Weak solution for granular model

Laurent Chupin (LMBP); Thierry Dubois (LMBP)

REVIEW 3 major objections 4 minor 29 references

Weak solution for granular model

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A simplified granular flow model coupling threshold rheology with dilatancy is shown to have a weak solution, with velocity and pressure uniquely determined.

desk verdict Genuinely new weak formulation and careful existence proof for a simplified granular model, but the approximate-solution step is cited rather than proved and the abstract overstates the scope. read the letter →

arxiv 2505.17588 v1 pith:S46WVIUN submitted 2025-05-23 math.AP physics.class-ph

Laurent Chupin (LMBP) , Thierry Dubois (LMBP) This is my paper · ORCID

classification math.APphysics.class-ph MSC 35D3035Q3576T25

keywords granularflowthresholdrheologydilatancyweaksolutionexistenceanduniquenesspressurepositivitynon-Newtonianviscositymu-I

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 establishes an existence and uniqueness theory for a PDE model of dense granular flow that combines threshold rheology with dilatancy. The central result, Theorem 1, shows that for a square-integrable initial velocity and a force field in a natural dual space, the simplified model (9)–(11) has a weak solution on a bounded three-dimensional domain, and that the velocity and pressure are unique. The mechanism is that the coupling of the pressure-dependent yield threshold with the dilatancy law $\mathrm{div}\,u = 2|Su| - \sqrt{p}$ produces a dissipated-energy inequality strong enough to control the pressure, keep it non-negative, and identify all nonlinear weak limits. The same strategy extends to a model with evolving volume fraction, provided a small regularizing term is added; the original inviscid granular model is not covered.

What carries the argument

The load-bearing object is the energy inequality obtained by testing the momentum equation with $u$ and the dilatancy equation with $p$. The nonlinear viscous term $\mathrm{div}(2|Du|Du)$ makes the strain-rate tensor live in $L^3$, giving enough regularity to give meaning to products such as $p|Su|$ and to run the compactness argument. The threshold relation is encoded through the regularized stress $\sigma_\varepsilon = p_\varepsilon Su_\varepsilon/(|Su_\varepsilon|+\varepsilon)$, while the square-root pressure law is approximated by concave functions $V_\varepsilon$; a convexity lemma and a flux-identification lemma then pin down the weak limits, and the dilatancy constraint $\mathrm{div}\,u = 2|Su|-\sqrt{p}$ is recovered exactly.

What would settle it

Find a weak solution of (9)–(12) in the sense of Definition 1 whose pressure is negative on a set of positive measure. Because the energy inequality together with $|\sigma|\le p$ forces $\sigma:Su = 2p|Su|$ almost everywhere, such a solution would contradict Proposition 1 and therefore Theorem 1.

Watch

Extended reading notes

Core claim

For the simplified system (9)–(12), the authors prove that a weak solution $(u,p,\sigma)$ exists whenever $u_{\rm init}\in L^2(\Omega)$ and $f\in L^{3/2}(0,T;W^{-1,3/2}(\Omega))$, and that $u$ and $p$ are unique. The weak formulation (13)–(15) is deliberately built so that the full threshold rheology $\sigma:Su = 2p|Su|$, $|\sigma|\le p$ is recovered from the energy inequality alone, even though it is not imposed directly in the definition of weak solution. Existence is obtained by passing to the limit in a regularized system, with a convex-analysis and monotonicity argument identifying the weak limits of $|Du_\varepsilon|Du_\varepsilon$, $V_\varepsilon(p_\varepsilon)$, and $|Su_\varepsilon|$. A notable part of the proof is that the pressure is shown to be non-negative almost everywhere, a property the model needs physically.

Load-bearing premise

The proof hinges on the nonlinear viscosity term $\mathrm{div}(2|Du|Du)$, which is added for technical reasons and is absent from the original physical granular model; without it, the $L^3$ regularity of the strain rate and the identification of the nonlinear terms fail.

Editorial extensions

If this is right

If the theorem is correct, the simplified granular model is well-posed for the stated data class, giving numerical discretizations a target problem with existence and uniqueness guarantees.
The proof delivers non-negative pressure as a theorem, not an assumption, which is physically natural and rarely proved rigorously for such models.
The same energy argument works for a volume-fraction model when a small regularizing term $\xi(\Delta\varphi - \varphi\sqrt{p})$ is added, and the paper shows that the simplified system is the leading-order approximation of the full $\mu(I)$-rheology model in a specific regime.
The uniqueness result is restricted to velocity and pressure; the stress is not claimed unique, which suggests that numerical methods should treat stress as a derived quantity.
The inviscid limit and the case $\xi=0$ are explicitly left open, so the theorem marks a first step rather than a full theory of the original physical model.

Reading between the lines

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

The technical nonlinear viscosity $\mathrm{div}(2|Du|Du)$ is the real price of the theorem: the physical model from the paper's starting point has no such term, and the proof gives no indication how to replace the resulting $L^3$ regularity with a weaker mechanism.
The pressure-as-Lagrange-multiplier viewpoint noted in Remark 4 points to a variational route: treating the dilatancy relation as a constraint in a minimization problem could yield the stress relation without the viscous regularization, if the constraint can be handled in the non-smooth regime.
The same energy-dissipation strategy is likely transferable to other threshold rheologies, such as $\mu(J)$-rheology for immersed granular flows, but the dilation closure and stability conditions would need to be verified case by case.
A concrete testable extension is to build a numerical scheme that preserves the discrete analogue of the energy inequality (15) and the bounds on $\varphi$; if such a scheme exists, the theorem's structural assumptions are strong enough for practical simulation.

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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

Read the paper. The genuinely new thing is Definition 1: putting the threshold rheology into an energy inequality instead of the constitutive relation, then recovering σ:Su = 2p|Su| from the weak form via Proposition 1. That is a clever and useful trick, and the existence proof built around it — Galerkin approximations, uniform L^3 estimates, convexity/Zhikov identification of weak limits, pressure positivity — is mostly careful and correct. The uniqueness argument via monotonicity is also fine. Credit where due: this is the first existence/uniqueness theory for a Drucker-Prager-type granular model with dilatancy coupling, and the authors are honest that the nonlinear viscosity is a technical device (Remark 3).

The soft spots are real but not fatal. First, Proposition 2 — existence for the ε-approximate system — is asserted on a citation to [2] plus a single energy identity. That is load-bearing: Theorem 1 needs a rigorous sequence (uε,pε,σε), and the paper does not show the Galerkin construction or the uniform-in-dimension estimates required before passing to a solution of the approximating system. It may be standard, but it is not demonstrated here. A referee should ask for a complete proof or a very precise adaptation of [2]. Second, the scope is overstated. The abstract says 'real breakthrough' and 'granular model', but the theorem is for a simplified system with a technical viscosity; the physically original model has no viscosity, and the authors say the L^3 framework depends on it. The extension to the full model (Theorem 2) only works with a positive ξ regularization, and the limit ξ→0 is explicitly left open. These are honest limitations — they are flagged — but they should be reflected in the abstract and introduction. Third, the regularity framework is tight: products like p|Su| make sense only in L^{3/2}, and any weakening of the viscous term would break the argument. That is fine as a theorem, but it narrows the claim.

Bottom line: worth a serious referee. With a filled-in Proposition 2 and a toned-down abstract, it becomes a solid contribution to the analysis of non-isochoric granular models. I would send it to review rather than desk reject, but the referee should be told to verify the Galerkin step.

Referee Report

3 major / 4 minor

Summary. The paper proposes to prove existence and uniqueness of weak solutions for a simplified dense granular flow model that couples a pressure-dependent threshold rheology (Drucker-Prager type) with a dilatancy law. In Section 3, the authors define a weak formulation (Definition 1) that incorporates an energy inequality, show in Proposition 1 that this weak formulation recovers the full rheological relation, and then state Theorem 1: for initial data in L² and forcing in L^{3/2}(0,T;W^{-1,3/2}), there exists a weak solution, with u and p uniquely determined. The proof constructs a sequence of approximate problems (17)--(20) involving a nonlinear viscosity |Du|Du, a regularized yield law σε = pε Suε/(|Suε|+ε), and an ε-heat equation for the pressure. After establishing uniform estimates (Proposition 3) and compactness (Proposition 4), the authors pass to the limit using convexity arguments, Zhikov's lemma, and careful weak-strong identifications. Section 4 sketches an extension to a model with variable volume fraction (Theorem 2) and discusses the relation of the simplified system to the µ(I)-rheology via an asymptotic expansion.

Significance. If the gaps noted below are closed, the main theorem would be a valuable contribution: the proof strategy is innovative in exploiting the coupling between rheology and dilatancy to obtain dissipated energy, and the weak-strong limit identification via convexity and Zhikov’s lemma is elegant. The paper is largely self-contained for the simplified model and gives a complete uniqueness proof. However, two load-bearing issues remain: the existence proof of the ε-approximate system is only sketched by reference to a Galerkin method, and the identification V(p)=√p is not fully established on the zero set of p. The result is also conditional on the technical nonlinear viscosity |Du|Du, which is not part of the original inviscid granular model; the authors acknowledge this in Remark 3, but the abstract’s claim of a “real breakthrough” should be tempered accordingly. Overall, the paper is promising and the main ideas are sound, but the proof as written is not yet complete.

major comments (3)

[§3.3, Proposition 2 (approximate existence)] The proof of Proposition 2 is only a sketch: it derives the energy identity (24) and states that existence follows from a classical Galerkin method, citing [2]. This is load-bearing because Theorem 1’s limit passage in §3.5–3.6 requires a fully constructed sequence (uε,pε,σε). The Galerkin passage for the coupled system (22)–(23) needs to be demonstrated: one must show uniform bounds at the Galerkin level, pass to the limit in the finite-dimensional approximations, and identify the nonlinear term σε = pε Suε/(|Suε|+ε). A bare reference to [2] is not sufficient unless the applicability to this specific coupled system is spelled out. Please either provide the full Galerkin argument or state precisely which theorem in [2] covers the present system and why.
[§3.6, Corollary 1 and the paragraph after (33)] The conclusion “equality B = 0 implies V(p) = √p” is not justified on the set where p = 0. Corollary 1 gives V(p) ≤ √p and pV(p) ≥ p√p; since p ≥ 0, these imply V(p) = √p only on {p > 0}. On {p = 0} they only give V(p) ≤ 0. The term B = ∫(pV(p) − pV(p)) vanishes identically on {p = 0}, because both products contain the factor p, so B = 0 imposes no constraint on V(p) there. Since Vε(0) = 0 but Vε(x) = x/ε for x < 0, the weak limit V(p) could in principle carry a negative contribution supported on {p = 0} (e.g. if pε ≈ −εχE on a set E of positive measure). Without V(p) = √p a.e., equation (29) is not the target equation (14). An additional argument is needed, for instance a sign or monotonicity estimate showing that such negative contributions vanish, or a modification of the approximation so that Vε is bounded below so that V(p) is forced to be nonnegative.
[§4.1, Theorem 2 and Remark 6] Theorem 2 is stated as a theorem, but its proof is only sketched in “Ideas for the proof” and relies on formal energy estimates for regular solutions (Propositions 5 and 6). The passage to weak solutions for the ξ-regularized system (37)–(40) is not carried out; in particular, the strong compactness of φε and the identification of the nonlinear terms (φmax − φ)√p and Dt(φ,u) are not demonstrated. Since Remark 6 already notes that the argument breaks down when ξ = 0, the status of Theorem 2 should be clarified: either supply the missing compactness and limit-identification details, or state Theorem 2 as a formal extension rather than a proved result.

minor comments (4)

[§3.6, notation in (31) and (33)] The notation for the weak limits is ambiguous: the weak limit of pεVε(pε) and the product of p with the weak limit of Vε(pε) are both written as “pV(p)”, differing only by an overline that is lost in the displayed equations. Please use distinct symbols, e.g. \overline{pV(p)} and p\overline{V(p)}, to avoid confusion in the argument.
[Abstract and Introduction] The phrase “This work represents a real breakthrough” is stronger than the mathematical content warrants, given that the main theorem requires the technical nonlinear viscosity (Remark 3) and that the extension Theorem 2 is only sketched. I recommend tempering this claim and stating explicitly in the abstract that the existence result is proved for a regularized simplified model.
[§4.2, asymptotic derivation] The derivation of the simplified system (55)–(57) from the µ(I)-rheology model is purely formal. The text should state clearly that this is a formal asymptotic reduction, not a rigorous singular-limit result, to avoid giving the impression that Theorem 1 applies to the original inviscid model.
[§3.3, Proposition 2 uniqueness] The last sentence of the proof of Proposition 2 refers to Subsection 3.7 for uniqueness, but Subsection 3.7 uses Proposition 1, which is specific to the limiting rheology (10), not to the regularized relation (18). Either give a direct uniqueness argument for the approximate system or omit the uniqueness claim from Proposition 2, since existence is all that is needed for the main theorem.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the existence proof is self-contained; [14] is contextual and [2] is an external Galerkin reference.

full rationale

Walking the derivation of Theorem 1: the approximate system (17)-(20) is asserted to have solutions by a Galerkin argument referenced to [2] (external authors, not a self-citation) combined with the energy identity (24); the uniform estimates in Proposition 3 follow from that identity; weak compactness in Proposition 4 and the convex-analysis Lemmas 1-2 identify the nonlinear limits. None of these steps is equivalent to the conclusion. In particular, the energy inequality (15) in Definition 1 is not assumed in order to force the rheology: it is derived from the approximate energy estimates and passed to the limit, and Proposition 1 only shows that the inequality |sigma| <= p together with the energy inequality forces the exact flow rule sigma:Su = 2p|Su| by comparing the energy identity with (15). The nonlinear viscosity is explicitly declared 'essentially technical' in Remark 3, so this is a transparent modeling assumption rather than a hidden ansatz. The self-citation [14] provides the physical model and stability context, but no proof step in Theorem 1 relies on [14] for a mathematical fact. The proof of Proposition 2 is admittedly sketched with 'see [2]', and Theorem 2 is proved through formal estimates, but these are exposition/rigor gaps rather than circular reductions: they do not make the claimed result equal to its inputs. Score 1 reflects only the minor contextual self-citation, which is not load-bearing.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central theorem rests on standard PDE tools (Korn, Aubin-Lions, Zhikov, convexity) plus a set of modeling choices that are not independently derived: the normalized constants, the technical non-linear viscosity, and, in Theorem 2, the artificial regularization xi H. No new physical entities are postulated, but the physically relevant zero-regularization limit and the inviscid model are not covered.

free parameters (2)

normalized coefficients in (9)-(11) = alpha0 = beta0 = nu0 = phi0 rho0 = 1
The model is studied only in the normalized form; the paper asserts the normalization 'does not qualitatively alter' the result but does not provide an explicit rescaling argument for arbitrary positive physical coefficients.
regularization strength xi in (37)-(40) = xi > 0, arbitrary
Theorem 2 requires a strictly positive artificial regularization term xi H with H = Delta phi - phi sqrt(p) to ensure compactness and pressure bounds; the physical case xi=0 is explicitly left open (Remark 6).

assumptions (8)

standard math Korn's inequality in W^{1,3} for bounded domains
Used to control ||u||_{1,3} by ||Du||_3 in the energy estimate (Proposition 3, Step 1), cited to [20].
standard math Aubin-Lions-Simon compactness lemma
Used to obtain strong convergence of u_epsilon in L^3(0,T;L^2) (Proposition 4), cited to [11, p.102].
standard math Zhikov lemma on weak convergence of fluxes (Lemma 2)
Central to identifying the weak limit of |Du_epsilon|Du_epsilon in Section 3.6, cited to [29, Lemma 1] and [17, p.1715 and p.1730].
standard math Convex and concave weak lower and upper semicontinuity (Lemma 1)
Used to compare weak limits of convex and concave functions of Du_epsilon, p_epsilon, etc., cited to [13, p.38].
standard math Existence and uniqueness for the Galerkin approximate system (17)-(20)
Proposition 2 is asserted as 'classical' via Galerkin citing [2]; no proof is given in the paper.
domain assumption Drucker-Prager rheology and Roux-Radjai dilatancy law as constitutive relations
The model (1)-(4) is taken from the authors' prior work [14] and from [27]; these phenomenological laws are assumed, not derived.
ad hoc to paper Non-linear viscosity 2|Du|Du is sufficient for the L^3 framework
Remark 3 states this choice is 'essentially technical'; the proof's functional setting depends on it.
ad hoc to paper Artificial regularization xi H and modified derivative D_t in the volume-fraction model
Theorem 2 uses H = Delta phi - phi sqrt(p) and the averaged derivative to obtain compactness and energy estimates; these are not part of the physical model and the xi to 0 limit is open.

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Cite this review

Pith. "Pith review of Weak solution for granular model." pith.science (2026). https://pith.science/paper/S46WVIUN

@misc{pith2026250517588,
  author       = {Pith},
  title        = {Pith review of: Weak solution for granular model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S46WVIUN}},
  note         = {Machine review of arXiv:2505.17588}
}

read the original abstract

This article is devoted to questions concerning the existence of solutions for partial differential equation problems modeling granular flows. The models studied take into account the complex threshold rheology of these flows, as well as the dilatance effects. It is the coupling of these two physical phenomena that ensures stability and the existence of dissipated energy. The key point of the article is to understand how this energy can ensure the existence of a weak solution. We first establish a complete result on a simplified model, then demonstrate how it can be extended to more general cases. This work represents a real breakthrough in the mathematical analysis of this type of models for complex flows.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

Abbatiello, T

A. Abbatiello, T. Los, J. M´ alek, and O. Sou vcek. Three-dimensional ﬂows of pore pressure-activated bingham ﬂuids. Mathematical Models and Methods in Applied Sciences , 29(11):2089–2125, 2019

work page 2019

[2] [1]

Abbatiello, M

A. Abbatiello, M. Bul ´ ıvcek, T. Los, J. M´ alek, and O. Sou vcek. On unsteady ﬂows of pore pressure- activated granular materials. Zeitschrift f¨ ur angewandte Mathematik und Physik, 72(1):6, 2021

work page 2021

[3] [3]

Andreotti, Y

B. Andreotti, Y. Forterre, and O. Pouliquen. Les milieux granulaires-entre ﬂuide et solide: Entre ﬂuide et solide . EDP sciences, 2012

work page 2012

[4] [4]

Barker, J

T. Barker, J. Gray, D. Schaeﬀer, and M. Shearer. Well-posedn ess and ill-posedness of single-phase models for suspensions. J. Fluid Mech. , 954:A17, 2023

work page 2023

[5] [5]

Barker, D

T. Barker, D. Schaeﬀer, M. Shearer, and J. Gray. Well-posed c ontinuum equations for granular ﬂow with compressibility and µ(I)-rheology. Proc. R. Soc. Lond. A , 473(2201):20160846, 2017. 18

work page 2017

[6] [6]

Barker, D

T. Barker, D. G. Schaeﬀer, P. Boh´ orquez, and J. Gray. Well-p osed and ill-posed behaviour of the µ(I)-rheology for granular ﬂow. J. Fluid Mech. , 779:794–818, 2015

work page 2015

[7] [7]

A. Beck. First-order methods in optimization . SIAM, 2017

work page 2017

[8] [8]

Bouchut, E

F. Bouchut, E. D. Fernandez-Nieto, A. Mangeney, and G. Narb ona-Reina. A two-phase shallow debris ﬂow model with energy balance. ESAIM: Mathematical Modelling and Numerical Analysis , 49(1):101– 140, 2015

work page 2015