On local Type I singularities of the Navier-Stokes equations and Liouville theorems

Abstract

We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.

arXiv commentPublished in Journal of Mathematical Fluid Mechanics

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Cited by in corpus (2)

• Scale-invariant estimates and vorticity alignment for Navier-Stokes in the half-space with no-slip boundary conditions
• Mild criticality breaking for the Navier-Stokes equations