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Effect of random noises on pathwise solutions to the high-dimensional modified Euler-Poincaré system

Lei Zhang

Abstract

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincaré system (MEP2) on the torus $\mathbb{T}^d$, $d\geq 1$. We first establish a local well-posedness framework in the sense of Hadamard for the MEP2 driven by general nonlinear multiplicative noises. Then two kinds of global existence and uniqueness results are demonstrated: One indicates that the MEP2 perturbed by nonlocal-type random noises with proper intensity admits a unique large global strong solution; The other one infers that, if the initial data is sufficiently small, then the MEP2 perturbed by linear multiplicative noise has a unique global solution with high probability. In the case of one dimension, we find that the stochastic MEP2 will break down in finite time when the initial data meets appropriate shape condition.

Effect of random noises on pathwise solutions to the high-dimensional modified Euler-Poincaré system

Abstract

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincaré system (MEP2) on the torus , . We first establish a local well-posedness framework in the sense of Hadamard for the MEP2 driven by general nonlinear multiplicative noises. Then two kinds of global existence and uniqueness results are demonstrated: One indicates that the MEP2 perturbed by nonlocal-type random noises with proper intensity admits a unique large global strong solution; The other one infers that, if the initial data is sufficiently small, then the MEP2 perturbed by linear multiplicative noise has a unique global solution with high probability. In the case of one dimension, we find that the stochastic MEP2 will break down in finite time when the initial data meets appropriate shape condition.

Paper Structure

This paper contains 20 sections, 19 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deterministic background
  4. Stochastic setting
  5. Assumptions and main results
  6. Plan of the paper
  7. Local well-posedness
  8. Proof of Theorem \ref{['th1']}(2)
  9. Regularization of SMEP2
  10. Momentum estimates
  11. Smooth strong solutions
  12. Martingale solutions
  13. Pathwise uniqueness
  14. Regularized pathwise solution
  15. Proof of Theorem \ref{['th1']}(1)
  16. ...and 5 more sections

Key Result

Theorem 1.4

Let $s>1+\frac{d}{2}$, $d\geq1$, and $\textbf{y}_0$ be a $\mathbb{H}^s$-valued $\mathcal{F}_0$-measurable random variable such that $\mathbb{E}\|\textbf{y}_0\|_{\mathbb{H}^s}^2<\infty$. Under the Assumption assume, we obtain the following conclusions:

Theorems & Definitions (40)

  • Definition 1.1
  • Remark 1.3
  • Theorem 1.4: Local existence with general noises
  • Remark 1.5
  • Theorem 1.6: I. Global existence with large initial data
  • Remark 1.7
  • Theorem 1.8: II. Global existence with small initial data
  • Remark 1.9
  • Theorem 1.10: Blow-up criteria
  • Remark 1.11
  • ...and 30 more