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arxiv: 2604.13330 · v1 · submitted 2026-04-14 · 🧮 math.AP

Derivation of effective kinetic equations describing oscillations in viscoelasticity and in compressible Navier-Stokes

Athanasios E. Tzavaras This is my paper

Pith reviewed 2026-05-10 14:12 UTC · model grok-4.3

classification 🧮 math.AP
keywords viscoelasticityNavier-Stokeskinetic formulationpersistent oscillationsnonlinear homogenizationhyperbolic-parabolic systemsphase transitionsbarotropic fluids
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The pith

Kinetic formulation techniques from conservation laws derive effective equations consisting of a kinetic equation coupled to the macroscopic flow for oscillating solutions in viscoelasticity and compressible Navier-Stokes.

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

The paper studies hyperbolic-parabolic systems that support persistent oscillations, using two mechanical examples: Kelvin-Voigt viscoelasticity with double-well strain energies for phase transitions, and the barotropic compressible Navier-Stokes equations. It first constructs solutions that maintain these oscillations over time. For the nonlinear homogenization problem in one-dimensional Lagrangian viscoelasticity and in barotropic fluids, the work adapts ideas from the kinetic formulation of conservation laws to obtain an effective description. The resulting system pairs a kinetic equation that tracks the distribution of oscillations with the averaged macroscopic flow equations.

Core claim

For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids, ideas from the kinetic formulation of conservation laws can be used to derive effective equations consisting of a kinetic equation coupled with the macroscopic flow.

What carries the argument

The effective kinetic equation coupled to the macroscopic flow, obtained by extending kinetic formulation methods to capture persistent oscillations in these hyperbolic-parabolic systems.

Load-bearing premise

That kinetic formulation techniques developed for conservation laws extend directly to these systems once solutions with persistent oscillations have been constructed, without extra restrictions on double-well potentials or barotropic pressure laws.

What would settle it

A concrete oscillating solution of the one-dimensional viscoelasticity system whose macroscopic averages fail to satisfy the derived kinetic equation coupled to the flow.

Figures

Figures reproduced from arXiv: 2604.13330 by Athanasios E. Tzavaras.

Figure 1
Figure 1. Figure 1: Deformation gradient for oscillating solutions, F+ = F0 + a ⊗ ν, F− = F0 + b ⊗ ν. defined on [1, 2] × R d that jumps across the steady interfaces x · ν = k and x · ν = k + θ, k ∈ Z. The deformation gradient and stretching are given respectively by ∇y(t, x) = ( t(F0 + a ⊗ ν) k < x · ν < k + θ t(F0 + b ⊗ ν) k + θ < x · ν < k + 1 k ∈ Z ∇v(t, x) = ( F0 + a ⊗ ν k < x · ν < k + θ F0 + b ⊗ ν k + θ < x · ν < k + 1… view at source ↗
Figure 2
Figure 2. Figure 2: density jumps at moving interfaces - dimension d = 2 is an exact solution of (5.14). Indeed,  ∂ ∂t + ˆuj ∂ ∂yj  yi t = 0 ∂t ρ0 t d + div ρ0 t d y t  = 0 For a, b positive constants and 0 < θ < 1 the function (ρ, u) defined on the domain Q = [1, 2]×R d by ρ(t, y) = ( a t d kt < |y| < (k + θ)t b t d (k + θ)t < |y| < (k + 1)t , k ∈ N0 = {0, 1, 2, ...} , u(t, y) = y t . (5.16) consists of a smooth velocity… view at source ↗
Figure 3
Figure 3. Figure 3: Placement of initial data relative to the constitutive function On the other hand if the initial data are F(x0, ξ) =    0 ξ < η1 B(ξ) η1 < ξ < η2 1 η2 ≤ ξ where η1 < b < η2 and B(ξ) is continuous and increasing from B(η1) = 0 to B(η2) = 1, see Fig.3. The solution then splits into two parts that eventually in infinite time develop to two distinct phases, and F(t, x0, ξ) will converge to the sum of two … view at source ↗
read the original abstract

These lecture notes are devoted to solutions of hyperbolic-parabolic systems with persistent oscillations. We consider two examples both from mechanics: (i) The system of viscoelasticity of Kelvin-Voigt type with strain energies involving double well potentials, as employed in phase transitions. (ii) The compressible Navier-Stokes equations for a barotropic gas. For each system we construct solutions with persistent oscillations. In a later part we consider the nonlinear homogenization problem. For the systems of viscoelasticity in one-space dimension in Lagrangian coordinates, and for the compressible Navier-Stokes system for barotropic fluids we show how ideas from the kinetic formulation of conservation laws can be used to derive effective equations. The effective equation consists by a kinetic equation coupled with the macroscopic flow.

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

2 major / 1 minor

Summary. The manuscript constructs solutions exhibiting persistent oscillations for the Kelvin-Voigt viscoelasticity system with double-well strain energies in one space dimension (Lagrangian coordinates) and for the barotropic compressible Navier-Stokes equations. It then applies ideas from the kinetic formulation of conservation laws to derive effective equations for the nonlinear homogenization problem; these effective equations consist of a kinetic equation coupled to the macroscopic flow.

Significance. If the constructions of oscillating solutions and the subsequent kinetic derivations are carried out with full rigor and appropriate compactness/entropy controls, the work would offer a potentially useful bridge between kinetic methods and hyperbolic-parabolic systems arising in phase transitions and compressible fluid mechanics, providing a framework for effective descriptions of oscillations.

major comments (2)
  1. [Abstract] Abstract: the central claim that kinetic formulation ideas from conservation laws directly yield a closed effective kinetic equation coupled to the macroscopic flow is load-bearing, yet the abstract supplies no equations, no statement of the kinetic measure, no error estimates, and no indication of how nonlinear terms are closed; this prevents verification that the extension preserves the necessary entropy dissipation and compactness properties.
  2. [Nonlinear homogenization] Nonlinear homogenization section: the derivation assumes that oscillating solutions for arbitrary double-well potentials (viscoelasticity) and barotropic pressures (Navier-Stokes) automatically supply the BV compactness or entropy production required to extract and control the kinetic measure, but the hyperbolic-parabolic structure does not guarantee this without additional growth or convexity restrictions; the manuscript must either impose such conditions explicitly or prove new compactness results at this step.
minor comments (1)
  1. [Abstract] The phrase 'consists by a kinetic equation' is grammatically incorrect and should read 'consists of a kinetic equation'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our lecture notes. We address the major comments point by point below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that kinetic formulation ideas from conservation laws directly yield a closed effective kinetic equation coupled to the macroscopic flow is load-bearing, yet the abstract supplies no equations, no statement of the kinetic measure, no error estimates, and no indication of how nonlinear terms are closed; this prevents verification that the extension preserves the necessary entropy dissipation and compactness properties.

    Authors: We agree that the abstract is concise and could better signal the technical content. We will revise it to include a brief statement of the kinetic measure (a Young measure supported on the oscillation variables) and the schematic form of the effective kinetic equation coupled to the macroscopic flow. Detailed closure of the nonlinear terms via the kinetic formulation, together with the entropy dissipation and compactness arguments, are developed in the main text (Sections 3 and 4). As this is a theoretical derivation rather than a quantitative approximation result, explicit error estimates are not part of the contribution and are not included in the abstract. revision: partial

  2. Referee: [Nonlinear homogenization] Nonlinear homogenization section: the derivation assumes that oscillating solutions for arbitrary double-well potentials (viscoelasticity) and barotropic pressures (Navier-Stokes) automatically supply the BV compactness or entropy production required to extract and control the kinetic measure, but the hyperbolic-parabolic structure does not guarantee this without additional growth or convexity restrictions; the manuscript must either impose such conditions explicitly or prove new compactness results at this step.

    Authors: The constructions in the manuscript are not for arbitrary potentials or pressures; they are carried out for specific classes of double-well energies and barotropic pressures that admit explicit oscillating solutions with the required a priori bounds. In the viscoelasticity case (one-dimensional Lagrangian coordinates), the viscous damping term together with the growth conditions on the double-well potential directly yield uniform BV bounds on the strain, which are used to extract the kinetic measure. For the barotropic Navier-Stokes system, the entropy dissipation identity and the structure of the pressure law provide the necessary compactness to control the kinetic measure. We will add an explicit remark at the beginning of the nonlinear homogenization section stating the precise growth and convexity assumptions on the potentials and pressures under which the constructions hold, thereby making the compactness controls transparent without invoking new general theorems. revision: partial

Circularity Check

0 steps flagged

No circularity: external kinetic ideas applied to constructed oscillating solutions

full rationale

The paper first constructs solutions with persistent oscillations for the Kelvin-Voigt viscoelasticity system (double-well potentials) and barotropic compressible Navier-Stokes. It then imports kinetic formulation techniques developed for conservation laws to obtain an effective kinetic equation coupled to the macroscopic flow. This step does not reduce any derived quantity to a fitted parameter, self-definition, or self-citation chain; the kinetic measure extraction and closure rely on external conservation-law results (entropy dissipation, compactness) rather than redefining the target effective equation in terms of itself. No load-bearing uniqueness theorem or ansatz is smuggled via self-citation. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, ad-hoc axioms, or new entities are stated. The work implicitly relies on standard existence theory for hyperbolic-parabolic systems and on the applicability of kinetic averaging techniques.

axioms (2)
  • domain assumption Existence of weak or measure-valued solutions with persistent oscillations for the viscoelasticity and compressible Navier-Stokes systems
    Required to construct the oscillating solutions before homogenization can be performed
  • domain assumption Kinetic formulation techniques developed for scalar conservation laws extend to the present hyperbolic-parabolic systems
    Central to the derivation of the effective equations

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Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    Andrews and J

    G. Andrews and J. M. Ball. Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelas- ticity.J. Differential Equations,44(2) (1982), 306–341

    work page 1982

  2. [2]

    J. M. Ball and R. D. James. Fine phase mixtures as minimizers of energy.Arch. Rational Mech. Anal.,100(1) (1987), 13–52

    work page 1987

  3. [3]

    J. M. Ball, A version of the fundamental theorem for Young measures, inPDEs and Continuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys.344, Springer, Berlin, 1989, pp. 207–215

    work page 1988

  4. [4]

    Brenier, R´ esolution d’ equations d’ ´ evolution quasilin´ eaires en dimensionNd’espace ` a l’aide d´ equations lin´ eaires en dimensionN+ 1,J

    Y. Brenier, R´ esolution d’ equations d’ ´ evolution quasilin´ eaires en dimensionNd’espace ` a l’aide d´ equations lin´ eaires en dimensionN+ 1,J. Differential Equations50(1983), 375-390

    work page 1983

  5. [5]

    Bresch and P.E

    D. Bresch and P.E. Jabin. Global existence of weak solutions for compressible Navier-Stokes equations: Thermo- dynamically unstable pressure and anisotropic viscous stress tensor.Annals of Math.,188(2) (2018), 577-684

    work page 2018

  6. [6]

    G.-Q. G. Chen and B. Perthame, Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire20(2003), no. 4, 645–668

    work page 2003

  7. [7]

    C. M. Dafermos. Can dissipation prevent the breaking of waves? InTransactions of the Twenty-Sixth Conference of Army Mathematicians (Hanover, N.H., 1980), volume 1 ofARO Rep. 81, pages 187–198. U.S. Army Res. Office, Research Triangle Park, NC, 1981

    work page 1980

  8. [8]

    C. M. Dafermos. Contemporary Issues in the Dynamic Behavior of Continuous Media, LCDS Lecture Notes # 85-1, 1985

    work page 1985

  9. [9]

    C. M. Dafermos.Hyperbolic conservation laws in continuum physics, volume 325 ofGrundlehren der mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, fourth edition, 2016

    work page 2016

  10. [10]

    Dellacherie and P.-A

    C. Dellacherie and P.-A. Meyer,Probabilit´ es et Potentiel, Hermann, 1975

    work page 1975

  11. [11]

    Demoulini

    S. Demoulini. Weak solutions for a class of nonlinear systems of viscoelasticity.Arch. Ration. Mech. Anal.,155(4) (2000), 299–334

    work page 2000

  12. [12]

    Demoulini, D

    S. Demoulini, D. M. A. Stuart and A. E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics,Arch. Ration. Mech. Anal.205(2012), 927–961

    work page 2012

  13. [13]

    Dunford and J.T

    N. Dunford and J.T. Schwartz,Linear Operators, Part I, Interscience, 1967

    work page 1967

  14. [14]

    Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable,Comment

    E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable,Comment. Math. Univ. Carolin.42(2001), no. 1, 83–98

    work page 2001

  15. [15]

    Feireisl, A

    E. Feireisl, A. Novotn´ y and H. Petzeltov´ a, On the domain dependence of solutions to the compressible Navier- Stokes equations of a barotropic fluid, Math. Methods Appl. Sci.25(2002), no. 12, 1045–1073

    work page 2002

  16. [16]

    Feireisl.Dynamics of viscous compressible fluids, volume 26 ofOxford Lecture Series in Mathematics and its Applications

    E. Feireisl.Dynamics of viscous compressible fluids, volume 26 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004

    work page 2004

  17. [17]

    G. B. Folland.Real analysis. Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, second edition, 1999. Modern techniques and their applications, A Wiley-Interscience Publication. OSCILLATIONS AND HOMOGENIZATION 57

    work page 1999

  18. [18]

    Friesecke and G

    G. Friesecke and G. Dolzmann. Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy.SIAM J. Math. Anal.,28(2) (1997), 363–380

    work page 1997

  19. [19]

    Hillairet

    M. Hillairet. Propagation of density-oscillations in solutions to the barotropic compressible Navier-Stokes system. J. Math. Fluid Mech.,9(3) (2007) 343–376

    work page 2007

  20. [20]

    D. Hoff. Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data.Proc. Roy. Soc. Edinburgh Sect. A,103(3-4) (1986), 301–315

    work page 1986

  21. [21]

    Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontin- uous initial data,J

    D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontin- uous initial data,J. Differential Equations120(1995), no. 1, 215–254

    work page 1995

  22. [22]

    Hwang and A

    S. Hwang and A. E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: application to relaxation and diffusion-dispersion approximations,Comm. Partial Differential Equations27(2002), no. 5-6, 1229–1254

    work page 2002

  23. [23]

    R. D. James. Finite deformation by mechanical twinning.Arch. Rational Mech. Anal.,77(2) (1981), 143–176

    work page 1981

  24. [24]

    Koumatos, C

    K. Koumatos, C. Lattanzio, S. Spirito, and A. E. Tzavaras. Existence and uniqueness for a viscoelastic Kelvin- Voigt model with nonconvex stored energy.J. Hyperbolic Differ. Equ.,20(2) (2023), 433–474

    work page 2023

  25. [25]

    Lions, Compacit´ e des solutions des ´ equations de Navier-Stokes compressibles isentropiques, C

    P.-L. Lions, Compacit´ e des solutions des ´ equations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris S´ er. I Math.317(1993), no. 1, 115–120

    work page 1993

  26. [26]

    Lions.Mathematical topics in fluid mechanics

    P.-L. Lions.Mathematical topics in fluid mechanics. Vol. 2, volume 10 ofOxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998. Compressible models, Oxford Science Publications

    work page 1998

  27. [27]

    Lions, B

    P.-L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations,J. Amer. Math. Soc.7(1994), no. 1, 169–191

    work page 1994

  28. [28]

    Lions, B

    P.-L. Lions, B. Perthame, and E. Tadmor. Kinetic formulation of the isentropic gas dynamics andp-systems. Comm. Math. Phys.,163(2) (1994), 415–431

    work page 1994

  29. [29]

    Liu and Y

    T.-P. Liu and Y. Zeng. Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws.Mem. Amer. Math. Soc.,125(1997), no. 599, viii+120 pp

    work page 1997

  30. [30]

    Liu, Regularization of solution operators for quasilinear hyperbolic-parabolic partial differential equations, J

    T.-P. Liu, Regularization of solution operators for quasilinear hyperbolic-parabolic partial differential equations, J. Partial Differential Equations10(4): 347–354, 1997

    work page 1997

  31. [31]

    Novotn´ y and I

    A. Novotn´ y and I. Straˇ skraba,Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications, 27, Oxford Univ. Press, Oxford, 2004

    work page 2004

  32. [32]

    Pedregal,Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and Their Applications30, Birkh¨ auser, Basel, 1997

    P. Pedregal,Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and Their Applications30, Birkh¨ auser, Basel, 1997

    work page 1997

  33. [33]

    Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and Stability,Arch

    R.L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: Admissibility and Stability,Arch. Rational Mech. Anal.97(1987), 353-394

    work page 1987

  34. [34]

    Perthame.Kinetic formulation of conservation laws, volume 21 ofOxford Lecture Series in Mathematics and its Applications

    B. Perthame.Kinetic formulation of conservation laws, volume 21 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002

    work page 2002

  35. [35]

    Perthame and A

    B. Perthame and A. E. Tzavaras. Kinetic formulation for systems of two conservation laws and elastodynamics. Arch. Ration. Mech. Anal.,155(1) (2000), 1–48

    work page 2000

  36. [36]

    Plotnikov and J

    P.I. Plotnikov and J. Sokolowski.Compressible Navier-Stokes equations. Theory and shape optimization.Mono- grafie Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk, New Series 73, Birkh¨ auser, Basel, 2012

    work page 2012

  37. [37]

    D. Serre. Variations de grande amplitude pour la densit´ e d’un fluide visqueux compressible.Phys. D,48(1) (1991), 113–128

    work page 1991

  38. [38]

    Tartar, Compensated compactness and applications to partial differential equations

    L. Tartar, Compensated compactness and applications to partial differential equations. InNonlinear Analysis and Mechanics, R.J. Knops, ed., Heriot–Watt Symposium, Vol. IV, Pitman Research Notes in Mathematics, Pitman, Boston, 1979, pp.136–192

    work page 1979

  39. [39]

    Tzavaras

    A.E. Tzavaras. Sustained oscillations in hyperbolic-parabolic systems.Arch. Rational Mech. Anal.,248(2024): Art 51, 29pp

    work page 2024

  40. [40]

    Tzavaras Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems.J

    A.E. Tzavaras Oscillations in Compressible Navier-Stokes and Homogenization in Phase Transition problems.J. Hyperbolic Differ. Equ.,21(3) (2024), 827 - 843. (Athanasios E. Tzavaras) Computer, Electrical and Mathematical Science and Engineering Division King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Saudi Arabia Email address...

    work page 2024