Opuscula Mathematica
Opuscula Math. 46, no. 5 (2026), 623-639

Pure-swirl loss of boundedness under \(L^1_tL^2_x\) forcing: exact integrability of an explicit family

Hugo Beirão da Veiga
Jiaqi Yang
Abstract. 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.
Keywords: Navier-Stokes equations, external force, loss of boundedness, pure swirl, mixed norms, Leray-Hopf solution.
Mathematics Subject Classification: 35Q30, 76D03, 35A01.
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About the authors ▼
Hugo Beirão da Veiga
https://orcid.org/0000-0003-0063-8349
Pisa University, Department of Mathematics, Pisa, Italy; Academia das Ciências de Lisboa (The Portuguese Academy of Science), Portugal
Jiaqi Yang (corresponding author)
Northwestern Polytechnical University, School of Mathematics and Statistics, Xi'an, China
About the article ▼
Communicated by Daniele Cassani.
Received: 2026-08-10.
Revised: 2026-10-12.
Accepted: 2026-10-13.
Published online: 2026-10-28.
Cite this article as ▼
Hugo Beirão da Veiga, Jiaqi Yang, Pure-swirl loss of boundedness under \(L^1_tL^2_x\) forcing: exact integrability of an explicit family, Opuscula Math. 46, no. 5 (2026), 623-639, https://doi.org/10.7494/OpMath.202610131.
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