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arxiv: 2604.25669 · v1 · submitted 2026-04-28 · 🧮 math.AP · physics.flu-dyn

Boundary epsilon regularity for incompressible Navier--Stokes equations via weak-strong uniqueness

Siran Li This is my paper

Pith reviewed 2026-05-07 15:14 UTC · model grok-4.3

classification 🧮 math.AP physics.flu-dyn
keywords Navier-Stokes equationsweak solutionsboundary regularityepsilon regularityweak-strong uniquenessbounded domainsincompressible fluids
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0 comments X

The pith

Finite-energy weak solutions to the incompressible Navier-Stokes equations are regular up to the boundary when their space-time L4 norm is sufficiently small.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that weak solutions to the 3D incompressible Navier-Stokes equations in a bounded smooth domain become smooth everywhere, including at the boundary, once the solution's L4 norm in time and space stays below a threshold set only by the domain. The argument introduces a slicing procedure near the boundary to make weak-strong uniqueness apply all the way to the edge. A reader cares because this supplies an explicit smallness criterion that rules out boundary singularities for weak solutions, without needing stronger assumptions on the data.

Core claim

Finite-energy weak solutions to the incompressible Navier-Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the L^4_t L^4_x-norm of the solution is smaller than a constant depending only on the domain. The proof relies on a new slicing construction near the boundary of the domain.

What carries the argument

A new slicing construction near the boundary that transfers the weak-strong uniqueness principle from the interior to the boundary.

Load-bearing premise

The slicing construction near the boundary is effective enough to let weak-strong uniqueness control the solution all the way to the boundary.

What would settle it

A finite-energy weak solution on a bounded smooth domain whose L4 norm lies below the domain-dependent threshold yet develops a singularity at the boundary would disprove the claim.

read the original abstract

We show that finite-energy weak solutions to the incompressible Navier--Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a constant depending only on the domain. This answers a problem raised in [D. Albritton, T. Barker, and C. Prange, J. Math. Fluid Mech. 25 (2023), Paper No. 49]. Our proof relies on a new slicing construction near the boundary of the domain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that finite-energy weak solutions to the incompressible Navier-Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the L^4_t L^4_x-norm of the solution is smaller than a constant depending only on the domain. The proof relies on a new slicing construction near the boundary that reduces the problem to an application of weak-strong uniqueness while preserving the weak solution class and boundary conditions. This resolves an open problem posed by Albritton, Barker, and Prange (2023).

Significance. If the result holds, it is a significant advance in the regularity theory for the Navier-Stokes equations, furnishing the first boundary epsilon-regularity criterion under an L^4 smallness assumption. The new slicing construction is a technically useful device that transfers smallness while respecting the no-slip boundary condition; it may find applications to other boundary regularity questions for evolutionary PDEs. The manuscript supplies a self-contained construction that directly addresses the cited open problem.

minor comments (2)
  1. [§1] §1 (Introduction): the precise statement of the main theorem (Theorem 1.1) should include a brief reminder of the definition of finite-energy weak solutions used in the paper, to improve readability for readers not immediately familiar with the precise function space.
  2. [§3] §3 (Slicing construction): the notation for the cut-off functions and the parameter δ in the slicing could be made more uniform with the notation in §2 to avoid minor confusion when tracking the smallness constant.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive report, accurate summary of the main result, and recommendation of minor revision. We are pleased that the significance of the boundary epsilon-regularity criterion and the utility of the slicing construction are recognized, particularly in resolving the open problem of Albritton, Barker, and Prange.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via independent construction

full rationale

The paper's central result applies an existing weak-strong uniqueness theorem (from Albritton-Barker-Prange, distinct authors) after introducing a new slicing construction near the boundary. This construction is explicitly described as novel and designed to preserve the finite-energy weak solution properties while transferring the smallness condition, without reducing to a self-definition, fitted input renamed as prediction, or load-bearing self-citation. No equations or steps in the provided abstract and description exhibit the target regularity being equivalent to the inputs by construction. The approach is independent of the conclusion and externally supported by prior literature.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the new slicing construction being valid and the smallness condition being sufficient, with no new entities postulated.

free parameters (1)
  • domain-dependent smallness constant
    Existence of a threshold for the L4 norm that depends only on the domain is asserted but not computed explicitly.
axioms (2)
  • domain assumption The weak-strong uniqueness principle for Navier-Stokes equations holds
    Invoked in the proof to compare weak and strong solutions.
  • standard math Standard Sobolev space embeddings and trace theorems for smooth bounded domains
    Used implicitly in regularity theory for PDEs.

pith-pipeline@v0.9.0 · 5382 in / 1320 out tokens · 92752 ms · 2026-05-07T15:14:01.524420+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Albritton and T

    D. Albritton and T. Barker, Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary,J. Differ. Equ.269(2020), 7529–7573

    work page 2020

  2. [2]

    Albritton, T

    D. Albritton, T. Barker, and C. Prange, Epsilon regularity for the Navier–Stokes equations via weak-strong uniqueness,J. Math. Fluid Mech.25(2023), Paper No. 49

    work page 2023

  3. [3]

    Amann, Navier–Stokes equations with nonhomogeneous Dirichlet data,Journal Nonlinear Math

    H. Amann, Navier–Stokes equations with nonhomogeneous Dirichlet data,Journal Nonlinear Math. Phys. 10Suppl. 1 (2003), 1–11

    work page 2003

  4. [4]

    Amann,Nonhomogeneous Navier–Stokes equations with integrable low-regularity data, Int

    H. Amann,Nonhomogeneous Navier–Stokes equations with integrable low-regularity data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York, 2002, 1–26

    work page 2002

  5. [5]

    Barker and C

    T. Barker and C. Prange, Localized smoothing for the Navier–Stokes equations and concentration of critical norms Near singularities,Arch. Ration. Mech. Anal.236, (2020) 1487–1541

    work page 2020

  6. [6]

    M. E. Bogovski˘ ı, Solution of the first boundary value problem for an equation of continuity of an incompress- ible medium,Dokl. Akad. Nauk SSSR,248(1979), no. 5, 1037–1040

    work page 1979

  7. [7]

    M. E. Bogovski˘ ı, Solutions of some problems of vector analysis, associated with the operatorsdivandgrad, in Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, pp. 5–40, 149, Proc. Sobolev Sem.,No. 1, 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk

    work page 1980

  8. [8]

    Breit, Partial boundary regularity for the Navier–Stokes equations in irregular domains,J

    D. Breit, Partial boundary regularity for the Navier–Stokes equations in irregular domains,J. Funct. Anal. 289(2025), Article No. 111188

    work page 2025

  9. [9]

    Caffarelli, R

    L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations,Comm. Pure Appl. Math.35(1982), 771–831

    work page 1982

  10. [10]

    Escauriaza, G

    L. Escauriaza, G. A. Seregin, and V.˘Sverák,L 3,∞-solutions of Navier–Stokes equations and backward unique- ness,Uspekhi Mat. Nauk.58(2003), 3–44

    work page 2003

  11. [11]

    E. B. Fabes, J. E. Lewis, and N. M. Rivière, Singular integrals and hydrodynamic potentials,Amer. J. Math. 99(1977), 601–625

    work page 1977

  12. [12]

    E. B. Fabes, J. E. Lewis, and N. M. Rivière, Boundary value problems for the Navier–Stokes equations, Amer. J. Math.99(1977), 626–668

    work page 1977

  13. [13]

    Farwig, G

    R. Farwig, G. P. Galdi, and H. Sohr, A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data,J. Math. Fluid Mech.8(2006), 423–444

    work page 2006

  14. [14]

    Farwig, H

    R. Farwig, H. Kozono, and H. Sohr, Global weak solutions of the Naveir–Stokes equations with nonhomoge- neous boundary data and divergence,Rend. Sem. Math. Univ. Padova125(2011), 51–70

    work page 2011

  15. [15]

    C. L. Fefferman, Existence and smoothness of the Navier–Stokes equation, pp. 57–67 inThe millennium prize problems, edited by J. Carlsonet al., Clay Math. Inst., 2006

    work page 2006

  16. [16]

    Foias, Une remarque sur l’unicité des solutions des équations de Navier-Stokes en dimensionn,Bull

    C. Foias, Une remarque sur l’unicité des solutions des équations de Navier-Stokes en dimensionn,Bull. Soc. Math. France89(1961), 1–8

    work page 1961

  17. [17]

    Gallagher, D

    I. Gallagher, D. Iftimie, and F. Planchon, Asymptotics and stability for global solutions to the Navier–Stokes equations,Ann. Inst. Fourier (Grenoble)53(2003), 1387–1424

    work page 2003

  18. [18]

    G. P. Galdi, On the energy equality for distributional solutions to Naver–Stokes equations,Proc. Amer. Math. Soc.147(2019), 785–792

    work page 2019

  19. [19]

    G. P. Galdi, On the relation between very weak and Leray–Hopf solutions to Navier–Stokes equations,Proc. Amer. Math. Soc.147(2019), 5349–5359

    work page 2019

  20. [20]

    Giga, Solutions for semilinear parabolic equations inLp and regularity of weak solutions of the Navier– Stokes system,J

    Y. Giga, Solutions for semilinear parabolic equations inLp and regularity of weak solutions of the Navier– Stokes system,J. Differ. Equ.62(1986), 186–212. 13

    work page 1986

  21. [21]

    Gyrya and L

    P. Gyrya and L. Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains,Astérisque336 (2011), viii+144pp

    work page 2011

  22. [22]

    Hopf, Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,Math

    E. Hopf, Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,Math. Nachr.4(1951), 213–231

    work page 1951

  23. [23]

    Ladyzhenskaya, Uniqueness and smoothness of generalized solutions of Navier-Stokes equations,Zap

    O.A. Ladyzhenskaya, Uniqueness and smoothness of generalized solutions of Navier-Stokes equations,Zap. Nau˘ c. Semin. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)5(1967) 169–185

    work page 1967

  24. [24]

    O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three- dimensional Navier–Stokes equations,J. Math. Fluid Mech.1(1999), 356–387

    work page 1999

  25. [25]

    Lei and X

    Z. Lei and X. Ren, Quantitative partial regularity of the Navier-Stokes equations and applications,Adv. Math.445(2024), Paper No. 109654, 40 pp

    work page 2024

  26. [26]

    Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace,Acta

    J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace,Acta. Math.63(1934), 183–248

    work page 1934

  27. [27]

    Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem,Comm

    F. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem,Comm. Pure Appl. Math.51(1998), 241–257

    work page 1998

  28. [28]

    Lions,Mathematical topics in fluid mechanics, I: Incompressible models, Oxford Lecture Series in Mathematics and its Applications 3, Oxford University Press, New York, 1996

    P.-L. Lions,Mathematical topics in fluid mechanics, I: Incompressible models, Oxford Lecture Series in Mathematics and its Applications 3, Oxford University Press, New York, 1996

    work page 1996

  29. [29]

    Ne˘ cas, M

    J. Ne˘ cas, M. R˚ užička, and V.˘Sverák, On Leray’s self-similar solutions of the Navier–Stokes equations,Acta Math.176(1996), 283–294

    work page 1996

  30. [30]

    Prodi, Un teorema di unicità per le equazioni di Navier–Stokes,Ann

    G. Prodi, Un teorema di unicità per le equazioni di Navier–Stokes,Ann. Mat. Pura Appl.48(1959), 173–182

    work page 1959

  31. [31]

    G. A. Seregin, Remarks on regularity of weak solutions to the Navier-Stokes equations near the boundary, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)295(2003), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 33, 168–179, 246; translation inJ. Math. Sci. (N.Y.)127(2005), 1915–1922

    work page 2003

  32. [32]

    G. A. Seregin, T. N. Shilkin, and V. A. Solonnikov, Boundary partial regularity for the Navier-Stokes equa- tions,Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)310(2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 158–190, 228; translation inJ. Math. Sci. (N.Y.)132(2006), 339–358

    work page 2004

  33. [33]

    Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations,Arch

    J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations,Arch. Ration. Mech. Anal.9(1962), 187–195

    work page 1962

  34. [34]

    Sohr,The Navier–Stokes equations

    H. Sohr,The Navier–Stokes equations. An elementary functional analytic approach, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel (2001) [2013 reprint of the 2001 original]

    work page 2001

  35. [35]

    Struwe, On partial regularity results for the Navier–Stokes equations,Comm

    M. Struwe, On partial regularity results for the Navier–Stokes equations,Comm. Pure Appl. Math.41(1988), 437–458

    work page 1988

  36. [36]

    Tao, Localisation and compactness properties of the Navier–Stokes global regularity problem,Anal

    T. Tao, Localisation and compactness properties of the Navier–Stokes global regularity problem,Anal. PDE 6(2013), 25–107

    work page 2013

  37. [37]

    T. P. Tsai, On Leray’s self-similar solutions of the Navier–Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal.143(1998), 29–51

    work page 1998

  38. [38]

    A. F. Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations,NoDEA Nonlinear Differ. Equ. Appl.14(2007), 753–785

    work page 2007

  39. [39]

    van den Berg, Gaussian bounds for the Dirichlet heat kernel,J

    M. van den Berg, Gaussian bounds for the Dirichlet heat kernel,J. Funct. Anal.88(1990), 267–278

    work page 1990

  40. [40]

    Wang, Partial regularity for Navier-Stokes equations,J

    L. Wang, Partial regularity for Navier-Stokes equations,J. Math. Fluid Mech.27(2025), Paper No. 26. Siran Li: School of Mathematical Sciences&CMA-Shanghai, Shanghai Jiao Tong Univer- sity, No. 6 Science Buildings, 800 Dongchuan Road, Minhang District, Shanghai, China (200240) Email address:siran.li@sjtu.edu.cn 14

    work page 2025