Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On blow up NLS with a multiplicative noise

Chenjie Fan, Junzhe Wang

TL;DR

The paper analyzes the 3D focusing cubic SNLS with multiplicative, space-colored and time-white noise in Stratonovich form, proving a large deviation type bound on blow-up probability for small times. The authors employ a truncation approach to obtain globally well-posed auxiliary problems and reduce blow-up to controlling a stochastic convolution $J_R u_R$; they prove a deterministic stability estimate and a stochastic BDG-based bound, obtaining $\log \mathbb{P}(\text{blow-up in }[0,T]) \lesssim -T^{-1/4}$ for small $T$, with $\beta=1/4$. This provides a stability‑oriented view contrasting earlier results suggesting noise can accelerate blow-up, and it extends to subcritical models via a similar framework. The work leverages Strichartz dispersive estimates and sharp stochastic integrability bounds, contributing a quantitative probabilistic control on blow-up in stochastic NLS and informing the interplay between random perturbations and nonlinear dispersive dynamics.

Abstract

It is of significant interest to understand whether a noise will speed up or prevent blow up. Under certain nondegenerate conditions, \cite{dD2005Blowup} proved a multiplicative noise will speed up blow up of NLS, in the sense that, blow up can happen in any short time with positive probability. We prove that such probability is indeed quite small, and provide a large deviation type upper bound.

On blow up NLS with a multiplicative noise

TL;DR

The paper analyzes the 3D focusing cubic SNLS with multiplicative, space-colored and time-white noise in Stratonovich form, proving a large deviation type bound on blow-up probability for small times. The authors employ a truncation approach to obtain globally well-posed auxiliary problems and reduce blow-up to controlling a stochastic convolution ; they prove a deterministic stability estimate and a stochastic BDG-based bound, obtaining for small , with . This provides a stability‑oriented view contrasting earlier results suggesting noise can accelerate blow-up, and it extends to subcritical models via a similar framework. The work leverages Strichartz dispersive estimates and sharp stochastic integrability bounds, contributing a quantitative probabilistic control on blow-up in stochastic NLS and informing the interplay between random perturbations and nonlinear dispersive dynamics.

Abstract

It is of significant interest to understand whether a noise will speed up or prevent blow up. Under certain nondegenerate conditions, \cite{dD2005Blowup} proved a multiplicative noise will speed up blow up of NLS, in the sense that, blow up can happen in any short time with positive probability. We prove that such probability is indeed quite small, and provide a large deviation type upper bound.
Paper Structure (14 sections, 7 theorems, 63 equations)

This paper contains 14 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Background
  4. More technical background
  5. Notations
  6. Preliminaries
  7. Stochastic preliminary
  8. Dispersive preliminary
  9. Overview
  10. Truncated SNLS Equation
  11. Overview of the proof of Theorem \ref{['thm:main']}
  12. Proof of two Lemmas
  13. Proof of Lemma \ref{['lem:deterministic']}
  14. Proof of Lemma \ref{['lem:stochastic']}

Key Result

Theorem 1.1

For any initial data $u_0 \in H^1(\mathbb{R}^3)$, there exists $T_0(\|u_0\|_{H^1})>0$, and a universal constant $\beta > 0$ such that the local solution $u$ of SNLS satisfies for any $0<T\le T_0$. In particular we can take $\beta=\frac{1}{4}$ here.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1: BDG inequaility
  • Lemma 3.2: dispersive estimate
  • Lemma 3.3: Strichartz estimate in $\mathbb{R}^3$
  • Proposition 4.1
  • proof : Proof of Thm \ref{['thm:main']} asssuming Prop \ref{['prop:main']}
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Prop \ref{['prop:main']} asssuming lemma \ref{['lem:deterministic']} and lemma \ref{['lem:stochastic']}
  • ...and 4 more