Pith. sign in

REVIEW 2 major objections 4 minor 65 references

Weak solutions to the 2D or 3D stochastic NSCHEs

T0 review · 2 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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

Pith's one-line read Transport-noise Navier–Stokes–Cahn–Hilliard mixtures admit global weak martingale solutions in 2D and 3D, with pathwise uniqueness in 2D.

desk verdict Solid first existence theory for NSCHEs with genuine transport noise, via a carefully executed hybrid of Mikulevicius-Rozovskii and Brzeźniak-Motyl; the coercivity gap is real but standard. read the letter →

arxiv 2607.10486 v1 pith:CCCDFUST submitted 2026-07-11 math.PR

Z. Brzeźniak , A. Ndongmo Ngana This is my paper · ORCID
classification math.PR MSC 60H1535Q3035R6076D05
keywords stochasticNavier–Stokes–Cahn–HilliardtransportnoiseweakmartingalesolutionspathwiseuniquenessdiffuseinterfacemodelGalerkinapproximationproblem
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

The paper studies a diffuse-interface model of two incompressible fluids whose velocity is driven by transport-type noise. It proves that, under a short list of abstract coercivity and growth conditions on the noise coefficients and with a regular Landau free-energy potential, global weak martingale solutions exist in both two and three space dimensions. In two dimensions the same solutions are pathwise unique (hence strong) once the multiplicative noise is Lipschitz. The argument unifies two previously separate approaches to stochastic PDEs: a Galerkin approximation that produces tight laws, followed by a direct identification of the limiting measure as a solution of a martingale problem on a non-metric path space, without changing the underlying probability space. The result supplies the first rigorous existence theory for this physically natural class of stochastic diffuse-interface models.

What carries the argument

A Galerkin scheme whose laws are tight in a carefully chosen non-metric space Z_T of paths; the limiting measure is identified as a solution of the associated martingale problem by a generalization of Prohorov’s theorem that avoids the Jakubowski–Skorokhod representation.

What would settle it

Exhibit a smooth, divergence-free transport field for which the coercivity constant δ₀ vanishes and show that the corresponding Galerkin sequence loses its uniform energy bound, so that tightness fails.

Watch

Extended reading notes

Core claim

Under the abstract assumptions of Section 5 (coercivity of the transport noise relative to viscosity, linear growth of the multiplicative coefficients, Carathéodory regularity, and the Landau potential), the stochastic Navier–Stokes–Cahn–Hilliard system possesses at least one global weak martingale solution on any finite time horizon; when the spatial dimension is two the solution is pathwise unique.

Load-bearing premise

The transport noise must be strictly weaker than the viscous dissipation (a spectral-gap condition that keeps the energy estimate closed).

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the stochastic Navier–Stokes–Cahn–Hilliard system (NSCHEs) with transport (gradient) noise on a bounded domain in R^d, d=2 or 3. Under abstract assumptions (coercivity of the noise relative to viscosity, linear growth of the multiplicative coefficients, Carathéodory regularity, Landau potential), the authors prove existence of a global weak martingale solution on any finite horizon T (Theorem 5.10) and, when d=2, pathwise uniqueness of the strong solution (Theorem 5.11 / 11.1). The argument proceeds by Galerkin approximation, uniform energy estimates, tightness of laws on a non-metric path space Z_T, weak convergence of measures via a Prohorov-type result, and identification of the limit as a solution of the martingale problem; the construction unifies the Mikulevicius–Rozovskii and Brzeźniak–Motyl frameworks.

Significance. Transport noise is physically natural for binary fluid mixtures, yet previous stochastic NSCHE analyses treated only additive or non-gradient multiplicative noise. Establishing global weak martingale solutions in both 2D and 3D, together with 2D pathwise uniqueness, under a single abstract set of hypotheses is a genuine advance. The hybrid tightness–martingale-problem method is carefully executed and may be reusable for other coupled fluid–phase-field systems. The energy estimates, compactness criteria, and uniqueness argument are written out in detail and appear self-contained under the stated assumptions.

major comments (2)
  1. The coercivity condition (3.33)/(5.11) that forces the transport-noise intensity strictly below viscosity (δ₀>0) is load-bearing: without it the a-priori estimate (6.10) and all subsequent Galerkin tightness collapse. The paper correctly invokes the condition throughout Sections 5–6 and Lemma 5.7, but the physical range of admissible noise amplitudes is never quantified. A short remark on how restrictive (3.33) is for typical transport-noise models, or a pointer to literature where the spectral-gap condition is known to hold, would strengthen the applicability claim.
  2. In the identification of the limit (Theorem 8.13 and Part 4 of the proof of Theorem 5.10), the representation of the martingale M via the cylindrical Wiener process (9.35)–(9.37) relies on Lemma B.8. The lemma is stated for the Gelfand triple (U,H,U'), yet the verification that the quadratic variation process satisfies the integrability needed for the lemma is only sketched via (9.20) and (8.46). A more explicit check that the stopped processes remain square-integrable under the Landau-potential growth would remove residual doubt about the passage from the local martingale problem to the Itô equation.
minor comments (4)
  1. The arXiv identifier in the header is 2607.10486; the year 2607 is almost certainly a typographical error and should be corrected before publication.
  2. Notation for the chemical potential switches between μ, μ(ϕ) and μ_n without a uniform convention; a short glossary at the beginning of Section 2 would help the reader.
  3. Several lengthy technical estimates are deferred to Appendices A–E. Cross-references in the main text (e.g., “by Lemma A.2”) are accurate, but a one-sentence roadmap of what each appendix contains would improve readability.
  4. In the uniqueness section the process Y₂(t) defined in (11.4) is quite involved; a brief explanation of the origin of each term would make the Schmalfuss-trick argument easier to follow.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: pure existence/uniqueness via energy estimates, Galerkin, tightness and martingale identification under stated abstract assumptions; self-citations are methodological tools only.

  1. self citation load bearing [Introduction, p. 4 and Concluding Remarks Sec 12]
    "This is the first result addressing a unified framework derived from the works by Brzeźniak & Motyl and Mikulevicius & Rozovskii... The main contribution of this work is the development of a unified framework that combines two complementary approaches, building on the results [46] and [11]."

    The authors cite their own prior methodological papers as the source of the hybrid tightness/martingale-problem technique. This is not load-bearing for the existence/uniqueness claims themselves (which are proved from scratch via energy estimates and Galerkin), but it is the only self-referential element; the concrete estimates for the NSCHE system are independent.

full rationale

The paper derives global weak martingale solutions (Thm 5.10) and 2-D pathwise uniqueness (Thm 5.11/11.1) from first-principles a-priori estimates (Prop 6.1, 6.4, 6.8), Galerkin approximations (6.4), tightness on the non-metric space Z_T (Lem 7.14 via Aldous-Rebolledo), Prohorov-type weak convergence of laws (Prop 8.1), and identification of the martingale problem via stopped processes M^{n,z} and the representation (9.35)–(9.37). All estimates close under the explicit abstract hypotheses of Sec 5 (coercivity (5.11), linear growth (5.10), Carathéodory regularity, Landau potential). There are no fitted parameters, no data predictions, no self-definitional loops, and no uniqueness imported as an external black-box fact that forces the result. Self-citations to Brzeźniak–Motyl [11] and Mikulevicius–Rozovskii [46] appear only as sources of the hybrid methodological framework that is then re-worked and applied to the new NSCHE system; the concrete estimates and the identification argument are carried out in full in the present text. Minor self-citation of methodological tools does not raise the score above 1.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure-existence theorem in infinite-dimensional stochastic analysis. All free parameters are model constants fixed a priori; the axioms are standard functional-analytic and probabilistic background plus the structural assumptions needed for the energy estimates.

assumptions (4)
  • domain assumption Coercivity of the transport noise: ν|ξ|² - ½∑_k |σ_k·ξ|² ≥ δ₀|ξ|² (Assumption 3.16 / (5.11))
    Essential for the basic energy estimate that closes the a-priori bounds; without it the Galerkin sequence is not tight.
  • domain assumption Landau potential ψ(s)=¼(1-s²)² (or any C² potential satisfying the growth (2.64))
    Used to obtain the L¹ bound on ψ(ϕ) and the cancellation identities involving ψ'(ϕ).
  • standard math Domain O is bounded of class C³; filtration satisfies the usual conditions; cylindrical Wiener process on ℓ²
    Standard setting for stochastic Navier-Stokes theory; needed for Sobolev embeddings and Stokes operator regularity.
  • domain assumption Linear growth and Carathéodory regularity of the multiplicative noise coefficients G and Σ (Assumptions 5.3)
    Guarantees that the stochastic integrals are well-defined martingales and that the Galerkin approximations converge.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weak solutions to the 2D or 3D stochastic NSCHEs." pith.science (2026). https://pith.science/paper/CCCDFUST

@misc{pith2026260710486,
  author       = {Pith},
  title        = {Pith review of: Weak solutions to the 2D or 3D stochastic NSCHEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCCDFUST}},
  note         = {Machine review of arXiv:2607.10486}
}
read the original abstract

We consider a diffuse interface model for the mixture of two incompressible fluids driven by transport noise. Under suitable abstract assumptions, we prove the existence of a global weak martingale solution as well as the pathwise uniqueness of a global strong solution in the two dimensional case. This is the first result addressing a unified framework derived from the works by Brze{\'z}niak \& Motyl and Mikulevicius \& Rozovskii, on the study of stochastic partial differential equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 1 linked inside Pith

  1. [1]

    Abels,On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities.Arch

    H. Abels,On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities.Arch. Rational Mech. Anal.194(2009), 463–506

    2009

  2. [2]

    Abels, D

    H. Abels, D. Depner, and H. Garcke,Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities.J. Math. Fluid Mech.15(2013), 453–480

    2013

  3. [3]

    Aldous,Stopping times and tightness.Ann

    D. Aldous,Stopping times and tightness.Ann. Probab.6(4) (1978), 335–340

    1978

  4. [4]

    Boyer and P

    F. Boyer and P. Fabrie, MATHEMATICAL TOOLS FOR THE STUDY OF THE INCOMPRESSIBLENAVIER-STOKES EQUATIONS AND RELATED MODELS.183, Springer, 2012

    2012

  5. [5]

    Brezis, FUNCTIONALANALYSIS, SOBOLEVSPACES ANDPARTIALDIFFERENTIALEQUATIONS

    H. Brezis, FUNCTIONALANALYSIS, SOBOLEVSPACES ANDPARTIALDIFFERENTIALEQUATIONS. Springer Science & Business Media, LLC 2011

    2011

  6. [6]

    Brezis, Analyse fonctionnelle, Masson, 1983

    H. Brezis, Analyse fonctionnelle, Masson, 1983

    1983

  7. [7]

    Brze ´zniak, B

    Z. Brze ´zniak, B. Ferrario, and M. Zanella,Invariant measures for a stochastic nonlinear and damped 2D Schr ¨odinger equation. Nonlinearity37(1), id.015001, 67 pp., 2024

    2024

  8. [8]

    Brze ´zniak, F

    Z. Brze ´zniak, F. Hornung, and U. Manna,Weak martingale solutions for the stochastic nonlinear Schr¨odinger equation driven by pure jump noise. Stoch. Partial Differ. Equ., Anal. Comput.,8(1), 1–53 (2020)

    2020

Show all 65 references
  1. [9]

    Brze ´zniak, T

    Z. Brze ´zniak, T. Kosmala, E. Motyl, and P. Razafimandimby,Weak solutions of Navier-Stokes Equation with purely discontinuous L ´evy Noise, Electron. J. Probab.31, 1-90 (2026)

    2026

  2. [10]

    Brze ´zniak, B

    Z. Brze ´zniak, B. Maslowski, and J. Seidler,Stochastic nonlinear beam equations.Probab. Theory Relat. Fields132(2005), 119–149. 106

    2005

  3. [11]

    Brze ´zniak and E

    Z. Brze ´zniak and E. Motyl,Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D domains. J. Differential Equations254(4) (2013), 1627–1685

    2013

  4. [12]

    Brze ´zniak and E

    Z. Brze ´zniak and E. Motyl,The existence of martingale solutions to the stochastic Boussinesq equations. Global and Stochastic Analysis1(2) (2014), 1–42

    2014

  5. [13]

    Brze ´zniak, E

    Z. Brze ´zniak, E. Motyl, and M. Ondrej ´at,Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains.Ann. Probab.45(5) (2017), 3145–3201

    2017

  6. [14]

    Brze ´zniak and S

    Z. Brze ´zniak and S. Peszat,Stochastic two dimensional Euler equations. Ann. Probab.29(4) (2013), 1796–1832

    2013

  7. [15]

    P. C. Hohenberg and B. I. Halperin,Theory of dynamic critical phenomena.Rev. Mod. Phys.49(1977), 435–479

    1977

  8. [16]

    Da Prato and A

    G. Da Prato and A. Debussche,Stochastic Cahn-Hilliard equations.Nonlinear Anal.26(2) (1996), 241–263

    1996

  9. [17]

    Da Prato and J

    G. Da Prato and J. Zabczyk, STOCHASTICEQUATIONS ININFINITEDIMENSIONS(2nd ed.), vol.152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2014

    2014

  10. [18]

    Deugou ´e, A

    G. Deugou ´e, A. Ndongmo Ngana, and T. Tachim Medjo,Strong solutions for the stochastic Cahn- Hilliard-Navier-Stokes system.J. Differential Equations275(2021), 27–76

    2021

  11. [19]

    Deugou ´e, A

    G. Deugou ´e, A. Ndongmo Ngana, and T. Tachim Medjo,Some convergences results on the stochastic Cahn-Hilliard-Navier-Stokes equations with multiplicative noise.Potential Anal.59(2023), 263–282

    2023

  12. [20]

    Deugou ´e and T

    G. Deugou ´e and T. Tachim Medjo,Convergence of the solutions of the stochastic 3D globally modified Cahn-Hilliard-Navier-Stokes equations.J. Differential Equations265(2) (2018), 545–592

    2018

  13. [21]

    Deugou ´e and T

    G. Deugou ´e and T. Tachim Medjo,The exponential behavior of a stochastic globally modified Cahn-Hilliard-Navier-Stokes model with multiplicative noise.J. Math. Anal. Appl.460(1) (2018), 140–163

    2018

  14. [22]

    Di Primio, L

    A. Di Primio, L. Scarpa and M. Zanella,Existence, uniqueness and asymptotic stability of invariant measures for the stochastic Allen-Cahn-Navier-Stokes system with singular potential, arXiv:2501.06174

    arXiv

  15. [23]

    R. M. Dudley, REAL ANALYSIS AND PROBABILITY. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002

    1989

  16. [24]

    S. N. Ethier and T. G. Kurtz, MARKOVPROCESSES: CHARACTERIZATION ANDCONVERGENCE, Wiley, 1986

    1986

  17. [25]

    C. G. Gal and M. Grasselli,Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Ann. Inst. H. Poincar ´e Anal. Non Lin ´eaire27(1) (2010), 401–436

    2010

  18. [26]

    C.G. Gal, M. Grasselli, and A. Miranville,Cahn-Hilliard-Navier-Stokes systems with moving contact lines. Calc. Var. Partial Differ. Equ.55(50) (2016), 1–47

    2016

  19. [27]

    Gilbarg and N.S

    D. Gilbarg and N.S. Trudinger, ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS OF SECOND ORDER(2nd ed.), vol.224of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, 1983

    1983

  20. [28]

    Giorgini, A

    A. Giorgini, A. Miranville, R. Temam,Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system.SIAM J. Math. Anal.51(2019), 2535–2574

    2019

  21. [29]

    Giorgini, A

    A. Giorgini, A. Ndongmo Ngana, T. Tachim Medjo, and R. Temam,Existence and regularity of strong solutions to a nonhomogeneous Kelvin-Voigt-Cahn-Hilliard system.J. Differential Equations372(2023), 612–656

    2023

  22. [30]

    Giorgini and R

    A. Giorgini and R. Temam,Weak and strong solutions to the nonhomogeneous incompressible Navier- Stokes-Cahn-Hilliard system. J. Math. Pures Appl.144(2020), 194–249

    2020

  23. [31]

    M. E. Gurtin, D. Polignone, and J. Vi ˜nals,Two-phase binary fluids and immiscible fluids described by an order parameter.Math. Models Methods Appl. Sci.6(1996), 815–831

    1996

  24. [32]

    Ikeda and S

    N. Ikeda and S. Watanabe, STOCHASTICDIFFERENTIALEQUATIONS ANDDIFFUSIONPROCESSES(2nd ed.). North-Holland/Kodansha, 1989

    1989

  25. [33]

    Jacod and A

    J. Jacod and A. N. Shiryaev, LIMIT THEOREMS FOR STOCHASTIC PROCESSES(2nd ed.). Springer, 2002

    2002

  26. [34]

    Jakubowski,The almost sure Skorokhod representation for subsequences in nonmetric spaces.Theory Probab

    A. Jakubowski,The almost sure Skorokhod representation for subsequences in nonmetric spaces.Theory Probab. Appl.42(1) (1998) 167–174

    1998

  27. [35]

    Jakubowski,On the Skorokhod topology.Ann

    A. Jakubowski,On the Skorokhod topology.Ann. Inst. H. Poincar ´e Probab. Statist.22(3) (1986), 263–285

    1986

  28. [36]

    Karatzas and S

    I. Karatzas and S. E. Shreve, BROWNIAN MOTION AND STOCHASTIC CALCULUS(2nd ed.), vol.113, Springer-Verlag, 1991

    1991

  29. [37]

    J. L. Kelley, GENERALTOPOLOGY. Graduate Texts in Mathematics, New York: D. Van Nostrand Company, 1995

    1995

  30. [38]

    Kato, PERTURBATION THEORY FOR LINEAR OPERATORS

    T. Kato, PERTURBATION THEORY FOR LINEAR OPERATORS. Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.xxii+619 pp. ISBN: 3-540-58661-X 107

    1980

  31. [39]

    L. D. Landau and E. M. Lifshitz, COURSE OFTHEORETICALPHYSICSSTATISTICALPHYSICS. V ol.5, Pergamon Press, Oxford-Edinburgh-New York, 1968

    1968

  32. [40]

    Lions and E

    J.-L. Lions and E. Magenes, NON-HOMOGENEOUS BOUNDARY VALUE PROBLEMS AND APPLICATIONS. V ol. I. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972

    1972

  33. [41]

    Liu and J

    C. Liu and J. Shen,A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method.Physica D179(3-4) (2003), 211–228

    2003

  34. [42]

    Liu and N

    C. Liu and N. J. Walkington,An Eulerian description of fluids containing visco-hyperelastic particles. Arch. Rat. Mech. Anal.159(2001), 229–252

    2001

  35. [43]

    M ´etivier, SEMIMARTINGALES

    M. M ´etivier, SEMIMARTINGALES. ACOURSE ON STOCHASTIC PROCESSES. De Gruyter Studies in Mathematics, 2. Walter de Gruyter & Co., Berlin-New York, 1982.xi+287 pp. ISBN: 3-11-008674-3

    1982

  36. [44]

    M ´etivier, STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN INFINITE-DIMENSIONAL SPACES

    M. M ´etivier, STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN INFINITE-DIMENSIONAL SPACES. With a preface by G. Da Prato; Scuola Normale Superiore di Pisa. Quaderni. Scuola Normale Superiore, Pisa, 1988

    1988

  37. [45]

    M ´etivier, J

    M. M ´etivier, J. Pellaumail, STOCHASTIC INTEGRATION, Probability and Mathematical Statistics. Academic Press New York-London-Toronto, 1980.xii+196 pp. ISBN: 0-12-491450-0

    1980

  38. [46]

    Mikulevicius and B

    R. Mikulevicius and B. L. Rozovskii,GlobalL 2-solutions of stochastic Navier-Stokes equations. Ann. Probab.33(1), 137–176 (2005)

    2005

  39. [47]

    Mikulevicius and B

    R. Mikulevicius and B. L. Rozovskii, MARTINGALE PROBLEMS FOR STOCHASTICPDE’S. IN STOCHASTIC PARTIALDIFFERENTIALEQUATIONS: Six Perspectives (R. Carmona and B. L. Rozovskii, eds.) Mathematical Surveys and Monographs 64 243–325. Amer. Math. Soc., Providence, RI, 1998

    1998

  40. [48]

    Øksendal, STOCHASTICDIFFERENTIALEQUATIONS: ANINTRODUCTION WITHAPPLICATIONS

    B. Øksendal, STOCHASTICDIFFERENTIALEQUATIONS: ANINTRODUCTION WITHAPPLICATIONS. 6th ed., Springer-Verlag, Berlin, 2003

    2003

  41. [49]

    Pardoux,Stochastic partial differential equations and filtering of diffusion processes.Stochastics3 (1979), 127–167

    E. Pardoux,Stochastic partial differential equations and filtering of diffusion processes.Stochastics3 (1979), 127–167

    1979

  42. [50]

    K. R. Parthasarathy, PROBABILITY MEASURES ON METRIC SPACES. PROBABILITY ANDMATHEMATICAL STATISTICS, No. 3. Academic Press, Inc., New York-London, 1967

    1967

  43. [51]

    P. E. Protter, Stochastic integration and differential equations (2nd ed.), vol.21of Applications of Mathematics. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004

    2004

  44. [52]

    Prevot and M

    C. Prevot and M. R ¨ockner,A concise course on stochastic partial differential equations. Springer LN in Math. 1905, Berlin (2007)

    1905

  45. [53]

    Reed and B

    M. Reed and B. Simon, METHODS OF MODERN MATHEMATICAL PHYSICS. I. FUNCTIONAL ANALYSIS (2nd ed.), Academic Press, Inc. New York, 1980

    1980

  46. [54]

    Revuz and M

    D. Revuz and M. Yor, CONTINUOUSMARTINGALES ANDBROWNIANMOTION, 3rd ed., Springer, 1999

    1999

  47. [55]

    Rudin, FUNCTIONAL ANALYSIS

    W. Rudin, FUNCTIONAL ANALYSIS. INTERNATIONALSERIES INPURE ANDAPPLIEDMATHEMATICS. McGraw-Hill, Inc., New York, second edition, 1991

    1991

  48. [56]

    W. A. Strauss,On continuity of functions with values in various Banach spaces.Pacific J. Math.19 (3) (1966), 543–555

    1966

  49. [57]

    D. W. Stroock and S.R.S. Varadhan, MULTIDIMENSIONAL DIFFUSION PROCESSES. Reprint of the 1997 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2006

    1997

  50. [58]

    Tachim Medjo,On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model.J

    T. Tachim Medjo,On the existence and uniqueness of solution to a stochastic 2D Cahn-Hilliard-Navier-Stokes model.J. Differential Equations263(2017), 1028–1054

    2017

  51. [59]

    Tachim Medjo, C

    T. Tachim Medjo, C. Tone, and F. Tone,Maximum principle of optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints.Optim Control Appl Meth.42(2021), 807–832

    2021

  52. [60]

    Temam, NAVIER-STOKES EQUATIONS

    R. Temam, NAVIER-STOKES EQUATIONS. Theory and numerical analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001

    1984

  53. [61]

    Temam, NAVIER-STOKESEQUATIONS

    R. Temam, NAVIER-STOKESEQUATIONS. North-Holland, Amsterdam, 1979

    1979

  54. [62]

    Temam, INFINITE-DIMENSIONALDYNAMICALSYSTEMS INMECHANICS ANDPHYSICS, Springer-Verlag, New York, 1997

    R. Temam, INFINITE-DIMENSIONALDYNAMICALSYSTEMS INMECHANICS ANDPHYSICS, Springer-Verlag, New York, 1997

    1997

  55. [63]

    M. J. Vishik, A. V . Fursikov, MATHEMATICAL PROBLEM OF STATISTICAL HYDROMECHANICS, kluwer Academic Publishers, Dordrecht, 1988

    1988

  56. [64]

    Williams, PROBABILITY WITHMARTINGALES, Cambridge University Press, Cambridge, 1991

    D. Williams, PROBABILITY WITHMARTINGALES, Cambridge University Press, Cambridge, 1991

    1991

  57. [65]

    Zhao,Strong solutions to the density-dependent incompressible Cahn-Hilliard-Navier-Stokes system.J

    L. Zhao,Strong solutions to the density-dependent incompressible Cahn-Hilliard-Navier-Stokes system.J. Hyperbolic Differ. Equ.16(2019), 701–742. 108

    2019

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.