A Blow up Solution of the Navier-Stokes Equations with a Super Critical Forcing Term

Abstract

A forced solution $v$ of the axially symmetric Navier-Stokes equation in a finite cylinder $D$ with suitable boundary condition is constructed. The forcing term, whose order of scaling is slightly worse than the critical order $−2,$ is in the mildly super critical space $L^q_tL^1_x$ for all $q>1.$ The velocity, which is smooth until its final blow up moment, is in the energy space throughout.