Pure-swirl loss of boundedness under $L^1_tL^2_x$ forcing: exact mixed-norm ranges

Abstract

We construct an explicit pure-swirl solution of the forced three-dimensional Navier--Stokes equations in a circular cylinder. For every $T>0$ and $1\le p,q<\infty$ with $1/p+1/q>1$, the force belongs to $L^q(0,T;L^p(D))$ and is smooth for $t<T$. Moreover, the same construction always yields the additional energy-class property $f\in L^1(0,T;L^2(D))$. The solution is classical on $[0,T)$, extends strongly in $L^2(D)$ to the unique Leray--Hopf solution at time $T$, and satisfies the energy equality, while $\|v(t)\|_{L^\infty(D)}\to\infty$ as $t\uparrow T$. We determine the exact mixed-norm ranges of both the force and the velocity and obtain two-sided rates for the force, the velocity supremum norm and the enstrophy. The construction refines Zhang's profile by an annular cancellation that preserves smoothness at the symmetry axis. It is a direct, self-contained sharpening of the $k=1$ part of our previous weighted construction and supersedes its endpoint discussion. Since the convection term is absorbed by the pressure, the same example applies to the forced Stokes system.