On Singularity Formation of a Nonlinear Nonlocal System

TL;DR

This paper rigorously proves finite time singularity formation and global regularity for a nonlinear nonlocal system inspired by 3D Navier-Stokes equations, supported by numerical evidence of self-similar blow-up.

Contribution

It introduces a simplified nonlocal model replacing the Riesz operator with the Hilbert transform and establishes conditions for singularity formation and regularity.

Findings

Proves finite time singularity for large classes of initial data.

Establishes global regularity for certain smooth initial data.

Numerical results show asymptotically self-similar blow-up similar to 3D models.

Abstract

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible Navier-Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier-Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove the global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.