Pith. sign in

REVIEW 3 major objections 4 minor 37 references

Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · 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 new mixed Lebesgue-space controls on Leray solutions of the 3D incompressible Navier–Stokes equations, including second derivatives in $L^{\tilde r}_T L^r$ for arbitrarily large $r$.

desk verdict The claimed second-derivative bounds are new, but Lemma 1 is false — an explicit counterexample — so the paper's main results are unproved. read the letter →

arxiv 2411.13980 v1 pith:H546HKO7 submitted 2024-11-21 math.AP

Igor Honoré (UCBL , ICJ , PSPM) This is my paper · ORCID
classification math.AP MSC 35Q3076D0376D0526D10
keywords IncompressibleNavier-StokesequationLeraysolutionintegralinequalityLebesguespaceestimatessecond-orderderivativesBihari-LaSalleDuhamelformulaheatkernelregularization
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

This paper proves new a priori bounds for Leray solutions of the 3D incompressible Navier–Stokes equations without assuming smallness of the data. The main theorem gives explicit mixed Lebesgue-space controls for the solution, its gradient, and its Hessian: for $k=2$ and every $r\ge 2$, the relation $1/\tilde r + 6/r = 9$ holds, so $\nabla^2 u \in L^{\tilde r}_T L^r$ with $\tilde r = r/(9r-6)$, a regime of arbitrarily large spatial integrability that earlier second-derivative results (covering only $r<3/2$) did not reach. The proof works through a Duhamel formula around a heat equation corrected by the flow of $u$, combined with a new Bihari–LaSalle inequality that converts nonlinear time-integral bounds into Lebesgue norms. A fifth section adds weighted singular controls $\sup_{t\in[0,T]}\int_0^t (t-s)^{-\theta}\|\nabla^k u(s)\|_{L^r}\,ds < \infty$ under the condition $\theta < (3-kr)/(2r)$, $k\in\{0,1,2\}$, $1

What carries the argument

The engine is a Duhamel formula written around a heat equation whose drift is frozen along the flow $\theta_{s,t}(x)$ of $u$: after mollification, $u$ solves $\partial_t u + u(t,\theta_{t,\tau}(\xi))\cdot\nabla u - \nu\Delta u = u^\Delta_{[\tau,\xi]}\cdot\nabla u + \Xi[u\cdot\nabla u] + Pf$, with the Gaussian kernel $\hat p_{\tau,\xi}(s,t,x,y)$ centered at the frozen trajectory. Choosing the freezing point $(\tau,\xi)=(t,x)$ gives a representation of the vorticity $\omega=\nabla\times u$ in which the nonlinear term becomes a double difference $[u(s,\theta_{s,t}(x))-u(s,y)]^{\otimes 2}$ against a derivative of the heat kernel. That difference is converted into the Sobolev–Slobodeckij norm $[u(s,\cdot)]_{W^{\gamma,2r}}$, which interpolation bounds by the known $L^\infty_T L^2$ and $L^2_T \dot H^1$ energy controls, leaving an integrable time singularity $(t-s)^{-1+\gamma}$. The new Bihari–LaSalle lemma (Lemma 1) then turns the resulting inequality $\phi(t) \le a(t) + \int_0^t (t-s)^{-1+\gamma}\psi(s)\phi^\beta(s)\,ds$, $\beta\in[0,1)$, into $L^p$ bounds using the symmetric increasing rearrangement and the Hardy–Littlewood–Sobolev inequality.

What would settle it

Test the monotonicity assertion inside Lemma 1 with elementary data: take $T=1$, $\gamma=\beta=1/2$, $a(t)=1$, $\psi(s)=1+s\sin(10s)$, form $K_t(\tilde s) = (a(t)^{1-\beta} + \int_0^{\tilde s}(t-\sigma)^{-1+\gamma}\psi(\sigma)\,d\sigma)^{1/(1-\beta)}$, and compute whether $s \mapsto \tilde K_s^{*\beta}(s)$ is non-decreasing as the proof requires; a single decrease at any pair $s_1<s_2$ invalidates the lemma, and with it the proof of Theorem 1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for a Leray solution $u$ of (1.2) with $u_0\in L^2(\mathbb{R}^3)\cap \dot B^{2(\tilde r-1)/\tilde r}_{r,\tilde r}(\mathbb{R}^3)$ and $f\in L^\infty_T L^2 \cap L^\infty_T \dot B^2_{r,\infty}(\mathbb{R}^3)$, the norm $\|\nabla^k u\|_{L^{\tilde r}_T L^r}$ is finite whenever the indices satisfy the stated relations. The main extension is the branch $k=2$, $r\ge 2$, where $\tilde r = r/(9r-6)$, equivalently $1/\tilde r + 6/r = 9$; this puts $\nabla^2 u$ in $L^{r/(9r-6)}_T L^r$ for every finite $r$, an arbitrarily large spatial integrability that was previously out of reach, since the classical Constantin–Lions bound $2/\tilde r + 3/r = 4$ covered only $r \in (1,3/2)$. Theorem 5 in the same paper establishes weighted singular controls $\int_0^T (T-t)^{-\theta}\|\nabla^k u(t)\|_{L^r}\,dt < \infty$ for $\theta < (3-kr)/(2r)$, $k\in\{0,1,2\}$, $1<r<3/k$. The author's aim is to establish these bounds as unconditional statements about Leray solutions; the novelty is the extension to large $r$ through the heat-flow Duhamel identity and the Bihari–LaSalle argument.

Load-bearing premise

The load-bearing premise is that the new Bihari–LaSalle lemma (Lemma 1, Section 2.4) is correct; its proof assumes that the rearranged comparison function is non-decreasing in $s$ so it can be pulled out of the integral, and that monotonicity is asserted without proof.

Editorial extensions

If this is right

  • For every finite $r\ge 2$, $\|\nabla^2 u\|_{L^{r/(9r-6)}_T L^r} < \infty$; in particular, as $r\to\infty$ the time exponent $\tilde r$ tends to $1/9$, a quantitative bound far stronger in spatial integrability than all previous second-derivative controls.
  • The same framework yields $\|\nabla u\|_{L^{\tilde r}_T L^r} < \infty$ for arbitrarily large $r$ via Sobolev embedding from the $k=2$ branch, and for $k=0$ recovers the known control $u\in L^1_T L^\infty$ at $r=\infty$.
  • The weighted singular estimates $\sup_{t\in[0,T]}\int_0^t (t-s)^{-\theta}\|\nabla^k u(s)\|_{L^r}\,ds < \infty$ are valid for all $\theta < (3-kr)/(2r)$, $k\in\{0,1,2\}$, $1<r<3/k$, giving a handle on the time singularities that appear in mild formulations.
  • Interpolating the new branches with the older $r<3/2$ controls produces a larger admissible region for $\nabla^2 u$ and $\nabla u$ (Figures 1–3); the equality case $(r,\tilde r)=(4/3,4/3)$ remains the only unresolved endpoint.
  • These bounds do not reach the Prodi–Serrin scaling $2/\tilde r + 3/r \le 1+k$, so they do not by themselves imply smoothness or rule out finite-time blow-up; they tighten the constraints on any hypothetical singularity.

Reading between the lines

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

  • Beyond the paper: the freeze-the-flow Duhamel identity might be pushed to $k\ge 3$, but the author notes that the Sobolev indices would force $r<1$; a genuinely new interpolation idea would be needed, so this is an obstacle explicitly acknowledged in the text rather than a proven impossibility.
  • Beyond the paper: the line $1/\tilde r + 6/r = 9$ has a scaling-invariant look, and it would be natural to test sharpness by constructing near-extremal Leray profiles (self-similar or numerically optimized) that saturate the mixed-norm bound.
  • Beyond the paper: because each new $L^{\tilde r}_T L^r$ control on $\nabla^2 u$ restricts possible blow-up profiles, combining this estimate with existing regularity criteria could sharpen conditional results on the critical norm $\|\nabla u\|_{L^1_T L^\infty}$.
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

3 major / 4 minor

Summary. The manuscript claims new estimates for Leray solutions of the 3D incompressible Navier-Stokes equations, controlling u, ∇u, and ∇²u in spaces L^{~ r}_T L^r under explicit relations between ~ r and r, together with weighted singular-in-time estimates. The method is a Duhamel formula around a convected heat kernel, combined with a new Bihari-LaSalle type lemma based on nondecreasing rearrangements. The paper also interpolates these estimates with earlier results and applies them to second and first derivatives.

Significance. If correct, the k=2 estimates would genuinely extend known derivative controls to arbitrarily large spatial integrability exponents, and the explicit relations (for instance 1/~ r + 6/r = 9 for r ≥ 2) are concrete and falsifiable. The manuscript is self-contained in its auxiliary inequalities and introduces no fitted parameters. However, the central auxiliary lemma is not valid, and one of the main ranges in Theorem 1 is internally inconsistent; the claimed results are therefore not established.

major comments (3)
  1. [Section 2.4, Lemma 1] The proof of Lemma 1 rests on the assertion, immediately after (2.23), that s ↦ K_s^β(s) is non-decreasing. This assertion is false. For T=2, a≡0, β=γ=1/2 and ψ(s)=1_{[0,1]}(s), one has K_1(1)=4 while K_2(2)=4(√2-1)^2≈0.686. If φ solves the corresponding Volterra equality, then φ(s)=π²s/4 on [0,1] and φ(t)=(π/2)∫_0^1 (t-s)^{-1/2}√s ds for t>1, which is decreasing on [1,2]; hence φ_*(2)=φ(1)=π²/4 > K_2^*(2). Multiplying φ by a constant c∈(0,1) close to 1 makes the strict inequality assumed in Lemma 1 hold while preserving the violation. Although the displayed ψ is discontinuous, replacing it by a continuous bump supported in [0,1] and close to 1 on [0,1] preserves the failure of the monotonicity claim; continuity is not what validates the step. Since this monotonicity step is the only mechanism behind Lemma 1, the subsequent applications of the lemma in Theorem 1, Corollary 2, and Theorem 5 are unsupported.
  2. [Theorem 1 and Lemma 3, Eq. (3.9)] The displayed formula ~ r = r(r-4)/(4r²-13r+6) is claimed for r ∈ [3,6]. This is impossible: at r=3 the formula gives ~ r=-1, and for every r ∈ (3,4) the numerator is negative while the denominator is positive, so ~ r < 0; at r=4 the expression vanishes. A Lebesgue time exponent must be strictly positive. The proof itself obtains the formula only for r>4, see Eq. (3.35), and the following paragraph treating 3<r≤4 incorrectly concludes ~ r ≥ 0. Thus the statement of Lemma 3 and the k=1 part of Theorem 1 are invalid as written, and the interpolations in Section 4 built on the [3,6] segment need to be revisited.
  3. [Section 3.2.1 and Eq. (3.25)] The passage from the mollified equation to the Leray solution is not proved. The flow in (3.11) is not uniquely defined for the non-smooth velocity field, and after defining the regularized problem the author suppresses the regularization index and states that the limit is direct, without a convergence argument for the flow-dependent kernel ^np_{t,x} or for the nonlinear term in (3.25). The available convergence L^∞_T L² ∩ L²_T H¹ does not by itself justify passing to the limit in this kernel. Consequently, even apart from Lemma 1, the estimates for ∇u and ∇²u are not rigorously established for general Leray solutions.
minor comments (4)
  1. [Throughout] The notation k ∈ [[0,2]] is nonstandard; it should be written k ∈ {0,1,2}.
  2. [Corollary 2] The formula for ~ r, written as “1/~ r = 1−β − γr / r”, is ambiguous; it should be parenthesized and checked against the Hardy-Littlewood-Sobolev condition (2.19).
  3. [Section 3.2.4] The text first says “If r>3, and if r ∈ (4,6)” but then treats 3 < r ≤ 4 separately; this organization should be corrected, and the sign of (3.35) on [3,4] should be checked explicitly.
  4. [Figures 1–3] The captions and panels are difficult to read because several curves and labels overlap; separate panels or higher-resolution figures would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the estimates are derived from fixed energy inputs and a self-contained Bihari-LaSalle lemma; the main proof gap is a correctness issue, not circularity.

full rationale

The derivation chain starts from Leray's energy estimates (1.3)-(1.4), standard heat-kernel and interpolation estimates, and a new Bihari-LaSalle lemma that is proved in the paper rather than imported from prior work. Theorem 1's integrability exponents are obtained by explicit Hölder, Hardy-Littlewood-Sobolev, and interpolation bookkeeping; no parameter is fitted to the target data, and no numerical constant is chosen after the fact to force the conclusion. The author cites no prior work of his own in a load-bearing way; references such as Leray, Constantin, Lions, and Vasseur are external and provide standard or comparison results. The only notable internal risk is in the proof of Lemma 1 in Section 2.4: the proof relies on the assertion that s ↦ K*_s^β(s) is non-decreasing, flagged by the footnote 'This is exactly this argument which imposes us to use the monotonous rearrangements.' That monotonicity is asserted rather than proved, and the lemma's conclusion would not follow if it fails. However, this is a correctness gap in a self-contained auxiliary lemma, not circularity: the assertion is not equivalent to Theorem 1's conclusion, is not derived from that conclusion, and does not reduce the target estimate to its own input. I therefore find no significant circularity.

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

The paper introduces no fitted constants or new entities. It relies on standard inequalities and the classical energy estimates. The main risk is the new Bihari-LaSalle lemma and the flow limit, which are internal to the paper and not independently verified.

assumptions (4)
  • domain assumption Energy estimates (1.3)-(1.4) for Leray solutions: ||u||_{L^infinity_T L^2} + ||nabla u||_{L^2_T L^2} < infinity.
    Classical results (Leray 1934) used as the starting point for all estimates in the paper.
  • standard math Hardy-Littlewood-Sobolev inequality (2.19) and rearrangement inequalities (2.7)-(2.8).
    Used throughout, including the proof of Lemma 1 and the final norm transfer.
  • standard math Gagliardo-Nirenberg and Brezis-Mironescu interpolation inequalities (2.15)-(2.22).
    Used to relate L^r norms of u and nabla u, and to optimize exponents.
  • domain assumption Well-posedness of the mollified flow (3.11) and passage to the limit for Leray solutions.
    The paper mollifies the equation to define the flow and then claims uniform estimates allowing limit passage; this is sketched, not fully proven.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces." pith.science (2026). https://pith.science/paper/H546HKO7

@misc{pith2026241113980,
  author       = {Pith},
  title        = {Pith review of: Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H546HKO7}},
  note         = {Machine review of arXiv:2411.13980}
}
abstract

We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that $\sup_{t \in [0,T]} \int_{0}^t (t-s)^{-\theta} \|\nabla^k u(s,\cdot)\|_{L^r} ds<+ \infty$, for all $\theta< \frac{3-kr}{2r}$, $k \in [0,2]$, and $1<r<\frac{3}{k}$.

Figures

Figures reproduced from arXiv: 2411.13980 by the authors.

Figure 1
Figure 1. ∇2u ∈ L r˜ ([0, T], Lr (R 3 , R 3 )) The paper is organised as following. In Section 2, we set some useful notations and results for our analysis. The proof of Theorem 1 is developed in Section 3. We perform, in Section 4, interpolations between the already known controls and with our new results. We also provide, in Section 5, a counterpart in weighted Lebesgue space of Theorem 5. 2 Definitions and useful results W… view at source ↗
Figure 2
Figure 2. ∇2u ∈ L r˜ ([0, T], Lr (R 3 , R 3 )) 25 [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. ∇u ∈ L r˜ ([0, T], Lr (R 3 , R 3 )) Proof of Corollary 4 and interpolation [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 34 canonical work pages

  1. [1]

    G. Alberti. Some remarks about a notion of rearrangement. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 29(2):457--472, 2000

    work page 2000

  2. [2]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations , volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, Heidelberg, 2011

    work page 2011

  3. [3]

    Berestycki and T

    H. Berestycki and T. Lachand-Robert. Some properties of monotone rearrangement with applications to elliptic equations in cylinders. Math. Nachr. , 266:3--19, 2004

    work page 2004

  4. [4]

    Brezis and P

    H. Brezis and P. Mironescu. Gagliardo- N irenberg inequalities and non-inequalities: the full story. Ann. Inst. H. Poincar\' e C Anal. Non Lin\' e aire , 35(5):1355--1376, 2018

    work page 2018

  5. [5]

    H. Brezis. Analyse fonctionnelle . Dunod, 1999

    work page 1999

  6. [6]

    G. Carbou. Unicit\'e et minimalit\'e des solutions d'une \'equation de G inzburg- L andau. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire , 12(3):305--318, 1995

    work page 1995

  7. [7]

    D. Chae. Some a priori estimates for weak solutions of the 3 - D N avier- S tokes equations. J. Math. Anal. Appl. , 167(1):236--244, 1992

    work page 1992

  8. [8]

    J.-Y. Chemin. Perfect incompressible fluids , volume 14 of Oxford Lecture Series in Mathematics and its Applications . The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie

    work page 1998

Show all 37 references
  1. [9]

    Constantin

    P. Constantin. Navier- S tokes equations and area of interfaces. Comm. Math. Phys. , 129(2):241--266, 1990

    1990

  2. [10]

    Constantin

    P. Constantin. Remarks on the N avier- S tokes equations. In New perspectives in turbulence ( N ewport, RI , 1989) , pages 229--261. Springer, New York, 1991

    1989

  3. [11]

    C. R. Doering and C. Foias. Energy dissipation in body-forced turbulence. J. Fluid Mech. , 467:289--306, 2002

    2002

  4. [12]

    S. S. Dragomir. The G ronwall type lemmas and applications , volume 29 of Monografii Matematice (Timi soara) [Mathematical Monographs (Timi soara)] . Universitatea din Timi soara, Facultatea de Stiin te ale Naturii, Sec tia Matematic a, Timi soara, 1987

    1987

  5. [13]

    G. F. D. Duff. Derivative estimates for the N avier- S tokes equations in a three-dimensional region. Acta Math. , 164(3-4):145--210, 1990

    1990

  6. [14]

    C. L. Fefferman. Existence and smoothness of the N avier- S tokes equation. In The millennium prize problems , pages 57--67. Clay Math. Inst., Cambridge, MA, 2006

    2006

  7. [15]

    Foia s , C

    C. Foia s , C. Guillop\' e , and R. Temam. New a priori estimates for N avier- S tokes equations in dimension 3 . Comm. Partial Differential Equations , 6(3):329--359, 1981

    1981

  8. [16]

    D. Henry. Geometric theory of semilinear parabolic equations , volume 840 of Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1981

    1981

  9. [17]

    B. Kawohl. Rearrangements and convexity of level sets in PDE , volume 1150 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1985

    1985

  10. [18]

    O. A. Ladyzhenskaya . The mathematical theory of viscous incompressible flow. New York - London - Paris: Gordon and Breach Science Publishers. XVIII, 224 p. (1969). , 1969

    1969

  11. [19]

    J. Leray. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. , 63:193--248, 1934

    1934

  12. [20]

    P.L. Lions. Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models . Mathematical Topics in Fluid Mechanics. Oxford University Press, Incorporated, 1996

    1996

  13. [21]

    E. H. Lieb and M. Loss. Analysis , volume 14 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, second edition, 2001

    2001

  14. [22]

    Lemari\'e-Rieusset

    P.-G. Lemari\'e-Rieusset. Recent developments in the Navier-Stokes problem . CRC Press, 2002

    2002

  15. [23]

    P. G. Lemari\' e -Rieusset. The N avier- S tokes equations in the critical M orrey- C ampanato space. Rev. Mat. Iberoam. , 23(3):897--930, 2007

    2007

  16. [24]

    P. G. Lemari\' e -Rieusset. The N avier- S tokes problem in the 21st century . CRC Press, Boca Raton, FL, 2016

    2016

  17. [25]

    A. J. Majda and A. L. Bertozzi. Vorticity and incompressible flow , volume 27 of Cambridge Texts in Applied Mathematics . Cambridge University Press, Cambridge, 2002

    2002

  18. [26]

    D. S. Mitrinovi\'c, J. E. Pe cari\'c, and A. M. Fink. Inequalities involving functions and their integrals and derivatives , volume 53 of Mathematics and its Applications (East European Series) . Kluwer Academic Publishers Group, Dordrecht, 1991

    1991

  19. [27]

    C. W. Oseen. Sur les formules de green g\' e n\' e ralis\' e es qui se pr\' e sentent dans l'hydrodynamique et sur quelquesunes de leurs applications. Acta Math. , 34(1):205--284, 1911

    1911

  20. [28]

    C. W. Oseen. Sur les formules de G reen g\' e n\' e ralis\' e es qui se pr\' e sentent dans l'hydrodynamique et sur quelquesunes de leurs applications. Acta Math. , 35(1):97--192, 1912

    1912

  21. [29]

    G. Prodi. Un teorema di unicit\`a per le equazioni di N avier- S tokes. Ann. Mat. Pura Appl. (4) , 48:173--182, 1959

    1959

  22. [30]

    J. Serrin. On the interior regularity of weak solutions of the N avier- S tokes equations. Arch. Rational Mech. Anal. , 9:187--195, 1962

    1962

  23. [31]

    E. M. Stein. Singular integrals and differentiability properties of functions . Princeton university press, 1970

    1970

  24. [32]

    T. Tao. Localisation and compactness properties of the N avier- S tokes global regularity problem. Anal. PDE , 6(1):25--107, 2013

    2013

  25. [33]

    L. Tartar. Topics in nonlinear analysis , volume 13 of Publications Math\'ematiques d'Orsay 78 . Universit\'e de Paris-Sud, D\'epartement de Math\'ematiques, Orsay, 1978

    1978

  26. [34]

    R. Temam. Navier-Stokes Equations: Theory and Numerical Analysis . Studies in mathematics and its applications. North-Holland, 1979

    1979

  27. [35]

    H. Triebel. Theory of function spaces , volume 78 of Monographs in Mathematics . Birkh\" a user Verlag, Basel, 1983

    1983

  28. [36]

    A. Vasseur. Higher derivatives estimate for the 3 D N avier- S tokes equation. Ann. Inst. H. Poincar\' e C Anal. Non Lin\' e aire , 27(5):1189--1204, 2010

    2010

  29. [37]

    Vasseur and J

    A. Vasseur and J. Yang. Second derivatives estimate of suitable solutions to the 3 D N avier- S tokes equations. Arch. Ration. Mech. Anal. , 241(2):683--727, 2021

    2021

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.