The Probabilistic Method and large initial data for Generalized Navier-Stokes systems

TL;DR

This paper introduces a probabilistic method to demonstrate the existence of large initial data in various norms that lead to global regular solutions for generalized Navier-Stokes systems, highlighting most such data produce regular solutions.

Contribution

It presents a novel probabilistic approach to establish the existence of large initial data resulting in global solutions for generalized Navier-Stokes equations.

Findings

Most large initial data in specified norms yield global regular solutions.

The probabilistic method identifies a large subset of initial data leading to regularity.

The approach broadens understanding of initial data size and solution regularity in fluid dynamics.

Abstract

In this paper we introduce a probabilistic approach to show the existence of initial data with arbitrarily large $L^2(\mathbb{R}^3)$, $\dot{H}^{1/2}(\mathbb{R}^3)$ and $\mathcal{PM}^2$-norms for which a Generalized Navier-Stokes system generate a global regular solution. More precisely, we show that from a certain family of possible large initial data most of them give raise to global regular solutions to a given Generalized Navier-Stokes system.