Theorem · version 1
If a solution u of the three-dimensional Navier–Stokes system satisfies ∫₀ᵀ ‖u(t)‖_{L^q}^p dt < ∞ with 2/p + 3/q ≤ 1 and q > 3, then u is smooth on [0, T].
A conditional result: it applies only to solutions already known to satisfy the bound.
Proved
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Kang–Protas · Ladyzhenskaya–Prodi–Serrin conditions · Abstract; conditions (9)–(10) →
Observed Oct 21, 2026
In this problem · publication v8
Conditional result · Proved
Scope: Control of a scale-invariant (or better) norm of the velocity implies smoothness.
Limit: No global bound of this strength is known for general solutions; the energy bounds stop at 2/p + 3/q = 3/2.
A distinct scope, not an additive fraction of the problem solved.
Relationship not independently reviewed. This concerns its use in this problem.
The Ladyzhenskaya–Prodi–Serrin conditions: if the time integral of the L^q norm to the power p is bounded, with 2/p + 3/q ≤ 1 and q > 3, the solution is smooth on [0, T].
paraphrase · No upstream full text is republished by this record.
Di Kang · Bartosz Protas · Published · Publication date Sep 6, 2022
Published in Journal of Nonlinear Science 32, article 81 (2022), DOI 10.1007/s00332-022-09832-7. Statements here use the pinned arXiv v1: the abstract and estimate (12).
Source revision: arxiv-v1. Observed Oct 21, 2026.
The cited work does not report AI involvement; absence of attribution is not evidence of absence.
Source updated Oct 11, 2021.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →