pith. sign in

arxiv: 2604.16110 · v1 · submitted 2026-04-17 · 🧮 math.NA · cs.NA

Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions

Jan Giesselmann , Philipp \"Offner , Robert Sauerborn This is my paper

Pith reviewed 2026-05-10 07:57 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Navier-Stokes-Kortewegfinite volume schemedissipative weak solutionsconditional convergencestructure-preserving discretizationnumerical analysis for fluids
0
0 comments X

The pith

A structure-preserving finite volume scheme for the Navier-Stokes-Korteweg system converges conditionally to dissipative weak solutions.

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

The paper introduces dissipative weak solutions for the Navier-Stokes-Korteweg equations and proves that a recently developed finite volume scheme converges to them when the scheme is consistent and stable. This convergence rests on the scheme's exact preservation of mass, momentum, and a discrete energy dissipation inequality, which pass to the limit without needing additional compactness arguments. The result extends earlier convergence theorems for the Euler and Navier-Stokes equations to a model that includes capillary forces. A reader cares because it supplies a rigorous justification for using the scheme in regimes where classical smooth solutions cease to exist.

Core claim

The central claim is that any sequence generated by the structure-preserving finite volume scheme that remains bounded in suitable norms converges, up to subsequence, to a dissipative weak solution of the Navier-Stokes-Korteweg system. A dissipative weak solution satisfies the integral form of mass and momentum conservation together with an energy inequality that accounts for viscous dissipation and capillary contributions. The proof proceeds by showing that the scheme's conservation and dissipation properties are inherited by the limit, together with consistency of the numerical fluxes.

What carries the argument

dissipative weak solutions, which are weak solutions augmented by a global energy dissipation inequality that the finite volume scheme preserves at the discrete level and passes to the limit.

If this is right

  • Consistency and stability of the finite volume scheme are together sufficient for convergence to a dissipative weak solution.
  • The same proof strategy applies to other structure-preserving discretizations of capillary fluid models.
  • Existence of dissipative weak solutions for the Navier-Stokes-Korteweg system follows from the existence of a stable consistent scheme.
  • The numerical method remains reliable even after singularities form in the solution.

Where Pith is reading between the lines

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

  • The result suggests that existence proofs for the continuous Navier-Stokes-Korteweg system can be obtained by constructing and analyzing suitable numerical approximations.
  • Similar conditional convergence statements may hold for higher-order or adaptive versions of the scheme once their discrete dissipation properties are verified.
  • One could test the practical convergence rate by monitoring how closely the discrete energy dissipation tracks the continuous inequality on refined meshes.

Load-bearing premise

The scheme produces a uniformly stable sequence of approximate solutions whose consistency errors vanish in the appropriate sense, and the limit object satisfies the definition of a dissipative weak solution.

What would settle it

A concrete sequence of initial data and mesh refinements for which the scheme is consistent and produces bounded solutions, yet the limit fails to satisfy the energy dissipation inequality.

read the original abstract

We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency.

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 paper introduces a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and proves conditional convergence of a structure-preserving finite volume scheme to such solutions. The argument relies on the scheme's discrete conservation properties, a discrete energy-dissipation inequality that accounts for the Korteweg capillary term, and consistency estimates to pass to the limit and recover the DW solution definition. This extends prior results on DW solutions for the Euler and Navier-Stokes equations to the NSK model.

Significance. If the result holds, the work provides a rigorous justification for convergence of numerical schemes for the NSK system in the absence of strong solutions, extending the Lax equivalence theorem framework to capillary fluid models. The structure-preserving finite volume scheme and the tailored DW solution concept are strengths, as they directly leverage conservation and dissipation to obtain compactness and weak convergence without additional ad-hoc assumptions.

minor comments (3)
  1. The introduction should explicitly cite the specific reference for the 'recently proposed FV scheme' mentioned in the abstract to allow readers to trace the discrete scheme definition.
  2. Notation for the discrete capillary energy term in the dissipation inequality could be clarified to avoid confusion with the continuous Korteweg stress tensor.
  3. The consistency estimates in the limit passage would benefit from a dedicated subsection summarizing the key bounds used for the capillary contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no listed major comments, so we have no specific points requiring rebuttal or detailed response. We will incorporate any minor editorial or technical suggestions in the revised version of the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard compactness arguments

full rationale

The paper proposes a definition of dissipative weak solutions for the NSK system that extends the Lax equivalence theorem, then establishes conditional convergence of a given structure-preserving FV scheme by verifying discrete conservation, a discrete energy-dissipation inequality (including capillary terms), and consistency estimates, followed by standard weak-convergence and compactness passage to the limit. These steps rely on explicit a-priori bounds and consistency error control rather than redefining the target solution in terms of the scheme outputs or fitting parameters to the result. The argument builds on referenced prior work for Euler/NS cases but does not reduce the central claim to a self-citation chain or tautological renaming; the DW concept is independently motivated and the convergence proof supplies independent content through the scheme-specific estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The abstract provides insufficient detail to enumerate free parameters or specific axioms; the main addition appears to be the definition of dissipative weak solutions and the conditional convergence statement.

axioms (1)
  • domain assumption Existence of dissipative weak solutions for the NSK system that are compatible with the scheme's discrete conservation and dissipation properties
    The convergence result is stated toward such solutions; their existence is presupposed for the conditional statement to be meaningful.
invented entities (1)
  • Dissipative weak (DW) solutions no independent evidence
    purpose: Generalized solution concept that incorporates energy dissipation to enable convergence proofs for the NSK system
    Newly proposed in the paper as an extension of weak solutions tailored to the numerical analysis setting.

pith-pipeline@v0.9.0 · 5417 in / 1331 out tokens · 52394 ms · 2026-05-10T07:57:32.193046+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Abgrall, M

    R. Abgrall, M. Luk´ aˇ cov´ a-Medvid’ov´ a, and P.¨Offner. Convergence of residual distribution schemes for the Euler equations via dissipative weak solutions. Math. Models Methods Appl. Sci., 01(33), 2023

    work page 2023

  2. [2]

    Antonelli and S

    P. Antonelli and S. Spirito. Global existence of weak solutions to the navier–stokes–korteweg equations.Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 2022

    work page 2022

  3. [3]

    Chertock, P

    A. Chertock, P. Degond, and J. Neusser. An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations.J. Comput. Phys., 335:387–403, 2017

    work page 2017

  4. [4]

    Chikami and T

    N. Chikami and T. Kobayashi. Global well-posedness and time-decay estimates of the compressible Navier-Stokes-Korteweg system in critical Besov spaces.J. Math. Fluid Mech., 21(2):32, 2019. Id/No 31

    work page 2019

  5. [5]

    Dhaouadi and M

    F. Dhaouadi and M. Dumbser. A structure-preserving finite volume scheme for a hyperbolic reformulation of the Navier–Stokes–Korteweg equations.Mathe- matics, 11(4), 2023

    work page 2023

  6. [6]

    R. J. Diperna. Measure-valued solutions to conservation laws.Arch. Ration. Mech. Anal., 88:223–270, 1985

    work page 1985

  7. [7]

    Donatelli, E

    D. Donatelli, E. Feireisl, and P. Marcati. Well/ill posedness for the Euler- Korteweg-Poisson system and related problems.Commun. Partial Differ. Equations, 40(7):1314–1335, 2015

    work page 2015

  8. [8]

    Eiter, J

    T. Eiter, J. Giesselmann, R. Lasarzik, P. ¨Offner, and R. Sauerborn. Conver- gence analysis for a finite-volume scheme for the Euler- and Navier-Stokes- Korteweg system via energy-variational solutions, 2026

    work page 2026

  9. [9]

    Feireisl, P

    E. Feireisl, P. Gwiazda, A. ´Swierczewska-Gwiazda, and E. Wiedemann. Dis- sipative measure-valued solutions to the compressible Navier-Stokes system. Calc. Var. Partial Differ. Equ., 55(6):20, 2016. Id/No 141

    work page 2016

  10. [10]

    Feireisl, M

    E. Feireisl, M. Luk´ aˇ cov´ a-Medviˇdov´ a, H. Mizerov´ a, and B. She. On the conver- gence of a finite volume method for the Navier-Stokes-Fourier system.IMA J. Numer. Anal., 41(4):2388–2422, 2021

    work page 2021

  11. [11]

    Feireisl, M

    E. Feireisl, M. Luk´ aˇ cov´ a-Medvid’ov´ a, H. Mizerov´ a, and B. She.Numerical analysis of compressible fluid flows. Springer, 2021

    work page 2021

  12. [12]

    Gallenm¨ uller, P

    D. Gallenm¨ uller, P. Gwiazda, A. ´Swierczewska-Gwiazda, and J. Wo´ znicki. Cahn-hillard and Keller-Segel systems as high-friction limits of Euler-Korteweg and Euler-Poisson equations.Calc. Var. Partial Differ. Equ., 63(2):27, 2024. Id/No 47

    work page 2024

  13. [13]

    Giesselmann, C

    J. Giesselmann, C. Lattanzio, and A. E. Tzavaras. Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics.Arch. Ration. Mech. Anal., 223(3):1427–1484, 2017

    work page 2017

  14. [14]

    Giesselmann, P

    J. Giesselmann, P. ¨Offner, and R. Sauerborn. A structure preserving finite volume scheme for the Navier-Stokes-Korteweg equations, 2026. 20

    work page 2026

  15. [15]

    Hajduk, D

    H. Hajduk, D. Kuzmin, G. Lube, and P. ¨Offner. Locally energy-stable finite element schemes for incompressible flow problems: design and analysis for equal-order interpolations.Comput. Fluids, 294:14, 2025. Id/No 106622

    work page 2025

  16. [16]

    Luk´ aˇ cov´ a-Medvid’ov´ a and P.¨Offner

    M. Luk´ aˇ cov´ a-Medvid’ov´ a and P.¨Offner. Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutions.Appl. Math. Comput., 436:22, 2023. Id/No 127508

    work page 2023

  17. [17]

    Luk´ aˇ cov´ a-Medvid’ov´ a and A

    M. Luk´ aˇ cov´ a-Medvid’ov´ a and A. Sch¨ omer. Existence of dissipative solutions to the compressible Navier-Stokes system with potential temperature transport. J. Math. Fluid Mech., 24(3):31, 2022. Id/No 82

    work page 2022

  18. [18]

    Luk´ aˇ cov´ a-Medviˇdov´ a, B

    M. Luk´ aˇ cov´ a-Medviˇdov´ a, B. She, and Y. Yuan. Convergence and error es- timates of a penalization finite volume method for the compressible Navier- Stokes system.IMA J. Numer. Anal., 45(2):1054–1101, 2025

    work page 2025

  19. [19]

    Noble and J.-P

    P. Noble and J.-P. Vila. Stability theory for difference approximations of Euler- Korteweg equations and application to thin film flows.SIAM J. Numer. Anal., 52(6):2770–2791, 2014

    work page 2014

  20. [20]

    M. Slemrod. Hilbert’s sixth problem and the failure of the Boltzmann to Euler limit.Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci., 376(2118):12,

    work page

  21. [21]

    L. Tian, Y. Xu, Y. Yang, and J. J. W. Van der Vegt. An energy stable lo- cal discontinuous Galerkin method for the isothermal Navier-Stokes-Korteweg equations.J. Sci. Comput., 104(2):33, 2025. Id/No 63

    work page 2025

  22. [22]

    J. Yang, H. Wang, and Q. Shi. Dissipative measure-valued solutions of the compressible Navier–Stokes–Korteweg system.Mathematical Methods in the Applied Sciences, 49(2):851–860, 2026. 21

    work page 2026