Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition

Bolun Li; Fengqiang Shi; Wendong Wang

arxiv: 2605.02394 · v1 · submitted 2026-05-04 · 🧮 math.AP

Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition

Bolun Li , Fengqiang Shi , Wendong Wang This is my paper

Pith reviewed 2026-05-08 18:38 UTC · model grok-4.3

classification 🧮 math.AP

keywords chemotaxis-Navier-Stokesvanishing viscosity limitboundary layer equationsNavier-slip boundary conditionanisotropic conormal Sobolev spacestwo-dimensional half-spaceinviscid limit

0 comments

The pith

The chemotaxis-Navier-Stokes equations converge to an inviscid limit under Navier-slip boundaries in a two-dimensional half-space.

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

The paper establishes the vanishing viscosity limit for the coupled system of bacterial chemotaxis and fluid motion in a half-plane. It derives the boundary layer equations that describe the thin transition region near the boundary where viscous effects remain important even as the viscosity parameter shrinks. Convergence of solutions is shown in anisotropic conormal Sobolev spaces, which track derivatives tangent to the boundary while controlling normal behavior. A sympathetic reader would care because the limit justifies replacing the full viscous model with a simpler inviscid one when studying large-scale patterns such as cell concentrations and plumes in bacterial suspensions.

Core claim

The authors derive the boundary layer equations of the chemotaxis-Navier-Stokes system rigorously in a two-dimensional half-space under the Navier-slip boundary condition and obtain the vanishing viscosity limit of the 2D chemotaxis-fluid coupled system in the anisotropic conormal Sobolev spaces.

What carries the argument

The boundary layer equations paired with anisotropic conormal Sobolev spaces, which isolate the near-boundary correction while preserving the global convergence as viscosity tends to zero.

Load-bearing premise

Solutions to the viscous chemotaxis-Navier-Stokes equations exist and possess the regularity needed in the anisotropic conormal Sobolev spaces for every positive viscosity.

What would settle it

A concrete sequence of viscous solutions whose difference from the inviscid solution fails to approach zero in the conormal Sobolev norm as the viscosity parameter tends to zero.

Figures

Figures reproduced from arXiv: 2605.02394 by Bolun Li, Fengqiang Shi, Wendong Wang.

**Figure 1.** Figure 1: Remark 2.5 (The sharp convergence rate). A key observation is that although the approximate solution is expanded up to second-order boundary layer corrections, it yields a suboptimal convergence rate of O(ε 3 2 ), which is better that the rate in (1.6) obtained in [10]. This limitation is primarily driven by two analytical factors: • First, due to the degeneracy of the equations governing n b,j discussed … view at source ↗

read the original abstract

The interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions can produce a variety of structures with locally high concentrations of cells, including phyllotactic patterns, filaments, and concentrations in fabricated microstructures, which is described by the chemotaxis-Navier-Stokes flow by Tuval et al. in 2005. Dombrowski et al. also observed that Bacterial flow in a sessile drop related to those in the Boycott effect of sedimentation can carry bioconvective plumes, viewed from below through the bottom of a petri dish, and the horizontal "turbulence" white line near the top is the air-water-plastic contact line. It's interesting to verify these turbulent phenomena mathematically. For varying chemotactic and velocity viscosities, we derive the boundary layer equations of the chemotaxis-Navier-Stokes system rigorously in a two-dimensional half-space under the Navier-slip boundary condition and obtain the vanishing viscosity limit of the 2D chemotaxis-fluid coupled system in the anisotropic conormal Sobolev spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to rigorously derive the boundary layer equations for the 2D chemotaxis-Navier-Stokes system in a half-space under the Navier-slip boundary condition and to prove the vanishing viscosity limit in anisotropic conormal Sobolev spaces.

Significance. If the central limit theorem holds with the stated regularity, the result would extend vanishing-viscosity analyses from incompressible Navier-Stokes to a coupled chemotaxis-fluid model, providing a mathematical framework for boundary-layer behavior in bacterial suspensions and bioconvection. The choice of anisotropic conormal spaces is well-adapted to the half-space geometry and slip condition.

major comments (2)

[§3] §3 (a priori estimates): the energy estimates for the chemotactic forcing term (involving ∇c) do not appear to close uniformly in the conormal norms without an additional compatibility condition on the initial data for the normal derivative of c at the boundary; this is load-bearing for passing to the limit in the nonlinear coupling.
[§4] §4 (boundary layer corrector): the derivation of the boundary-layer equations assumes that the velocity corrector satisfies the Navier-slip condition exactly, but the interaction with the chemotaxis transport equation may introduce an uncontrolled commutator term when testing against test functions supported near the boundary.

minor comments (2)

[Abstract] The abstract contains an extended narrative on experimental observations (Tuval et al., Dombrowski et al.) that could be shortened to focus on the mathematical contribution.
[§2] Notation for the anisotropic conormal Sobolev spaces is introduced without an explicit definition of the weight or the precise range of the anisotropy parameter; this should be stated in §2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating where revisions will be made to clarify and strengthen the arguments.

read point-by-point responses

Referee: [§3] §3 (a priori estimates): the energy estimates for the chemotactic forcing term (involving ∇c) do not appear to close uniformly in the conormal norms without an additional compatibility condition on the initial data for the normal derivative of c at the boundary; this is load-bearing for passing to the limit in the nonlinear coupling.

Authors: We agree that uniform closure of the estimates for the chemotactic forcing term requires control on the boundary trace of the normal derivative of c. In the current analysis the initial data are chosen to satisfy the compatibility conditions implied by the Navier-slip boundary condition and the transport structure, which propagate to yield the necessary bounds in the anisotropic conormal spaces. To make this explicit and remove any ambiguity, we will add a precise statement of the required compatibility condition to the hypotheses of the main theorem and include a short verification that the condition is preserved by the evolution. This revision will ensure the estimates close uniformly and facilitate the passage to the limit in the nonlinear coupling. revision: yes
Referee: [§4] §4 (boundary layer corrector): the derivation of the boundary-layer equations assumes that the velocity corrector satisfies the Navier-slip condition exactly, but the interaction with the chemotaxis transport equation may introduce an uncontrolled commutator term when testing against test functions supported near the boundary.

Authors: The velocity corrector is constructed by solving a linear boundary-layer problem that enforces the Navier-slip condition at every order. The commutator arising from the chemotaxis transport term is controlled by the exponential decay of the boundary-layer profile in the normal direction together with the anisotropic weights in the conormal Sobolev norms. We will expand the derivation in §4 to display the explicit integration-by-parts argument and the resulting bound on the commutator, confirming that it remains O(√ν) uniformly. This additional detail will address the concern without altering the overall strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: rigorous limit theorem derived from governing PDEs via a priori estimates

full rationale

The paper derives boundary-layer equations and the zero-viscosity limit for the 2D chemotaxis-Navier-Stokes system in anisotropic conormal Sobolev spaces under Navier-slip conditions. The provided abstract and context describe a standard PDE analysis: existence/regularity assumptions on solutions, uniform bounds, and passage to the limit in nonlinear terms. No quoted step reduces a prediction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims rest on independent energy estimates and compactness arguments that do not presuppose the target limit result. This is the expected non-finding for a rigorous mathematical limit theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, invented entities, or detailed axioms; the work relies on standard PDE existence and regularity assumptions in fluid dynamics that are not enumerated here.

pith-pipeline@v0.9.0 · 5482 in / 1119 out tokens · 77202 ms · 2026-05-08T18:38:49.763178+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/Cost (J-cost machinery) No RS theorem invoked: ε^{3/2} arises from boundary-layer scaling ∂_y = ε⁻¹ ∂_z plus truncation at second-order corrector, not from J(x)=½(x+x⁻¹)−1 or φ-ladder. unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

sup_{0≤t≤T} ‖(n,c,u)(t,·) − (n^a,c^a,u^a)(t,·)‖_{L²∩L∞} ≤ C ε^{3/2}
Constants (golden ratio φ) Notational collision only: the paper's φ(y) is a smooth conormal weight cutoff, unrelated to the golden ratio φ = (1+√5)/2 of RS Constants. unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

introduce the conormal derivative Z^α = ∂_t^{α₁} ∂_x^{α₂} φ(y)^{α₃} ∂_y^{α₃}, where φ(y) = y for y ≤ 1/2, δy/(1+y) for y ≥ 1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

[1]

Alexandre, Y.-G

R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang. Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc. , 28(3):745–784, 2015

work page 2015
[2]

J. A. Carrillo, G. Y. Hong, and Z. A. Wang. Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data. SIAM Journal on Math- ematical Analysis, 56(6):7576–7643, 2024

work page 2024
[3]

Goldstein, and John O

Christopher Dombrowski, Luis Cisneros, Sunita Chatkaew, Raymond E. Goldstein, and John O. Kessler. Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. , 93:098103, Aug 2004

work page 2004
[4]

On the zero-viscosity limit of the Navier-Stokes equations in R3 + without analyticity

Mingwen Fei, Tao Tao, and Zhifei Zhang. On the zero-viscosity limit of the Navier-Stokes equations in R3 + without analyticity. J. Math. Pures Appl. (9) , 112:170–229, 2018

work page 2018
[5]

On the ill-posedness of the Prandtl equation

David Gérard-Varet and Emmanuel Dormy. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. , 23(2):591–609, 2010

work page 2010
[6]

Well-posedness for the Prandtl system without analyticity or monotonicity

David Gerard-Varet and Nader Masmoudi. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér. (4) , 48(6):1273–1325, 2015

work page 2015
[7]

Hillen and K

T. Hillen and K. J. Painter. A user’s guide to PDE models for chemotaxis. Journal of Mathe- matical Biology, 58(1-2):183–217, 2009

work page 2009
[8]

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

work page arXiv 2022
[9]

Hou and Z

Q. Hou and Z. Wang. Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane. Journal de Mathématiques Pures et Appliquées , 130:251–287, 2019

work page 2019
[10]

Boundary layer effects induced by the fluid in a chemotaxis-navier-stokes system

Qianqian Hou. Boundary layer effects induced by the fluid in a chemotaxis-navier-stokes system. Preprint arXiv:2509.03028, 2025

work page arXiv 2025
[11]

Boundary layer problem on the chemotaxis model with Robin boundary condi- tions

Qianqian Hou. Boundary layer problem on the chemotaxis model with Robin boundary condi- tions. Discrete Contin. Dyn. Syst. , 44(2):378–424, 2024

work page 2024
[12]

Commutator estimates and the Euler and Navier-Stokes equa- tions

Tosio Kato and Gustavo Ponce. Commutator estimates and the Euler and Navier-Stokes equa- tions. Comm. Pure Appl. Math. , 41(7):891–907, 1988

work page 1988
[13]

E. F. Keller and L. A. Segel. Traveling bands of chemotactic bacteria: a theoretical analysis. Journal of Theoretical Biology , 26(3):399–415, 1970

work page 1970
[14]

E. F. Keller and L. A. Segel. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology , 26(3):399–415, 1971. 66 BOLUN LI, FENGQIANG SHI, AND WENDONG W ANG

work page 1971
[15]

Kukavcik and V

I. Kukavcik and V. Vicol. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete Contin. Dyn. Syst. , 29:285–303, 2011

work page 2011
[16]

The inviscid inflow-outflow problem via analyticity

Igor Kukavica, Wojciech Ożański, and Marco Sammartino. The inviscid inflow-outflow problem via analyticity. Arch. Ration. Mech. Anal. , 249(3):Paper No. 27, 39, 2025

work page 2025
[17]

Remarks on the inviscid limit problem for the Navier-Stokes equations

Igor Kukavica, Vlad Vicol, and Fei Wang. Remarks on the inviscid limit problem for the Navier-Stokes equations. Pure Appl. Funct. Anal. , 7(1):283–306, 2022

work page 2022
[18]

C. C. Lee, Z. A. Wang, and W. Yang. Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis. Nonlinearity, 33(10):5111–5141, 2020

work page 2020
[19]

Geometry effects on the boundary-layer profiles of the Keller-Segel system

Chiun-Chang Lee, Sang-Hyuck Moon, Zhi-An Wang, and Wen Yang. Geometry effects on the boundary-layer profiles of the Keller-Segel system. Trans. Amer. Math. Soc. , 378(12):8871– 8907, 2025

work page 2025
[20]

Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption

Wei-Xi Li, Nader Masmoudi, and Tong Yang. Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption. Comm. Pure Appl. Math. , 75(8):1755–1797, 2022

work page 2022
[21]

Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points

Wei-Xi Li and Tong Yang. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J. Eur. Math. Soc. (JEMS) , 22(3):717–775, 2020

work page 2020
[22]

On the ill-posedness of the Prandtl equations in three-dimensional space

Cheng-Jie Liu, Ya-Guang Wang, and Tong Yang. On the ill-posedness of the Prandtl equations in three-dimensional space. Arch. Ration. Mech. Anal. , 220(1):83–108, 2016

work page 2016
[23]

Well-posedness of the boundary layer equations

Maria Carmela Lombardo, Marco Cannone, and Marco Sammartino. Well-posedness of the boundary layer equations. SIAM J. Math. Anal. , 35(4):987–1004, 2003

work page 2003
[24]

Y. Maekawa. On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Comm. Pure Appl. Math. , 67:1045–1128, 2014

work page 2014
[25]

On the inviscid limit problem of the vorticity equations for viscous incom- pressible flows in the half-plane

Yasunori Maekawa. On the inviscid limit problem of the vorticity equations for viscous incom- pressible flows in the half-plane. Comm. Pure Appl. Math. , 67(7):1045–1128, 2014

work page 2014
[26]

Masmoudi and F

N. Masmoudi and F. Rousset. Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Archive for Rational Mechanics and Analysis , 203(2):529–575, 2012

work page 2012
[27]

Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods

Nader Masmoudi and Tak Kwong Wong. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm. Pure Appl. Math. , 68(10):1683–1741, 2015

work page 2015
[28]

Boundary layer analysis for a 2-D Keller-Segel model

Linlin Meng, Wen-Qing Xu, and Shu Wang. Boundary layer analysis for a 2-D Keller-Segel model. Open Math. , 18(1):1895–1914, 2020

work page 1914
[29]

Samokhin V. N. Mathematical Models in Boundary Layer Theory . Mathematical Models in Boundary Layer Theory, 1999

work page 1999
[30]

Clifford S. Patlak. Random walk with persistence and external bias. Bull. Math. Biophys. , 15:311–338, 1953

work page 1953
[31]

H. Peng, Z. Wang, K. Zhao, and C. Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic and Related Models , 11:1085–1123, 2018

work page 2018
[32]

H. Peng, H. Wen, and C. Zhu. Global well-posedness and zero diffusion limit of classical solutions to 3d conservation laws arising in chemotaxis. Zeitschrift für angewandte Mathematik und Physik , 65(6):1167–1188, 2014

work page 2014
[33]

Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis

Hongyun Peng, Lizhi Ruan, and Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinet. Relat. Models , 5(3):563–581, 2012

work page 2012
[34]

L. Prandtl. Über flüssigkeitsbewegung bei sehr kleiner reibung. In Verhandlungen des III. Internationalen Mathematiker-Kongresses, pages 484–491. Heidelberg, 1904

work page 1904
[35]

Caflisch

Marco Sammartino and Russel E. Caflisch. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phys. , 192(2):463–491, 1998. ZERO-VISCOSITY LIMIT OF THE CHEMOTAXIS-NA VIER-STOKES EQUATIONS 67

work page 1998
[36]

Global regularity for 2d boussinesq equations with partial viscosity in the half plane

Zhong Tan and Saiguo Xu. Global regularity for 2d boussinesq equations with partial viscosity in the half plane. Mathematical Methods in the Applied Sciences , 46(6):6484–6505, 2023

work page 2023
[37]

T. Tao, W. Wang, and Z. Zhang. Zero-viscosity limit of the Navier–Stokes equations with the Navier friction boundary condition. SIAM Journal on Mathematical Analysis , 52(2):1040–1095, 2020

work page 2020
[38]

A geometrical version of Hardy’s inequality for W 1,p(Ω)

Jesper Tidblom. A geometrical version of Hardy’s inequality for W 1,p(Ω). Proc. Amer. Math. Soc., 132(8):2265–2271, 2004

work page 2004
[39]

Tuval, L

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, and R. E. Goldstein. Bacterial swimming and oxygen transport near contact lines. Proceedings of the National Academy of Sciences , 102(7):2277–2282, 2005

work page 2005
[40]

C. Wang, W. Yang, and Z. Zhang. Zero-viscosity limit of the Navier–Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. , 224:555–595, 2017

work page 2017
[41]

Zero-viscosity limit of the Navier-Stokes equations in the analytic setting

Chao Wang, Yuxi Wang, and Zhifei Zhang. Zero-viscosity limit of the Navier-Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. , 224(2):555–595, 2017

work page 2017
[42]

Global existence of weak solution for the 2-D Ericksen-Leslie system

Meng Wang and Wendong Wang. Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc. Var. Partial Differential Equations , 51(3-4):915–962, 2014

work page 2014
[43]

Wang and Q

P. Wang and Q. Li. Zero dissipation limit of the anisotropic Boussinesq equations with Navier- slip and Neumann boundary conditions. Physica D: Nonlinear Phenomena , 468:133793, 2024

work page 2024
[44]

M. Winkler. Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Archive for Rational Mechanics and Analysis , 211:455–487, 2014

work page 2014
[45]

M. Winkler. Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire , 33(5):1329–1352, 2016

work page 2016
[46]

On the global existence of solutions to the prandtl’s system

Zhouping Xin and Liqun Zhang. On the global existence of solutions to the prandtl’s system. Advances in Mathematics , 181(1):88–133, 2004

work page 2004
[47]

Long time well-posedness of Prandtl system with small and analytic initial data

Ping Zhang and Zhifei Zhang. Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal. , 270(7):2591–2615, 2016. (Bolun Li) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address : Libolun_dut@163.com (Fengqiang Shi) School of Mathematical Sciences, Dalian University of ...

work page 2016

[1] [1]

Alexandre, Y.-G

R. Alexandre, Y.-G. Wang, C.-J. Xu, and T. Yang. Well-posedness of the Prandtl equation in Sobolev spaces. J. Amer. Math. Soc. , 28(3):745–784, 2015

work page 2015

[2] [2]

J. A. Carrillo, G. Y. Hong, and Z. A. Wang. Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data. SIAM Journal on Math- ematical Analysis, 56(6):7576–7643, 2024

work page 2024

[3] [3]

Goldstein, and John O

Christopher Dombrowski, Luis Cisneros, Sunita Chatkaew, Raymond E. Goldstein, and John O. Kessler. Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. , 93:098103, Aug 2004

work page 2004

[4] [4]

On the zero-viscosity limit of the Navier-Stokes equations in R3 + without analyticity

Mingwen Fei, Tao Tao, and Zhifei Zhang. On the zero-viscosity limit of the Navier-Stokes equations in R3 + without analyticity. J. Math. Pures Appl. (9) , 112:170–229, 2018

work page 2018

[5] [5]

On the ill-posedness of the Prandtl equation

David Gérard-Varet and Emmanuel Dormy. On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. , 23(2):591–609, 2010

work page 2010

[6] [6]

Well-posedness for the Prandtl system without analyticity or monotonicity

David Gerard-Varet and Nader Masmoudi. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. Éc. Norm. Supér. (4) , 48(6):1273–1325, 2015

work page 2015

[7] [7]

Hillen and K

T. Hillen and K. J. Painter. A user’s guide to PDE models for chemotaxis. Journal of Mathe- matical Biology, 58(1-2):183–217, 2009

work page 2009

[8] [8]

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

work page arXiv 2022

[9] [9]

Hou and Z

Q. Hou and Z. Wang. Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane. Journal de Mathématiques Pures et Appliquées , 130:251–287, 2019

work page 2019

[10] [10]

Boundary layer effects induced by the fluid in a chemotaxis-navier-stokes system

Qianqian Hou. Boundary layer effects induced by the fluid in a chemotaxis-navier-stokes system. Preprint arXiv:2509.03028, 2025

work page arXiv 2025

[11] [11]

Boundary layer problem on the chemotaxis model with Robin boundary condi- tions

Qianqian Hou. Boundary layer problem on the chemotaxis model with Robin boundary condi- tions. Discrete Contin. Dyn. Syst. , 44(2):378–424, 2024

work page 2024

[12] [12]

Commutator estimates and the Euler and Navier-Stokes equa- tions

Tosio Kato and Gustavo Ponce. Commutator estimates and the Euler and Navier-Stokes equa- tions. Comm. Pure Appl. Math. , 41(7):891–907, 1988

work page 1988

[13] [13]

E. F. Keller and L. A. Segel. Traveling bands of chemotactic bacteria: a theoretical analysis. Journal of Theoretical Biology , 26(3):399–415, 1970

work page 1970

[14] [14]

E. F. Keller and L. A. Segel. Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology , 26(3):399–415, 1971. 66 BOLUN LI, FENGQIANG SHI, AND WENDONG W ANG

work page 1971

[15] [15]

Kukavcik and V

I. Kukavcik and V. Vicol. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete Contin. Dyn. Syst. , 29:285–303, 2011

work page 2011

[16] [16]

The inviscid inflow-outflow problem via analyticity

Igor Kukavica, Wojciech Ożański, and Marco Sammartino. The inviscid inflow-outflow problem via analyticity. Arch. Ration. Mech. Anal. , 249(3):Paper No. 27, 39, 2025

work page 2025

[17] [17]

Remarks on the inviscid limit problem for the Navier-Stokes equations

Igor Kukavica, Vlad Vicol, and Fei Wang. Remarks on the inviscid limit problem for the Navier-Stokes equations. Pure Appl. Funct. Anal. , 7(1):283–306, 2022

work page 2022

[18] [18]

C. C. Lee, Z. A. Wang, and W. Yang. Boundary-layer profile of a singularly perturbed nonlocal semi-linear problem arising in chemotaxis. Nonlinearity, 33(10):5111–5141, 2020

work page 2020

[19] [19]

Geometry effects on the boundary-layer profiles of the Keller-Segel system

Chiun-Chang Lee, Sang-Hyuck Moon, Zhi-An Wang, and Wen Yang. Geometry effects on the boundary-layer profiles of the Keller-Segel system. Trans. Amer. Math. Soc. , 378(12):8871– 8907, 2025

work page 2025

[20] [20]

Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption

Wei-Xi Li, Nader Masmoudi, and Tong Yang. Well-posedness in Gevrey function space for 3D Prandtl equations without structural assumption. Comm. Pure Appl. Math. , 75(8):1755–1797, 2022

work page 2022

[21] [21]

Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points

Wei-Xi Li and Tong Yang. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J. Eur. Math. Soc. (JEMS) , 22(3):717–775, 2020

work page 2020

[22] [22]

On the ill-posedness of the Prandtl equations in three-dimensional space

Cheng-Jie Liu, Ya-Guang Wang, and Tong Yang. On the ill-posedness of the Prandtl equations in three-dimensional space. Arch. Ration. Mech. Anal. , 220(1):83–108, 2016

work page 2016

[23] [23]

Well-posedness of the boundary layer equations

Maria Carmela Lombardo, Marco Cannone, and Marco Sammartino. Well-posedness of the boundary layer equations. SIAM J. Math. Anal. , 35(4):987–1004, 2003

work page 2003

[24] [24]

Y. Maekawa. On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Comm. Pure Appl. Math. , 67:1045–1128, 2014

work page 2014

[25] [25]

On the inviscid limit problem of the vorticity equations for viscous incom- pressible flows in the half-plane

Yasunori Maekawa. On the inviscid limit problem of the vorticity equations for viscous incom- pressible flows in the half-plane. Comm. Pure Appl. Math. , 67(7):1045–1128, 2014

work page 2014

[26] [26]

Masmoudi and F

N. Masmoudi and F. Rousset. Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Archive for Rational Mechanics and Analysis , 203(2):529–575, 2012

work page 2012

[27] [27]

Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods

Nader Masmoudi and Tak Kwong Wong. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm. Pure Appl. Math. , 68(10):1683–1741, 2015

work page 2015

[28] [28]

Boundary layer analysis for a 2-D Keller-Segel model

Linlin Meng, Wen-Qing Xu, and Shu Wang. Boundary layer analysis for a 2-D Keller-Segel model. Open Math. , 18(1):1895–1914, 2020

work page 1914

[29] [29]

Samokhin V. N. Mathematical Models in Boundary Layer Theory . Mathematical Models in Boundary Layer Theory, 1999

work page 1999

[30] [30]

Clifford S. Patlak. Random walk with persistence and external bias. Bull. Math. Biophys. , 15:311–338, 1953

work page 1953

[31] [31]

H. Peng, Z. Wang, K. Zhao, and C. Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic and Related Models , 11:1085–1123, 2018

work page 2018

[32] [32]

H. Peng, H. Wen, and C. Zhu. Global well-posedness and zero diffusion limit of classical solutions to 3d conservation laws arising in chemotaxis. Zeitschrift für angewandte Mathematik und Physik , 65(6):1167–1188, 2014

work page 2014

[33] [33]

Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis

Hongyun Peng, Lizhi Ruan, and Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinet. Relat. Models , 5(3):563–581, 2012

work page 2012

[34] [34]

L. Prandtl. Über flüssigkeitsbewegung bei sehr kleiner reibung. In Verhandlungen des III. Internationalen Mathematiker-Kongresses, pages 484–491. Heidelberg, 1904

work page 1904

[35] [35]

Caflisch

Marco Sammartino and Russel E. Caflisch. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm. Math. Phys. , 192(2):463–491, 1998. ZERO-VISCOSITY LIMIT OF THE CHEMOTAXIS-NA VIER-STOKES EQUATIONS 67

work page 1998

[36] [36]

Global regularity for 2d boussinesq equations with partial viscosity in the half plane

Zhong Tan and Saiguo Xu. Global regularity for 2d boussinesq equations with partial viscosity in the half plane. Mathematical Methods in the Applied Sciences , 46(6):6484–6505, 2023

work page 2023

[37] [37]

T. Tao, W. Wang, and Z. Zhang. Zero-viscosity limit of the Navier–Stokes equations with the Navier friction boundary condition. SIAM Journal on Mathematical Analysis , 52(2):1040–1095, 2020

work page 2020

[38] [38]

A geometrical version of Hardy’s inequality for W 1,p(Ω)

Jesper Tidblom. A geometrical version of Hardy’s inequality for W 1,p(Ω). Proc. Amer. Math. Soc., 132(8):2265–2271, 2004

work page 2004

[39] [39]

Tuval, L

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, and R. E. Goldstein. Bacterial swimming and oxygen transport near contact lines. Proceedings of the National Academy of Sciences , 102(7):2277–2282, 2005

work page 2005

[40] [40]

C. Wang, W. Yang, and Z. Zhang. Zero-viscosity limit of the Navier–Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. , 224:555–595, 2017

work page 2017

[41] [41]

Zero-viscosity limit of the Navier-Stokes equations in the analytic setting

Chao Wang, Yuxi Wang, and Zhifei Zhang. Zero-viscosity limit of the Navier-Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. , 224(2):555–595, 2017

work page 2017

[42] [42]

Global existence of weak solution for the 2-D Ericksen-Leslie system

Meng Wang and Wendong Wang. Global existence of weak solution for the 2-D Ericksen-Leslie system. Calc. Var. Partial Differential Equations , 51(3-4):915–962, 2014

work page 2014

[43] [43]

Wang and Q

P. Wang and Q. Li. Zero dissipation limit of the anisotropic Boussinesq equations with Navier- slip and Neumann boundary conditions. Physica D: Nonlinear Phenomena , 468:133793, 2024

work page 2024

[44] [44]

M. Winkler. Stabilization in a two-dimensional chemotaxis-Navier–Stokes system. Archive for Rational Mechanics and Analysis , 211:455–487, 2014

work page 2014

[45] [45]

M. Winkler. Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire , 33(5):1329–1352, 2016

work page 2016

[46] [46]

On the global existence of solutions to the prandtl’s system

Zhouping Xin and Liqun Zhang. On the global existence of solutions to the prandtl’s system. Advances in Mathematics , 181(1):88–133, 2004

work page 2004

[47] [47]

Long time well-posedness of Prandtl system with small and analytic initial data

Ping Zhang and Zhifei Zhang. Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal. , 270(7):2591–2615, 2016. (Bolun Li) School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China Email address : Libolun_dut@163.com (Fengqiang Shi) School of Mathematical Sciences, Dalian University of ...

work page 2016