Pith. sign in

REVIEW 4 major objections 5 minor 26 references

Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read This paper proves that in a wedge of half-angle $\alpha$ every no-slip self-similar flow type exists exactly up to a critical flux, computes the critical flux from elliptic integrals, proves uniqueness for pure outflow, pure inflow, and…

desk verdict Rigorous classification of Jeffery-Hamel flows in a wedge is mostly solid, but the headline non-uniqueness theorem is only proved for one special angle and m=1, with the general cases explicitly omitted. read the letter →

arxiv 2412.07283 v2 pith:M4MRKXGS submitted 2024-12-10 math.AP

Jeaheang Bang , Changfeng Gui , Hao Liu , Yun Wang , Chunjing Xie This is my paper · ORCID
classification math.AP MSC 35Q3076D0533E0534B1535B40
keywords self-similarsolutionsNavier-StokesequationsJeffery-Hamelflowsno-slipboundaryconditionellipticintegralsfluxthresholdnon-uniquenessaperturedomain
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

The paper gives a complete existence classification for self-similar radial flows in a two-dimensional wedge with no-slip walls and a prescribed flux. For each flow type $(m_+,m_-)$ it proves a maximum flux $\Phi_{\max}^{(m_+,m_-)}(\alpha)$ exists, with solutions of that type below the threshold and none above. It proves uniqueness of the pure outflow, pure inflow, and $(m,m)$ flows, and it uncovers a new non-uniqueness: fluxes between the $(m,m)$ and $(m,m+1)$ thresholds carry at least two $(m,m+1)$ solutions. The results confirm parts of the 1940 numerical study [23] and correct others, for example the maximum flux for type $(1,1)$ at $\alpha=\pi/2$ is $0$, not $0.5$. As an application, the small-flux solution in an aperture domain is shown to have a type $(0,1)$ leading term upstream and a type $(1,2)$ leading term downstream.

What carries the argument

The argument rests on the reduction of the Navier-Stokes system to the boundary-value problem $f''=-f^2-4f+b$, $f(-\alpha)=f(\alpha)=0$, with the flux as the integral of $f$; the azimuthal velocity vanishes and the pressure is eliminated. Multiplying by $f'$ gives the first integral $(f')^2=Q(f)$, where $Q$ is the cubic $-\frac23(f-e_1)(f-e_2)(f-e_3)$ with roots summing to $-6$. The angle and flux become elliptic integrals $I(e_1,e_2)$ and $J(e_1,e_2)$ over the interval between consecutive roots, and for $(1,1)$ flows they combine into the identity $\alpha^2+\alpha\Phi/4=H(\bar\gamma)$ with $H(\bar\gamma)=[(\bar\gamma^2-2)K(\bar\gamma)+3E(\bar\gamma)]K(\bar\gamma)$. The strict monotonicity of $H$ and of the root maps reduces existence and uniqueness to a one-parameter level-set analysis, and non-uniqueness appears when the level set of the angle integral has two branches.

What would settle it

For a wedge half-angle such as $\alpha=\pi/4$ that is not the special angle $I_+(1,0)$, trace the level set $I_{1,2}(e_1,e_2)=\alpha$ numerically and plot the flux $J_{1,2}$ along it: if for some $\alpha$ the second branch ends before $\Phi_{\max}^{(1,2)}(\alpha)$, the claimed non-uniqueness interval for type $(1,2)$ shrinks; if the same failure occurs for some $m\ge2$, the non-uniqueness statement would need revision.

Watch

Extended reading notes

Core claim

The central claim is that the set of no-slip self-similar solutions in a sector is organised by flux thresholds: for every admissible type with $|m_+-m_-|\le 1$ the admissible fluxes form a half-line bounded above by $\Phi_{\max}^{(m_+,m_-)}(\alpha)$, and the extremal solution often degenerates to a neighbouring type. The paper establishes the threshold values through complete and incomplete elliptic integrals, proves that the maximum-flux curves are monotone in the angle in the stated ranges, and shows that the flux interval $(\Phi_{\max}^{(m,m)}(\alpha),\Phi_{\max}^{(m,m+1)}(\alpha))$ contains two distinct solutions of type $(m,m+1)$. This gives the first rigorous justification of some entries in [23] and shows that other tabulated entries are numerically inaccurate.

Load-bearing premise

The non-uniqueness part of the classification assumes that the two-branch level-set analysis carried out at one special wedge angle repeats for every angle in $(0,\pi/2)$ and for every $m\ge2$ without essential change; the paper states this extension but does not display the general argument.

Editorial extensions

If this is right

  • For half-angle $\alpha\ge\pi/2$, no pure outflow exists; for $\alpha<\pi/2$ the pure outflow is unique and its maximum flux satisfies $\Phi_{\max}^{(1,0)}(\alpha)=8(\pi/2-\alpha)+o(\pi/2-\alpha)$ near the borderline.
  • Pure inflow exists for every negative flux when $\alpha\le\pi/2$, and only up to a negative threshold when $\alpha>\pi/2$; that threshold equals the $(1,1)$ maximum flux.
  • The $(m,m)$ maximum flux is $m\Phi_{\max}^{(1,0)}(\alpha/m)$, so periodic flows inherit the half-angle restriction of the pure outflow problem.
  • For every $m\ge1$ and every flux in $(\Phi_{\max}^{(m,m)}(\alpha),\Phi_{\max}^{(m,m+1)}(\alpha))$, there are at least two $(m,m+1)$ self-similar solutions.
  • In the aperture domain with small flux, the far-field leading term is a type $(0,1)$ inflow upstream and a type $(1,2)$ outflow downstream, fixing the missing type information in the prior existence theorem [9].

Reading between the lines

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

  • Beyond the paper, the same level-set calculation could be run numerically for any intermediate angle to test whether the two-branch structure for $(1,2)$ flows persists; this would convert the asserted 'similar' extension at the end of Section 6.1 from an assumption into a checked fact.
  • If the threshold mechanism is robust, it should apply to other scale-invariant settings, such as rotated self-similar solutions or sector flows with Navier-slip boundary conditions, where the ODE gets an extra parameter.
  • The numerical corrections flagged in [23] suggest that other entries of the 1940 tables, especially fluxes at which outflow regions merge, could be recomputed with the same elliptic-integral expressions.
  • The two solutions sharing one flux are a candidate for flow hysteresis in wedge-shaped channels: the same imposed flux may be reached by different velocity profiles depending on the path of the flux.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. This paper studies self-similar (Jeffery-Hamel) solutions of the steady Navier-Stokes equations in a two-dimensional sector with no-slip boundary conditions. After reducing the PDE to the ODE f'' = -f^2 - 4f + b with boundary and flux constraints, the authors express the angle and flux as elliptic integrals of the roots e1,e2,e3 and classify existence by flow type (m+,m-). The main results are: necessary and sufficient flux thresholds Φmax(m+,m-)(α) for existence; uniqueness of pure outflow, pure inflow, and type (m,m) flows; non-uniqueness of type (m,m+1) flows for fluxes between the (m,m) and (m,m+1) thresholds; endpoint characterizations at maximal flux; asymptotics of the thresholds; and an application identifying the leading-order terms of the small-flux aperture-domain solution as type (0,1) upstream and type (1,2) downstream. The proofs rely on monotonicity properties of complete and incomplete elliptic functions, many of which are established in the paper.

Significance. If the full statements hold, the paper gives the first rigorous classification of no-slip Jeffery-Hamel flows in a sector, corrects Rosenhead's numerical table (e.g., Φmax(1,1)(π/2)=0 rather than 0.5), and identifies the previously unspecified aperture-domain asymptotics. The derivations are parameter-free, with independent numerical benchmarks, and the analytic machinery (monotonicity of H(γ), level-set analysis) is carefully developed for the cases that are proved. However, the headline non-uniqueness result is, as written, rigorously established only for one special angle and m=1; the extension to all α and m≥2 is asserted by 'similar' arguments that are not supplied. This is the main weakness.

major comments (4)
  1. [Section 6.1, proof of Proposition 6.1(2) (m=1)] The non-uniqueness proof is carried out only for the special angle ¯α = I+(1,0). Lemma 6.6 computes the derivative of J1,2 along the level set at e1=1 using the explicit cancellations in (52)-(54), and the text then states 'The general case α∈(0,π/2) is similar and omitted.' This is load-bearing for Theorem 1.3(1): for a general α the branch point e1*(α) is defined implicitly by I+(e1*(α),0)=α, and the sign of the analogous derivative at (e1*(α),0) is precisely what guarantees that every Φ in (Φ(1,1)max(α), Φ(1,2)max(α)) is attained on two distinct branches. Without this computation the non-uniqueness statement is not established for α≠¯α.
  2. [Section 6.1, final paragraph] The assertion that the case m≥2 in Proposition 6.1 'follows the same lines' is load-bearing for Theorem 1.3(2) and Theorem 1.2(3). No analogue of Lemmas 6.2-6.6 is stated for Im,m+1, and the topology of the level sets {Im,m+1=α} is not described. In particular, the inequalities Φ(m,m)max(α) < Φ(m,m+1)max(α) ≤ (m/(m+1))Φ(m+1,m+1)max(α) are asserted for m≥2 without proof. The authors should either provide the omitted details or state a version of the theorem limited to the cases actually proved.
  3. [Section 6.1, proof of Proposition 6.1(1)] The dichotomy in Part (1) depends on the assertion that the attainable fluxes along the level set form an interval (-∞, Φ(1,2)max(α)]. The proof gives an upper bound (49) and the limit -∞ from Lemma 6.3, but it does not explicitly state the continuity of J1,2(e1,e2(e1)) on (e1*(α),∞) nor the resulting intermediate-value argument. This is standard and likely repairable, but it should be part of the proof.
  4. [Section 7, Theorem 7.1(5) and (7)] The endpoint cases at maximal flux are used in the conventions of Theorem 1.3 and in Remark 1.5 to correct Rosenhead's table. Part (5) (type (m,0) solution at Φ=Φ(m,m)max for m≥2) is dismissed with 'similar to Part (4)', and Part (7) is likewise not proved. These are not mere repetitions: the proof of Part (4) relies on the special structure of e1=0 for α≥π/2, which does not directly transfer to the m≥2 case. Please provide the proof or explicitly label these statements as conjectural.
minor comments (5)
  1. [Section 4, proof of Proposition 4.1] The word 'abd' in 'Using (33), (34), abd (35)' should be 'and'.
  2. [Remark 7.1] The expression 'α∈(0, pi/2)' uses 'pi' instead of the symbol π.
  3. [Section 6.2, proof of Lemma 6.9] The displayed condition 'I2,1(e1,e♯ 2(e1) = inf' is missing a closing parenthesis and should read 'I2,1(e1,e♯2(e1)) = inf'.
  4. [Section 6.2, proof of Proposition 6.7] The reference '(65) and (6.2)' appears to be a typo; equation (6.2) is not defined, and the intended reference is likely to the displayed inequality for ∂²I+/∂e2² above (66).
  5. [Section 6.1] The phrases 'The general case α∈(0,π/2) is similar and omitted' and 'follows the same lines, so we omit the details' should be replaced by actual proofs or precise reductions, for the reasons given in the major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: thresholds derived from ODE/elliptic integrals; external benchmarks are independent, and the only flagged omission is an unproved extension, not a self-referential reduction.

full rationale

The paper's derivation chain is self-contained and not circular. The central objects Φ(m+,m−)max(α) are defined by solving the angle constraint I(e1,e2)=α and then maximizing the flux integral J(e1,e2) on that level set; e.g., Φ(1,0)max(α)=2J+(e∗1(α),0) with e∗1(α) defined by I+(e∗1(α),0)=α (equations (16)-(17)). Existence, uniqueness, and non-existence are then obtained by monotonicity, intermediate-value arguments, and asymptotic estimates, not by fitting parameters to the quantities being predicted. The monotonicity of H(γ) is quoted from Guillod and Wittwer [10], an independent external result; the aperture-domain input from Galdi-Padula-Solonnikov [9] is likewise external and is used only to supply the solution whose leading-order type is then identified by exclusion using the classification proved in the paper. Comparisons to Rosenhead's numerics are benchmarks, not inputs. There is no load-bearing self-citation: reference [2] is cited only as contextual literature. The one flagged limitation is an omitted proof, not circularity: the two-branch non-uniqueness argument in Section 6.1 is carried out for the special angle α¯=I+(1,0) and m=1, with the text stating 'The general case α∈(0,π/2) is similar and omitted' and 'The proof of Proposition 6.1 when m≥2 follows the same lines, so we omit the details.' This is a gap in the written proof of the full generality of Theorem 1.3(1)-(2), but it does not make any conclusion equivalent to its input by construction. No fitted constant is renamed as a prediction, and no uniqueness theorem is imported from the authors' own prior work to force the classification. Accordingly, no circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted free parameters appear: e1, e2, e3 are integration constants fixed by boundary and flux equations, and all Phi_max values are derived quantities rather than fits. The analysis imports standard elliptic-function facts, the prior aperture existence theorem, and the self-similar ansatz as its domain of study. No new physical entities are invented.

assumptions (4)
  • domain assumption Self-similar ansatz: u = f(theta)/r e_r + g(theta)/r e_theta, p = p(theta)/r^2.
    This is the definition of the class under study, introduced in Section 2 as equation (7). The no-slip boundary condition then forces g = 0, giving the reduced ODE problem (9).
  • standard math Elliptic integral identities and monotonicity: H(gamma) = ((gamma^2 - 2)K(gamma) + 3E(gamma))K(gamma) is strictly decreasing on [0,1), proved in Guillod-Wittwer [10, Appendix A]; derivative formula dK/dgamma = E/(gamma(1-gamma^2)) - K/gamma from Whittaker-Watson [26, p.501].
    Used in Section 4 to rewrite the type-(1,1) flux condition as equation (30) and to select a unique gamma for each flux. The paper does not re-prove these analytic facts.
  • standard math External existence theorem of Galdi, Padula, and Solonnikov [9, Lemma 5.1, Theorem 5.1]: for small flux there is a unique solution in the aperture domain with a Jeffery-Hamel leading term satisfying bounds (74)-(75).
    Theorem 1.4 and Proposition 8.2 take this theorem as input and only identify the type of the leading term. Failure of this prior result would affect the aperture application, not the sector classification.
  • standard math Classical Jeffery-Hamel classification of self-similar solutions in the whole plane, Theorem 1.1 from [12,13,24].
    Stated at the start of Section 1 as background and comparison. The sector proofs are self-contained from the ODE reduction and do not depend on this classification, but the framing and the comparison with Rosenhead's numerical table do.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector." pith.science (2026). https://pith.science/paper/M4MRKXGS

@misc{pith2026241207283,
  author       = {Pith},
  title        = {Pith review of: Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4MRKXGS}},
  note         = {Machine review of arXiv:2412.07283}
}
read the original abstract

This paper is concerned with self-similar solutions of the steady Navier-Stokes system in a two-dimensional sector with the no-slip boundary condition. We give necessary and sufficient conditions in terms of the angle of the sector and the flux to guarantee the existence of self-similar solutions of a given type. We also investigate the uniqueness and non-uniqueness of flows with a given type, which not only give rigorous justifications for some statements in \cite{Rosenhead40} but also show that some numerical computations in \cite{Rosenhead40} may not be precise. The non-uniqueness result is a new phenomenon for these flows. As a consequence of the classification of self-similar solutions in the half-space, we characterize the leading order term of the steady Navier-Stokes system in an aperture domain when the flux is small. The main approach is to study the ODE system governing self-similar solutions, where the detailed properties of both complete and incomplete elliptic functions have been investigated.

Figures

Figures reproduced from arXiv: 2412.07283 by the authors.

Figure 1
Figure 1. Pure outflow and pure inflow [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Type (1,1) and type (2,1) flow [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Type (2,2) and type (3,2) flow Now, let us state our first main result, which is mainly about the existence and non￾existence of a self-similar solution of each type. Theorem 1.2. Consider a self-similar (SS) solution u to the Navier-Stokes equations (1) with the no-slip boundary condition (2) and the flux condition (4) in a sector K, which is defined in (3) with angle 2α (α ∈ (0, π)). The following statements hold.… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Plots for Q(f) and a plot for velocity of pure outflow The above fact together with (10) tells that given the angle α and the flux Φ, we should have the following two equations, I+(e1, e2) := Z e1 0 df p Q(f) = Z e1 0 df » − 2 3 (f − e1)(f − e2)(f − e3) = α, (12) J+(e1…
Figure 5
Figure 5. Figure 5: Type (1,1) flow and type (2,2) flow-case I (A) Type (1,1) flow (B) Type (2,2) flow [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: Type (1,1) flow and type (2,2) flow-case II and J(e1, e2) := Z e1 e2 f df » − 2 3 (f − e1)(f − e2)(f − e3) = Φ 2 . (26) Let K(γ) and E(γ) be the complete elliptic functions of the first kind and the second kind, respectively, i.e., K(γ) = Z 1 0 dt p (1 − t 2 )(1 − γ 2 …
Figure 7
Figure 7. Figure 7: The plot of I2,1(e1, e2) and level set of I2,1(e1, e2) = α Hence one has lim e2→0− ∂I2,1 ∂e2 = lim e2→0− ∂I ∂e2 + lim e2→0− ∂I+ ∂e2 = +∞. On the other hand, as e2 → (−3 − 1 2 e1) +, e2 − e3 → 0, and thus ¯γ → 1. It then follows from (64) and (66) that lim e2→(−3− 1 2 e…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages

  1. [1]

    Abernathy

    F. Abernathy. Film notes for fundamentals of boundary layers. National Committee for Fluid Mechanics Film, Encyclopedia Britannica Educational Corporation, 425, 1970

    work page 1970

  2. [2]

    J. Bang, C. Gui, H. Liu, Y. Wang, and C. Xie. Rigidity of steady solutions to the Navier-Stokes equations in high dimensions. arXiv preprint arXiv:2306.05184 , 2023

    arXiv 2023

  3. [3]

    W. H. H. Banks, P. G. Drazin, and M. B. Zaturska. On perturbations of Jeffery-Hamel flow. J. Fluid Mech., 186:559–581, 1988

    work page 1988

  4. [4]

    Borchers, G

    W. Borchers, G. P. Galdi, and K. Pileckas. On the uniqueness of Leray-Hopf solutions for the flow through an aperture. Arch. Rational Mech. Anal., 122(1):19–33, 1993

    work page 1993

  5. [5]

    Borchers and K

    W. Borchers and K. Pileckas. Existence, uniqueness and asymptotics of steady jets. Arch. Rational Mech. Anal., 120(1):1–49, 1992

    work page 1992

  6. [6]

    P. G. Drazin and N. Riley. The Navier-Stokes equations: a classification of flows and exact solutions , vol- ume 334 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006

    work page 2006

  7. [7]

    L. E. Fraenkel. Laminar flow in symmetrical channels with slightly curved walls. I. On the Jeffery-Hamel solutions for flow between plane walls. Proc. Roy. Soc. London Ser. A , 267:119–138, 1962. SELF-SIMILAR SOLUTIONS WITH PHYSICAL BOUNDARIES 49

    work page 1962

  8. [8]

    G. P. Galdi. An introduction to the mathematical theory of the Navier-Stokes equations . Springer Mono- graphs in Mathematics. Springer, New York, second edition, 2011. Steady-state problems

    work page 2011

Show all 26 references
  1. [9]

    G. P. Galdi, M. Padula, and V. A. Solonnikov. Existence, uniqueness and asymptotic behaviour of solutions of steady-state Navier-Stokes equations in a plane aperture domain. Indiana Univ. Math. J. , 45(4):961–995, 1996

    1996

  2. [10]

    Guillod and P

    J. Guillod and P. Wittwer. Generalized scale-invariant solutions to the two-dimensional stationary Navier-Stokes equations. SIAM J. Math. Anal. , 47(1):955–968, 2015

    2015

  3. [11]

    A. Gusarov. Entrainment flow of a jet emerging into a half-space with the no-slip boundary condition. Physics of Fluids , 32(8), 2020

    2020

  4. [12]

    G. Hamel. Spiralf¨ ormige bewegungen z¨ aher fl¨ ussigkeiten.Jahresbericht der deutschen mathematiker- vereinigung, 25:34–60, 1917

    1917

  5. [13]

    G. B. Jeffery. The two-dimensional steady motion of a viscous fluid. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science , 29(172):455–465, 1915

    1915

  6. [14]

    Jia and V

    H. Jia and V. ˇSver´ ak. Asymptotics of stationary navier stokes equations in higher dimensions. Acta Mathematica Sinica, English Series , 34(4):598–611, dec 2017

    2017

  7. [15]

    Kobayashi

    T. Kobayashi. Jeffery-Hamel’s flows in the plane. J. Math. Sci. Univ. Tokyo , 21(1):61–77, 2014

    2014

  8. [16]

    Kobayashi

    T. Kobayashi. Steady Navier-Stokes equations with Poiseuille and Jeffery-Hamel flows in R2. Rocky Mountain J. Math. , 49(6):1909–1929, 2019

    1909

  9. [17]

    Korolev and V

    A. Korolev and V. ˇSver´ ak. On the large-distance asymptotics of steady state solutions of the Navier- Stokes equations in 3D exterior domains. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 28(2):303–313, 2011

    2011

  10. [18]

    L. Li, Y. Li, and X. Yan. Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. I. One singularity. Arch. Ration. Mech. Anal., 227(3):1091–1163, 2018

    2018

  11. [19]

    Miura and T.-P

    H. Miura and T.-P. Tsai. Point singularities of 3D stationary Navier-Stokes flows. J. Math. Fluid Mech., 14(1):33–41, 2012

    2012

  12. [20]

    S. A. Nazarov, A. Sequeira, and J. H. Videman. Steady flows of Jeffrey-Hamel type from the half- plane into an infinite channel. I. Linearization on an antisymmetric solution. J. Math. Pures Appl. (9) , 80(10):1069–1098, 2001

    2001

  13. [21]

    S. A. Nazarov, A. Sequeira, and J. H. Videman. Steady flows of Jeffery-Hamel type from the half- plane into an infinite channel. II. Linearization on a symmetric solution. J. Math. Pures Appl. (9) , 81(8):781–810, 2002

    2002

  14. [22]

    Rivkind and V

    L. Rivkind and V. A. Solonnikov. Jeffery-Hamel asymptotics for steady state Navier-Stokes flow in domains with sector-like outlets to infinity. J. Math. Fluid Mech. , 2(4):324–352, 2000

    2000

  15. [23]

    Rosenhead

    L. Rosenhead. The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 175(963):436– 467, 1940

    1940

  16. [24]

    ˇSver´ ak

    V. ˇSver´ ak. On Landau’s solutions of the Navier-Stokes equations. volume 179, pages 208–228. 2011. Problems in mathematical analysis. No. 61

    2011

  17. [25]

    Tian and Z

    G. Tian and Z. Xin. One-point singular solutions to the Navier-Stokes equations. Topol. Methods Non- linear Anal., 11(1):135–145, 1998. 50 JEAHEANG BANG, CHANGFENG GUI, HAO LIU, YUN W ANG, AND CHUNJING XIE

    1998

  18. [26]

    E. T. Whittaker and G. N. Watson. A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions . Cambridge University Press, New York, fourth edition, 1962. Institu...

    1962

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.