Quantitative estimates for the forced Navier-Stokes equations and applications

In this paper, we prove a localisation of a slightly supercritical (Orlicz) regularity criterion for the 3D incompressible Navier-Stokes equations. This is a refinement to the recent partial positive answer to Tao’s conjecture [Tao21] as given in [BP21b]. The proof requires new quantitative estimates for critically bounded solutions of the forced Navier-Stokes equations, where the forcing is induced by the localisation. A by-product of these new estimates is an application to the Boussinesq equations, where we prove a quantitative blow-up rate for the critical $L^3$ norm of the velocity. We prove these quantitative estimates using Carleman inequalities as in [Tao21], and subsequently in [BP21a], with an additional forcing term. An obstacle to doing this is that, in the Carleman inequalities, the forcing term is amplified on large scales. Additionally, the low regularity of the forcing requires the addition of Caccioppoli-type estimates to deal with the Carleman inequalities appropriately.

💡 Research Summary

The paper develops new quantitative regularity estimates for the three‑dimensional incompressible Navier–Stokes equations (NSE) when a forcing term is present. The motivation stems from Tao’s 2021 breakthrough, which replaced qualitative blow‑up arguments with quantitative Carleman‑inequality based estimates, and from the subsequent work of Buckmaster–Ponce (BP) that proved a slightly super‑critical Orlicz‑type blow‑up criterion. However, both of those results dealt with the unforced NSE; the presence of a forcing term, which naturally appears when one localises a blow‑up criterion, creates two major difficulties: (1) the forcing term is amplified in the Carleman inequalities, especially on large spatial scales, and (2) the forcing is only known in low‑regularity spaces, so higher‑order derivatives of the velocity are not directly controllable.

To overcome these obstacles the authors combine several tools:

1. Carleman inequalities with forcing. Two Carleman estimates are used: one that yields a Gaussian lower bound (large‑scale estimate) and another that provides backward uniqueness. When a forcing term (F) is added, each inequality acquires an extra term of the form
   \