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arxiv: 2606.11720 · v1 · pith:B3ADGNMWnew · submitted 2026-06-10 · 🧮 math.AP

Strict 2.5D Shadows for One-Component Navier-Stokes Regularity

Runlong Yu This is my paper

Pith reviewed 2026-06-27 09:18 UTC · model grok-4.3

classification 🧮 math.AP MSC 35Q30
keywords Navier-Stokes equationsone-component regularity2.5D shadowspressure quotientconditional reductionweak solutionsvertical momentumcovariance stress
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The pith

A conditional reduction derives a lower bound on the regularity radius for one-component Navier-Stokes solutions from the smallness of the vertical component.

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

The paper formulates a conditional finite-scale reduction theorem for the local one-component regularity problem of suitable weak solutions to the three-dimensional Navier-Stokes equations. Starting from a scale-invariant bound Phi(1) less than or equal to M and smallness of the critical vertical component C_3(1) equal to delta, the argument compares the solution with a strict two-and-a-half-dimensional shadow class in the harmonic-pressure quotient. The Reynolds commutator from coarse graining is absorbed as a positive covariance stress, while genuinely vertical residuals carry a positive power of delta. Under the listed structural inputs the theorem derives a lower bound r_reg(0,0) greater than or equal to c times the absolute value of log C_3(1) to the power minus sigma over 3. This reduction links the three-dimensional vertical momentum equation to the elimination of flat trace obstructions.

Core claim

The central claim is a theorem-driven reduction: under the explicitly listed structural inputs of prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate, one obtains r_reg(0,0) greater than or equal to c sub M,theta times the absolute value of log C_3(1) to the power minus sigma over 3. The comparison is performed in the harmonic-pressure quotient, the Reynolds commutator is treated as positive covariance stress absorbed by an unresolved-variance buffer, and the the

What carries the argument

Strict two-and-a-half-dimensional shadow class compared in the harmonic-pressure quotient, which absorbs the Reynolds commutator as positive covariance stress while passing vertical residuals through a positive power of the smallness parameter delta.

If this is right

  • Strict-shadow selection failure reduces to a finite-mode flat trace obstruction.
  • The obstruction is eliminated conditionally by vertical duality from the full three-dimensional vertical momentum equation.
  • Genuinely vertical residuals carry a positive power of delta and may pass through finite-stage exponential constants.
  • The contribution is a reduction theorem rather than an unconditional resolution of the logarithmic one-component regularity problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the structural inputs hold in practice, the reduction would convert the regularity question into a question about the absence of flat traces in the shadow class.
  • The same shadow comparison technique might apply to related partial regularity questions for other incompressible fluid systems.
  • Numerical checks of the admissibility conditions on known approximate solutions could indicate whether the reduction applies to concrete flows.

Load-bearing premise

The solution satisfies the listed structural inputs including prepared pressure-covariance closure and vertical-duality active-residual estimate.

What would settle it

A suitable weak solution with Phi(1) bounded by M, C_3(1) small, all structural inputs satisfied, yet with regularity radius strictly smaller than the derived lower bound.

read the original abstract

We formulate and prove a conditional finite-scale reduction theorem for the local one-component regularity problem for suitable weak solutions of the three-dimensional Navier--Stokes equations. Starting from a scale-invariant bound Phi(1) <= M and smallness of the critical vertical component C_3(1) = delta, the argument compares the solution with a strict two-and-a-half-dimensional shadow class. The comparison is made in the harmonic-pressure quotient, which is the natural local topology for pressure compactness. The Reynolds commutator produced by coarse graining is treated as a positive covariance stress and is absorbed by an unresolved-variance buffer; consequently this stress contributes additively, while the genuinely vertical residuals carry a positive power of delta and may pass through finite-stage exponential constants. The theorem is deliberately stated as a reduction theorem. Under the explicitly listed structural inputs--prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate--we derive r_reg(0,0) >= c_{M,theta} |log C_3(1)|^{-sigma/3}. The paper does not constitute an unconditional resolution of the logarithmic one-component regularity problem. Its contribution is a theorem-driven reduction: strict-shadow selection failure is reduced to a finite-mode flat trace obstruction, and that obstruction is eliminated, conditionally, by vertical duality forced by the full three-dimensional vertical momentum equation.

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 formulates and proves a conditional finite-scale reduction theorem for the local one-component regularity problem for suitable weak solutions of the three-dimensional Navier-Stokes equations. Starting from a scale-invariant bound Φ(1) ≤ M and smallness of the critical vertical component C_3(1) = δ, the solution is compared with a strict two-and-a-half-dimensional shadow class in the harmonic-pressure quotient. The Reynolds commutator is treated as positive covariance stress absorbed by an unresolved-variance buffer. Under the explicitly listed structural inputs (prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate), the theorem derives the bound r_reg(0,0) ≥ c_{M,θ} |log C_3(1)|^{-σ/3}. The paper explicitly disclaims constituting an unconditional resolution of the logarithmic one-component regularity problem.

Significance. If the listed structural inputs are verified, this provides a useful reduction of the one-component regularity question to the elimination of a finite-mode flat trace obstruction via vertical duality. The manuscript's deliberate framing as a reduction theorem, with clear statement that it does not resolve the problem unconditionally, is a strength that clarifies the scope of the contribution. The approach of absorbing the Reynolds commutator as positive stress while vertical residuals carry powers of δ is technically sound within the conditional framework.

minor comments (2)
  1. [Abstract] The lengthy sentence listing the structural inputs would benefit from being reformatted as a bulleted list for improved clarity.
  2. Several novel terms such as 'strict two-and-a-half-dimensional shadow class', 'harmonic-pressure quotient', and 'unresolved-variance buffer' are used in the abstract without immediate definition; providing short explanations would help readers unfamiliar with the specific framework.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The referee's summary accurately captures the conditional nature of the finite-scale reduction theorem and the explicit structural assumptions under which the logarithmic lower bound on the regularity radius is derived. We appreciate the recognition that the deliberate framing as a reduction theorem (rather than an unconditional resolution) is a strength, and that the technical treatment of the Reynolds commutator and vertical residuals is sound within the stated framework. No major comments were raised, so the point-by-point responses section is empty. We will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

Conditional reduction with no detectable circularity

full rationale

The paper explicitly frames its central result as a conditional reduction theorem: under a listed set of structural inputs (prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate), the bound r_reg(0,0) >= c_{M,theta} |log C_3(1)|^{-sigma/3} follows from comparison with the strict 2.5D shadow class. No step is shown to define any input in terms of the output regularity radius, to rename a fitted quantity as a prediction, or to rely on a self-citation chain that itself reduces to the target claim. The paper disclaims any unconditional resolution, confirming the derivation does not purport to be self-contained beyond the stated assumptions. No load-bearing self-definitional, fitted-input, or uniqueness-imported steps appear in the provided text.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 3 invented entities

The central claim rests on seven explicitly named structural assumptions treated as domain inputs plus several new technical objects introduced for the comparison argument; no numerical fitting to data is described.

free parameters (4)
  • M
    Upper bound on the scale-invariant quantity Phi(1)
  • delta
    Smallness parameter for the vertical component C_3(1)
  • sigma
    Exponent appearing in the derived regularity radius bound
  • theta
    Parameter entering the constant c_{M,theta}
axioms (7)
  • domain assumption prepared pressure-covariance closure
    Listed structural input required for the reduction
  • domain assumption weak horizontal-defect admissibility
    Listed structural input required for the reduction
  • domain assumption sharp admissible-time trace tightness
    Listed structural input required for the reduction
  • domain assumption singular-stratum tangent-cone inputs
    Listed structural input required for the reduction
  • domain assumption strict limiting smoothing and decay
    Listed structural input required for the reduction
  • domain assumption finite-window trace-cost/Newton solvability
    Listed structural input required for the reduction
  • domain assumption vertical-duality active-residual estimate
    Listed structural input required for the reduction
invented entities (3)
  • strict two-and-a-half-dimensional shadow class no independent evidence
    purpose: Comparison class for the solution in the regularity argument
    Introduced to enable the finite-scale reduction
  • harmonic-pressure quotient no independent evidence
    purpose: Topology in which the comparison with the shadow class is performed
    Described as the natural local topology for pressure compactness
  • unresolved-variance buffer no independent evidence
    purpose: Device to absorb the Reynolds commutator as positive covariance stress
    Used to treat the stress additively in the argument

pith-pipeline@v0.9.1-grok · 5803 in / 1894 out tokens · 30693 ms · 2026-06-27T09:18:19.149104+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 37 canonical work pages · cited by 3 Pith papers · 1 internal anchor

  1. [1]

    Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace,Acta Mathematica63(1934), 193–248

    J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace,Acta Mathematica63 (1934), 193–248. DOI:https://doi.org/10.1007/BF02547354

    work page doi:10.1007/bf02547354 1934

  2. [2]

    Hopf, ¨Uber die Anfangswertaufgabe f¨ ur die hydrodynamischen Grundgleichungen,Mathema- tische Nachrichten4(1951), 213–231

    E. Hopf, ¨Uber die Anfangswertaufgabe f¨ ur die hydrodynamischen Grundgleichungen,Mathema- tische Nachrichten4(1951), 213–231. DOI:https://doi.org/10.1002/mana.3210040121

    work page doi:10.1002/mana.3210040121 1951

  3. [3]

    Prodi, Un teorema di unicità per le equazioni di Navier–Stokes,Annali di Matematica Pura ed Applicata(4)48(1959), 173–182

    G. Prodi, Un teorema di unicit` a per le equazioni di Navier–Stokes,Annali di Matematica Pura ed Applicata48(1959), 173–182. DOI:https://doi.org/10.1007/BF02410664

    work page doi:10.1007/bf02410664 1959

  4. [4]

    Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations,Archive for Rational Mechanics and Analysis9(1962), 187–195

    J. Serrin, On the interior regularity of weak solutions of the Navier–Stokes equations,Archive for Rational Mechanics and Analysis9(1962), 187–195. DOI: https://doi.org/10.1007/ BF00253344

    1962

  5. [5]

    Scheffer, Partial regularity of solutions to the Navier–Stokes equations,Pacific Journal of Mathematics66(1976), no

    V. Scheffer, Partial regularity of solutions to the Navier–Stokes equations,Pacific Journal of Mathematics66(1976), no. 2, 535–552. DOI:https://doi.org/10.2140/pjm.1976.66.535

    work page doi:10.2140/pjm.1976.66.535 1976

  6. [6]

    Scheffer, Hausdorff measure and the Navier–Stokes equations,Communications in Mathe- matical Physics55(1977), 97–112

    V. Scheffer, Hausdorff measure and the Navier–Stokes equations,Communications in Mathe- matical Physics55(1977), 97–112. DOI:https://doi.org/10.1007/BF01626512

    work page doi:10.1007/bf01626512 1977

  7. [7]

    Caffarelli, R

    L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations,Communications on Pure and Applied Mathematics35(1982), no. 6, 771–831. DOI:https://doi.org/10.1002/cpa.3160350604

    work page doi:10.1002/cpa.3160350604 1982

  8. [8]

    Sohr and W

    H. Sohr and W. von Wahl, On the regularity of the pressure of weak solutions of Navier–Stokes equations,Archiv der Mathematik46(1986), no. 5, 428–439. DOI: https://doi.org/10. 1007/BF01210782

    1986

  9. [9]

    Simon, Compact sets in the spaceLp(0,T ;B),Annali di Matematica Pura ed Applicata146 (1986), 65–96

    J. Simon, Compact sets in the space Lp(0, T; B),Annali di Matematica Pura ed Applicata146 (1986), 65–96. DOI:https://doi.org/10.1007/BF01762360

    work page doi:10.1007/bf01762360 1986

  10. [10]

    Struwe, On partial regularity results for the Navier–Stokes equations,Communications on Pure and Applied Mathematics41(1988), no

    M. Struwe, On partial regularity results for the Navier–Stokes equations,Communications on Pure and Applied Mathematics41(1988), no. 4, 437–458. DOI: https://doi.org/10.1002/ cpa.3160410404

    1988

  11. [11]

    Constantin, W

    P. Constantin, W. E, and E. S. Titi, Onsager’s conjecture on the energy conservation for solutions of Euler’s equation,Communications in Mathematical Physics165(1994), 207–209. DOI:https://doi.org/10.1007/BF02099744

    work page doi:10.1007/bf02099744 1994

  12. [12]

    Kozono and H

    H. Kozono and H. Sohr, Regularity criterion of weak solutions to the Navier–Stokes equations, Advances in Differential Equations2(1997), no. 4, 535–554. DOI: https://doi.org/10. 57262/ade/1366741147

    arXiv 1997

  13. [13]

    Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem,Communications on Pure and Applied Mathematics51(1998), no

    F.-H. Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem,Communications on Pure and Applied Mathematics51(1998), no. 3, 241–257. DOI: https://doi.org/10.1002/(SICI) 1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. 169

    work page doi:10.1002/(sici 1998

  14. [14]

    O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier–Stokes equations,Journal of Mathematical Fluid Mechanics1 (1999), 356–387. DOI:https://doi.org/10.1007/s000210050015

    work page doi:10.1007/s000210050015 1999

  15. [15]

    H. J. Choe and J. L. Lewis, On the singular set in the Navier–Stokes equations,Journal of Functional Analysis175(2000), no. 2, 348–369. DOI: https://doi.org/10.1006/jfan.2000. 3582

    work page doi:10.1006/jfan.2000 2000

  16. [16]

    G. A. Seregin and V. ˇSver´ ak, Navier–Stokes equations with lower bounds on the pressure, Archive for Rational Mechanics and Analysis163(2002), no. 1, 65–86. DOI: https://doi. org/10.1007/s002050200199

    work page doi:10.1007/s002050200199 2002

  17. [17]

    Escauriaza, G

    L. Escauriaza, G. Seregin, and V. ˇSver´ ak,L3,∞-solutions of Navier–Stokes equations and backward uniqueness,Russian Mathematical Surveys58(2003), no. 2, 211–250. DOI: https: //doi.org/10.1070/RM2003v058n02ABEH000609

    work page doi:10.1070/rm2003v058n02abeh000609 2003

  18. [18]

    Penel and M

    P. Penel and M. Pokorn´ y, Some new regularity criteria for the Navier–Stokes equations containing gradient of the velocity,Applications of Mathematics49(2004), no. 5, 483–493. DOI:https://doi.org/10.1023/B:APOM.0000048124.64244.7e

    work page doi:10.1023/b:apom.0000048124.64244.7e 2004

  19. [19]

    Duzaar and G

    F. Duzaar and G. Mingione, The p-harmonic approximation and the regularity of p-harmonic maps,Calculus of Variations and Partial Differential Equations20(2004), 235–256. DOI: https://doi.org/10.1007/s00526-003-0233-x

    work page doi:10.1007/s00526-003-0233-x 2004

  20. [20]

    Kukavica and M

    I. Kukavica and M. Ziane, One component regularity for the Navier–Stokes equations,Nonlin- earity19(2006), no. 2, 453–469. DOI:https://doi.org/10.1088/0951-7715/19/2/012

    work page doi:10.1088/0951-7715/19/2/012 2006

  21. [21]

    & K¨ ohn, A

    I. Kukavica and M. Ziane, Navier–Stokes equations with regularity in one direction,Journal of Mathematical Physics48(2007), no. 6, 065203, 10 pp. DOI: https://doi.org/10.1063/1. 2395919

    work page doi:10.1063/1 2007

  22. [22]

    G. A. Seregin, Estimates of suitable weak solutions to the Navier–Stokes equations in critical Morrey spaces,Journal of Mathematical Sciences143(2007), no. 2, 2961–2968. DOI: https: //doi.org/10.1007/s10958-007-0178-2

    work page doi:10.1007/s10958-007-0178-2 2007

  23. [23]

    G. A. Seregin, Local regularity for suitable weak solutions of the Navier–Stokes equations, Russian Mathematical Surveys62(2007), no. 3, 595–614. DOI: https://doi.org/10.1070/ RM2007v062n03ABEH004415

    2007

  24. [24]

    Gustafson, K

    S. Gustafson, K. Kang, and T.-P. Tsai, Interior regularity criteria for suitable weak solutions of the Navier–Stokes equations,Communications in Mathematical Physics273(2007), 161–176. DOI:https://doi.org/10.1007/s00220-007-0214-6

    work page doi:10.1007/s00220-007-0214-6 2007

  25. [25]

    Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations,Nonlinear Differential Equations and Applications14(2007), 753–785

    A. Vasseur, A new proof of partial regularity of solutions to Navier–Stokes equations,Nonlinear Differential Equations and Applications14(2007), 753–785. DOI: https://doi.org/10.1007/ s00030-007-6001-4

    2007

  26. [26]

    Zhou and M

    Y. Zhou and M. Pokorn´ y, On a regularity criterion for the Navier–Stokes equations involving gradient of one velocity component,Journal of Mathematical Physics50(2009), no. 12, 123514, 11 pp. DOI:https://doi.org/10.1063/1.3268589. 170

    work page doi:10.1063/1.3268589 2009

  27. [27]

    Cao and E

    C. Cao and E. S. Titi, Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor,Archive for Rational Mechanics and Analysis202 (2011), 919–932. DOI:https://doi.org/10.1007/s00205-011-0439-6

    work page doi:10.1007/s00205-011-0439-6 2011

  28. [28]

    Jia and V

    H. Jia and V. ˇSver´ ak, Local-in-space estimates near initial time for weak solutions of the Navier–Stokes equations and forward self-similar solutions,Inventiones Mathematicae196 (2014), 233–265. DOI:https://doi.org/10.1007/s00222-013-0468-x

    work page doi:10.1007/s00222-013-0468-x 2014

  29. [29]

    G. A. Seregin,Lecture Notes on Regularity Theory for the Navier–Stokes Equations, World Scientific, Hackensack, NJ, 2015. DOI:https://doi.org/10.1142/9314

    work page doi:10.1142/9314 2015

  30. [30]

    Chemin and P

    J.-Y. Chemin and P. Zhang, On the critical one component regularity for 3-D Navier–Stokes system,Annales Scientifiques de l’ ´Ecole Normale Sup´ erieure49(2016), no. 1, 131–167. DOI: https://doi.org/10.24033/asens.2278

    work page doi:10.24033/asens.2278 2016

  31. [31]

    Chemin, P

    J.-Y. Chemin, P. Zhang, and Z. Zhang, On the critical one component regularity for 3-D Navier–Stokes system: general case,Archive for Rational Mechanics and Analysis224(2017), no. 3, 871–905. DOI:https://doi.org/10.1007/s00205-017-1089-0

    work page doi:10.1007/s00205-017-1089-0 2017

  32. [32]

    Guevara and N

    C. Guevara and N. C. Phuc, Local energy bounds and epsilon-regularity criteria for the 3D Navier–Stokes system,Calculus of Variations and Partial Differential Equations56(2017), no. 3, article no. 68, 16 pp. DOI:https://doi.org/10.1007/s00526-017-1151-7

    work page doi:10.1007/s00526-017-1151-7 2017

  33. [33]

    Kukavica, W

    I. Kukavica, W. Rusin, and M. Ziane, An anisotropic partial regularity criterion for the Navier–Stokes equations,Journal of Mathematical Fluid Mechanics19(2017), 123–133. DOI: https://doi.org/10.1007/s00021-016-0278-1

    work page doi:10.1007/s00021-016-0278-1 2017

  34. [34]

    Wolf, On the local pressure of the Navier–Stokes equations and related systems,Advances in Differential Equations22(2017), no

    J. Wolf, On the local pressure of the Navier–Stokes equations and related systems,Advances in Differential Equations22(2017), no. 5/6, 305–338. DOI: https://doi.org/10.57262/ade/ 1489802453

    work page doi:10.57262/ade/ 2017

  35. [35]

    B. Han, Z. Lei, D. Li, and N. Zhao, Sharp one component regularity for Navier–Stokes,Archive for Rational Mechanics and Analysis231(2019), 939–970. DOI: https://doi.org/10.1007/ s00205-018-1292-7

    2019

  36. [36]

    Barker and C

    T. Barker and C. Prange, Quantitative regularity for the Navier–Stokes equations via spatial concentration,Communications in Mathematical Physics385(2021), 717–792. DOI: https: //doi.org/10.1007/s00220-021-04122-x

    work page doi:10.1007/s00220-021-04122-x 2021

  37. [37]

    Kang and D

    K. Kang and D. D. Nguyen, Local regularity criteria in terms of one velocity component for the Navier–Stokes equations,Journal of Mathematical Fluid Mechanics25(2023), article no. 10, 15 pp. DOI:https://doi.org/10.1007/s00021-022-00754-8

    work page doi:10.1007/s00021-022-00754-8 2023

  38. [38]

    Albritton, T

    D. Albritton, T. Barker, and C. Prange, Epsilon regularity for the Navier–Stokes equations via weak–strong uniqueness,Journal of Mathematical Fluid Mechanics25(2023), article no. 49, 12 pp. DOI:https://doi.org/10.1007/s00021-023-00780-0

    work page doi:10.1007/s00021-023-00780-0 2023

  39. [39]

    Finite-Scale One-Component Regularity via Harmonic Pressure for the 3D Navier-Stokes Equations

    R. Yu, Finite-scale one-component regularity via harmonic pressure for the 3D Navier–Stokes equations, arXiv:2606.08352 [math.AP], 2026. DOI: https://doi.org/10.48550/arXiv.2606. 08352

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606 2026

  40. [40]

    Bierstone and P

    E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets,Publications Math´ ematiques de l’IH ´ES67(1988), 5–42. DOI:https://doi.org/10.1007/BF02699126. 171

    work page doi:10.1007/bf02699126 1988

  41. [41]

    Amann,Linear and Quasilinear Parabolic Problems

    H. Amann,Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory, Birkh¨ auser, Basel, 1995. DOI:https://doi.org/10.1007/978-3-0348-9221-6

    work page doi:10.1007/978-3-0348-9221-6 1995

  42. [42]

    Lunardi,Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995

    A. Lunardi,Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh¨ auser, Basel, 1995. DOI:https://doi.org/10.1007/978-3-0348-9234-6

    work page doi:10.1007/978-3-0348-9234-6 1995

  43. [43]

    Kurdyka, On gradients of functions definable in o-minimal structures,Annales de l’Institut Fourier48(1998), no

    K. Kurdyka, On gradients of functions definable in o-minimal structures,Annales de l’Institut Fourier48(1998), no. 3, 769–783. DOI:https://doi.org/10.5802/aif.1638

    work page doi:10.5802/aif.1638 1998

  44. [44]

    S. G. Krantz and H. R. Parks,A Primer of Real Analytic Functions, 2nd ed., Birkh¨ auser, Boston, 2002. DOI:https://doi.org/10.1007/978-0-8176-8134-0

    work page doi:10.1007/978-0-8176-8134-0 2002

  45. [45]

    Kielh¨ ofer,Bifurcation Theory: An Introduction with Applications to Partial Differ- ential Equations, 2nd ed., Springer, New York, 2012

    H. Kielh¨ ofer,Bifurcation Theory: An Introduction with Applications to Partial Differ- ential Equations, 2nd ed., Springer, New York, 2012. DOI: https://doi.org/10.1007/ 978-1-4614-0502-3. 172

    2012