Effect of random noises on pathwise solutions to the high-dimensional modified Euler-Poincar\'{e} system

TL;DR

This paper investigates the well-posedness and global existence of solutions to a high-dimensional stochastic modified Euler-Poincaré system under various noise conditions, revealing both stability and finite-time breakdown scenarios.

Contribution

It establishes local well-posedness and demonstrates global existence results for the stochastic MEP2 under different noise types and initial conditions, including finite-time blow-up in one dimension.

Findings

Global strong solutions exist under nonlocal noise with proper intensity.

Small initial data lead to high-probability global solutions with linear noise.

Finite-time breakdown occurs in one dimension with certain initial shapes.

Abstract

In this paper, we study the Cauchy problem for the stochastically perturbed high-dimensional modified Euler-Poincar\'{e} 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.