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Global unique solution to the perturbation of the Burgers' equation forced by derivatives of space-time white noise

Kazuo Yamazaki

TL;DR

This work analyzes the 1D Burgers' equation forced by a fractional derivative of order $\frac{1}{2}$ of space-time white noise, establishing global well-posedness for both mild and HL weak solutions. Building on the Anderson Hamiltonian framework of Allez–Chouk and the global approach to stochastic NS by Hairer–Rosati, the authors adapt these techniques to a derivative-noise regime in one dimension, introducing enhanced noise and paracontrolled calculus to manage ill-defined nonlinearities. The paper proves a local mild-solution result (Theorem 2.2), a global HL weak-solution result (Theorem 2.3), and constructs a robust 1D Anderson-type operator (Proposition 2.4) via a resolvent approach, including convergence and renormalization controls. A key feature is the use of logarithmic renormalization bounds and a careful low/high frequency decomposition to achieve global-in-time control in a highly singular SPDE setting, providing a rigorous framework for endpoint derivative-noise forcing in low dimensions with potential KPZ-related implications.

Abstract

We consider the one-dimensional Burgers' equation forced by fractional derivative of order $\frac{1}{2}$ applied on space-time white noise. Relying on the approaches of Anderson Hamiltonian from Allez and Chouk (2015, arXiv:1511.02718 [math.PR]) and two-dimensional Navier-Stokes equations forced by space-time white noise from Hairer and Rosati (2024, Annals of PDE, \textbf{10}, pp. 1--46), we prove the global-in-time existence and uniqueness of its mild and weak solutions.

Global unique solution to the perturbation of the Burgers' equation forced by derivatives of space-time white noise

TL;DR

This work analyzes the 1D Burgers' equation forced by a fractional derivative of order of space-time white noise, establishing global well-posedness for both mild and HL weak solutions. Building on the Anderson Hamiltonian framework of Allez–Chouk and the global approach to stochastic NS by Hairer–Rosati, the authors adapt these techniques to a derivative-noise regime in one dimension, introducing enhanced noise and paracontrolled calculus to manage ill-defined nonlinearities. The paper proves a local mild-solution result (Theorem 2.2), a global HL weak-solution result (Theorem 2.3), and constructs a robust 1D Anderson-type operator (Proposition 2.4) via a resolvent approach, including convergence and renormalization controls. A key feature is the use of logarithmic renormalization bounds and a careful low/high frequency decomposition to achieve global-in-time control in a highly singular SPDE setting, providing a rigorous framework for endpoint derivative-noise forcing in low dimensions with potential KPZ-related implications.

Abstract

We consider the one-dimensional Burgers' equation forced by fractional derivative of order applied on space-time white noise. Relying on the approaches of Anderson Hamiltonian from Allez and Chouk (2015, arXiv:1511.02718 [math.PR]) and two-dimensional Navier-Stokes equations forced by space-time white noise from Hairer and Rosati (2024, Annals of PDE, \textbf{10}, pp. 1--46), we prove the global-in-time existence and uniqueness of its mild and weak solutions.

Paper Structure

This paper contains 16 sections, 33 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Motivation from physics and real-world applications
  3. Past relevant results and mathematical motivation
  4. Approach of HR24
  5. Statement of main results
  6. Preliminaries
  7. Besov spaces and Bony's paraproducts
  8. Proof of Theorem \ref{['Theorem 2.2']}
  9. Proof of Theorem \ref{['Theorem 2.3']}
  10. Proof of Proposition \ref{['Proposition 2.4']}
  11. Further preliminaries
  12. Further details
  13. Proof of Lemma \ref{['Product estimate lemma']}
  14. Proof of Proposition \ref{['Proposition 4.11']}
  15. Proof of Proposition \ref{['Proposition 4.12']}
  16. ...and 1 more sections

Key Result

Proposition 2.1

There exists a null set $\mathcal{N} \subset \Omega$ such that for all $\omega \in \Omega \setminus \mathcal{N}$ and $\kappa > 0$, the following holds. For any $\theta^{\text{in}} \in \mathcal{C}^{-1+ \kappa}(\mathbb{T})$ that is mean-zero, there exists a $T^{\max} (\omega, \theta^{\text{in}}) \in (

Theorems & Definitions (64)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.1
  • Lemma 3.3
  • ...and 54 more