Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

Hideo Kozono; Matthias Hieber; Patrick Tolksdorf; Sylvie Monniaux

REVIEW 2 major objections 5 minor 1 cited by

Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read On bounded Lipschitz domains in two and three dimensions, the Keller-Segel-Navier-Stokes system and its chemotaxis-consumption variant admit unique local strong solutions in critical Besov spaces, global strong solutions near equilibria…

desk verdict The local well-posedness framework on Lipschitz domains is genuinely new and mostly solid, but the stability theorem as stated is false for f=0 because it omits a necessary smallness condition on |u_s|. read the letter →

arxiv 2505.04503 v1 pith:CJYOEWZ3 submitted 2025-05-07 math.AP math.FA

Matthias Hieber , Hideo Kozono , Sylvie Monniaux , Patrick Tolksdorf This is my paper · ORCID

classification math.APmath.FA MSC 35Q3035Q9292C17

keywords Keller-Segel-Navier-Stokessystemchemotaxis-consumptionmodelLipschitzdomainscriticalBesovspacesmaximalL^pregularitystrongwell-posednessexponentialstabilitypositivity

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 proves that the Keller-Segel-Navier-Stokes system, which couples bacterial chemotaxis to an incompressible buoyancy-driven fluid, is strongly well-posed on bounded Lipschitz domains in dimensions two and three. For initial data in scaling-critical Besov spaces, a unique local strong solution exists. Near nontrivial equilibria, sufficiently small data produce global solutions that converge exponentially to the equilibrium, and for smoother data the solutions are globally bounded in space and time while preserving nonnegativity of the population density. The same results are established for the chemotaxis-consumption variant. The significance is that the theory works without any smoothness of the boundary, which matters for realistic containers and for domains with corners.

What carries the argument

The argument recasts the system as a semilinear evolution equation ∂_t U + A U = Φ(U,U) on a product space whose components are a negative-order Sobolev space for the density, an L^q space for the chemoattractant, and the solenoidal L^q space for the fluid, with A a triangular matrix built from the Neumann Laplacian, a shifted Neumann Laplacian, and the Stokes operator. The load-bearing identity is the square-root domain characterization D((−Δ_q)^{1/2}) = $W^{{1,q}}$(Ω) and D($A^{{1/2}}$) = $W^{{1,q}}$_{0,σ}(Ω), valid for q in a narrow interval controlled by a geometric constant that depends on the Lipschitz boundary. This identifies the half-interpolation space as L^q × $W^{{1,q*}}$ × $W^{{1,q}}$_{0,σ}, which makes the bilinear estimates for the nonlinearity Φ possible through Sobolev embeddings. Time-weighted maximal L^p regularity in the critical-space framework turns those estimates into a contraction argument; for the global result, the linearization about a stationary solution is shown to inherit maximal regularity under a smallness condition on the product of the mean density and the coupling force.

What would settle it

For a bounded Lipschitz domain with a reentrant corner, compute explicitly the domains D((−Δ_q)^{1/2}) and D($A^{{1/2}}$) for q satisfying the paper's admissible interval; if either is not the corresponding $W^{{1,q}}$ space, then the half-interpolation identification fails and the Besov-space trace computation in Theorem 2.1 does not follow.

Watch

Extended reading notes

Core claim

The central claim is that the Lipschitz geometry of the domain does not obstruct the strong well-posedness theory of these chemotaxis-fluid models. Theorem 2.1 states that for any bounded Lipschitz domain in two or three dimensions, any admissible exponents p and q, and any coupling force f in L^n, every initial datum in the critical Besov spaces $B^{{n/q−2}}$_{q,p,0}(Ω) × $B^{{n/q*}}$_{q*,p}(Ω) × $B^{{n/q−1}}$_{q,p,0,σ}(Ω) gives rise to a unique local strong solution in a time-weighted maximal-regularity space. Theorem 2.3 adds that if the data lie in a small ball around a stationary solution whose fluid part solves a stationary Navier-Stokes system driven by the mean population density times f, then the solution exists globally and converges exponentially to the equilibrium. Theorem 2.4 shows that sufficiently regular small data give solutions that are globally bounded in space and time and that a nonnegative initial density stays nonnegative. Theorem 7.1 transfers all of these statements to the chemotaxis-consumption-Navier-Stokes system.

Load-bearing premise

The proof rests on the external result that, on bounded Lipschitz domains, the square roots of the Neumann Laplacian and the Stokes operator have first-order Sobolev spaces as their domains for q in a narrow interval set by the Lipschitz geometry; if that characterization fails for an admissible q, the interpolation computation and the entire contraction argument collapse.

Editorial extensions

If this is right

Local strong well-posedness holds in scaling-critical Besov spaces, extending the critical-space theory of the Navier-Stokes equations to the coupled chemotaxis-fluid system on arbitrary bounded Lipschitz domains.
Nontrivial equilibria are exponentially stable in the critical functional setting whenever the data are close enough and the coupling force is small in the stated sense.
For smoother data, solutions are globally bounded in space and time and preserve positivity of the population density, making the solutions suitable for biological interpretation.
The same existence, stability, boundedness, and positivity results cover both the Keller-Segel-Navier-Stokes and the chemotaxis-consumption-Navier-Stokes systems, with arbitrary coupling forces such as buoyancy.

Reading between the lines

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

The method suggests that other chemotaxis-fluid couplings with the same quadratic structure, for example models with logistic growth or altered boundary conditions, should inherit the same well-posedness pattern as long as the square-root domain characterizations remain available; this is an extension the paper does not pursue.
The admissible range of q is tied to a geometric constant of the Lipschitz domain, so the range of spaces in which the theory works may shrink as corners become sharper; checking whether the interval becomes empty for some domain would reveal whether the critical-space statement is optimal rather than purely technical.
Because positivity is proved through a linear comparison principle that is independent of the smallness condition, the positivity conclusion may extend to the local strong solutions of Theorem 2.1 whenever they exist, even far from equilibrium; this is a direct but unstated corollary of the proof structure.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read.

The genuinely new thing is Theorem 2.1: local strong well-posedness for the KSNS and CCNS systems in critical Besov spaces on bounded Lipschitz domains. That is a real step beyond smooth-domain results, and the way the authors use square-root domain characterizations to get the interpolation spaces is elegant. The bilinear estimates in Lemmas 3.3 and 3.4 look right, modulo details a referee will want spelled out, and the local theory hangs together. I also think the treatment of the CCNS variant as a corollary is honest.

The soft spot is Theorem 2.3. The statement only assumes the product smallness in Assumption 2.2, i.e. |u_s| times a norm of f. But Lemma 4.2, which the proof invokes, requires |u_s| itself to be small: the condition there is |u_s|∥f∥_{L^2} + |u_s| ≤ δ. For f ≡ 0, Assumption 2.2 is vacuous. Take any large constant equilibrium (u_s, u_s, 0). Linearizing about this state on a Neumann eigenfunction of eigenvalue λ_k gives a 2×2 matrix with determinant λ_k(λ_k+1 − u_s), which is negative when u_s > λ_1+1. So the equilibrium is linearly unstable, yet Theorem 2.3 as written claims exponential stability for all f satisfying Assumption 2.2. That is not a nitpick; it is a load-bearing gap in the main stability statement. The fix looks straightforward—add |u_s| ≤ δ (or the version from Remark 4.3) to the assumptions of Theorem 2.3 and Theorem 7.1. The proof of Theorem 2.3 already says 'let δ be small enough that Lemma 4.2 applies', so the authors clearly intended that condition; they just omitted it from the statement. With that repair, the global and stability results should go through.

Second-tier concerns: Lemma 3.4 has a terse R-boundedness step (the independence of ν in the Kalton-Weis application) that needs more detail. The dependence on deep external results—square-root domains, maximal regularity on Lipschitz domains—is acceptable because they are published, but a referee must check the parameter ranges (2.7) carefully. The positivity argument in Section 5 is fine.

Bottom line: the paper deserves a serious referee. The local theory and the framework are valuable, and the stability gap is fixable. Whoever referees it should flag the missing |u_s| smallness and ask for a corrected statement. I would not desk-reject.

Referee Report

2 major / 5 minor

Summary. The manuscript develops a maximal-regularity framework for the Keller-Segel-Navier-Stokes system (KSNS) and the chemotaxis-consumption variant (CCNS) on bounded Lipschitz domains in dimensions 2 and 3. The main results are: local strong well-posedness for initial data in critical Besov spaces (Theorem 2.1); global strong well-posedness and exponential stability of nontrivial stationary solutions under a smallness condition (Theorem 2.3); global boundedness and positivity for smoother data (Theorem 2.4); and analogous statements for CCNS (Theorem 7.1). The proofs rely on square-root domain characterizations for the Neumann Laplacian and Stokes operator on Lipschitz domains, time-weighted maximal regularity, and a parametric contraction argument.

Significance. If the stability statement is corrected, this would be a substantial contribution: it extends critical-space well-posedness for chemotaxis-fluid systems from smooth or convex domains to general bounded Lipschitz domains, and it uses the deep Lipschitz-domain operator theory in a careful and mostly explicit parameter regime. The smallness constants are universal rather than fitted, and the bilinear estimates are presented in detail. The local existence, boundedness, and positivity parts appear coherent and well supported by the cited operator-theoretic results. However, the global stability theorem as stated is false, so the paper cannot be accepted without amendment.

major comments (2)

[Section 4, Theorem 2.3 (with Assumption 2.2 and Lemma 4.2)] The global stability statement is false as stated because Assumption 2.2 does not control |u_s|. For f = 0 the assumption is vacuous, so the theorem applies to constant equilibria (u_s,u_s,0) for arbitrarily large u_s. Linearizing (KSNS) about such an equilibrium on a Neumann eigenfunction of -Δ with eigenvalue λ_k gives the system a' = -λ_k a + u_s λ_k b, b' = a - (λ_k+1)b, whose 2x2 matrix has determinant λ_k(λ_k+1-u_s). For any u_s larger than λ_k+1 for some Neumann eigenvalue λ_k, the determinant is negative and a positive eigenvalue exists, so the equilibrium is linearly unstable and cannot be exponentially stable. The proof gap is that Lemma 4.2, which the proof of Theorem 2.3 invokes, requires |u_s| ||f||_{L^2} + |u_s| ≤ δ, and Remark 4.3 explicitly retains the +|u_s| term, but this condition is never transferred to Assumption 2.2. The theorem must be amended by adding |u_s| ≤ δ (or the full sum) to Assumption 2.2 and by aligning the norms used there with those used in Lemma 4.2.
[Theorems 2.4 and 7.1] The missing smallness on |u_s| propagates to Theorem 2.4 and to Theorem 7.1(2)-(3), since both statements depend on the stationary construction and on the δ from Theorem 2.3. The same correction must be made consistently in these statements, otherwise the boundedness, positivity, and CCNS stability results inherit the counterexample described above.

minor comments (5)

[Throughout] The manuscript contains several typographical and OCR-like artifacts (e.g., 'NA VIER-STOKES' in the title, 'underyling', 'arbritray', 'ch emotaxis') that should be cleaned before publication.
[Theorem 7.1] The opening phrase 'Let all parameters be chosen as in Theorems 2.1, 2.2, 2.3 and 2.4' refers to Assumption 2.2, which is not a theorem; please renumber to avoid confusion.
[Section 8, proof of Proposition 8.1] In the interpolation equality following 'Using [39, Thm. 3.5]', the pair is written as (H^{s0,q}_0, H^{s0,q}_0) but should be (H^{s0,q}_0, H^{s1,q}_0).
[Lemma 3.4] The assertion that the R-bound of f(z) = z^{1-β}(1+A)^β (νz + ν^{1-1/β}(1+A))^{-1} is independent of ν is stated without proof; a short homogeneity argument (e.g., setting w = z/(ν^{-1/β}(1+A))) would make the verification transparent.
[Remark 4.3] Remark 4.3 already contains the corrected condition with the +|u_s| term; the authors should either promote this condition to Assumption 2.2 or explain explicitly why the theorem's assumptions imply it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation rests on independent prior operator-theoretic theorems; no fitted constant is repackaged as a prediction.

full rationale

The paper is not circular in any of the enumerated patterns. Its load-bearing analytic input is the square-root characterization (2.6), D((−Δ_q)^{1/2}) = W^{1,q}(Ω) and D(A^{1/2}) = W^{1,q}_{0,σ}(Ω), cited to Jerison–Kenig [18] for the Neumann Laplacian and to Gabel–Tolksdorf [11] and Tolksdorf [36] for the Stokes operator. Although [11] and [36] are coauthored by one of the present authors, these are published, externally proved operator-theoretic results whose assumptions concern only Lipschitz geometry and admissible L^q ranges, not the Keller–Segel–Navier–Stokes system; they are therefore independent evidence under the review rules. The maximal-regularity and H∞-calculus inputs from [8], [21], [15], and [32] are likewise prior theorems with proofs not assuming the target well-posedness. The bilinear contraction estimates in Lemmas 3.3 and 3.4 use only these embeddings and universal constants; no empirical parameter is fitted, and no smallness threshold is matched to the data by construction. The stability proof in Section 4 is a genuine perturbation argument about the stationary solution, and the exponential-decay conclusion is not already contained in Assumption 2.2. No known result is merely renamed, and no ansatz is smuggled in via self-citation. I note separately, as a non-circular correctness risk rather than circularity, that Theorem 2.3's Assumption 2.2 omits the |u_s| ≤ δ term that Lemma 4.2 and Remark 4.3 explicitly require; this is a hypothesis mismatch, not a derivation that reduces to its own input.

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

No free parameters are fitted to data and the paper introduces no new physical or mathematical entities. The central claim rests on established theorems about operators on Lipschitz domains: square-root domain characterizations, bounded H-infinity-calculus, maximal Lp-regularity, and regularity of the stationary Stokes problem. These are external results with published proofs; they are listed as axioms because the present paper uses them without reproving them, but they are not assumptions whose content includes the target theorem.

assumptions (6)

standard math Square-root domain characterization for the Neumann Laplacian: D((-Delta_q)^{1/2}) = W^{1,q}(Omega) in bounded Lipschitz domains for q satisfying (2.3).
Inherited from Jerison-Kenig [18, Thm 1]; used to compute X_{1/2} and the embeddings in Lemma 2.5 and Lemma 3.3; entry point Eq. (2.6).
standard math Square-root domain characterization for the Stokes operator: D(A^{1/2}) = W^{1,q}_{0,sigma}(Omega).
Inherited from Gabel-Tolksdorf [11, Thm 1.3] and Tolksdorf [36, Thm 1.1]; used in Eq. (2.6), Prop. 2.6, and Lemma 3.3.
standard math Bounded H-infinity-calculus and maximal Lp-regularity for the Neumann Laplacian and the Stokes operator on bounded Lipschitz domains for q in (2.3).
Used in Prop. 2.6 and Lemma 4.2; sourced from Kunstmann-Weis [21, Thm 16], Egert [8, Thm 1.3], and Hieber-Pruess [15, Thm 3.1].
standard math Regularity improvement for the stationary Stokes problem in Lipschitz domains, implying the stationary velocity w_s lies in D(A) subset L^infty.
Used in Lemma 4.1 to ensure the stationary solution is bounded enough for the perturbation operator B_s and for the Y0-setting in Theorem 2.4; sourced from Brown-Shen [3, Thm 2.12] and Mitrea-Wright [29, Thm 10.6.2].
standard math Weighted Sobolev embeddings of Meyries-Veraar and the Kalton-Weis R-bounded functional calculus.
Used in Lemma 3.4 to control the bilinear estimate in time-weighted spaces; cited to [26, Cor. 1.4] and [32, Thm 4.5.6].
standard math Interpolation theory for Bessel potential and Besov scales on Lipschitz domains, including solenoidal versions.
Used in the appendix (Proposition 8.1) and in the trace-space computation in Theorem 2.1; sourced from Triebel [38, 39] and Mitrea-Monniaux [28].

how reviews work

0 comments

Cite this review

Pith. "Pith review of Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains." pith.science (2026). https://pith.science/paper/CJYOEWZ3

@misc{pith2026250504503,
  author       = {Pith},
  title        = {Pith review of: Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CJYOEWZ3}},
  note         = {Machine review of arXiv:2505.04503}
}

abstract

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The traveling wave solutions of the 1D hyperbolic Keller-Segel equations
math.AP 2026-08 conditional novelty 6.0 of 10

Explicit families of entropy-admissible stationary and traveling wave solutions are constructed and classified for the 1D hyperbolic Keller-Segel system with quorum sensitivity.

Reference graph

Works this paper leans on

41 extracted references · 39 canonical work pages · cited by 1 Pith paper

[1]

H. Amann. Parabolic evolution equations in interpolati on and extrapolation spaces. J. Funct. Anal. 78 (2), 233–270 (1988)

work page 1988
[2]

H. Amann. Nonhomogeneous linear and quasilinear ellipt ic and parabolic boundary value problems. In: Func- tion Spaces, Diﬀerential Operators and Nonlinear Analysis . H Schmeisser, H. Triebel (eds.), Teubner, Stuttgart, 1993, 9–126

work page 1993
[3]

Brown and Z

R. Brown and Z. Shen. Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44 (4), 1183–1206 (1995)

work page 1995
[4]

M. Cannone. Ondelettes, paraproduits et Navier-Stokes . Diderot Editeur , Paris, 1995

work page 1995
[5]

M. Cannone. Chapter 3 - Harmonic analysis tools for solvi ng the incompressible Navier–Stokes equations. Handbook of mathematical ﬂuid dynamics , Editor(s): S. Friedlander, D. Serre, North-Holland 3, 161–244 (2004)

work page 2004
[6]

Cl´ ement and S.G

Ph. Cl´ ement and S.G. Li. Abstract parabolic quasilinea r equations and applications to a groundwater ﬂow problem. Advances Math. Sciences Appl . 3, 17-32 (1993)

work page 1993
[7]

R. Denk, M. Hieber and J. Pr¨ uss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (3), viii+114 pp. (2003)

work page 2003
[8]

M. Egert. L p-estimates for the square root of elliptic systems with mixe d boundary conditions. J. Diﬀerential Equations 265 (4), 1279–1323 (2018)

work page 2018

Show all 41 references

[9]

Fabes, O

E. Fabes, O. Mendez and M. Mitrea. Boundary layers on Sobo lev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159 (2), 323–368 (1998)

1998
[10]

Fuest and L

M. Fuest and L. Lankeit. Corner and collaps: some simple observations concerning critical mass and boundary blow-up in the fully parabolic Keller-Segel system. Appl. Math. Lett. 146, paper no. 108788, 9 pp. (2023)

2023
[11]

Gabel and P

F. Gabel and P. Tolksdorf. The Stokes operator in two-di mensional bounded Lipschitz domains. J. Diﬀerential Equations 340, 227–272 (2022)

2022
[12]

Geng and Z

J. Geng and Z. Shen. Resolvent Estimates in L ∞ for the Stokes Operator in nonsmooth domains. Available at arXiv: https://arxiv.org/abs/2408.03844
[13]

Gong and S

Y. Gong and S. He. On the 8 π -critical-mass threshold of a Patlak-Keller-Navier-Stok es system. SIAM J. Math. Anal. 53 (2), 2925–2956 (2021)

2021
[14]

Hieber, K

M. Hieber, K. Kress and C. Stinner. The Keller-Segel sys tem on bounded convex domains in critical spaces. Partial Diﬀer. Equ. Appl. 2, paper no 38, (2021)

2021
[15]

Hieber and J

M. Hieber and J. Pr¨ uss. Heat kernels and maximal L p − Lq estimates for parabolic evolution equations. Comm. Partial Diﬀerential Equations 22 (9-10), 1647–1669 (1997)

1997
[16]

Horstmann, H

D. Horstmann, H. Meinlschmidt and J. Rehberg. The full K eller–Segel model is well-posed on nonsmooth domains. Nonlinearity 31 (4), 1560–1592 (2018)

2018
[17]

Z. Hu, A. Kiselev and Y. Yao. Suppression of chemotactic singularity by buoyancy. Available at arXiv: https://arxiv.org/abs/2305.01036

arXiv
[18]

Jerison and C

D. Jerison and C. Kenig. The functional calculus for the Laplacian on Lipschitz domains. Journ´ ees´Equations aux d´ eriv´ ees partielles1–10 (1989)

1989
[19]

Jerison and C

D. Jerison and C. Kenig. The inhomogeneous Dirichlet pr oblem in Lipschitz domains. J. Funct. Anal. 130 (1), 161–219 (1995)

1995
[20]

Kiselev, F

A. Kiselev, F. Nazarov, L. Ryzhik and Y. Yao. Chemotaxis and reactions in biology. J. Eur. Math. Soc. 25 (7), 2641–2696 (2023)

2023
[21]

P. C. Kunstmann and L. W eis. New criteria for the H ∞ -calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. 17 (1), 387–409 (2017)

2017
[22]

Lankeit and M

J. Lankeit and M. Winkler. Facing low regularity in chem otaxis systems. Jahresber. Dtsch. Math.-Ver. 122, 35–64 (2020)

2020
[23]

Lankeit and M

J. Lankeit and M. Winkler. Depleting the signal: Analys is of chemotaxis-consumption models - a survey. Stud. Appl. Math. 151 (4), 1197–1229 (2023)

2023
[24]

Lemari´ e-Rieusset.Recent developments in the Navier-Stokes problem

P.-G. Lemari´ e-Rieusset.Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC Res. Notes Math., 431 Chapman & Hall/CRC, Boca Raton, FL, xiv+395 pp. (2002)

2002
[25]

Li and T

Z. Li and T. Zhou. Finite-time blowup for Keller-Segel- Navier-Stokes system in three dimensions. Available at arXiv: https://arxiv.org/abs/2404.17228

arXiv
[26]

Meyries and M

M. Meyries and M. Veraar. Sharp embedding results for sp aces of smooth functions with power weights. Studia Math. 208 (3), 257–293 (2012)

2012
[27]

D. Mitrea. Layer potentials and Hodge decompositions i n two dimensional Lipschitz domains. Math. Ann. 322 (1), 75–101 (2002)

2002
[28]

Mitrea and S

M. Mitrea and S. Monniaux. The regularity of the Stokes o perator and the Fujita-Kato approach to the Navier- Stokes initial value problem in Lipschitz domains. J. Funct. Anal. 254 (6), 1522–1574 (2008)

2008
[29]

Mitrea and M

M. Mitrea and M. W right. Boundary value problems for the Stokes system in arbitrary L ipschitz domains . Ast´ erisque344, viii+241 pp. (2012). 24 MATTHIAS HIEBER, HIDEO KOZONO, SYL VIE MONNIAUX, AND PATR ICK TOLKSDORF

2012
[30]

Nagai, T

T. Nagai, T. Senba and K. Yoshida. Application of the Tru dinger-Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40 (3), 411–433 (1997)

1997
[31]

Pedely, J

T. Pedely, J. Kessler. Hydrodynamic phenomena in suspe nsions of swimming microorganisms. Annu. Rev. Fluid Mech. 92, 313–358 (1992)

1992
[32]

Pr¨ uss and G

J. Pr¨ uss and G. Simonett. Moving Interfaces and Quasilinear Parabolic Evolution Equ ations. Monogr. Math., 105 Birkh¨ auser/Springer, xix+609 pp. (2016)

2016
[33]

Pr¨ uss, G

J. Pr¨ uss, G. Simonett and M. Wilke. Critical spaces for quasilinear parabolic evolution equations and applica- tions. J. Diﬀerential Equations 264 (3), 2028–2074 (2018)

2018
[34]

Z. Shen. Resolvent estimates in L p for the Stokes operator in Lipschitz domains. Arch. Ration. Mech. Anal. 205 (2), 395–424 (2012)

2012
[35]

R. Temam. Navier-Stokes Equations. Stud. Math. Appl., 2 North-Holland Publishing Co., Amsterdam-New York, x+519 pp. (1979)

1979
[36]

Tolksdorf

P. Tolksdorf. On the L p-theory of the Navier-Stokes equations on three-dimension al bounded Lipschitz do- mains. Math. Ann. 371 (1-2), 445–460 (2018)

2018
[37]

Tolksdorf

P. Tolksdorf. The Stokes resolvent problem: optimal pr essure estimates and remarks on resolvent estimates in convex domains. Calc. Var. Partial Diﬀerential Equations 59 (5), paper no. 154, 40 pp. (2020)

2020
[38]

H. Triebel. Interpolation theory, function spaces, diﬀerential opera tors. North-Holland Math. Library, 18 North-Holland Publishing Co., Amsterdam-New York, 528 pp. (1978)

1978
[39]

H. Triebel. Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers. Rev. Mat. Complut. 15 (2), 475–524 (2002)

2002
[40]

Tuval, L

I. Tuval, L. Cisneros, C. Dombrowski, C.W. W olgemuth, J .O. Kessler and R.E. Goldstein. Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102 (2005), 2277–2282

2005
[41]

M. Winkler. Global weak solutions in a three-dimension al chemotaxis-Navier-Stokes system. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire33 (5), 1329–1352 (2016). TU Darmstadt, Angew andte Analysis, Schlossgartenstraße 764 289 Darmstadt, Germany Email address : hieber@mathema...

2016

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

H. Amann. Parabolic evolution equations in interpolati on and extrapolation spaces. J. Funct. Anal. 78 (2), 233–270 (1988)

work page 1988

[2] [2]

H. Amann. Nonhomogeneous linear and quasilinear ellipt ic and parabolic boundary value problems. In: Func- tion Spaces, Diﬀerential Operators and Nonlinear Analysis . H Schmeisser, H. Triebel (eds.), Teubner, Stuttgart, 1993, 9–126

work page 1993

[3] [3]

Brown and Z

R. Brown and Z. Shen. Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44 (4), 1183–1206 (1995)

work page 1995

[4] [4]

M. Cannone. Ondelettes, paraproduits et Navier-Stokes . Diderot Editeur , Paris, 1995

work page 1995

[5] [5]

M. Cannone. Chapter 3 - Harmonic analysis tools for solvi ng the incompressible Navier–Stokes equations. Handbook of mathematical ﬂuid dynamics , Editor(s): S. Friedlander, D. Serre, North-Holland 3, 161–244 (2004)

work page 2004

[6] [6]

Cl´ ement and S.G

Ph. Cl´ ement and S.G. Li. Abstract parabolic quasilinea r equations and applications to a groundwater ﬂow problem. Advances Math. Sciences Appl . 3, 17-32 (1993)

work page 1993

[7] [7]

R. Denk, M. Hieber and J. Pr¨ uss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (3), viii+114 pp. (2003)

work page 2003

[8] [8]

M. Egert. L p-estimates for the square root of elliptic systems with mixe d boundary conditions. J. Diﬀerential Equations 265 (4), 1279–1323 (2018)

work page 2018