TY  - JOUR
ID  - BeirãodaVeiga2026
LB  - BeirãodaVeiga2026
AU  - Beirão da Veiga, Hugo
AU  - Yang, Jiaqi
TI  - Pure-swirl loss of boundedness under \(L^1_tL^2_x\) forcing: exact integrability of an explicit family
ST  - Pure-swirl loss of boundedness under \(L^1_tL^2_x\) forcing: exact integrability of an explicit family
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2026
VL  - 46
IS  - 5
SP  - 623
EP  - 639
DO  - https://doi.org/10.7494/OpMath.202610131
UR  - https://doi.org/10.7494/OpMath.202610131
AB  - We give an explicit pure-swirl family of forced Navier-Stokes solutions in a bounded circular cylinder with no slip on the lateral wall and free slip on the horizontal faces. For each \(T\gt 0\) and each finite pair \(p,q\geq 1\) with \(1/p+1/q\gt 1\), we construct a corresponding force in \(L^q(0,T;L^p(D))\cap L^1(0,T;L^2(D))\) whose velocity loses \(L^\infty\) boundedness as \(t \uparrow T\). The construction refines the \(k=1\) case of our previous weighted ansatz: we correct the mixed-norm calculation and the convection identity, and impose an annular cancellation to ensure smoothness at the axis. For this family we prove exact mixed-norm membership criteria, including the spatial force endpoint \(p=1\), and two-sided rates for the force, velocity supremum norm and enstrophy. The velocity has a strong \(L^2\) trace at \(T\) and, with this trace, is the unique Leray-Hopf solution on \([0,T]\) and satisfies the energy equality. The same velocity and force solve Stokes with zero pressure.
KW  - Navier-Stokes equations
KW  - external force
KW  - loss of boundedness
KW  - pure swirl
KW  - mixed norms
KW  - Leray-Hopf solution
SN  - 1232-9274
ER  -