Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

TL;DR

This paper systematically searches for extreme or singular behaviors in fluid models, finding no evidence of finite-time blow-up in 3D Navier-Stokes flows through computational PDE optimization, and providing insights into energy estimates.

Contribution

It introduces a systematic computational approach to search for singularities in fluid models, extending previous work and analyzing multiple systems to assess the sharpness of energy estimates.

Findings

No singularities found in 3D Navier-Stokes flows using the proposed method.

Singularities are ruled out in 1D Burgers and 2D Navier-Stokes systems.

Provides insights into the sharpness of energy estimates for these models.

Abstract

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008), we sought such extreme behavior by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in 3D. We also discuss the results obtained for 1D Burgers and 2D Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed.