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Regularization by noise for the energy- and mass-critical nonlinear Schrödinger equations

Martin Spitz, Deng Zhang, Zhenqi Zhao

TL;DR

The paper addresses the challenge of blow-up and non-scattering in the focusing energy- and mass-critical stochastic NLS by introducing non-conservative multiplicative noise. It converts the SNLS to a random NLS via a Doss–Sussmann-type transform $u(t)=e^{\hat{\mu}t-W(t)}X(t)$ with $W(t)=\sum_k \phi_k\beta_k(t)$ and a geometric Brownian factor $h_c$, proving a pivotal decay estimate for $h_c$ after time $\|c\|^{-1}$. Using a two-step construction that separates a short-time dispersive regime from a long-time regime, the authors obtain global existence and forward scattering with probability tending to 1 as $\|c\|\to\infty$ in both the energy- and mass-critical cases. This establishes a robust regularization-by-noise phenomenon for critical dispersive SPDEs, extending prior subcritical results and highlighting the potential of noise to suppress singular behavior in focusing NLS.

Abstract

In this article we prove a regularization by noise phenomenon for the energy-critical and mass-critical nonlinear Schrödinger equations. We show that for any deterministic data, the probability that the corresponding solution exists globally and scatters goes to one as the strength of the non-conservative noise goes to infinity. The proof relies on the rescaling transform and a new observation on the rapid uniform decay of geometric Brownian motions after short time.

Regularization by noise for the energy- and mass-critical nonlinear Schrödinger equations

TL;DR

The paper addresses the challenge of blow-up and non-scattering in the focusing energy- and mass-critical stochastic NLS by introducing non-conservative multiplicative noise. It converts the SNLS to a random NLS via a Doss–Sussmann-type transform with and a geometric Brownian factor , proving a pivotal decay estimate for after time . Using a two-step construction that separates a short-time dispersive regime from a long-time regime, the authors obtain global existence and forward scattering with probability tending to 1 as in both the energy- and mass-critical cases. This establishes a robust regularization-by-noise phenomenon for critical dispersive SPDEs, extending prior subcritical results and highlighting the potential of noise to suppress singular behavior in focusing NLS.

Abstract

In this article we prove a regularization by noise phenomenon for the energy-critical and mass-critical nonlinear Schrödinger equations. We show that for any deterministic data, the probability that the corresponding solution exists globally and scatters goes to one as the strength of the non-conservative noise goes to infinity. The proof relies on the rescaling transform and a new observation on the rapid uniform decay of geometric Brownian motions after short time.
Paper Structure (4 sections, 8 theorems, 68 equations)

This paper contains 4 sections, 8 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Probabilistic tools
  3. The energy-critical case
  4. The mass-critical case

Key Result

Theorem 1.1

Let $d \geq 3$, $\alpha = 1 + \frac{4}{d-2}$, $c_k = \operatorname{Re}(\phi_k)$ for $k \in {\mathbb{N}}$ such that $c \in \ell^2$. Let $X_0 \in H^1({\mathbb R}^d)$. Then where $X$ denotes the unique solution of eq:StochasticNLS with initial data $X(0) = X_0$ and "scatters forward in time" means that there exists $X_+ \in H^1({\mathbb R}^d)$ such that where $\hat{\mu} = \frac{1}{2}\sum_{k=1}^\inf

Theorems & Definitions (17)

  • Theorem 1.1: Noise regularization for energy-critical SNLS
  • Theorem 1.2: Noise regularization for mass-critical SNLS
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1: Strichartz estimates KT98
  • Lemma 3.2: Estimates for the nonlinearity
  • ...and 7 more