Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations

TL;DR

This paper examines a simplified one-dimensional model inspired by the Navier-Stokes equations, demonstrating how the structure of nonlinear terms influences solution blow-up for large initial data.

Contribution

It adapts Montgomery-Smith's model to 2D and 3D, preserving divergence-free conditions, and shows that large initial data can cause blow-up, highlighting the importance of nonlinear structure.

Findings

Large initial data can lead to blow-up in the toy model.

The nonlinear term's structure is crucial for solution behavior.

Adaptation preserves divergence-free condition in 2D and 3D.

Abstract

S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow up for the toy model -- meaning that the precise structure of the nonlinear term is crucial to understand the dynamics of large solutions to the Navier-Stokes equations.