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arxiv: 2604.18211 · v1 · submitted 2026-04-20 · 🧮 math.AP

Weak solutions and weak-strong uniqueness for a Cahn-Hilliard type model with chemotaxis

Robert Lasarzik , Elisabetta Rocca , Giulio Schimperna This is my paper

Pith reviewed 2026-05-10 04:07 UTC · model grok-4.3

classification 🧮 math.AP
keywords Cahn-Hilliard equationchemotaxisweak solutionsweak-strong uniquenesscross-diffusionphase separationcancer growth model
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The pith

A Cahn-Hilliard system with chemotaxis admits global weak solutions and weak-strong uniqueness.

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

The paper establishes global existence of weak solutions for a coupled system consisting of a Cahn-Hilliard equation governing a phase field variable and a nonlinear parabolic equation for an additional concentration variable that features a cross-diffusion term. These weak solutions are equipped with an energy imbalance and a logarithmic inequality for the concentration. The authors also prove that any such weak solution must coincide with the local strong solution whenever the latter exists. A sympathetic reader would care because the model describes phase separation processes influenced by chemical signals, as in tumor growth, and the results provide a framework for analyzing the system over long times despite the analytic challenges created by the cross-diffusion.

Core claim

We prove global-in-time existence for a very weak notion of solution to the coupled system, to which we add a suitable energy imbalance and a logarithmic inequality for the nutrient. Noting that the system also admits local-in-time strong solutions, we exhibit a weak-strong uniqueness result whose proof exploits in an essential way the entropy-type inequality satisfied by weak solutions.

What carries the argument

The entropy-type inequality together with the logarithmic inequality for the nutrient concentration, which together control the cross-diffusion term and enable comparison between weak and strong solutions.

If this is right

  • Global weak solutions permit the study of long-time phase separation dynamics under chemotactic influence.
  • Weak-strong uniqueness guarantees that the weak formulation is consistent with the strong formulation on the interval where strong solutions exist.
  • The inequalities provide a direct method to absorb the cross-diffusion term without additional regularization.
  • The framework applies directly to models of cancer growth where the nutrient concentration affects tumor phase evolution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same inequalities might be used to derive further regularity or to obtain convergence rates for approximations.
  • Similar weak-strong uniqueness arguments could apply to other phase-field systems coupled to Keller-Segel type equations.
  • The result opens the possibility of passing to the limit in numerical schemes that preserve the entropy inequality.
  • One could test whether the weak solutions exhibit the same asymptotic behavior as the strong solutions when the latter can be continued globally.

Load-bearing premise

Weak solutions satisfy the stated energy imbalance and logarithmic inequality for the nutrient.

What would settle it

A concrete weak solution that satisfies the equations in the distributional sense but violates the logarithmic inequality for the nutrient over a positive time interval.

read the original abstract

We prove existence of weak solutions and weak-strong uniqueness for a mathematical model which couples the evolution of a phase-parameter $\varphi$ satisfying a Cahn-Hilliard type relation with the one of an additional variable $\sigma$ influencing the phase separation process. The main application of the model refers to cancer growth processes, where $\sigma$ may represent the concentration of a chemical substance affecting the evolution of the tumor, and is governed by a nonlinear parabolic equation characterized by a cross-diffusion term alike that occurring in the Keller-Segel model for chemotaxis. This term is also responsible for the most relevant difficulties in the mathematical analysis of the system. Complementing previous results on the model, we prove here global in time existence for a very weak notion of solution to which a suitable energy imbalance and a logarithmic inequality for the nutrient are added. Noting that the system also admits local in time "strong" solutions, we can also exhibit a weak-strong uniqueness result whose proof exploits in an essential way the entropy-type inequality satisfied by weak solutions.

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

0 major / 3 minor

Summary. The manuscript proves global-in-time existence of weak solutions for a Cahn-Hilliard-chemotaxis system coupling a phase field φ with a nutrient concentration σ governed by a nonlinear parabolic equation containing a Keller-Segel-type cross-diffusion term. Weak solutions are defined to satisfy an energy imbalance together with a logarithmic inequality for σ; these are then used to control the cross-diffusion term and establish weak-strong uniqueness against local-in-time strong solutions. The model is motivated by applications to cancer growth.

Significance. If the central estimates hold, the result supplies a rigorous existence and uniqueness theory for a biologically motivated PDE system in which the cross-diffusion term creates substantial analytic difficulties. By augmenting the weak-solution notion with an entropy imbalance and a logarithmic inequality, the authors obtain global weak solutions and a weak-strong uniqueness statement that bridges the gap between very weak and strong regimes. This approach is technically demanding and, if verified, would constitute a useful addition to the literature on chemotaxis models and tumor-growth PDEs.

minor comments (3)
  1. [Section 2] The precise functional setting for the weak solutions (spaces for φ, σ, and the test functions) and the exact statement of the energy imbalance and logarithmic inequality should be collected in a single definition early in the paper (ideally Section 2) rather than scattered across the existence and uniqueness sections.
  2. [Section 4] In the uniqueness argument, the passage from the difference of the weak and strong formulations to the Gronwall-type estimate for the relative energy should be written with explicit constants; at present the dependence on the strong-solution norms is only sketched.
  3. [Introduction] A short remark on the relation between the present notion of weak solution and the notions used in the cited previous works on the same model would help readers assess the precise improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance. We are pleased that the referee recognizes the technical challenges posed by the cross-diffusion term and the value of the augmented weak-solution notion together with the weak-strong uniqueness result. The recommendation of minor revision is noted; we will incorporate any editorial or minor improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical analysis

full rationale

The paper establishes global existence of weak solutions (augmented by an entropy imbalance and logarithmic inequality) and weak-strong uniqueness for a coupled Cahn-Hilliard/chemotaxis system via standard PDE techniques: Galerkin approximation, a priori estimates, compactness arguments, and passage to the limit. These steps rely on the structure of the equations and the added inequalities incorporated into the weak-solution definition, which are then exploited directly for uniqueness against local strong solutions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; prior results are complemented but the central proofs are independently derived from the model equations and functional-analytic tools. The work is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard functional-analytic assumptions for the nonlinearities and diffusion coefficients plus the structural properties of the cross-diffusion term that enable the entropy and logarithmic inequalities.

axioms (1)
  • domain assumption The nonlinear potentials and mobility functions satisfy standard growth and convexity conditions typical for Cahn-Hilliard and Keller-Segel-type systems
    These conditions are invoked to close the a-priori estimates and obtain the energy imbalance.

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Abels, H

    H. Abels, H. Garcke, and G. Gr¨ un. Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities.Math. Models Methods Appl. Sci., 22(3):1150013, 40, 2012. 35

    work page 2012

  2. [2]

    Agosti and A

    A. Agosti and A. Signori. Analysis of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model with chemotaxis and angiogenesis.J. Differ. Equations, 403:308–367, 2024

    work page 2024

  3. [3]

    Barbu.Nonlinear semigroups and differential equations in Banach spaces

    V. Barbu.Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste Romˆ ania, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian

    work page 1976

  4. [4]

    Bergh and J

    J. Bergh and J. L¨ ofstr¨ om.Interpolation spaces. An introduction, volume 223 of Grundlehren Math. Wiss.Springer, Cham, 1976

    work page 1976

  5. [5]

    Brezis.Functional analysis, Sobolev spaces and partial differential equations

    H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universi- text. New York, NY: Springer, 2011

    work page 2011

  6. [6]

    Debussche and L

    A. Debussche and L. Dettori. On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal., 24(10):1491–1514, 1995

    work page 1995

  7. [7]

    Demengel and G

    F. Demengel and G. Demengel.Functional spaces for the theory of elliptic partial differ- ential equations. Transl. from the French by Reinie Ern´ e. Universitext. Berlin: Springer, 2012

    work page 2012

  8. [8]

    R. Denk, M. Hieber, and J. Pr¨ uss.R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, volume 788 ofMem. Am. Math. Soc.Providence, RI: American Mathematical Society (AMS), 2003

    work page 2003

  9. [9]

    R. Denk, M. Hieber, and J. Pr¨ uss. OptimalLp-L q-estimates for parabolic boundary value problems with inhomogeneous data.Math. Z., 257(1):193–224, 2007

    work page 2007

  10. [10]

    Diening, P

    L. Diening, P. Harjulehto, P. H¨ ast¨ o, and M. Ruˇ ziˇ cka.Lebesgue and Sobolev spaces with variable exponents, volume 2017 ofLecture Notes in Mathematics. Springer, Heidelberg, 2011

    work page 2017

  11. [11]

    Eiter and R

    T. Eiter and R. Lasarzik. Existence of energy-variational solutions to hyperbolic conserva- tion laws.Calc. Var. Partial Differ. Equ., 63(4):40, 2024. Id/No 103

    work page 2024

  12. [12]

    Emmrich and R

    E. Emmrich and R. Lasarzik. Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions.Discrete and Continuous Dynamical Systems, 38(9):4617– 4635, 2018

    work page 2018

  13. [13]

    Frigeri, K

    S. Frigeri, K. F. Lam, and E. Rocca. On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities. InSolvability, regularity, and optimal control of boundary value problems for PDEs. In honour of Prof. Gianni Gilardi, pages 217–254. Cham: Springer, 2017

    work page 2017

  14. [14]

    Garcke and K

    H. Garcke and K. F. Lam. Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis.Discrete and Continuous Dynamical Systems, 37(8):4277–4308, 2017

    work page 2017

  15. [15]

    Garcke and K

    H. Garcke and K. F. Lam. Well-posedness of a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport.European Journal of Applied Mathematics, 28(2):284–316, 2017

    work page 2017

  16. [16]

    Garcke and D

    H. Garcke and D. Trautwein. Numerical analysis for a Cahn-Hilliard system modelling tumour growth with chemotaxis and active transport.J. Numer. Math., 30(4):295–324, 2022

    work page 2022

  17. [17]

    Giorgini, M

    A. Giorgini, M. Grasselli, and A. Miranville. The Cahn-Hilliard-Oono equation with sin- gular potential.Math. Models Methods Appl. Sci., 27(13):2485–2510, 2017. 36

    work page 2017

  18. [18]

    Giorgini, J

    A. Giorgini, J. He, and H. Wu. Global weak solutions to a Navier–Stokes–Cahn–Hilliard system with chemotaxis and mass transport: Cross diffusion versus logistic degradation. Math. Mod. and Meth. Appl. Sci., 36(01):57–110, 2026

    work page 2026

  19. [19]

    H¨ omberg, R

    D. H¨ omberg, R. Lasarzik, and L. Plato. On the existence of generalized solutions to a spatio-temporal predator-prey system with prey-taxis.J. Evol. Equ., 23(1):44, 2023. Id/No 20

    work page 2023

  20. [20]

    Keller and L

    E. Keller and L. Segel. Model for chemotaxis.J. Theoret. Biol., 30:225–234, 1971

    work page 1971

  21. [21]

    Kenmochi, M

    N. Kenmochi, M. Niezg´ odka, and I. Paw low. Subdifferential operator approach to the Cahn-Hilliard equation with constraint.J. Differential Equations, 117(2):320–356, 1995

    work page 1995

  22. [22]

    Lasarzik

    R. Lasarzik. On the existence of energy-variational solutions in the context of multidi- mensional incompressible fluid dynamics.Math. Methods Appl. Sci., 47(6):4319–4344, 2024

    work page 2024

  23. [23]

    Lasarzik, E

    R. Lasarzik, E. Rocca, and R. Rossi. Existence and weak-strong uniqueness for damage systems in viscoelasticity.Nonlinearity, 38:125016, 2024

    work page 2024

  24. [24]

    Lasarzik, E

    R. Lasarzik, E. Rocca, and G. Schimperna. Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model.Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., 33(2):229–269, 2022

    work page 2022

  25. [25]

    Mainik and A

    A. Mainik and A. Mielke. Existence results for energetic models for rate-independent systems.Calc. Var. Partial Differential Equations, 22(1):73–99, 2005

    work page 2005

  26. [26]

    Miranville

    A. Miranville. The Cahn–Hilliard equation and some of its variants.AIMS Mathematics, 2(3):479–544, 2017

    work page 2017

  27. [27]

    Miranville and S

    A. Miranville and S. Zelik. Robust exponential attractors for Cahn-Hilliard type equations with singular potentials.Math. Methods Appl. Sci., 27(5):545–582, 2004

    work page 2004

  28. [28]

    Rocca, G

    E. Rocca, G. Schimperna, and A. Signori. On a Cahn-Hilliard-Keller-Segel model with generalized logistic source describing tumor growth.J. Differential Equations, 343:530– 578, 2023

    work page 2023

  29. [29]

    Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 of ISNM, Int

    T. Roub´ ıˇ cek.Nonlinear partial differential equations with applications, volume 153 of ISNM, Int. Ser. Numer. Math.Basel: Birkh¨ auser, 2nd ed. edition, 2013

    work page 2013

  30. [30]

    Schimperna

    G. Schimperna. On a modified Cahn-Hilliard-Brinkman model with chemotaxis and non- linear sensitivity. Preprint, arXiv:2411.12505 [math.AP] (2024), 2024

    work page arXiv 2024

  31. [31]

    Schimperna and I

    G. Schimperna and I. Paw l ow. On a class of Cahn-Hilliard models with nonlinear diffusion. SIAM J. Math. Anal., 45(1):31–63, 2013

    work page 2013

  32. [32]

    Schimperna and A

    G. Schimperna and A. Segatti. Global attractor for a cahn-hilliard-chemotaxis model with logistic degradation. Preprint, arXiv:2511.11363 [math.AP] (2025), 2025

    work page arXiv 2025

  33. [33]

    J. Simon. Compact sets in the spaceL p(0, T;B).Ann. Mat. Pura Appl. (4), 146:65–96, 1987. 37

    work page 1987