pith. sign in

arxiv: 2604.16000 · v1 · submitted 2026-04-17 · 🧮 math.AP

Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows

Rahul Barthwal , Philipp \"Offner , Christian Rohde This is my paper

Pith reviewed 2026-05-10 08:25 UTC · model grok-4.3

classification 🧮 math.AP
keywords systemsentropyexistencesolutionsthin-filmapproximateclassfirst-order
0
0 comments X

The pith

Global weak entropy solutions exist for a class of non-symmetric Keyfitz-Kranzer systems that includes thin-film flow models.

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

Thin liquid layers spreading on surfaces are modeled by systems of equations that can develop singularities or fail to exist over long times. This work focuses on a family of such systems known as non-symmetric Keyfitz-Kranzer type, which covers lubrication approximations. The authors construct a family of entropy pairs that work for both the original system and a smoothed version with added diffusion. They locate an invariant region that keeps solutions bounded, then pass to the limit as diffusion vanishes to obtain global weak solutions satisfying entropy conditions.

Core claim

We prove the existence of global weak entropy solutions for a class of non-symmetric Keyfitz-Kranzer type systems that includes lubrication models for thin-film flow.

Load-bearing premise

The existence of an invariant region in the state space that yields uniform L^∞ bounds for the approximate parabolic system, combined with the admissibility of the identified entropy pairs for the tailored second-order regularization.

read the original abstract

We prove the existence of global weak entropy solutions for a class of non-symmetric Keyfitz-Kranzer type systems that includes lubrication models for thin-film flow. We identify a family of entropy/entropy-flux pairs for these first-order systems, which is, in particular, admissible for a tailored second-order approximate system. The latter is motivated by higher-order dissipation operators in thin-film flow models. By identifying an invariant region in the state space, it is possible to derive a-priori $L^\infty$-bounds for the sequence of solutions to the approximate system. Exploiting the parabolic and transport structure of the equations associated with the Riemann invariants, we then rigorously justify the vanishing-diffusion limit and establish the existence of weak entropy solutions for the Cauchy problem for the first-order systems.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain-standard assumptions in hyperbolic PDE theory: the existence of suitable entropy pairs for the regularized system and the presence of an invariant region yielding L^∞ bounds. No free parameters or new entities are introduced.

axioms (2)
  • domain assumption A family of entropy/entropy-flux pairs exists that is admissible for the tailored second-order approximate system
    Invoked to justify the entropy condition in the vanishing-diffusion limit.
  • domain assumption An invariant region exists in state space that produces uniform L^∞ bounds for solutions of the approximate system
    Used to pass to the limit and obtain global solutions.

pith-pipeline@v0.9.0 · 5437 in / 1275 out tokens · 62680 ms · 2026-05-10T08:25:55.979580+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

28 extracted references · 28 canonical work pages · 1 internal anchor

  1. [1]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara.Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Oxford: Clarendon Press, 2000

    work page 2000

  2. [2]

    Barthwal and T

    R. Barthwal and T. Raja Sekhar. Two-dimensional non-self-similar Riemann solutions for a thin film model of a perfectly soluble anti-surfactant solution.Quarterly of Applied Mathe- matics, 80(4):717–738, 2022

    work page 2022

  3. [3]

    Barthwal, T

    R. Barthwal, T. Raja Sekhar, and G. P. Raja Sekhar. Construction of solutions of a two- dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solu- tion.Mathematical Methods in the Applied Sciences, 46(6):7413–7434, 2023

    work page 2023

  4. [4]

    Barthwal and C

    R. Barthwal and C. Rohde. A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant.arXiv preprint arXiv:2502.17205, 2025

    work page internal anchor Pith review arXiv 2025

  5. [5]

    Barthwal, C

    R. Barthwal, C. Rohde, and A. Sen. Existence and stability of the Riemann solutions for a non-symmetric Keyfitz–Kranzer type model.Nonlinearity, 39(3):035006, 2026

    work page 2026

  6. [6]

    Barthwal, C

    R. Barthwal, C. Rohde, and Y. Wang. A generalized Riemann problem solver for a hyperbolic model of two-layer thin film flow.Journal of Scientific Computing, 106(1):25, 2026

    work page 2026

  7. [7]

    Bianchini and A

    S. Bianchini and A. Bressan. Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Mathematics. Second Series, 161(1):223–342, 2005

    work page 2005

  8. [8]

    Chen and M

    G.-Q. Chen and M. Perepelitsa. Vanishing viscosity solutions of the compressible euler equations with spherical symmetry and large initial data.Communications in Mathemati- cal Physics, 338(2):771–800, 2015

    work page 2015

  9. [9]

    J. Conn, B. Duffy, D. Pritchard, S. Wilson, and K. Sefiane. Simple waves and shocks in a thin film of a perfectly soluble anti-surfactant solution.Journal of Engineering Mathematics, 107(1):167–178, 2017. 16

    work page 2017

  10. [10]

    C. M. Dafermos.Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin, 2000

    work page 2000

  11. [11]

    Diehl and C

    D. Diehl and C. Rohde. On the structure of MHD shock waves in diffusive-dispersive media. Journal of Mathematical Fluid Mechanics, 8(1):120–145, 2006

    work page 2006

  12. [12]

    R. J. DiPerna. Convergence of approximate solutions to conservation laws.Archive for Rational Mechanics and Analysis, 82(1):27–70, 1983

    work page 1983

  13. [13]

    L. C. Evans.Partial differential equations, volume 19 ofGrad. Stud. Math.Providence, RI: American Mathematical Society, 1998

    work page 1998

  14. [14]

    Freist¨ uhler

    H. Freist¨ uhler. Rotational degeneracy of hyperbolic systems of conservation laws.Archive for Rational Mechanics and Analysis, 113(1):39–64, 1991

    work page 1991

  15. [15]

    Freist¨ uhler

    H. Freist¨ uhler. On the Cauchy problem for a class of hyperbolic systems of conservation laws. Journal of Differential Equations, 112(1):170–178, 1994

    work page 1994

  16. [16]

    A. Heibig. Existence and uniqueness of solutions for some hyperbolic systems of conservation laws.Archive for Rational Mechanics and Analysis, 126(1):79–101, 1994

    work page 1994

  17. [17]

    James, Y.-J

    F. James, Y.-J. Peng, and B. Perthame. Kinetic formulation for chromatography and some other hyperbolic systems.Journal de Math´ ematiques Pures et Appliqu´ ees, 74(4):367, 1995

    work page 1995

  18. [18]

    B. L. Keyfitz and H. C. Kranzer. A system of non-strictly hyperbolic conservation laws arising in elasticity theory.Archive for Rational Mechanics and Analysis, 72(3):219–241, 1980

    work page 1980

  19. [19]

    Lu.Hyperbolic conservation laws and the compensated compactness method

    Y. Lu.Hyperbolic conservation laws and the compensated compactness method. Chapman and Hall/CRC, 2002

    work page 2002

  20. [20]

    Y.-G. Lu. Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz- Kranzer type.Journal of Functional Analysis, 261(10):2797–2815, 2011

    work page 2011

  21. [21]

    Y.-G. Lu. Existence of global entropy solutions to general system of Keyfitz-Kranzer type. Journal of Functional Analysis, 264(10):2457–2468, 2013

    work page 2013

  22. [22]

    Y.-G. Lu. Existence of global weak entropy solutions to some nonstrictly hyperbolic systems. SIAM Journal on Mathematical Analysis, 45(6):3592–3610, 2013

    work page 2013

  23. [23]

    Pandey, R

    A. Pandey, R. Barthwal, and T. R. Sekhar. Construction of solutions of the Riemann prob- lem for a two-dimensional Keyfitz-Kranzer type model governing a thin film flow.Applied Mathematics and Computation, 498:129378, 2025

    work page 2025

  24. [24]

    D. Serre. Solutions ` a variations born´ ees pour certains syst` emes hyperboliques de lois de conser- vation. (Solutions of bounded variations for certain hyperbolic systems of conservation laws). Journal of Differential Equations, 68:137–168, 1987

    work page 1987

  25. [25]

    C. Shen. Delta shock wave solution for a symmetric Keyfitz–Kranzer system.Applied Mathe- matics Letters, 77:35–43, 2018

    work page 2018

  26. [26]

    B. Temple. Systems of conservation laws with invariant submanifolds.Transactions of the American Mathematical Society, 280(2):781–795, 1983. 17

    work page 1983

  27. [27]

    Yang and Y

    H. Yang and Y. Zhang. New developments of delta shock waves and its applications in systems of conservation laws.Journal of Differential Equations, 252(11):5951–5993, 2012

    work page 2012

  28. [28]

    Yang and Y

    H. Yang and Y. Zhang. Delta shock waves with dirac delta function in both components for systems of conservation laws.Journal of Differential Equations, 257(12):4369–4402, 2014. 18

    work page 2014