Pith. sign in

REVIEW 3 major objections 5 minor 43 references

Boundary layer analysis for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part I: Well-posedness

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read For the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity and Navier-slip boundary conditions, the leading outer system — the supercritical chemotaxis Euler equations — is locally well-posed, and every boundary-layer profile…

desk verdict Technically rich boundary-layer program for the chemotaxis-Navier-Stokes system; the profile estimates are solid, but the Euler well-posedness proof is missing its contraction argument and may require more regularity. read the letter →

arxiv 2608.05940 v1 pith:64KD5UMD submitted 2026-08-06 math.AP

Hui Wang , Wendong Wang , Lingling Zhao This is my paper · ORCID
classification math.AP MSC 35Q9276N0535B4435Q3035Q35
keywords Chemotaxis-Navier-StokessystemNavier-slipboundaryconditionlayerLogarithmicsensitivityWell-posednessAsymptoticexpansionSupercriticalchemotaxisEulerequationsWeightedanisotropicSobolevspaces
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

This paper studies the two-dimensional chemotaxis-Navier-Stokes system with logarithmic sensitivity and small viscosity, under Navier-slip boundary conditions, and aims to establish the boundary-layer structure it develops as the viscosity tends to zero. Using the Cole-Hopf transformation $v=-\nabla\ln c$, the authors write solutions as matched asymptotic expansions in powers of $\sqrt{\varepsilon}$, coupling an outer part governed by the Euler-like limit with an inner part living in the stretched variable $z=y/\sqrt{\varepsilon}$. The paper proves two well-posedness results: the leading outer system (the supercritical chemotaxis Euler equations) has a unique local classical solution, and each derived boundary-layer profile equation has a unique solution in weighted anisotropic Sobolev spaces. These results give the analytical foundation for the boundary-layer convergence analysis announced for Part II, and they show that the near-wall distributions of fluid velocity, bacterial density, and oxygen concentration are determined jointly by viscous and chemotactic effects.

What carries the argument

The central object is the matched asymptotic expansion (2.3), which splits each unknown into an outer profile and an inner boundary-layer profile in $z=y/\sqrt{\varepsilon}$, together with the Cole-Hopf transformation $v=-\nabla\ln c$ that removes the logarithmic singularity. For the outer system, the load-bearing identity is the div-curl structure $\nabla\times v=0$, $\nabla\cdot u=0$, which yields the elliptic estimate $\|\nabla^{k+1}u\|_{L^2}\le C\|\nabla^k\omega\|_{L^2}$ and the cancellation $\int \partial_t v\cdot(\nabla\partial_t v)\cdot u=0$ that controls the supercritical nonlinear terms. For the inner profiles, the workhorse is the weighted anisotropic Sobolev space $H^{k,m,l}$ with norm $\|(1+z^{2k})^{1/2}\partial_x^\alpha\partial_z^\gamma g\|_{L^2}$, together with the observation that $z(1+z^{2\kappa_{\alpha,j}})\le C(1+z^{2\kappa_{\alpha-\beta,j}})$, which allows the unbounded factor $z$ to be absorbed by assigning larger polynomial weights to lower-order tangential derivatives.

What would settle it

A direct numerical experiment on the $\varepsilon>0$ system with smooth, compatible data: compute the weighted $L^2$ error between $n^\varepsilon$ and the truncated expansion $n_{I,0}+\sqrt{\varepsilon}\,n_{B,1}+\varepsilon\,n_{B,2}$ at a fixed time as $\varepsilon\to 0$; if the error does not vanish at the expected rate, the matched-expansion ansatz (2.3) is not the true structure of the solution, and Theorems 2.1-2.2 would describe profiles that do not arise in the actual limit.

Watch

Extended reading notes

Core claim

The central claim is that the matched asymptotic expansion is not merely formal: the leading outer profiles satisfy the chemotaxis-Euler system (2.4), which is locally well-posed for initial data in $H^m_{xy}\times H^m_{xy}\times H^{m+1}_{xy}$ satisfying compatibility and curl-free conditions; and the inner profiles $v_{B,0}^2, n_{B,1}, v_{B,1}^1, v_{B,1}^2, n_{B,2}, u_{B,1}^1, u_{B,2}^2, v_{B,2}^1, u_{B,2}^1, u_{B,3}^2, p_{B,2}$ defined by (2.5)-(2.15) each admit a unique solution with weighted anisotropic Sobolev regularity, provided the outer solution has sufficiently high tangential regularity. The main difficulties overcome are the loss of diffusion in the Euler limit, handled through the curl-free structure of $v$ and elliptic div-curl estimates, and the unbounded normal transport terms $z\,a(t,x)\partial_z f$ in the inner equations, handled through polynomial weights and boundary homogenization.

Load-bearing premise

The analysis assumes that the matched asymptotic expansion (2.3) genuinely describes solutions of the $\varepsilon>0$ system—that is, that the true solution is an outer flow plus a rapidly decaying layer of width $\sqrt{\varepsilon}$; the proof of that convergence is deferred to Part II, so the profile equations studied here would be meaningless if the ansatz were not realized by the actual system.

Editorial extensions

If this is right

  • Part II's convergence proof now has the ingredients it needs: the outer solution exists on $[0,T]$ with the stated Sobolev regularity, and every inner profile it feeds is a well-posed solution of its equation in the weighted spaces.
  • For data with $m\ge 11$, the first several boundary-layer profiles through order $\varepsilon$ in density and velocity and their integral corrections are uniquely determined, so the formal expansion identifies the actual asymptotic limit rather than a spurious candidate.
  • The regularity transfer $7\le m_1\le m-4$ quantifies how much outer regularity survives into the boundary layer: constructing the first-order inner profiles costs between three and four derivatives of the outer solution.
  • The supercritical chemotaxis Euler equations inherit a local well-posedness theory from the energy estimates: no diffusion is needed in the $v$-equation when the initial data are curl-free and compatible.
  • The Navier-slip boundary conditions translate into explicit boundary data for the inner profiles, such as $v_{B,0}^2|_{z=0}=-v_{I,0}^2$, so the boundary layer is forced by the mismatch between the outer trace and the slip condition, not by an arbitrary ansatz.

Reading between the lines

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

  • The same weighted-estimate cascade likely extends to all higher-order profiles $j\ge 2$: since each new profile solves a linear equation whose source terms are already controlled, an induction analogous to Corollary 4.1 should yield regularity with a loss of roughly three tangential derivatives per level, a claim the paper leaves implicit.
  • The curl-free hypothesis on $v_0$ is the structural hinge of Theorem 2.1: the cancellation that kills the worst supercritical terms uses $\nabla\times v_0=0$, so dropping it would likely destroy local well-posedness or force a much shorter time of existence; this suggests a natural test problem for ill-posedness.
  • The explicit integral formulas (2.6), (2.13)-(2.15) turn the boundary layer into a computable object: comparing direct simulations of the $\varepsilon>0$ system against $n_{I,0}+\sqrt{\varepsilon}\,n_{B,1}+\varepsilon\,n_{B,2}$ would give a quantitative check of the ansatz before Part II appears.
  • Because the leading inner corrections satisfy $n_{B,0}=0$ and $u_{B,0}=0$ while $v_{B,0}^2$ differs from the outer trace, the boundary layer at leading order is purely a vertical redistribution of the transformed chemotactic velocity; the cell density itself changes only at order $\varepsilon^{1/2}$, which predicts a stratification of the oxygen gradient rather than of cell count near the wall.
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 / 5 minor

Summary. This paper is described as Part I of a two-part study of boundary layers for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity in the half-space. After the Cole-Hopf transformation (1.2), the authors write down a matched asymptotic expansion (2.3) in powers of sqrt(ε), formally derive the leading-order outer system (2.4) and a hierarchy of inner profile equations (2.5)-(2.15), and state two main theorems: local well-posedness of the supercritical chemotaxis-Euler system (2.4) with data in H^m x H^m x H^{m+1}, m ≥ 2 (Theorem 2.1), and weighted anisotropic Sobolev regularity and uniqueness for the boundary-layer profiles (Theorem 2.2). Section 3 contains the formal expansion, Section 4 provides a priori estimates for smooth solutions of (2.4), and Section 5 proves weighted energy estimates for the linearized profile equations. The paper does not address convergence of the approximate solutions; that is explicitly deferred to Part II.

Significance. If the two theorems are fully proved, the paper would be a useful technical foundation for the boundary-layer analysis of a singular chemotaxis-fluid system: the profile equations are derived in detail, the weighted estimates for the z-dependent transport terms are nontrivial, and the regularity hierarchy in Theorem 2.2 is explicitly organized. The paper is also honest in stating that the actual convergence/error analysis is deferred to Part II. The main strength is the systematic weighted anisotropic energy framework for the inner-layer equations with unbounded coefficients. The main weakness is that the existence part of Theorem 2.1 is not written down, and the linear profile problems in Section 5 are treated by a priori estimates rather than by a constructed existence argument. There are no fitted parameters and the estimates are deterministic, which is a positive feature. In its current state, the manuscript is a set of substantial technical estimates rather than a completed well-posedness proof.

major comments (3)
  1. [Section 4, Proof of Theorem 2.1] The proof of the central existence and uniqueness statement is a single sentence: "together with the contraction mapping argument and difference energy estimates, we establish Theorem 2.1." No contraction map, no solution space, and no difference estimate is written. This is load-bearing because Lemmas 4.4, 4.5 and Corollary 4.1 are a priori estimates for a given smooth solution and do not by themselves produce a solution. Moreover, for the difference of two solutions, δv satisfies ∂_t δv + ∇(u_1·δv + δu·v_2) − ∇δn = 0; after two derivatives and testing against ∇^2δv, the term ∫ ∇^2δu · ∇^3 v_2 · ∇^2δv arises, together with an analogous boundary trace term in the n-equation. Since Theorem 2.1 assumes only v_2 ∈ H^2 in the case m = 2, ∇^3 v_2 is not in L^2, so the difference estimate does not close at the stated regularity unless a cancellation is exhibited. Unless the contraction argument is supplied with m ≥ 3 or an additional cancellation is shown, Theorem 2.1 is not established as stated.
  2. [Section 5, Propositions 5.1 and 5.2] These propositions are phrased as "Then the solution f_B satisfies..." and are the basis for concluding that each profile equation "admits a unique solution" in Lemmas 5.1-5.6. However, the proofs are weighted energy estimates for a putative solution; no existence argument is given. The model problems (5.1) and (5.20) are linear parabolic equations with the unbounded transport term z a(t,x)∂_z f and, in (5.20), nonhomogeneous boundary data ∂_z g_B|_{z=0} = µ. Existence and uniqueness do not follow automatically from the displayed estimates. The paper should either provide a semigroup, Galerkin, or approximation argument, or cite a theorem that covers this unbounded-coefficient setting. As written, the uniqueness assertions in Theorem 2.2 are not proved.
  3. [Section 4, m-range in Theorem 2.1] Theorem 2.1 states m ≥ 2, but Lemma 4.4 treats H^2, Lemma 4.5 treats H^3, and Corollary 4.1 starts its induction at m ≥ 4. The proof of Theorem 2.1 gives no explanation of how the intermediate cases 2 ≤ m ≤ 3 are covered, nor which regularity and compatibility conditions the contraction mapping would require. At minimum, the statement should be restricted to the range for which the a priori estimates are actually proved, or the missing argument should be supplied.
minor comments (5)
  1. [Abstract and Introduction] There are several typographical issues, including "chemotacxis" in the abstract, "Lebegue spaces" in Section 2.1, and "eatimate" in Section 5; a careful copy-edit is needed.
  2. [Corollary 4.1] The inductive statement (4.54) uses the shorthand H_m in some places and H_mxy in others; the norms should be written consistently throughout the proof.
  3. [Section 3] The formal derivation is very long and many displayed identities are verified only by phrases such as "due to the rapid decay"; a table or diagram showing which profile feeds into which equation would substantially improve readability.
  4. [Theorem 2.2 and Remark 2.2] The restrictions m ≥ 11 and 7 ≤ m1 ≤ m-4 are stated but not derived in the proof of Theorem 2.2; the proof ends with a one-line reference to the preceding lemmas. A short derivation of these index restrictions should be added.
  5. [Abstract and Introduction] The paper repeatedly states that the convergence analysis and higher-order profiles are deferred to Part II; this should be explicitly identified as a limitation of Part I in the introduction, since the abstract's phrase 'boundary layer convergence' could be misread as a result established here.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the outer/inner well-posedness arguments are self-contained; the sole self-citation is contextual and non-load-bearing.

full rationale

The derivation chain is not circular. The Cole-Hopf transformation is introduced via prior external references [19,22], and the outer system (2.4) is simply the ε=0 specialization of the transformed problem; Theorem 2.1 rests on the a priori estimates in Lemmas 4.4-4.5 and Corollary 4.1, with the contraction step asserted rather than written out (a completeness gap, not a circularity). The boundary-layer profile equations (2.5)-(2.15) are obtained in Section 3 by direct matched-asymptotic expansion from the ansatz (2.3), and Theorem 2.2 follows by applying the linear well-posedness Propositions 5.1-5.2 to those profile equations with coefficients inherited from the outer solution; no fitted parameter or predicted quantity appears. The only self-citation is [21] (Li-Shi-Wang, co-authored by W. Wang), cited in the introduction and Remark 2.1 as comparable work; none of the estimates or equations in the paper reduce to that citation, and no uniqueness theorem is imported from the authors' prior work. The validity and convergence of the asymptotic ansatz are explicitly deferred to Part II, which is a stated limitation rather than circular reasoning. No equation in the paper reduces by construction to an input assumption.

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

The central claims rest on the formal asymptotic expansion ansatz (2.3) and on standard half-plane PDE estimates. No numerical parameters are fitted to data, and no new physical entities are introduced. The main unproven input is the validity of the boundary layer expansion itself, whose convergence is deferred to Part II.

assumptions (6)
  • domain assumption The Cole-Hopf transformation v = -∇ ln c = -∇c/c is valid, requiring c > 0 in the domain.
    Invoked in Section 1, equation (1.2), to transform the singular chemotaxis term into a transport-type system (1.4). If c touches zero, the transformation and the boundary condition (1.6) break down; no proof of positivity is given.
  • domain assumption The solution of (1.4)-(1.6) for ε>0 admits the matched asymptotic expansion (2.3) with inner profiles rapidly decreasing in z = y/√ε.
    The expansion is the starting point of Section 2.2 and Section 3; convergence justifying this ansatz is deferred to Part II, so the profile equations are derived under this unproven structural assumption.
  • domain assumption The outer limit system is exactly the ε=0 system (2.4), and the Navier-slip conditions (1.6) transfer to the boundary conditions in (2.4) and the inner layer boundary data (e.g., vB,0_2|z=0 = -vI,0_2).
    Used when matching orders in Section 3; this is part of the formal boundary layer framework.
  • standard math Lemma 4.1 (div-curl estimate ∥∇^{k+1}u∥_{L2} ≤ C∥∇^k ω∥_{L2} for divergence-free u with u2|y=0=0) holds in the half-plane.
    Cited from [8,33] and used throughout Section 4 to control velocity by vorticity.
  • domain assumption The compatibility conditions (M1)-(M4) are satisfied by the initial data; these are needed for the temporal regularity estimates in Lemma 4.4 and Lemma 4.5.
    Imposed in Theorem 2.1; if the data do not satisfy these conditions, the local well-posedness statement is not claimed.
  • standard math The trace inequality (5.40), ∥f(t,x,0)∥_{L2_T H^ι_x} ≤ C∥f∥_{L2_T H^{ι+1}_{xy}}, and boundary estimates (5.55)-(5.56) are valid.
    Used in Lemma 5.1 and later to bound boundary terms in the inner layer estimates.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Boundary layer analysis for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part I: Well-posedness." pith.science (2026). https://pith.science/paper/64KD5UMD

@misc{pith2026260805940,
  author       = {Pith},
  title        = {Pith review of: Boundary layer analysis for the 2D chemotaxis-Navier-Stokes system with logarithmic sensitivity, Part I: Well-posedness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64KD5UMD}},
  note         = {Machine review of arXiv:2608.05940}
}
abstract

This is the first part of a two-part work concerning the boundary layer convergence for chemotaxis-Navier-Stokes system in a two-dimensional half-space. In this paper, we investigate the chemotacxis-Navier-Stokes system with the logarithmic singularity under Navier-slip boundary conditions. More precisely, we perform an exact asymptotic expansion for the chemotaxis Navier-Stokes system with viscous coefficient $\varepsilon>0$, and obtain partial boundary layer profiles, establishing the well-posedness of the corresponding boundary layer profiles. Specially, we also establish the local well-posedness of solutions to the supercritical chemotaxis Euler equation (with $\varepsilon=0$) by overcoming the difficulty from the disappearance of diffusion terms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 41 canonical work pages

  1. [21]

    B. Li, F. Shi, W. Wang, Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition. arXiv preprint arXiv:2605.02394, 2026

    work page Pith review arXiv 2026

  2. [1]

    Adams, J.J.F

    A. Adams, J.J.F. Fournier, Sobolev Space, 2nd edn. Pure Appl. Math. 140 (2009)

    work page 2009

  3. [2]

    Braukhoff, B.Q

    M. Braukhoff, B.Q. Tang. Global solutions for chemotaxis-Navier-Stokes system with Robin boundary conditions. J. Differ. Equ. 269 (12) (2020) 10630-10669

    work page 2020

  4. [3]

    Carrillo, G

    J.A. Carrillo, G. Hong, Z. Wang, Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data. SIAM J. Math. Anal. 56 (6) (2024) 7576-7643. BOUNDARY LAYER OF THE 2D CHEMOTAXIS-NA VIER-STOKES SYSTEM 63

    work page 2024

  5. [4]

    Carrillo, J.Y

    J.A. Carrillo, J.Y. Li, and Z.A. Wang, Boundary spike-layer solutions of the singular Keller- Segel system: existence and stability. Proc. Lond. Math. Soc. 122 (3) (2021) 42-68

    work page 2021

  6. [5]

    Chen, J.M

    H. Chen, J.M. Li, K. Wang, On the vanishing viscosity limit of a chemotaxis model. Discrete Contin. Dyn. Syst. Ser. A. 40 (3) (2020) 1963-1987

    work page 2020

  7. [6]

    M. Chae, K. Kang, J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations. Commun. Partial Differ. Equations. 39 (7) (2014) 1205-1235

    work page 2014

  8. [7]

    Dombrowski, L

    C. Dombrowski, L. Cisneros, S. Chatkaew, R. Goldstein, and J. Kessler, Selfconcentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93 (2004) 098103

    work page 2004

Show all 43 references
  1. [8]

    Gilbarg, N.S

    D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition, Classics in Mathematics: SpringerVerlag; 2001

    1998

  2. [9]

    Hou, C.J

    Q. Hou, C.J. Liu, Y.G. Wang, Z. Wang, Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: one dimensional case. SIAM J. Math. Anal. 50 (3) (2018) 3058-3091

    2018

  3. [10]

    Hou, Boundary layer problem on chemotaxis-Navier-Stokes system with Robin boundary conditions

    Q. Hou, Boundary layer problem on chemotaxis-Navier-Stokes system with Robin boundary conditions. arXiv preprint arXiv:2205.08049, 2022

    2022 arXiv

  4. [11]

    Hou, Boundary layer effects induced by the fluid in a chemotaxis-Navier-Stokes system

    Q. Hou, Boundary layer effects induced by the fluid in a chemotaxis-Navier-Stokes system. arXiv preprint arXiv:2509.03028, 2025

    2025 arXiv

  5. [12]

    Q. Hou, Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane. J. Math. Pures Appl. 130 (2019) 251-287

    2019

  6. [13]

    Q. Hou, Z. Wang, K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis. J. Differ. Equ. 261 (2016) 5035-5070

    2016

  7. [14]

    Jin, Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions

    C.H. Jin, Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions. P. Roy. Soc. Edinb. A. 154 (2) (2024) 445-482

    2024

  8. [15]

    Keller, L.A

    E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (3) (1970) 399-415

    1970

  9. [16]

    Keller, L.A

    E.F. Keller, L.A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol. 30 (1971) 377-380

    1971

  10. [17]

    D. Li, T. Li, K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis. Math. Models Methods Appl. Sci. 21 (2011) 1631-1650

    2011

  11. [18]

    T. Li, R. Pan, K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemo- taxis. SIAM J. Appl. Math. 72 (1) (2012) 417-443

    2012

  12. [19]

    Levine, B.D

    H.A. Levine, B.D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997) 683-730

    1997

  13. [20]

    Levine, B.D

    H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton, A mathematical model for the roles of per- icytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis, Math. Biosci. 168 (2000) 71-115. 64 HUI W ANG, WENDONG W ANG, AND ...

    2000

  14. [22]

    T. Li, Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J. Differ. Equ. 250 (3) (2011) 1310-1333

    2011

  15. [23]

    Lee, Z.A

    C.C. Lee, Z.A. Wang, W. Yang, Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis. Nonlinearity, 33 (2020) 5111-5141

    2020

  16. [24]

    Liu, Large-time behavior in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with signal absorption

    J. Liu, Large-time behavior in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with signal absorption. J. Evol. Equations. 21 (2021) 5135-5170

    2021

  17. [25]

    Liu, Boundedness and asymptotic stabilization in a two-dimensional chemotaxis- Navier¨CStokes system with sub-logarithmic sensitivity

    J. Liu, Boundedness and asymptotic stabilization in a two-dimensional chemotaxis- Navier¨CStokes system with sub-logarithmic sensitivity. NoDEA. 32 (3) (2025) 42

    2025

  18. [26]

    H. Li, K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis. J. Differ. Equ. 258 (2) (2015) 302-338

    2015

  19. [27]

    Meng, W.Q

    L. Meng, W.Q. Xu, S. Wang, On the vanishing viscosity limit for a 3-D system arising from the Keller-Segel model. Math. Methods Appl. Sci. 43 (2) (2020) 920-938

    2020

  20. [28]

    Meng, W.Q

    L. Meng, W.Q. Xu, S. Wang, Boundary layer analysis for a 2-D Keller-Segel model. Open Math. 18 (1) (2020) 1895-1914

    2020

  21. [29]

    Nossal, Boundary movement of chemotactic bacterial populations, Math

    R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci. 13 (1972) 397-406

    1972

  22. [30]

    Peng, Z.A

    H. Peng, Z.A. Wang, K. Zhao, et al., Boundary layers and stabilization of the singular Keller- Segel system. Kinet. Rel. Models. 11 (5) (2018) 1085-1123

    2018

  23. [31]

    L.G Rebholz, D. Wang, Z. Wang, et al., Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete Contin. Dyn. Syst. Ser. A. 39 (7) (2019) 3789¨C3838

    2019

  24. [32]

    Tuval, L

    I. Tuval, L. Cisneros, C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines. P Natl Acad Sci Usa. 102 (7) (2005) 2277-2282

    2005

  25. [33]

    Z. Tan, S. Xu, Global regularity for 2D Boussinesq equations with partial viscosity in the half plane. Math. Methods Appl. Sci. 46 (6) (2023) 6484-6505

    2023

  26. [34]

    Weibull, Movement

    C. Weibull, Movement. In: The Bacteria. Academic Press, New York. 1 (1960) 153-205

    1960

  27. [35]

    Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system

    M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system. Arch. Ration. Mech. Anal. 211 (2) (2014) 455-487

    2014

  28. [36]

    Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops

    M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops. Comm. Partial Differential Equations. 37 (2) (2012) 319-351

    2012

  29. [37]

    Y.L. Wang„ M. Winkler, Z.Y. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity. Ann Scuola Norm-Sci. (2018) 421-466. BOUNDARY LAYER OF THE 2D CHEMOTAXIS-NA VIER-STOKES SYSTEM 65

    2018

  30. [38]

    Z. Wang, Z. Xiang, P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis. J. Differ. Equ. 260 (2016) 2225-2258

    2016

  31. [39]

    Zhang, Blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equa- tions

    Q. Zhang, Blowup criterion of smooth solutions for the incompressible chemotaxis-Euler equa- tions. ZAMM-Z. Angew. Math. Me. 96 (4) (2016) 466-476

    2016

  32. [40]

    Zheng, Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity

    J. Zheng, Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Ann. Mat. Pura Appl. 201 (1) (2022) 243-288

    2022

  33. [41]

    Zhang, Y

    Q. Zhang, Y. Zhang, On the global well-posedness for the 2D incompressible Keller-Segel- Navier-Stokes equations. ZAMM-Z. Angew. Math. Me. 99 (11) (2019)

    2019

  34. [42]

    Y. Zeng, K. Zhao, Recent results for the logarithmic Keller-Segel-Fisher/KPP system. Bol. Soc. Parana. Mat. 38 (3) (2020) 37-48

    2020

  35. [43]

    Zhang, X.X

    Q. Zhang, X.X. Zheng, Global well-posedness of axisymmetric solution to the 3D axisymmetric chemotaxis-Navier-Stokes equations with logistic source. J. Differ. Equ. 274 (2021) 576-612. (Hui Wang) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024,...

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.