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Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus

Hao Shen, Rongchan Zhu, Xiangchan Zhu

Abstract

This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on $\mathbb{R}^+\times \mathbb{T}^2$ within the framework of paracontrolled calculus \cite{GIP15}. The model is given by the equation: \begin{equation*} (\partial_t-Δ) u=F(u)η \end{equation*} where $η\in C^{-1-κ}$ with $1/6>κ>0$, and $F\in C_b^2(\mathbb{R})$. Assume that $η\in C^{-1-κ}$ and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work \cite{CFW24} by A.Chandra, G.L. Feltes and H.Weber to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument).

Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus

Abstract

This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on within the framework of paracontrolled calculus \cite{GIP15}. The model is given by the equation: \begin{equation*} (\partial_t-Δ) u=F(u)η \end{equation*} where with , and . Assume that and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work \cite{CFW24} by A.Chandra, G.L. Feltes and H.Weber to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument).
Paper Structure (11 sections, 12 theorems, 121 equations)

This paper contains 11 sections, 12 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Paracontrolled Solution
  4. Proof of Theorem \ref{['th:1']}
  5. Decomposition
  6. Bounds for $u_1^\sharp$ in $C_TC^{1-\kappa}$ and ${\mathbb C}_T^{2\alpha,\gamma}$
  7. Time regularity of $u$
  8. Bound for $u_2^\sharp$ in ${\mathbb L}^\infty_T$ and ${\mathbb S}^{2-\varepsilon/2,\gamma}_T$
  9. Appendix
  10. Schauder estimates
  11. Paraproduct estimates

Key Result

Theorem 1.2

Let $\eta\in L^\infty_TC^{-1-\kappa}$ can be lifted to an enhanced noise with $0<\kappa<\bar{\kappa}$. For every given initial condition $u_0\in L^\infty$, there exists a unique global-in-time solution to equ in a paracontrolled sense (see Definition def:sol).

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 12 more