A blow up solution of the Navier-Stokes equations with a critical force

TL;DR

This paper constructs a specific blow-up solution to the Navier-Stokes equations under critical forces, demonstrating potential singularity formation in physically relevant force regimes, including point-source-like forces.

Contribution

It provides the first explicit construction of blow-up solutions with critical and subcritical forces in Navier-Stokes, advancing understanding of singularity formation under physically realistic forces.

Findings

Constructed a smooth solution that blows up in finite time under critical force.

Shows singularity formation is possible with physically relevant forces like Coulomb and Yukawa.

Includes explicit force examples demonstrating the phenomenon.

Abstract

A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up moment, is in the energy space through out. Since most physical forces from a point source in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this result indicates possible singularity formation under these kind of forces. The result even holds for some log subcritical forces or some forces in the standard critical space $L^\infty_t L^{3/2}_x$, including the explicit force: $F=- \delta \frac{e^{-|x|^2}}{(|x|^2 + T-t) \,[1+ | \ln (|x|^2 + T-t)|]} (1, 0, 0) $ for any small $\delta>0$. The result can also be considered as a step in Scheffer's plan.