$ such that the equation is ill-posed in $H_x^γ$ for any $γ\in \mathbb{R}$. Conversely, when $\frac d2 \leq r \leq \infty$, the expected optimal regularity is given by $$H_x^{γ_*}, \quad γ_*=\mbox{min}\{2+\frac d2-\frac dr, 2\}.$$ We establish a comprehensive characterization of the regularity, with the exception of two dimensional endpoint case $d=2, r=1$. Our novel theoretical framework combines several fundamental ingredients: the construction of counterexamples, the proposal of splitting normal form method, and the iterative Duhamel construction. Furthermore, we briefly discuss the effect of the interaction between rough potentials and nonlinear terms on the regularity of solutions.

Regularization for the Schrödinger equation with rough potential: high-dimensional case

Abstract

In this work, we investigate the regularization mechanisms of the Schrödinger equation with a spatial potential where denotes a given spatial potential. The regularity of solutions constitutes one of the central problems in the theory of dispersive equations. Recent works \cite{Bai-Lian-Wu-2024, M-Wu-Z24} have established the sharp regularization mechanisms for this model in the whole space and on the torus , with being a rough potential. The present paper extends the line of research to the high-dimensional setting with rough potentials . More precisely, we first show that when , there exists some such that the equation is ill-posed in for any . Conversely, when , the expected optimal regularity is given by We establish a comprehensive characterization of the regularity, with the exception of two dimensional endpoint case . Our novel theoretical framework combines several fundamental ingredients: the construction of counterexamples, the proposal of splitting normal form method, and the iterative Duhamel construction. Furthermore, we briefly discuss the effect of the interaction between rough potentials and nonlinear terms on the regularity of solutions.

Paper Structure

This paper contains 30 sections, 30 theorems, 318 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. A discussion on the effects of nonlinearity
  4. Difficulty, novelty, ideas of proof.
  5. Organization of the paper
  6. Preliminary
  7. Notations
  8. Basic lemmas
  9. Supercritical case: the proof of Theorem \ref{['theorem-main 1']}
  10. Resonant and non-resonant decomposition
  11. Subcritical case: the proof of Theorem \ref{['theorem-one']}
  12. Low regularity for a subcritical index $\frac{d}{2}<r\leq 2$
  13. Boundary terms
  14. Resonance term and low frequency term
  15. High-order terms
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $d\geq 3$, $1\leq r<\frac{d}{2}$, there exists some $\eta \in L_x^{r}+L_x^{\infty}(\mathbb{R}^d)$ such that for any $\gamma\in \mathbb{R}$, eq:NLS is ill-posed in $H_x^{\gamma}(\mathbb{R}^d)$ .

Theorems & Definitions (57)

  • Definition 1.1: Well-posedness
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Remark 1.8
  • Lemma 2.1: Bernstein estimates
  • Lemma 2.2: Schur's test
  • ...and 47 more