Viscoelastic flows with conservation laws

Saint-Venant); S\'ebastien Boyaval (MATHERIALS

REVIEW 3 major objections 5 minor 68 references

Viscoelastic flows with conservation laws

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Pith's one-line read A new conservative Saint-Venant–Maxwell system is proved symmetric hyperbolic, so smooth shallow-water viscoelastic Cauchy problems are locally well-posed.

desk verdict A genuinely new conservative shallow-water Maxwell system with a plausible entropy, but the central symmetrizability proof is broken as printed: equation (21) does not follow from (20). read the letter →

arxiv 1908.03344 v1 pith:EVDYDEGZ submitted 2019-08-09 math.NA cs.NAmath.APphysics.class-phphysics.flu-dyn

Sébastien Boyaval (MATHERIALS , Saint-Venant) This is my paper · ORCID

classification math.NAcs.NAmath.APphysics.class-phphysics.flu-dyn MSC 35L6576A1065M0876M12

keywords Saint-Venant-MaxwellequationssymmetrichyperbolicsystemsconservationlawsviscoelasticflowsMaxwellfluidsshallow-waterfinitevolumemethodlocalwell-posedness

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

This paper tries to establish that Maxwell-type viscoelastic flows—fluids with memory described by one relaxation time—can be cast as a symmetric hyperbolic system of conservation laws in the shallow-water regime. The proposed Saint-Venant–Maxwell system introduces two new positive state variables, $A_h$ and $A_{cc}$, which track viscous deformation of the material microstructure, and rewrites mass, momentum, deformation, and internal-variable balance in conservative form. If the central claim is right, smooth Cauchy problems for the model are locally well-posed, giving the first conservative symmetric-hyperbolic formulation for these flows and a theoretical foundation for numerical simulation. The system contains the earlier Saint-Venant upper-convected Maxwell model as a closed subsystem, reduces to the classical Saint-Venant shallow-water equations as the elasticity modulus $G\to 0$, and reduces to thin-layer elastodynamics in the large-relaxation-time limit.

What carries the argument

The load-bearing device is the new pair of internal variables, a symmetric positive-definite $2\times2$ matrix $A_h$ and a scalar $A_{cc}>0$, interpreted as viscous deformations of the microstructure in the reference configuration, with relaxation equations $D_t A_h=(F_h^{-1}F_h^{-T}-A_h)/\lambda$ and $D_t A_{cc}=(H^{-2}-A_{cc})/\lambda$. Using these, the internal energy is written with strains $B_h=F_h A_h F_h^T$ and $B_{zz}=H^2 A_{cc}$ in a form whose change of conserved variables—$A_h^{-2}$ and $A_{cc}^{1/4}$—makes the total energy $H\tilde E$ strictly convex, the condition the Godunov–Mock theorem needs for symmetric hyperbolicity. The strict convexity of $\tilde E_2=\mathrm{tr}(F_h A_h F_h^T)$ in the pair $(F_h,A_h^{-2})$ is the convexity ingredient that carries the argument.

What would settle it

Take any two admissible pairs $(F_1,Y_1)$ and $(F_2,Y_2)$ with $Y=A_h^{-2}>0$ and test whether $E_2(\theta F_1+(1-\theta)F_2,\theta Y_1+(1-\theta)Y_2)<\theta E_2(F_1,Y_1)+(1-\theta)E_2(F_2,Y_2)$ for every $\theta\in(0,1)$; a single counterexample on $\{H>0,\ A_h>0,\ A_{cc}>0\}$ would refute the convexity step. Simpler: numerically evaluate the asserted inequality $\mathrm{tr}(H_\theta Y_\theta^{1/2}H_\theta^T)>\mathrm{tr}(F_\theta Y_\theta^{-1/2}F_\theta^T)$ in Eq. (23) for random matrices and report any failure.

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Extended reading notes

Core claim

The paper's central claim is Proposition 2.1: the quasilinear system of conservation laws, written for the conserved variables $(H, H U, H F_h, H A_{cc}^{1/4}, H A_h^{-2})$, is symmetric hyperbolic on the convex admissibility domain $\{H>0,\ A_h^{-1}=A_h^{-T}>0,\ A_{cc}^{-1}>0\}$, with mathematical entropy $H\tilde E = H(|U|^2+gH)/2 + G H\big(\mathrm{tr}(F_h A_h F_h^T)+H^2 A_{cc}\big)/2$. Symmetric hyperbolicity follows from the strict convexity of $\tilde E$ with respect to a full set of conserved variables via the Godunov–Mock theorem. This yields Corollary 2.1: smooth Cauchy problems are locally well-posed, strong solutions preserve the involution $H=|F_h|^{-1}$, and the companion free-energy relation holds with a thermodynamically compatible dissipation rate. The SVM system contains the upper-convected Maxwell model as a closed subsystem, and its formal limits reproduce Saint-Venant shallow-water flow as $G\to0$ and thin-layer elastodynamics as $\lambda\to\infty$.

Load-bearing premise

The entire well-posedness claim rests on the strict convexity of $\mathrm{tr}(F_h A_h F_h^T)$ in the pair $(F_h,A_h^{-2})$, which the proof of Proposition 2.1 asserts through a compressed matrix inequality around Eq. (23); if that convexity fails anywhere on the admissible domain, symmetric hyperbolicity—and with it local well-posedness—is unsupported.

Editorial extensions

If this is right

Smooth solutions of the SVM system exist locally in time from smooth initial data, so the model is a sound starting point for transient geophysical flow simulations.
The conservative form plus entropy inequality gives a target for Finite-Volume discretizations that preserve admissible states and dissipate free energy, as demonstrated by the proposed Riemann solver and four numerical test cases.
The limits $G\to0$ and $\lambda\to\infty$ recover Saint-Venant shallow-water flow and thin-layer elastodynamics, so a single model interpolates between liquid and solid behaviour in the shallow-water regime.
The closed SVUCM subsystem inherits a well-posedness statement for translation-invariant reductions, explaining previous 1D numerical observations.
The model channels Maxwell's relaxation idea into the standard theory of symmetrizable hyperbolic conservation laws, avoiding the non-conservative products that complicate other viscoelastic formulations.

Reading between the lines

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

If the convexity proof holds, the same change of variables may be tried on 3D and compressible Maxwell-type models, since the internal-variable idea is dimension-agnostic.
The strict-convexity step in Proposition 2.1 deserves a standalone verification; a fully expanded proof or a counterexample would settle whether the method is robust, and a numerical random search over admissible $(F_h,A_h^{-2})$ pairs could test the asserted matrix inequality directly.
The paper's interpretation of $A$ as a material-order parameter suggests a potential bridge to Reynolds-averaged turbulence closures, where a transported tensor would carry memory of flow-induced microstructure distortion.
A testable extension is to check whether the symmetry losses reported in the rotated and axisymmetric numerical tests disappear when the reconstruction preserves $H=|F_h|^{-1}$ exactly, which would isolate the involution-preservation step as the source of numerical anisotropy.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read it. The SVM system (20) with the internal variables A_h, A_cc is a real modeling contribution: it gives a conservative shallow-water Maxwell system that contains SVUCM and reduces to Saint-Venant and elastodynamics in the right limits, and the entropy (22) is a natural candidate. Remark 3 is honest about overlap with Peshkov–Romenski and Teshukov, so the abstract's "first" should be read as "first symmetric hyperbolic conservative shallow-water Maxwell system," not literally first hyperbolic viscoelastic model.

The problem is the central proof. Proposition 2.1 asserts symmetrizability of (20) after switching to HA_h^{-2}. But equation (21), as printed, is not the evolution of A_h^{-2} implied by (20). From (20), A_h satisfies dA_h/dt = (F_h^{-1}F_h^{-T} - A_h)/lambda; differentiating A_h^{-2} gives a source whose terms do not match the right-hand side of (21). In the scalar reduction the printed source has the wrong sign, a missing factor 2, and does not vanish at the relaxation equilibrium. So the system shown to be symmetric hyperbolic is not the SVM system (20), and Corollary 2.1's local well-posedness does not follow as written. This is central, not cosmetic. The convexity inequality around (23) is also too compressed: the notation D_theta, H_theta, and the chain of inequalities need a full rewrite before a referee can certify it.

The numerical section is ambitious but does not carry the theory: no code or data, only qualitative figures, and the paper itself reports symmetry loss and a smeared contact discontinuity. The source terms are modeled, not fitted, but they are postulated constitutive laws; that is acceptable if presented as modeling, and the paper mostly does that.

Net: the construction deserves serious consideration, and if the authors can fix (21) and give a clean convexity proof, this could be a useful paper. As it stands, the advertised well-posedness result is unsupported. I would send it to peer review because the idea is worth referee time, but I would flag the algebra explicitly. I would not rely on Proposition 2.1 in my own work until it is corrected.

Best

Referee Report

3 major / 5 minor

Summary. The paper introduces a Saint-Venant--Maxwell (SVM) system of hyperbolic balance laws for shallow-water viscoelastic flows, obtained by embedding the non-conservative upper-convected Maxwell shallow-water system into a larger conservation form through new positive-definite internal variables A_h and A_cc. The central theoretical claim is Proposition 2.1: that the SVM system, written in the conservative variables (H, HU, HF_h, H A_cc^{1/4}, H A_h^{-2}), is symmetric hyperbolic on the admissible domain, with the entropy H\tilde E given in Eq. (22). This yields local well-posedness (Corollary 2.1), containment of the SVUCM model as a closed subsystem, and the Saint-Venant and elastodynamic limits. The paper also proposes a finite-volume discretization based on a relaxation Riemann solver and reports four numerical test cases.

Significance. If the main theorem were correct, this would be a valuable contribution: the first conservative symmetric-hyperbolic formulation of a shallow-water Maxwell viscoelastic model, with potential applications to computational rheology and geophysical flows. The model construction is explicit and physically motivated, the embedding of SVUCM is clear, and the numerical experiments, though exploratory, illustrate the intended behavior of shear-wave propagation and vortex development. I give credit for the clear conservative extension strategy and the careful discussion of the limitations of the numerical scheme. However, the central well-posedness claim is not established as printed because the rewritten A_h-equation is algebraically inconsistent with the original SVM system and the convexity proof contains unverifiable notation. These are load-bearing issues that must be corrected before the main claims can be assessed.

major comments (3)

[§2.2, Eq. (21)] The printed equation for ∂t(H A_h^{-2}) is not equivalent to the SVM system (20). From (20) and mass conservation, D_t A_h = (F_h^{-1}F_h^{-T} - A_h)/λ, so with W = A_h^{-2}, direct differentiation gives D_t W = (2A_h^{-2} - A_h^{-1}F_h^{-1}F_h^{-T}A_h^{-2} - A_h^{-2}F_h^{-1}F_h^{-T}A_h^{-1})/λ. The right-hand side printed in (21) is instead (A_h^{-2} - A_h^{-2}F_h^{-1}F_h^{-T}A_h^{-1} + A_h^{-1}F_h^{-1}F_h^{-T}A_h^{-2})/λ. These expressions do not coincide in general. In the scalar reduction F_h = f, A_h = a, the correct source is 2a^{-2}(1 - a^{-1}f^{-2})/λ, whereas (21) gives a^{-2}/λ, which is nonzero at the relaxation equilibrium a = f^{-2}. Therefore the conservative system whose symmetrizability is claimed in Proposition 2.1 has not been shown to be the SVM system (20), and Corollary 2.1 is unsupported as stated.
[§2.2, proof of Proposition 2.1, Eq. (23)] The joint-convexity argument for \tilde E_2(F_h, A_h^{-2}) = tr(F_h A_h F_h^T) is not verifiable from the text. The objects H_θ and D_θ are introduced without a coherent definition, the displayed chain 'tr(H_θ Y_θ^{1/2} H_θ^T) > ... = tr(F_θ^T F_θ + θ(1−θ)D_θ^T D_θ) ≥ tr(F_θ^T F_θ)' does not transparently imply the claimed inequality θE_2(F_1,Y_1)+(1−θ)E_2(F_2,Y_2) > E_2(F_θ,Y_θ), and the final step 'since Y_θ^{-1/2} is symmetric positive definite' is unclear. Since strict convexity of the entropy in the conservative variables is the essential input to the Godunov--Mock theorem, this proof must be rewritten with all quantities defined and the inequalities justified. As printed, the strict convexity of \tilde E_2 is not established.
[§3 and Appendix A] The finite-volume section claims a fully admissible and entropy-consistent discretization for SVM, but Appendix A explicitly states that no 2D relaxation system can admit the proposed 1D Riemann solver as a particular solution and be consistent with all smooth solutions of the Lagrangian SVM system (29), see the discussion after Eq. (78). This limitation is acknowledged, but it should be reconciled with the statement of Proposition 3.5 that the Eulerian solver is fully admissible for SVM in the sense of Proposition 3.2, or Proposition 3.5 should be restricted to the reconstructed 1D sub-problem. The numerical claims are not central to the well-posedness theorem, but they are part of the paper's contribution and need a precise formulation.

minor comments (5)

[Throughout] There are numerous typos and inconsistencies: 'SVCUM' for 'SVUCM', 'Numebr' for 'Number', 'noet' for 'note', 'Corrolary' for 'Corollary', and 'entlightened' for 'enlightened'. These should be corrected in a revision.
[Proposition 2.1, Eq. (21)] The parenthetical '(A^{-1}_h is A^{-2}_h square-root matrix)' is garbled; the notation A_h^{-2} and the square-root convention used in the proof should be defined explicitly.
[Appendix A, Eqs. (67)--(68)] The same symbol is used for the relaxation time ε and for the sign parameter ϵ ∈ {+,−}, which makes the displays hard to follow. Please use distinct notation.
[§2.2, after Eq. (26)] The remark that no full set of conserved variables makes HE convex, while H\tilde E is used for symmetrizability, is an important caveat. The relation between the two entropies and the admissible weak solutions satisfying (26) should be clarified in a dedicated paragraph.
[§3.2, Proposition 3.4] The assumptions F^‖_f ≡ 0 and E^m_e F^⊥_f ≡ 1 are not invariant under a change of material basis; they should be stated as reconstruction conventions rather than generic assumptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SVM is constructed by postulating constitutive laws and then verifying strict convexity locally; self-citations are contextual, not load-bearing.

full rationale

The derivation chain for the central claim is self-contained. The paper postulates constitutive laws (16)-(17) for A_h and A_cc, chooses the internal energy (18), writes the conservative system (20), and then proves Proposition 2.1 by checking strict convexity of H\tilde E in the proposed variables, using the Godunov-Mock theorem from external literature. No parameter is fitted to a data subset and renamed a prediction; no uniqueness theorem is imported from the authors' prior work to forbid alternatives. The heavy citations to [9,10,11] motivate and compare models, but the symmetrizability argument is printed in the paper and does not reduce to those citations. The paper explicitly flags limitations, including the non-convexity of the physical free energy: "we have not been able to find a full set of conserved variables such that the free-energy HE is convex strictly on the whole admissible domain A when it is defined as in (14)... This is why we use H\tilde E rather than HE to show that the SVM system is symmetric hyperbolic" and the unresolved status of multidimensional entropy solutions. These are acknowledged limitations, not circular steps. A possible algebraic mismatch between Eq. (21) and Eq. (20) would be a correctness defect, not a circularity, because it does not make the claimed result equal to its own input by construction. Accordingly, no circularity is found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The SVM system introduces two new internal variables with postulated evolution laws; no fitted constants appear, but the physical status of these variables is the main burden. The central mathematical claims rely on standard theorems plus a compressed convexity proof.

assumptions (5)

standard math Godunov-Mock theorem: a system of conservation laws with a strictly convex entropy is symmetric hyperbolic.
Used in Prop 2.1 to conclude symmetric hyperbolicity from convexity of H\tilde E.
standard math Equivalence: H\tilde E is convex in (H,HU,HFh,HA_cc^{1/4},HA_h^{-2}) iff \tilde E is convex in (H^{-1},U,Fh,A_cc^{1/4},A_h^{-2}), cited from Bouchut [6].
Bridges the specific entropy to conserved variables; cited as [6] in Prop 2.1.
domain assumption The internal energy of SVUCM (18), with the logarithmic terms ln(det B_h)+ln B_zz, is the physically correct Helmholtz free energy for Maxwell fluids.
Taken from [67, 9]; needed for the dissipation computation D in Cor 2.1.
ad hoc to paper The new state variables A_h and A_cc satisfy the postulated relaxation laws (16)-(17): D_t A_h = (F_h^{-1}F_h^{-T}-A_h)/lambda, D_t A_cc = (H^{-2}-A_cc)/lambda.
These constitutive laws are the mechanism that embeds SVUCM into a conservative system; they are postulated, with only a formal stochastic interpretation in Remark 1, no independent experimental derivation.
domain assumption Smooth solutions preserve the involution H=|F_h|^{-1}, which is used to recover SVUCM and to interpret B_zz=H^2 A_cc.
Stated in Cor 2.1 'one can directly check'; standard for deformation-gradient-type equations but not proven in detail.

invented entities (2)

A_h (symmetric positive-definite 2x2 internal distortion tensor)
purpose: New state variable measuring viscous/microstructural deformation in the reference configuration; enables conservative reformulation of SVUCM.
No direct observable or falsifiable prediction is attached to A_h; it is observable only through B_h=F_h A_h F_h^T in smooth SVUCM solutions. The stochastic equations (34)-(35) are formal and not validated.
A_cc (scalar vertical distortion variable)
purpose: Accounts for vertical stretching of the microstructure, allowing a conservative formulation of B_zz.
Same status as A_h; no independent measurement proposed.

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Cite this review

Pith. "Pith review of Viscoelastic flows with conservation laws." pith.science (2026). https://pith.science/paper/EVDYDEGZ

@misc{pith2026190803344,
  author       = {Pith},
  title        = {Pith review of: Viscoelastic flows with conservation laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVDYDEGZ}},
  note         = {Machine review of arXiv:1908.03344}
}

read the original abstract

We propose in this work the first symmetric hyperbolic system of conservation laws to describe viscoelastic flows of Maxwell fluids, i.e. fluidswith memory that are characterized by one relaxation-time parameter. Precisely, the system of quasilinear PDEs is detailed for the shallow-water regime, i.e. for hydrostatic incompressible 2D flows with free surface under gravity. It generalizes Saint-Venant system to viscoelastic flows of Maxwell fluids, and encompasses previous works with F. Bouchut. It also generalizes the (thin-layer) elastodynamics of hyperelastic materials to viscous fluids, and to various rheologies between solid and liquid states that can be formulated using our new variable as material parameter.The new viscoelastic flow model has many potential applications, additionally to falling into the theoretical framework of (symmetric hyper-bolic) systems of conservation laws. In computational rheology, it offers a new approach to the High-Weissenberg Number Problem (HWNP). Fortransient geophysical flows, it offers perspectives of thermodynamically-compatible numerical simulations, with a Finite-Volume (FV) discretization say. Besides, one FV discretization of the new continuum model is proposed herein to precise our ideas incl. the physical meaning of the solutions. Perspectives are finally listed after some numerical simulations.

Figures

Figures reproduced from arXiv: 1908.03344 by the authors.

**Figure 1.** Test case 1: 1D slice of the variables H, Ux , Bzz, Bxx (from top to bottom, left to right) at times t = .1 and t = .2 along x ∈ [0, 8] mainly because of our mass-conforming projection of Fh in each cell at the end of the transport-projection method, whose implementation through re-balancing singular values in SVD decomposition does not preserve well the symmetry. One consequence is that the contact discontinuity is… view at source ↗

**Figure 2.** Test case 2: 1D slice of the variables H, Ux , Bzz, Bxx (from top to bottom, left to right) at times t = .1 and t = .2 along x ∈ [0, 8] [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗

**Figure 3.** Test case 2: 3D view of H (left) and 2D vector field (U x , Uy ) (right) at time t = .2 23 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Test case 3: 1D slice of the variables H, Ux , Bzz, Bxx (from top to bottom, left to right) at times t = .1 and t = .2 along a radial direction (θ fixed) [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: Test case 3: 3D view of H (left) and 2D vector field (U x , Uy ) (right) at time t = .2 24 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: Test case 4: 2D vector field (U x , Uy ) at times t = .5 and 10 (left/right) [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]

**Figure 7.** Figure 7: Test case 4: 3D view of H (top left), time evolution of the spaceaveraged energy (top right), and 1D slice of the final-time velocity components U x , Uy (bottom left and right) along x before and after the centerline y = 4 (hence the two curves for U x ). 25 [PITH_F…

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