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REVIEW 2 major objections 2 minor 46 references

Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Global strong solutions exist and are unique for the 1D compressible Navier-Stokes/Cahn-Hilliard system with vacuum without initial compatibility conditions.

desk verdict They get global strong solutions and uniqueness for the 1D NS/CH system with vacuum without compatibility conditions by using time-weighted estimates that close in Eulerian coordinates. read the letter →

arxiv 2606.29353 v1 pith:4JJCYBZM submitted 2026-06-28 math.AP

Shijin Ding , Yinghua Li , Yuanxiang Yan , Haoran Zheng This is my paper
classification math.AP
keywords Navier-StokesequationsCahn-Hilliardequationstrongsolutionsvacuumglobalexistenceuniqueness1Dcompressibleflowtime-weightedestimates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes global existence and uniqueness of strong solutions for the initial-boundary value problem of the 1D compressible Navier-Stokes/Cahn-Hilliard equations when the initial density may vanish. Time-weighted techniques are used to remove the requirement of any initial compatibility conditions, which produces a loss of regularity near the initial time and makes the uniqueness proof more difficult. Refined growth estimates together with singular-in-time weighted energy estimates are derived to obtain a Gronwall-type structure that closes the uniqueness argument directly in Eulerian coordinates rather than Lagrangian ones.

What carries the argument

Time-weighted techniques and singular-in-time weighted energy estimates that induce a Gronwall-type structure for uniqueness in Eulerian coordinates.

What would settle it

An explicit pair of distinct strong solutions for some initial data with vacuum that both satisfy the equations and boundary conditions for positive times.

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Extended reading notes

Core claim

The central claim is that the 1D compressible Navier-Stokes/Cahn-Hilliard system with vacuum admits a unique global strong solution for the initial-boundary value problem, obtained without imposing initial compatibility conditions by means of time-weighted techniques, with uniqueness established through refined growth estimates and singular-in-time weighted energy estimates that produce a Gronwall-type structure allowing closure in Eulerian coordinates.

Load-bearing premise

The specific structure of the one-dimensional system combined with the chosen weights allows the estimates to close without compatibility conditions.

Editorial extensions

If this is right

  • Strong solutions exist globally for initial data satisfying only basic integrability without extra compatibility requirements.
  • Uniqueness holds even though regularity may be lost near the initial time.
  • The proof remains in Eulerian coordinates and does not require a change to Lagrangian coordinates.
  • The estimates control the solution uniformly away from vacuum states while handling vacuum regions.

Reading between the lines

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

  • The weighted-energy method may apply to other one-dimensional fluid systems that permit vacuum states.
  • Numerical schemes for such equations could be initialized directly from vacuum data without artificial smoothing.
  • The loss of initial regularity might influence short-time behavior in approximation schemes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript proves global existence and uniqueness of strong solutions to the 1D compressible Navier-Stokes/Cahn-Hilliard initial-boundary value problem with vacuum. No initial compatibility conditions are imposed; time-weighted techniques are used, accepting a loss of regularity near t=0. Uniqueness is closed directly in Eulerian coordinates by deriving refined growth estimates together with singular-in-time weighted energy estimates that produce a Gronwall structure.

Significance. If the estimates close as claimed, the result would advance the theory of strong solutions for coupled compressible fluid-phase-field models by removing the standard compatibility requirement at vacuum. The time-weighted approach for handling initial singularities in 1D could be of interest for related systems where Lagrangian coordinates are inconvenient.

major comments (2)
  1. [§4] §4 (uniqueness argument): the claim that the singular-in-time weighted energies induce a closed Gronwall inequality requires explicit verification that every nonlinear term (including those from the Cahn-Hilliard chemical potential and the convective terms) remains integrable under the chosen weights near vacuum states and t=0; the current sketch does not display the coefficient bounds or absorption steps needed to confirm this control.
  2. [Theorem 1.1] Theorem 1.1 and the a-priori estimate section: the global existence statement asserts that the weighted energies remain finite for all t>0 without compatibility, yet the passage from local to global solutions via continuation relies on a uniform bound whose dependence on the initial data (especially the vacuum set) is not quantified; this bound is load-bearing for the global claim.
minor comments (2)
  1. [Abstract] The phrase 'No any initial compatibility conditions' in the abstract should be corrected to 'No initial compatibility conditions are required'.
  2. [§2] Notation for the weights (e.g., the precise form of the singular time factor) should be introduced once in §2 and used consistently thereafter to avoid repeated re-definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§4] §4 (uniqueness argument): the claim that the singular-in-time weighted energies induce a closed Gronwall inequality requires explicit verification that every nonlinear term (including those from the Cahn-Hilliard chemical potential and the convective terms) remains integrable under the chosen weights near vacuum states and t=0; the current sketch does not display the coefficient bounds or absorption steps needed to confirm this control.

    Authors: We agree that the uniqueness argument in Section 4 would benefit from more explicit verification. The manuscript derives refined growth estimates for solution differences followed by singular-in-time weighted energies that close via Gronwall, but the coefficient bounds and absorption for terms involving the chemical potential and convection are only sketched. In the revision we will insert the full integrability estimates under the chosen weights, confirming each nonlinear contribution is controlled near vacuum and t=0. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the a-priori estimate section: the global existence statement asserts that the weighted energies remain finite for all t>0 without compatibility, yet the passage from local to global solutions via continuation relies on a uniform bound whose dependence on the initial data (especially the vacuum set) is not quantified; this bound is load-bearing for the global claim.

    Authors: The a-priori estimates already yield a uniform bound controlled by the initial weighted norms, which encode the vacuum behavior. We acknowledge, however, that the explicit dependence on the measure of the initial vacuum set is not stated. We will revise the statement of Theorem 1.1 and the continuation argument to display this dependence explicitly. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct energy-method proof

full rationale

The paper presents a direct mathematical proof of global existence and uniqueness for the 1D NS/CH system via time-weighted energy estimates and Gronwall closure in Eulerian coordinates. No parameters are fitted to data, no predictions are made from subsets of results, and no self-citations are invoked as load-bearing uniqueness theorems or ansatzes. The derivation relies on the specific 1D structure and chosen weights to handle vacuum and loss of regularity, which are standard techniques in PDE analysis and do not reduce to self-definition or renaming of inputs. The argument is self-contained against external mathematical benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The proof relies on standard Sobolev embeddings, basic energy identities for the NS/CH system, and properties of 1D compressible flow with vacuum; no free parameters or invented entities are indicated in the abstract. Full details would list any specific a priori assumptions on the weights.

assumptions (1)
  • standard math Standard a priori estimates and Gronwall inequality apply to the weighted energies derived from the system.
    Invoked to close the estimates after deriving the singular-in-time bounds.

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Cite this review

Pith. "Pith review of Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum." pith.science (2026). https://pith.science/paper/4JJCYBZM

@misc{pith2026260629353,
  author       = {Pith},
  title        = {Pith review of: Global strong solutions for 1D compressible Navier-Stokes/Cahn-Hilliard equations with vacuum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JJCYBZM}},
  note         = {Machine review of arXiv:2606.29353}
}
read the original abstract

In this paper, we study the initial-boundary value problem of the 1D compressible Navier--Stokes/Cahn--Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem. No any initial compatibility conditions are required via time weighted techniques, which leads to a loss of regularity near the initial time. Therefore, the uniqueness of solutions obtained in this paper is even more challenging. To address this issue, we establish refined growth estimates and singular-in-time weighted energy estimates that induce a Gronwall-type structure, which ultimately allows us to close the uniqueness proof in Eulerian coordinates without passing to Lagrangian coordinates.

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Works this paper leans on

46 extracted references · 5 canonical work pages

  1. [1]

    Anderson, G.B

    D.M. Anderson, G.B. McFadden, A.A. Wheeler, Diffuse-interface methods in fluid mechanics,Annu. Rev. Fluid Mech., vol. 30, Annual Reviews, Palo Alto, CA, 1998, pp. 139–165

    1998

  2. [2]

    Abels, E

    H. Abels, E. Feireisl, On a diffuse interface model for a two-phase flow of compressible viscous fluids,Indiana Univ. Math. J.,57(2008), 659–698

    2008

  3. [3]

    Abels, Y

    H. Abels, Y. Liu and ˇS. Neˇ casov´ a, Low Mach number limit of a diffuse interface model for two-phase flows of compressible viscous fluids,GAMM-Mitteilungen.,47(2024), e202470008

    2024

  4. [4]

    Global weak solutions to a compressible Navier--Stokes/Cahn--Hilliard system with singular entropy of mixing

    D. Basari´ c, A. Giorgini, Global weak solutions to a compressible Navier–Stokes/Cahn–Hilliard system with singular entropy of mixing,arXiv preprint, arXiv:2506.07835, 2025

    work page Pith review arXiv 2025

  5. [5]

    Bresch, P

    D. Bresch, P. E. Jabin, Global existence of weak solutions for compressible Navier–Stokes equations: ther- modynamically unstable pressure and anisotropic viscous stress tensor,Ann. Math.(2)188(2018), no. 2, 577–684

    2018

  6. [6]

    Cherfils, E

    L. Cherfils, E. Feireisl, M. Mich´ alek, A. Miranville, M. Petcu and D. Praˇ z´ ak, The compressible Navier- StokesCahn-Hilliard equations with dynamic boundary conditions,Math. Models Methods Appl. Sci.,29 (2019), 2557–2584

    2019

  7. [7]

    M. Chen, X. Guo, Global large solutions for a coupled compressible Navier-Stokes/Allen-Cahn system with initial vacuum,Nonlinear Anal. Real World Appl.,37(2017), 350–373

    2017

  8. [8]

    Choe, and H

    H. Choe, and H. Kim, Strong solutions of the Navier–Stokes equations for isentropic compressible fluids,J. Differential Equations,190(2003), no. 2, 504–523

    2003

Show all 46 references
  1. [9]

    Y. Cho, H. J. Choe, and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids,J. Math. Pures Appl.,83(2004), no. 2, 243–275

    2004

  2. [10]

    Cho and H

    Y. Cho and H. Kim, Existence results for viscous polytropic fluids with vacuum,J. Differential Equations, 228(2006), no. 2, 377–411

    2006

  3. [11]

    S. Chen, C. Zhu, Blow-up criterion and the global existence of strong/classical solutions to Navier–Stokes/Allen–Cahn system,Z. Angew. Math. Phys.,72(2021), no. 1, Paper No. 14, 24 pp

    2021

  4. [12]

    S. Ding, Y. Li, Global solutions of a diffuse interface model for the two-phase flow of compressible viscous fluids in 1D,Commun. Math. Sci.,18(2020), no. 4, 1055–1086

    2020

  5. [13]

    S. Ding, Y. Li, W. Luo, Global solutions for a coupled compressible Navier-Stokes/Allen-Cahn system in 1D,J. Math. Fluid Mech.,15(2013), no. 2, 335–360

    2013

  6. [14]

    Danchin, P

    R. Danchin, P. B. Mucha, The incompressible Navier–Stokes equations in vacuum,Comm. Pure Appl. Math. 72(2019), no. 7, 1351–1385

    2019

  7. [15]

    ten Eikelder, E.H

    M.F.P. ten Eikelder, E.H. van Brummelen and D. Schillinger, Compressible N-phase fluid mixture models, arXiv preprint, arXiv:2503.24225, 2025

    2025

  8. [16]

    Elbar and A

    C. Elbar and A. Poulain, Analysis and numerical simulation of a generalized compressible Cahn–Hilliard–Navier–Stokes model with friction effects,ESAIM: M2AN,58(2024), 1989–2034

    2024

  9. [17]

    Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,Indiana Univ

    E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid,Indiana Univ. Math. J., 53(2004), no. 6, 1705–1738

    2004

  10. [18]

    M. Fei, X. Fei, D. Han and Y. Liu, Local-in-time existence of strong solutions to a quasi-incompressible Cahn–Hilliard–Navier–Stokes system,arXiv preprint, arXiv:2411.09455, 2024

    2024 arXiv

  11. [19]

    Feireisl, A

    E. Feireisl, A. Novotn´ y, H. Petzeltov´ a, On the existence of globally defined weak solutions to the Navier- Stokes equations,J. Math. Fluid Mech.,3(2001), no. 6, 358–392

    2001

  12. [20]

    Craig, X

    W. Craig, X. Huang, Y. Wang, Global wellposedness for the 3D inhomogeneous incompressible Navier–Stokes equations,J.Math. Fluid Mech.15(2013), 747–758

    2013

  13. [21]

    H. Gong, J. Li, X. Liu, X. Zhang, Local well-posedness of isentropic compressible Navier–Stokes equations with vacuum,Commun. Math. Sci.,18(2020), no. 7, 1891–1909

    2020

  14. [22]

    Huang, On local strong and classical solutions to the three-dimensional barotropic compressible Navier–Stokes equations with vacuum,Sci

    X. Huang, On local strong and classical solutions to the three-dimensional barotropic compressible Navier–Stokes equations with vacuum,Sci. China Math.,64(2021), no. 8, 1771–1788

    2021

  15. [23]

    Huang, J

    X. Huang, J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations,Arch. Ration. Mech. Anal.,227(2018), no. 3, 995–1059

    2018

  16. [24]

    Huang, J

    X. Huang, J. Li, and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations,Comm. Pure Appl. Math.,65 (2012), 549–585

    2012

  17. [25]

    Huang, Y

    X. Huang, Y. Wang, Global strong solution of 3D inhomogeneous Navier–Stokes equations with density- dependent viscosity,J.Differential Equations,259(2015), 1606–1627

    2015

  18. [26]

    Q. Jiu, M. Li, and Y. Ye, Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data,J. Differential Equations,257(2014), 311–350

    2014

  19. [27]

    Jiang, P

    S. Jiang, P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equa- tions,Comm. Math. Phys.,215(2001), no. 3, 559–581. 26 SHIJIN DING, YINGHUA LI, YUANXIANG YAN, AND HAORAN ZHENG

    2001

  20. [28]

    Jiang, P

    S. Jiang, P. Zhang, Axisymmetric solutions of the 3D Navier-Stokes equations for compressible isentropic fluids,J. Math. Pures Appl.,82(2003), no. 8, 949–973

    2003

  21. [29]

    Kotschote, R

    M. Kotschote, R. Zacher, Strong solutions in the dynamical theory of compressible fluid mixtures,Math. Models Methods Appl. Sci.25(2015), 1217–1256

    2015

  22. [30]

    Li, Local existence and uniqueness of strong solutions to the Navier–Stokes equations with nonnegative density,J.Differential Equations,263(2017), no

    J. Li, Local existence and uniqueness of strong solutions to the Navier–Stokes equations with nonnegative density,J.Differential Equations,263(2017), no. 10, 6512–6536

    2017

  23. [31]

    Li, Global well-posedness of the one-dimensional compressible Navier–Stokes equations with constant heat conductivity and nonnegative density,SIAM J

    J. Li, Global well-posedness of the one-dimensional compressible Navier–Stokes equations with constant heat conductivity and nonnegative density,SIAM J. Math. Anal.,51(2019), no. 5, 3666–3693

    2019

  24. [32]

    Li, Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D,Nonlin- earity33(2020), no

    J. Li, Global well-posedness of non-heat conductive compressible Navier–Stokes equations in 1D,Nonlin- earity33(2020), no. 5, 2181–2210

    2020

  25. [33]

    P. L. Lions, Mathematical topics in fluid mechanics. Vol. 2, Oxford Lecture Ser. Math. Appl., 10, The Clarendon Press, Oxford University Press, New York, 1998, xiv+348 pp

    1998

  26. [34]

    L. D. Landau and E. M. Lifshitz,Course of theoretical physics, Elsevier, 2013

    2013

  27. [35]

    Lowengrub, L

    J. Lowengrub, L. Truskinovsky, Quasi–incompressible Cahn–Hilliard fluids and topological transitions,Proc. Royal Soc. London A: Math. Phys. Eng. Sci.,454(1998), 2617–2654

    1998

  28. [36]

    Liang, D

    Z. Liang, D. Wang, Stationary Cahn–Hilliard–Navier–Stokes equations for the diffuse interface model of compressible flows,Math. Models Methods Appl. Sci.,30(2020), 2445–2486

    2020

  29. [37]

    Liang, D

    Z. Liang, D. Wang, Weak solutions to the stationary Cahn–Hilliard/Navier–Stokes equations for compressible fluids,J. Nonlinear Sci.,32(2022), 1–25

    2022

  30. [38]

    J. Li, Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the com- pressible Navier–Stokes equations with vacuum,Ann. PDE,5(2019), no. 1, 37 pp

    2019

  31. [39]

    Y. Li, M. Xie, Y. Yan, Exponential stability of a diffuse interface model of incompressible two-phase flow with phase variable dependent viscosity and vacuum,J. Differential Equations,414(2025), 842–889

    2025

  32. [40]

    Y. Li, M. Xie, Incompressible limit of strong solutions to the diffuse interface model for two-phase flows, arXiv preprint, arXiv: 2503.00857v1, 2025

    2025

  33. [41]

    Y. Li, H. Zheng, J. Zhou, Well-posedness and blow-up criterion for strong solutions of the compressible Navier-Stokes/Allen-Cahn system with vacuum,arXiv preprint, arXiv: 2605.24568, 2026

    2026 arXiv

  34. [42]

    S. Lai, H. Xu, J. Zhang, Well-posedness and exponential decay for the Navier–Stokes equations of viscous compressible heat-conductive fluids with vacuum,Math. Models Methods Appl. Sci.,32(2022), no. 9, 1725– 1784

    2022

  35. [43]

    Li and Y

    J. Li and Y. Zheng, Local existence and uniqueness of heat-conductive compressible Navier-Stokes equations in the presence of vacuum without initial compatibility conditions,J. Math. Fluid Mech.,25(2023), no. 1, 14

    2023

  36. [44]

    Salvi, I

    R. Salvi, I. Straˇ skraba, Global existence for viscous compressible fluids and their behavior ast→ ∞,J. Fac. Sci. Univ. Tokyo Sect. IA, Math.,40(1993), 17–51

    1993

  37. [45]

    Su, On global classical solutions to one-dimensional compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and vacuum.Bound

    M. Su, On global classical solutions to one-dimensional compressible Navier-Stokes/Allen-Cahn system with density-dependent viscosity and vacuum.Bound. Value Probl., 2021, Paper No. 92, 21 pp

    2021

  38. [46]

    Zhang, Global well-posedness for the incompressible Navier–Stokes equations with density-dependent viscosity coefficient,J.Differential Equations,259(2015), 1722–1742

    J. Zhang, Global well-posedness for the incompressible Navier–Stokes equations with density-dependent viscosity coefficient,J.Differential Equations,259(2015), 1722–1742. (S. Ding)School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Email ad...

    2015

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