Pith. sign in

REVIEW 2 major objections 8 minor 24 references

The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces

T0 review · 2 major / 8 minor · reviewed 2026-07-31 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Dual-scale memory viscosity makes the Navier-Stokes equations well-posed in a critical Besov space larger than the classical little-Besov closure, while remaining ill-posed below a sharp Lebesgue threshold.

desk verdict Sharp critical index and supercritical inflation look solid; the headline Besov well-posedness has a real low-frequency gap that hits exactly the dual-scale kernels the paper advertises. read the letter →

arxiv 2607.25009 v1 pith:MBHLJTO2 submitted 2026-07-27 math.AP

Bruno de Andrade This is my paper · ORCID
classification math.AP MSC 35Q3035R0976D0542B25
keywords Navier-Stokesequationsdual-scalememoryBesovspacesnorminflationpara-differentialcalculusanomalousdiffusionhereditaryviscositycritical
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Classical Navier-Stokes is known to be ill-posed in the largest scale-critical Besov space because high-frequency convective interactions can instantly dump energy into macroscopic modes. This paper replaces ordinary viscosity by a dual-scale hereditary memory kernel that encodes short-time elastic response and long-time fluid relaxation. The non-local dissipation breaks exact scaling, so the linear resolvent must be treated as a pseudo-differential operator whose spatial gradient exacts a fractional temporal penalty. The resulting critical Lebesgue index is p_c = N(1+α_∞)/(1−α_∞). Below that index the bilinear Picard iterate inflates instantly, proving ill-posedness. At the opposite extreme p o∞ the same memory damps the high-high resonant cascade strongly enough that small divergence-free data with high-frequency adherence produce unique global mild solutions in the critical space Ẋ^{-κ}_{∞,∞}, a regime strictly larger than the separable little-Besov subspace. The result therefore maps the precise topological boundary between well-posedness and collapse for this class of viscoelastic fluids.

What carries the argument

Asymmetric interpolation inside Bony’s paraproduct decomposition: one factor of the high-high residue is controlled by a fractional time weight t^γ while the other is controlled only by the base Besov norm. The resulting temporal singularity remains integrable, so the hereditary smoothing overpowers the convective cascade before it can inflate low-frequency modes.

What would settle it

Construct an explicit dual-scale kernel satisfying the stated high- and low-frequency asymptotics for which either the gradient resolvent fails the claimed L^q-L^p bound, or a sequence of high-frequency-adherent data of arbitrarily small Ẋ^{-κ}_{∞,∞} norm produces a mild solution that leaves every bounded set of that space in arbitrarily short time.

Watch

Extended reading notes

Core claim

For the incompressible Navier-Stokes system driven by a dual-scale admissible memory kernel, the Cauchy problem is globally well-posed in the Hadamard sense for small divergence-free data in the critical Besov space Ẋ^{-κ}_{∞,∞}(ℝ^N) that satisfy the high-frequency adherence condition lim_{j o+∞} 2^{-jκ}‖Δ_j u_0‖_{L^∞}=0, where κ=(1-α_∞)/(1+α_∞). This regime properly contains the little-Besov closure. At the same time, for every Lebesgue exponent 1<p<p_c with p_c=N(1+α_∞)/(1-α_∞) the data-to-solution map fails to be uniformly continuous at the origin by instantaneous norm inflation of the second Picard iterate.

Load-bearing premise

The short-time linear estimate that the gradient of the dual-scale resolvent maps L^q into L^p with a precise fractional time decay whenever 1/q-1/p is less than 1/N; if that decay rate is false, both the critical index and the contraction argument collapse.

Editorial extensions

If this is right

  • The classical Bourgain-Pavlović collapse is not universal for every non-local fluid model; dual-scale memory can suppress the low-frequency resonant cascade.
  • Well-posedness holds in a strictly larger set than the little-Besov space, so data whose high-frequency dyadic tail merely tends to zero (rather than belonging to the separable closure) are admissible.
  • The critical Lebesgue threshold p_c recovers the classical Kato space L^N when the short-time anomaly α_∞ vanishes and diverges as the fluid becomes strongly elastic.
  • Ill-posedness is confined to the non-separable high-frequency tail; any further extension of well-posedness must confront lacunary data that violate high-frequency adherence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same asymmetric-paraproduct device may apply verbatim to other hereditary or fractional parabolic systems whose linear symbols lie in S^{-2}_{1,0}.
  • If the open question on the non-adherent complement is settled negatively, dual-scale memory would give a complete topological dichotomy for this class of viscoelastic models.
  • Rheological measurements that fix the short-time exponent α_∞ would immediately translate into a concrete numerical value of the critical integrability threshold p_c for laboratory fluids.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 8 minor

Summary. The paper studies a Navier–Stokes system in which instantaneous viscosity is replaced by convolution against a 'dual-scale' memory kernel whose Laplace transform decays like |λ|^{-α∞} at high frequency and |λ|^{-α₀} at low frequency (hypotheses H1–H3). Three main results are claimed: (i) sharp short-time L^q–L^p estimates for the resolvent and its gradient, identifying a critical Lebesgue exponent p_c=N(1+α∞)/(1−α∞) (Lemma 4.1, Theorem 5.2); (ii) instantaneous norm inflation of the second Picard iterate, hence failure of uniform continuity of the flow map at the origin, for 1<p<p_c, proved by a frequency-modulation argument using compactly supported spectra and Bernstein inequalities in place of Hausdorff–Young (Theorem 5.3); and (iii) global Hadamard well-posedness for small divergence-free data in Ḃ^{−κ}_{∞,∞}, κ=(1−α∞)/(1+α∞), satisfying a one-sided high-frequency adherence condition — a class strictly broader than the little Besov closure — via an asymmetric paraproduct estimate in a temporally weighted path space (Lemma 6.2, Theorem 6.3).

Significance. If correct, the results cleanly delineate the well-posedness/ill-posedness boundary for a physically motivated class of hereditary-viscosity fluids. Strengths worth naming: the critical index p_c is derived parameter-free from the integrability condition ˜γ_{p/2,p}<1; the inflation argument bypasses Hausdorff–Young via compact spectra and Bernstein equivalence and makes a falsifiable prediction (inflation precisely for p<p_c=N/κ); the critical-regime Kato-space contraction (Thm 5.2(ii)) checks out by exact Beta-function bookkeeping; and the asymmetric paraproduct idea for taming the high-high cascade without non-integrable s^{-2γ} weights is a genuine technical contribution. However, the significance of the flagship Besov result is currently conditional: as written, Theorem 6.3 appears provable (by the natural repair) only under α₀≥α∞, which excludes the Cole–Cole and Prabhakar kernels used to motivate the entire 'dual-scale' program.

major comments (2)
  1. [§6, Lemma 6.2 / Theorem 6.3] §6, proof of Lemma 6.2, low-frequency step (p. 22). Two defects. (a) For j<0 the paper estimates only the high-high-into-low channel R_j, uses Ŝ≈1, and obtains 2^{-jκ}‖Δ_jB(t)‖ ≲ 2^{j(1-κ)}t^{1-γ}; since t^{1-γ}→∞ as t→∞, the claimed sup_{t>0}‖B(t)‖_{Ḃ^{-κ}}≤C‖u‖_X‖v‖_X is not established (the lemma statement requires the sup over all t>0). (b) The low-low paraproduct channel Δ_j(T_uv) for j→−∞ is never estimated. Its natural bound — ‖S_{j-1}u‖≲2^{jκ}‖u‖, ‖Δ_jv‖≲2^{jκ}‖v‖ (no t^γ weight exists on low blocks), gradient 2^j, resolvent lifetime ∫₀^∞|E_{1+α₀}(-c2^{2j}τ^{1+α₀})|dτ∼2^{-2j/(1+α₀)} — gives 2^{-jκ}‖Δ_jB‖≲2^{j[2/(1+α∞)-2/(1+α₀)]}. This is borderline for α₀=α∞ (Ex. 1), decays for α₀>α∞ (Ex. 2), but diverges as j→−∞ whenever α₀<α∞ — precisely the Cole–Cole kernel (Ex. 3, α₀=0) and Prabhakar kernel (Ex. 4) that the introduction advances as the motivation for dual-scale memory. Theore
  2. [§4, Lemma 4.1; dependence on [12]] The load-bearing linear estimates are imported from the author's concurrent unrefereed preprint [12] (arXiv:2607.17430): Lemma 4.1(i), the S^{-2}_{1,0} membership of the scaled resolvent symbol [12, Lemma 3.1] used in the proof of Lemma 4.1(ii), the low-frequency dyadic partition [12, Lemma 3.4] used in Lemma 4.2, the Mittag-Leffler profile bound on annuli C_j used in Lemma 6.2 and (6.6), and the uniform L^q bounds for the modulated operators K(s,λ) [12, Theorem 4.2] used in Theorem 5.3. Since p_c, the contraction, and the inflation argument all collapse if the multiplier-class membership or the temporal penalties fail, the manuscript is not currently verifiable on its own. At minimum the imported results should be stated precisely (with hypotheses) in an appendix; ideally the key multiplier estimates should be proved here.
minor comments (8)
  1. [Abstract / Theorem 5.3] Abstract and §5.2: 'confirms intrinsic ill-posedness' overstates Theorem 5.3, which proves failure of uniform continuity of the data-to-solution map at the origin via norm inflation of the second Picard iterate. Please align the wording with the statement.
  2. [§5.2] p. 17: 'Recall the dynamic scaling invariance of the high-frequency system' is misleading — the dual-scale system has no exact scale invariance (this is the paper's premise); only the high-frequency asymptotic profile is self-similar. Please rephrase.
  3. [§4, Lemma 4.1] Lemma 4.1(ii), p. 9: the claim that an S^{-1}_{1,0} symbol yields a convolution kernel with singularity O(|w|^{-(N-1)}) should carry a precise reference (e.g., Grafakos [15] or Hörmander) and an explicit statement that the kernel bounds are uniform in t∈(0,T] for the scaled family.
  4. [§5.2, §3.5] Notation: κ denotes the Besov index throughout but is reintroduced in §5.2 as 'the spatial scaling factor'; σ is used both for the temporal scaling modulation and for the symbol σ(P) of the Leray projector. Suggest renaming.
  5. [§6, Remark 6.4] Remark 6.4 uses the space ḃ^{-κ,+}_{∞,∞}, which is never defined (only ḃ^{ε,+}_{∞,∞} is). Please define or correct.
  6. [§6, Theorem 6.3] Theorem 6.3's statement should list which of (H1)-(H3) are assumed; as written it begins 'Let N≥2' with the hypotheses implicit, while the proof uses (H3) essentially in the K_low step.
  7. [§6, Theorem 6.3] To substantiate 'strictly broader than ˙b^{-κ}_{∞,∞}', an explicit example of u_0 satisfying (6.5) but failing the low-frequency adherence lim_{j→-∞}2^{-jκ}‖Δ_ju_0‖_∞=0 would help the reader.
  8. [§6, Lemma 6.2] p. 22, low-frequency step of Lemma 6.2: the assertion 'the resolvent lacks high-frequency decay within this regime (Ŝ≈1)' needs quantification via (H3) (regime of validity in |ξ|²t^{1+α₀}), especially if the t^{1-γ} bound is to be repaired.

Circularity Check

2 steps flagged · score 3.0 of 10

Load-bearing self-citation to concurrent preprint [12] for linear resolvent bounds; nonlinear analysis is independent, not definitional circularity.

  1. self citation load bearing [§4 Lemma 4.1 (pure resolvent); also Lemma 4.2, §5.2 citing [12, Thm 4.2]]
    "The pure resolvent estimate (i) was rigorously established in [12] by analyzing the Bromwich integral of the multiplier over the high-frequency spectrum; hence, our focus remains strictly on (ii). [...] Rather than relying on ad-hoc physical kernel decompositions, we explicitly leverage the pseudo-differential framework established in [12]. [...] the scaled pure resolvent symbol ˜Ψ(t, η)=bS(t, µ^{-1}η) belongs uniformly to the Hörmander class S^{-2}_{1,0} [12, Lemma 3.1]."

    Membership of the dual-scale symbol in S^{-2}_{1,0} and the pure-resolvent L^q–L^p decays are not re-proved; they are imported from the author’s concurrent preprint [12]. Every subsequent geometric barrier (1/q−1/p<1/N), the critical index p_c=N(1+α_∞)/(1−α_∞), and the temporal weights in the Besov path space X rest on that imported linear theory. Same-author unfinished upstream work is load-bearing for the linear foundation, though the nonlinear estimates themselves are derived here.

  2. self citation load bearing [§5.2 Proof of Theorem 5.3 (norm inflation scaling)]
    "As rigorously established in Section 4 and verified uniformly for the modulated family in [12, Theorem 4.2], the operators K(s, λ) are uniformly bounded multipliers in any L^q(R^N). [...] as λ→∞, the modulated operator K(s, λ) converges strongly to the limiting pure fractional resolvent S_{α_∞}(s)"

    Uniform boundedness and strong convergence of the frequency-modulated family K(s, λ) used to extract the non-trivial bilinear profile F_λ are justified by citation to [12, Theorem 4.2] rather than proved in-place. The inflation lower bound therefore inherits the same concurrent self-citation dependency as the linear theory, while the Bernstein/compact-support argument that converts spectral mass into an L^p lower bound is original to this paper.

full rationale

The paper’s nonlinear core (critical threshold from temporal integrability of ˜γ, frequency-modulated norm inflation via Bernstein, asymmetric Bony paraproduct contraction in the weighted path space X) is developed in-place and does not reduce by construction to its inputs. The only circularity-adjacent pattern is self-citation load-bearing: pure-resolvent L^q–L^p bounds, Hörmander-class membership S^{-2}_{1,0}, low-frequency dyadic partition, and modulated-operator uniformity are taken from the author’s concurrent arXiv:2607.17430 [12]. Those citations supply the linear foundation on which p_c and the Besov contraction rest, but they are upstream analytic claims rather than tautologies or fitted-parameter renamings. Hypotheses (H1)–(H3) are structural assumptions on the kernel, not encodings of the target theorems. No self-definitional loop, no data-fit-called-prediction, and no uniqueness theorem imported to forbid alternatives. Correctness gaps in the low-frequency channel of Lemma 6.2 (raised by the skeptic) are proof defects, not circularity. Score 3 reflects non-minor but non-central self-citation with independent nonlinear content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper is theorem-driven analysis. Load-bearing inputs are standard harmonic-analysis tools, the structural kernel hypotheses (H1)–(H3), and linear multiplier bounds imported from the author’s concurrent work. No empirical free parameters. The dual-scale admissible class and the asymmetric path space X are definitional setups, not new physical entities requiring external detection.

assumptions (5)
  • domain assumption Dual-scale kernel hypotheses (H1)–(H3): sectoriality with θ_∞<πα_∞/(1+α_∞), high-frequency |ĝ(λ)|∼|λ|^{−α_∞}, low-frequency |ĝ(λ)|∼|λ|^{−α_0}.
    Section 2.1; these fix the symbol class and the exponents γ, β, κ, p_c. Without them the Hörmander-class analysis and critical threshold do not hold.
  • domain assumption Short- and long-time L^q–L^p bounds for S(t) and ∇S(t)P, including membership of the scaled resolvent in S^{−2}_{1,0}.
    Lemmas 4.1–4.2; pure-resolvent parts cited from arXiv:2607.17430 [12]. Central for both p_c and the Duhamel contraction.
  • standard math Bernstein inequalities, Littlewood–Paley characterization of homogeneous Besov spaces, and Bony paraproduct decomposition.
    Section 3; used throughout §§5–6 for frequency localization and product estimates.
  • standard math Calderón–Zygmund / Marcinkiewicz boundedness of the Leray–Hopf projector on L^q for 1<q<∞, and dyadic L^∞ boundedness via Schwartz kernels.
    Subsection 3.5 and applications in §§4–6; needed because P is unbounded on L^∞ globally.
  • standard math Singular Gronwall / Beta-function convolution estimates and Banach fixed-point in Kato-type path spaces.
    Theorems 5.2 and 6.3; standard mild-solution machinery (Henry, Kato).
invented entities (2)
  • NSHV equations (Navier–Stokes with dual-scale hereditary viscosity) independent evidence
    purpose: Model incompressible flow with non-local dual-scale memory replacing instantaneous viscosity.
    Defined in §1–2; motivated by rheology citations but studied as a mathematical Cauchy problem.
  • Asymmetric path space X (base Ḃ^{−κ}_{∞,∞} plus high-frequency semi-norm t^γ ‖·‖_{ḃ^{ε,+}_{∞,∞}})
    purpose: Close the fixed-point argument while avoiding non-integrable s^{−2γ} singularities and low-frequency separability restrictions.
    Introduced before Lemma 6.2; a functional-analytic device, not a physical postulate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces." pith.science (2026). https://pith.science/paper/MBHLJTO2

@misc{pith2026260725009,
  author       = {Pith},
  title        = {Pith review of: The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBHLJTO2}},
  note         = {Machine review of arXiv:2607.25009}
}
abstract

This manuscript investigates the Cauchy problem for an incompressible fluid flow governed by a dual-scale hereditary memory, representing a viscoelastic variant of the classical Navier-Stokes equations that captures anomalous momentum transport. The non-local dissipation breaks exact global scale invariance, dictating a pseudo-differential analysis within the H\"{o}rmander class $S^{-2}_{1,0}$ where the spatial gradient induces a fractional temporal penalty. We rigorously establish $L^q-L^p$ decay estimates and identify the critical Lebesgue threshold $p_c = N (\frac{1+\alpha_\infty}{1-\alpha_\infty})$. In the supercritical regime $1 < p < p_c$, we bypass the loss of spatial localization by mapping the frequency-modulated bilinear flow directly into Fourier space; by utilizing Bernstein's inequalities, we prove instantaneous norm inflation at the origin and confirm intrinsic ill-posedness. Conversely, in the topological limit $p \to \infty$, we demonstrate that the dual-scale memory structurally prevents the collapse traditionally observed for classical fluids within the maximal critical Besov space $\dot{B}^{-1}_{\infty, \infty}$. By exploiting an asymmetric interpolation within Bony's para-differential calculus, we prove that the temporal smoothing overpowers the high-high convective resonant cascade. This delicate analytical balance confines ill-posedness to the non-separable high-frequency tail of the Besov topology, thereby establishing global-in-time Hadamard well-posedness for small initial data possessing high-frequency adherence within $\dot{B}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, a well-posedness regime strictly broader than the separable little Besov closure $\dot{b}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, where $\kappa = \frac{1-\alpha_\infty}{1+\alpha_\infty}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 1 linked inside Pith

  1. [12]

    de Andrade,Critical thresholds and instantaneous norm inflation for super-diffusive integro- differential equations, arXiv:2607.17430, 2026

    B. de Andrade,Critical thresholds and instantaneous norm inflation for super-diffusive integro- differential equations, arXiv:2607.17430, 2026

    arXiv 2026

  2. [1]

    J. M. Arrieta and A. N. Carvalho,Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc.352(2000), no. 1, 285–310

    2000

  3. [2]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin, and R. Danchin,Fourier Analysis and Nonlinear Partial Differential Equa- tions, Grundlehren der mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011

    2011

  4. [3]

    Barbu and S

    V. Barbu and S. S. Sritharan,Navier-Stokes equation with hereditary viscosity, Z. Angew. Math. Phys. 54(2003), no. 3, 449–461

    2003

  5. [4]

    Bony,Calcul symbolique et propagation des singularit´ es pour les ´ equations aux d´ eriv´ ees partielles non lin´ eaires, Ann

    J.-M. Bony,Calcul symbolique et propagation des singularit´ es pour les ´ equations aux d´ eriv´ ees partielles non lin´ eaires, Ann. Sci.´Ecole Norm. Sup. (4)14(1981), no. 2, 209–246

    1981

  6. [5]

    Bourgain and N

    J. Bourgain and N. Pavlovi´ c,Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal.255(2008), no. 9, 2233–2247

    2008

  7. [6]

    Cannone,A generalization of a theorem by Kato on Navier-Stokes equations, Rev

    M. Cannone,A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam.11 (1995), no. 3, 573–624

    1995

  8. [7]

    Cheskidov and R

    A. Cheskidov and R. Shvydkoy,Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proc. Amer. Math. Soc.138(2010), no. 3, 1059–1067

    2010

Show all 24 references
  1. [8]

    Christ, J

    M. Christ, J. Colliander, and T. Tao,Asymptotics, frequency modulation, and low regularity ill-posedness for canonical dispersive equations, Amer. J. Math.125(2003), no. 6, 1235–1293

    2003

  2. [9]

    de Andrade and A

    B. de Andrade and A. Viana,Abstract Volterra integrodifferential equations with applications to parabolic models with memory, Math. Ann.369(2017), no. 3-4, 1131–1175

    2017

  3. [10]

    de Andrade, C

    B. de Andrade, C. Silva, and A. Viana,L q-solvability for an equation of viscoelasticity in power type materials, Z. Angew. Math. Phys.72(2021), no. 1, Paper No. 10

    2021

  4. [11]

    de Andrade, C

    B. de Andrade, C. Cuevas, and J. Dantas,Navier-Stokes equation with hereditary viscosity and initial data in Besov-Morrey spaces, Z. Angew. Math. Phys.75(2024), no. 1, Paper No. 11

    2024

  5. [13]

    de Andrade and M

    B. de Andrade and M. G. de Santana,Abstract integrodifferential equations and applications, arXiv:2602.08691, 2026

    2026

  6. [14]

    Giusti, I

    A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Mainardi, M. Mentrelli, and Z. Tomovski,A practical guide to Prabhakar fractional calculus, Fract. Calc. Appl. Anal.,23(2020), 9–54

    2020

  7. [15]

    Grafakos,Classical Fourier Analysis, third ed., Graduate Texts in Math., vol

    L. Grafakos,Classical Fourier Analysis, third ed., Graduate Texts in Math., vol. 249, Springer, New York, 2014

    2014

  8. [16]

    Henry,Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol

    D. Henry,Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, 1981

    1981

  9. [17]

    Jaishankar and G

    A. Jaishankar and G. H. McKinley,Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations, Proc. Roy. Soc. Lond. Ser. A469(2013), no. 2149, 20120284

    2013

  10. [18]

    Kato,StrongL p-solutions of the Navier-Stokes equation inR m, with applications to weak solutions, Math

    T. Kato,StrongL p-solutions of the Navier-Stokes equation inR m, with applications to weak solutions, Math. Z.187(1984), no. 4, 471–480

    1984

  11. [19]

    Koch and D

    H. Koch and D. Tataru,Well-posedness for the Navier-Stokes equations, Adv. Math.157(2001), no. 1, 22–35. 28 B. DE ANDRADE

    2001

  12. [20]

    P. G. Lemari´ e-Rieusset,Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002

    2002

  13. [21]

    Mainardi,Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

    F. Mainardi,Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010

    2010

  14. [22]

    T. R. Prabhakar,A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J.,19(1971), 7–15

    1971

  15. [23]

    Sawano,Theory of Besov spaces, vol

    Y. Sawano,Theory of Besov spaces, vol. 56, Springer, 2018

    2018

  16. [24]

    D. V. Widder,The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N.J., 1941. (B. de Andrade)Departamento de Matem´atica, Universidade Federal de Sergipe, S˜ao Crist´ov˜ao - SE, Brazil. Email address:bruno@mat.ufs.br

    1941

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.