A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation

Abstract

The global regularity problem for the periodic Navier-Stokes systemfor u : R + × (R/Z) 3 → R 3 and p : R + × (R/Z) 3 → R asks whether to every smooth divergence-free initial datum u 0 : (R/Z) 3 → R 3 there exists a global smooth solution.In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function F : R + → R + for which one has a local-in-time a priori bound) ) for all 0 < T ≤ 1 and all smooth solutions u : [0, T ] × (R/Z) 3 → R 3 to the Navier-Stokes system.We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound.

Locations

• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF