Pith. sign in

REVIEW 2 major objections 6 minor 42 references

Suppression of blow-up in 3-D Keller-Segel system with fractional diffusion via Couette flow in whole space

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read This paper proves that a sufficiently large Couette shear flow suppresses finite-time blow-up in the three-dimensional Keller-Segel system with fractional diffusion of order α∈(1,2] on the whole space, with explicit Lp decay rates.

desk verdict A genuinely new Green's function technique for fractional Couette flow, but the k=2 mixed-derivative L1 bounds are asserted rather than proved—ask the authors for details before accepting. read the letter →

arxiv 2507.16160 v1 pith:435GBPCN submitted 2025-07-22 math.AP

Shijin Deng , Binbin Shi , Weike Wang , Yucheng Wang This is my paper · ORCID
classification math.AP MSC 35A0935E0535G5535M11
keywords Keller-SegelsystemCouetteflowfractionaldiffusionenhanceddissipationblow-upsuppressionGreen'sfunctionwholespaceLpdecay
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 asks whether a strong background shear flow can stop the finite-time blow-up that is known to occur in the Keller-Segel chemotaxis model when diffusion is fractional and the spatial domain is the whole of $\mathbb{R}^3$. It proves that the answer is yes: for fractional diffusion order $\alpha \in (1,2]$, every nonnegative initial datum in $W^{3,p}(\mathbb{R}^3) \cap L^1(\mathbb{R}^3)$ gives a unique global classical solution once the Couette flow amplitude $A$ is taken large enough, with explicit algebraic decay in every $L^p$ norm. The mechanism is dissipation enhancement: the Couette drift stretches high frequencies along the shear direction, effectively speeding up the fractional Laplacian so that the attractive nonlinearity can never concentrate enough mass to blow up. The whole-space setting matters because, unlike periodic boxes, the continuous spectrum leaves no spectral gap; the authors handle it with a Green's function whose $L^1$ estimate is finite at $t=0$.

What carries the argument

The carrying object is the Green's function $G$ of the linearized operator $\partial_t + Ay\partial_x + (-\Delta)^{\alpha/2}$. Its Fourier transform is explicit: $\hat{G}(\xi,\eta,\zeta,t;x',y',z') = \exp(-ix'\xi - iy'(\eta+At\xi) - iz'\zeta) \exp(-\int_0^t [\xi^2+(\eta+As\xi)^2+\zeta^2]^{\alpha/2}\, ds)$. The proof reduces everything to Lemma 3.4, which states that for $k = 1$ or $2$, $\|\partial_x^{k_1}\partial_y^{k_2}\partial_z^{k_3} G_2\|_{L^1} \leq C t^{-k/\alpha}(1+At)^{-k_1}$, with no $t=0$ singularity and no loss of the $(1+At)$ enhancement factor. Because the fractional symbol $|\Xi|^\alpha$ is not analytic, the authors cannot copy classical low/high frequency arguments; instead they introduce a space-frequency mixed decomposition, splitting space into regions where $x^2$ is comparable to $t^{2/\alpha}(1+At)^2$ and $y^2+z^2$ to $t^{2/\alpha}$, and control the $L^1$ norm through $H^2$ bounds on the symbol's derivatives. This is what turns enhanced dissipation into a singularity-free estimate robust enough for the Duhamel bootstrap.

What would settle it

For a fixed $\alpha\in(1,2)$, say $\alpha=3/2$, one could compute or sharply bound the mixed derivative norm $\|\partial_x\partial_y G_2(t)\|_{L^1}$ at $t$ near $A^{-\theta}$ and at $t$ near 1; if it scales worse than $t^{-2/\alpha}(1+At)^{-1}$ for large $A$, the estimates (4.14)-(4.23) would fail and the bootstrap would not close.

Watch

Extended reading notes

Core claim

The central claim is that a sufficiently large Couette flow suppresses blow-up for the 3-D generalized Keller-Segel system with fractional diffusion on the whole space. Concretely, Theorem 1.1 states that for $\alpha \in (1,2]$, $p \in [2,\infty)$, and any nonnegative $n_0$ in $W^{3,p}(\mathbb{R}^3) \cap L^1(\mathbb{R}^3)$, there exists $A_0 = A_0(\alpha, n_0)$ such that for all $A \geq A_0$ the system has a unique classical solution with $\|D^{\vartheta} n(t)\|_{L^p} \leq C(1+t)^{-(3/\alpha+1)(1-1/p)-|\vartheta|/\alpha}$ for $|\vartheta| \leq 3$. This extends the previously known $\alpha = 2$ case to fractional diffusion and, unlike periodic shear results, removes the dimension-dependent mass restriction: the shear flow suppresses blow-up regardless of the size of the initial mass. The essential observation is that a Fourier mode $(\xi,\eta,\zeta)$ under the Couette flow evolves its $y$-frequency into $\eta + As\xi$, so the time integral of the fractional symbol along the sheared trajectory produces extra $(1+At)$ factors; these factors are exactly the enhanced dissipation that the nonlinear estimates consume.

Load-bearing premise

The bootstrap rests on the $L^1$ bounds of Lemma 3.4 being exactly uniform in the Couette amplitude $A$ and carrying the stated $(1+At)$ factors for first and second derivatives; for the mixed second derivatives the proof says a 'corresponding modification' gives the bound, so the uniformity there is the main load-bearing assumption.

Editorial extensions

If this is right

  • For $\alpha \in (1,2]$, any nonnegative $n_0 \in W^{3,p} \cap L^1$ yields a unique global classical solution in 3-D whole space for large Couette amplitude, with the stated $L^p$ decay for all derivatives up to order 3.
  • The same Green's function structure works in any dimension $d \geq 2$, so the authors' method should give analogous whole-space suppression results for $\mathbb{R}^d$.
  • In contrast to periodic shear flows, where 3-D solutions with mass larger than $8\pi$ can still blow up, whole-space Couette flow suppresses blow-up for arbitrarily large mass.
  • The decay exponents in the theorem reflect enhanced dissipation: the $(1+At)$ factors in the Green's estimates enter the final rates, making the decay faster in the streamwise variable and independent of a spectral gap.
  • The regularity criterion in Theorem 4.1 means the $L^p$ bound alone guarantees continuation, so the bootstrap closes without needing pointwise control.

Reading between the lines

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

  • The same space-frequency decomposition should transfer to other aggregation or chemotaxis-fluid models with fractional diffusion, wherever the linearized operator is a shear flow plus a fractional Laplacian.
  • The restriction $\alpha > 1$ appears structural: the closing estimates need $\alpha - 1 > 0$ to absorb singular powers of $(t-s)$ and $A^{-\theta}$; a natural extrapolation is that the mechanism fails at $\alpha \leq 1$, where a different suppression route would be needed.
  • The $(1+At)$ factors suggest the effective dissipation rate grows linearly in $A$; one could test whether the optimal choice of $A_0$ in terms of $n_0$ and $\alpha$ is captured by the constants in Lemma 3.4, or whether a sharper $L^1$ estimate would lower the required amplitude.
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 / 6 minor

Summary. This paper considers the 3D Keller-Segel system with fractional diffusion (-Δ)^{α/2} (1<α≤2), an attractive nonlocal kernel B(n)=∇(-Δ)^{-1}n, and a large Couette background flow A y ∂_x n. The main theorem (Theorem 1.1) claims that for non-negative initial data in W^{3,p}(R^3)∩L^1(R^3), p∈[2,∞), and A sufficiently large, a unique classical solution exists globally and satisfies ∥D^ϑ n∥_{L^p} ≤ C(1+t)^{-(3/α+1)(1-1/p)-|ϑ|/α} for |ϑ|≤3. The proof is based on an explicit Fourier-space representation of the Green's function (Lemma 3.1), sharp L^1 and L^p estimates of the Green's function (Lemmas 3.2-3.5, Proposition 3.6), and a bootstrap argument (Lemma 4.2) combined with an induction for fractional derivatives (Lemma 4.6).

Significance. The result, if fully established, would be a meaningful advance: it extends blow-up suppression by shear flows in unbounded domains from the classical Laplacian case to fractional diffusion with α∈(1,2], and it provides quantitative algebraic decay rates. The Green's function approach with a space-frequency mixed decomposition is a genuine technical novelty, and the proof is self-contained in the sense that there are no fitted parameters or post-hoc exclusions; the large-amplitude condition on A appears only through the estimates of the enhanced dissipation factors (1+At). The paper also has a clear potential for further applications in higher dimensions. However, the proof is not yet complete in two load-bearing points: the second-order mixed-derivative L^1 estimates of the Green's function are only sketched, and the local well-posedness plus blow-up criterion are asserted without proof.

major comments (2)
  1. [Section 3.2, Lemma 3.4, k=2, Case 2] In the proof of Lemma 3.4 for k=2, only the bound for ∂_x^2 G_2 is established in detail; the remaining five second-order derivatives are dismissed with the statement that 'a corresponding modification of the estimates for k=1 yields' the claimed bounds. These bounds are load-bearing, not cosmetic: the fractional-derivative estimates (3.53) in Proposition 3.6 are obtained from the second-derivative L^1 bounds by Gagliardo-Nirenberg interpolation, and Lemma 4.6 uses the resulting bound ∥Λ^{1+γ}G(t-s)∥_{L^1} ≤ C (t-s)^{-(1+γ)/α} in the closing estimates (4.29) and (4.31). If any of the mixed-derivative bounds, especially those for ∂_x∂_y G_2 and ∂_x∂_z G_2, carried an extra factor of A or lost the (1+At)^{-1} factor, the smallness in A in the bootstrap (4.16)-(4.23) and in (4.28)-(4.31) would fail. The authors should provide the complete proof for all six second-order cases, with explicit control of the terms such as |η|H'_ξ and |η|H''_ξ in (3.26)-(3.28), or give a precise reduction showing exactly how the k=1 estimates imply the k=2 claims.
  2. [Section 4, Theorem 4.1] Theorem 4.1 asserts local well-posedness, non-negativity, and a blow-up criterion (4.1) for the system (1.1). The text says that non-negativity 'has already been proved in many references' and that local existence 'could be also proved by the standard method', but no proof or precise reference is given for the blow-up criterion. This criterion is used in the continuation argument in the proof of Theorem 1.1, where the solution is extended from T* to T** > T*. Since the equation contains the quadratic term n^2 (through ∇·(nB(n))) and a fractional diffusion of order α>1, the criterion is not immediate from standard semilinear theory and should be proved or explicitly located in the literature. This is a necessary step for the global existence claim.
minor comments (6)
  1. [Section 3.2, Proposition 3.6, Eq. (3.53)] The statement of (3.53) uses p on the right-hand side (t^{-(3/α)(1-1/p)-γ/α}(1+At)^{-(1-1/p)}) while the left-hand side is an L^q norm; the exponents should be written with q.
  2. [Section 4, proof of Theorem 1.1] The proof of the decay of integer derivatives D^ϑ n (|ϑ|≤3) from the fractional-derivative estimates in Lemma 4.6 is not written. The authors should add the standard argument: choose γ = |ϑ|/k with k > |ϑ|/(α-1), apply Lemma 4.6, and use the Mikhlin multiplier bound ∥D^ϑ f∥_{L^p} ≤ C∥Λ^{|ϑ|}f∥_{L^p} for p∈(1,∞).
  3. [Section 4, Lemma 4.4] In the proof of Lemma 4.4, the inequality '∥u(·,·,·,t)∥_{L∞} ≤ C∥n0(·,·,·,t)∥_{H2}' contains an extraneous t in the argument of n0; it should read ∥n0∥_{H^2}.
  4. [Introduction] The assertion that the method and estimates 'could also be applied for a general R^d (d≥2)' is stated without proof; if it is intended as a claim about future work, it should be phrased as a remark or conjecture rather than as a part of the proven contribution.
  5. [Throughout] The notation 9G(t)9_{L^p} for the maximum of the two L^p norms is nonstandard and hard to read; a more conventional notation (e.g., double bars with a subscript) would improve readability.
  6. [Title] There is a typo in the title on the arXiv page: 'SP ACE' should be 'SPACE'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives the Green's function estimates in-paper and closes the nonlinear argument by a discharged bootstrap, with self-citations only contextual.

full rationale

None of the paper's load-bearing steps reduces to its own inputs by construction. Lemma 3.1 solves the linearized Fourier problem (3.2) by direct substitution, and Lemmas 3.2-3.4 estimate this explicit representation using only standard external tools (Lemma 2.3 from [32], Lemma 2.4, Lemma 2.5, Young and Gagliardo-Nirenberg inequalities). The sharp (1+At)^{-k1} factors arise from the oscillatory integral for the Couette symbol, not from a fitted or assumed decay law. In Section 4, the bootstrap hypothesis (4.2) is a standard continuity argument: Lemma 4.2 improves the constant from 2δ to δ, thereby discharging the hypothesis, and Lemma 4.6 is an induction rather than a circular invocation of the desired bound. Self-citations to [10], [35], and [36] are contextual comparisons or methodological antecedents; the fractional Green's function estimates are proved in the present paper rather than imported from those works. The compressed second-derivative cases in Lemma 3.4, Case 2 ('a corresponding modification of the estimates for k=1 yields'), are a rigor or completeness concern, not circularity, because the asserted bounds are derived from the same Fourier representation rather than from Theorem 1.1's conclusion. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in solely via citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard harmonic analysis tools and on the specific Keller-Segel coupling. No new physical entities are introduced. The main unstated input is the local well-posedness theorem, which is asserted rather than proved. The Green's function is a mathematical construction, not an invented entity.

assumptions (6)
  • standard math Fourier representation of the fractional Laplacian (-Delta)^{alpha/2} for 0<alpha<=2
    Used to derive the exact Fourier representation of the Green's function in Lemma 3.1 via equation (1.2).
  • standard math Hardy-Littlewood-Sobolev inequality and L^p boundedness of homogeneous Fourier multipliers of degree 0
    Used in Lemma 4.3 to bound B(n)=nabla(-Delta)^{-1}n in L^r and its fractional derivatives.
  • standard math Kato-Ponce fractional Leibniz rule, Lemma 2.2
    Used in Lemma 4.6 to estimate products nB(n) under fractional derivatives.
  • standard math Gagliardo-Nirenberg interpolation inequalities for fractional derivatives
    Used in Proposition 3.6 to derive fractional-derivative Green's function estimates from integer-derivative bounds.
  • domain assumption The chemotaxis coupling B(n)=nabla(-Delta)^{-1}n is the appropriate attractive kernel for the 3D Keller-Segel model
    The whole paper studies this specific model, stated in (1.1)-(1.3); the result is conditional on this modeling choice.
  • domain assumption Local well-posedness and non-negativity of solutions, stated as Theorem 4.1
    The theorem is asserted as standard with no proof or specific reference, and it is needed to start the bootstrap argument.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Suppression of blow-up in 3-D Keller-Segel system with fractional diffusion via Couette flow in whole space." pith.science (2026). https://pith.science/paper/435GBPCN

@misc{pith2026250716160,
  author       = {Pith},
  title        = {Pith review of: Suppression of blow-up in 3-D Keller-Segel system with fractional diffusion via Couette flow in whole space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/435GBPCN}},
  note         = {Machine review of arXiv:2507.16160}
}
abstract

In this paper, we consider a Keller-Segel model with a fractional diffusion term in $\mathbb{R}^3$ in the background of a Couette flow. We show that when the background Couette flow is large enough, the dissipation enhancement induced could prevent the blow-up of solutions and thus prove the global existence and also obtain time decay rates of the solution in $L^p$ norm. The main tool of the proof is a corresponding Green's function and the key estimate is its $L^1$ estimate without singularities at $t=0$. To fulfill such an estimate, we meet great troubles caused by the fractional heat kernel together with the Couette flow in the model considered here and overcome the troubles by introducing a space-frequency mixed decomposition.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 41 canonical work pages

  1. [10]

    S. Deng, B. Shi, and W. Wang. Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space. J. Differential Equations, 432: 113265, 2025. SUPPRESSION OF BLOW-UP IN 3-D FRACTIONAL KELLER-SEGEL 21

    work page 2025

  2. [1]

    Arbon and J

    R. Arbon and J. Bedrossian. Quantitative hydrodynamic stability for Couette flow on unbounded domains with Navier boundary conditions. Comm. Math. Phys. , 406(6):Paper No. 129, 57, 2025

    work page 2025

  3. [2]

    Brezis and P

    H. Brezis and P. Mironescu. Gagliardo-Nirenberg, composition and products infractional Sobolev spaces. J. Evol. Equ. , 1(2001), no. 4, 387–404

    work page 2001

  4. [3]

    Bedrossian and S

    J. Bedrossian and S. He. Suppression of blow-up in Patlak-Keller-Segel via shear flows. SIAM J. Math. Anal. , 49(6):4722–4766, 2017

    work page 2017

  5. [4]

    Biler and G

    P. Biler and G. Karch. Blowup of solutions to generalized Keller-Segel model. J. Evol. Equ. , 10(2):247–262, 2010

    work page 2010

  6. [5]

    D. S. Christopher. Fourier Integrals in Classical Analysis, Cambridge: Cambridge Univ Press , 1993

    work page 1993

  7. [6]

    H. J. Choe, B. Lkhagvasuren, and M. Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Commun. Pure Appl. Anal. , 14(6):2453–2464, 2015

    work page 2015

  8. [7]

    Constantin, A

    P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoˇ s. Diffusion and mixing in fluid flow.Ann. of Math. (2) , 168(2):643– 674, 2008

    work page 2008

Show all 42 references
  1. [8]

    Coti Zelati and T

    M. Coti Zelati and T. Gallay. Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows. J. Lond. Math. Soc. (2) , 108(4):1358–1392, 2023

    2023

  2. [9]

    Cui and W

    S. Cui and W. Wang. Suppression of blow-up in multispecies Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow in a finite channel. SIAM J. Math. Anal. , 56(6):7683–7712, 2024

    2024

  3. [11]

    Ding and Z

    D. Ding and Z. Tan. Stability of the Couette flow for the two dimensional chemotaxis-Navier-Stokes system. Nonlinear Anal. Real World Appl. , 77:Paper No. 104058, 26, 2024

    2024

  4. [12]

    R. Duan, A. Lorz, and P. Markowich. Global solutions to the coupled chemotaxis-fluid equations. Comm. Partial Differential Equations, 35(9):1635–1673, 2010

    2010

  5. [13]

    Esfahani

    A. Esfahani. Anisotropic Gagliardo-Nirenberg inequality with fractional derivatives. Z. Angew. Math. Phys, , 2015, 66: 3345-3356

    2015

  6. [14]

    Y. Feng, B. Shi, and W. Wang. Dissipation enhancement of planar helical flows and applications to three-dimensional Kuramoto-Sivashinsky and Keller-Segel equations. J. Differential Equations , 313:420–449, 2022

    2022

  7. [15]

    Grafakos, S

    L. Grafakos, S. Oh. The Kato-Ponce Inequality, Comm Partial Differential Equations, 2014, 39: 1128-1157

    2014

  8. [16]

    S. He. Suppression of blow-up in parabolic-parabolic Patlak-Keller-Segel via strictly monotone shear flows. Nonlinearity, 31(8):3651–3688, 2018

    2018

  9. [17]

    S. He. Enhanced dissipation and blow-up suppression in a chemotaxis-fluid system. SIAM J. Math. Anal. , 55(4):2615– 2643, 2023

    2023

  10. [18]

    Hopf and J

    K. Hopf and J. L. Rodrigo. Aggregation equations with fractional diffusion: preventing concentration by mixing. Commun. Math. Sci. , 16(2):333–361, 2018

    2018

  11. [19]

    K. Kang, J. Lee, and M. Winkler. Global weak solutions to a chemotaxis-Navier-Stokes system in R3. Discrete Contin. Dyn. Syst. , 42(11):5201–5222, 2022

    2022

  12. [20]

    Kiselev and X

    A. Kiselev and X. Xu. Suppression of chemotactic explosion by mixing. Arch. Ration. Mech. Anal. , 222(2):1077–1112, 2016

    2016

  13. [21]

    Lafleche and S

    L. Lafleche and S. Salem. Fractional Keller-Segel equation: global well-posedness and finite time blow-up. Commun. Math. Sci. , 17(8):2055–2087, 2019

    2019

  14. [22]

    Li and J

    D. Li and J. Rodrigo. Finite-time singularities of an aggregation equation in Rn with fractional dissipation. Comm. Math. Phys. , 287(2):687–703, 2009

    2009

  15. [23]

    H. Li, N. Liu, and W. Zhao. Stability threshold of the two-dimensional Couette flow in the whole plane. arXiv:2501.10818, Jan. 2025

    2025 arXiv

  16. [24]

    H. Li, Z. Xiang, and X. Xu. Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Poiseuille flow. J. Differential Equations , 434:Paper No. 113301, 48, 2025

    2025

  17. [25]

    Li and J

    K. Li and J. Zheng. An optimal result for global classical and bounded solutions in a two-dimensional Keller-Segel- Navier-Stokes system with saturated sensitivity. Commun. Pure Appl. Anal. , 21(12):4147–4172, 2022

    2022

  18. [26]

    J. Liao, W. Wang, and T. Yang. Lp-convergence rates of planar waves for multi-dimensional Euler equations with damping. J. Differential Equations , 247(1):303-329, 2009

    2009

  19. [27]

    T.-P. Liu. Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. , 50(11):1113– 1182, 1997

    1997

  20. [28]

    Liu and W

    T.-P. Liu and W. Wang. The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi- dimensions. Comm. Math. Phys. , 196(1):145–173, 1998

    1998

  21. [29]

    Liu and Y

    T.-P. Liu and Y. Zeng. Time-asymptotic behavior of wave propagation around a viscous shock profile. Comm. Math. Phys., 290(1):23–82, 2009

    2009

  22. [30]

    Liu and Y

    T.-P. Liu and Y. Zeng. Shock waves in conservation laws with physical viscosity. Mem. Amer. Math. Soc. , 234(1105):vi+168, 2015

    2015

  23. [31]

    P. S. Marcus and W. H. Press. On green’s functions for small disturbances of plane couette flow. Journal of Fluid Mechanics, 79(3):525–534, 1977

    1977

  24. [32]

    Morimoto and C.-J

    Y. Morimoto and C.-J. Xu. Ultra-analytic effect of Cauchy problem for a class of kinetic equations. J. Differential Equations, 247(2):596–617, 2009

    2009

  25. [33]

    T. Nagai. Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. , 5(2):581–601, 1995

    1995

  26. [34]

    Nishihara, W

    K. Nishihara, W. Wang, and T. Yang. Lp-convergence rate to nonlinear diffusion waves for p-system with damping. J. Differential Equations, 161(1):191-218, 2000

    2000

  27. [35]

    B. Niu, B. Shi, and W. Wang. Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow. Discrete Contin. Dyn. Syst. doi:10.3934/dcds.2025014, 2025

    2025 doi

  28. [36]

    Shi and W

    B. Shi and W. Wang. Suppression of blow up by mixing in generalized Keller-Segel system with fractional dissipation. Commun. Math. Sci. , 18(5):1413–1440, 2020

    2020

  29. [37]

    Shi and W

    B. Shi and W. Wang. Global well-posedness for aggregation equation with time-space nonlocal operator and shear flow. Commun. Appl. Math. Comput. , 5(3):1274–1288, 2023

    2023

  30. [38]

    Souplet and M

    P. Souplet and M. Winkler. Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3. Comm. Math. Phys. , 367(2):665–681, 2019

    2019

  31. [39]

    E. Stein. Harmonic Analysis;Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton: Princeton Univ Press, 1993

    1993

  32. [40]

    Wang and Y

    W. Wang and Y. Wang. The Lp decay estimates for the chemotaxis-shallow water system. J. Math. Anal. Appl. , 474(1):640–665, 2019

    2019

  33. [41]

    S.-H. Yu. Nonlinear wave propagations over a Boltzmann shock profile. J. Amer. Math. Soc. , 23(4):1041–1118, 2010

    2010

  34. [42]

    L. Zeng, Z. Zhang, and R. Zi. Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Couette flow. J. Funct. Anal. , 280(10):108967, 2021. School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, P.R.China. Email addres...

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.