{"id":"0c6af735-123c-4059-96c3-f7c6e7c9dae2","arxiv_id":"2412.14316","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A long-term stable fully discrete scheme for stochastic non-Newtonian Stokes equations is proved; two empirical-measure sequences each achieve one of tightness or semigroup-invariance, merged only under a condition the paper's own numerics suggest fails for fixed time step.","lead":"Researchers propose a computer simulation scheme for a random fluid model used in turbulence research, the generalised Stokes equations with transport noise, and prove the scheme stays stable for arbitrarily long time horizons. It is a step toward computing the long-run equilibrium statistics of such turbulent fluid models, where random small-scale motions never die out.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 10's uniform Sobolev-stability of the L2-projection onto discretely divergence-free subspaces is unverified for any GD family; Theorem 11's horizon-uniform constants and the tightness argument in Lemma 21 depend on it.","rationale":"The reader's weakest assumption identifies exactly the same point, and I agree that it is the most load-bearing condition in the paper. The central proof, Theorem 11, is logically sound under Assumption 10; the issue is applicability. The theorem is conditional on a stability property that the authors themselves state is open for constrained subspaces, and no concrete GD family is shown to satisfy it. The numerical validation uses a single fixed mesh, so it cannot test uniformity. This does not make the paper internally inconsistent, but it does mean the headline claim of long-term stability for a broad class of discretisations is thinner than its conditional statement suggests. The check I propose would either verify the assumption for the practical Taylor-Hood family or expose that the theorem's constants deteriorate with mesh refinement. Since the reader already returned CONDITIONAL and my concern matches the reader's weakest assumption, the appropriate verdict is unchanged. I found no independent flaw in the stability proof, and I credit the paper for clearly stating the conditional nature of the invariant-measure construction and for providing reproducible code.","tokens_in":41631,"tokens_out":8429,"duration_ms":81944,"concrete_test":"Using the Firedrake code from the paper's repository, compute C^Sob_D(p) for the Taylor-Hood GD on a sequence of uniformly refined triangulations of the unit square (e.g., 13x13, 25x25, 49x49 vertices). For p=2 this is the operator norm of z -> epsilon_D K_D z from H^1 to L^2, obtainable as the largest singular value of the corresponding linear map; for p=1.5 and p=3, approximate it by maximising over a fixed set of high-frequency smooth test functions and monitor mesh independence. If the norm grows with refinement, Assumption 10 fails for the discretisation family actually used in Section 6, and the uniform-in-D constants in Theorem 11 are unavailable. If it stays bounded, the assumption is verified for Taylor-Hood and the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing concern is the uniform Sobolev-stability assumption on the L2-projection onto the discretely divergence-free subspace, Assumption 10 combined with Definition 8 and Eq. (3.5). In the proof of Theorem 11, this enters the estimate of Term II at inequality (5.6) through K = Gamma^3 ||sigma||_Linf ||(sigma . grad) g||_{W^{1,p'}}; without it, the time-global bound (3.9) carries a mesh-dependent constant, and the subsequent tightness of the second measure sequence (Lemma 21) and the construction of a limit measure lose their uniformity. The authors explicitly flag in Remark 9 that 'its stability for constrained subspaces, such as E_{D,0}, remains to be explored.' For the only discretisation used in the numerical experiments, Taylor-Hood on a fixed 13x13 mesh, the constant is finite by finite-dimensionality, but this says nothing about a refined or unstructured family. No example family satisfying Assumption 10 is provided, so the advertised 'broad class of particular Gradient Discretisations' is not demonstrated to have any member whose stability constants remain controlled across a discretisation sequence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a fully discrete numerical scheme for the generalised Stokes system driven by transport noise, based on the Gradient Discretisation Method and a Crank--Nicolson-type time stepping with Temam's antisymmetric noise discretisation. Under a set of structural assumptions on the gradient discretisation (Assumption 10), it proves long-term stability of the scheme with constants growing at most linearly in the time horizon (Theorem 11), establishes at most one invariant measure for the discrete transition semigroup (Theorem 18), and constructs two sequences of empirical measures with complementary properties: the first is asymptotically invariant (Lemma 19) and the second has a tightness-based limit (Lemma 21). A sufficient condition (4.2) is given to merge existence and invariance, and the homogeneous-boundary case is fully characterised by convergence to the Dirac measure at zero (Example 25). Two numerical experiments on Taylor--Hood elements illustrate the influence of transport noise on dissipation, mixing, and vortex structure, and investigate the validity of condition (4.2).","tokens_in":41723,"tokens_out":6285,"duration_ms":56653,"significance":"If the results are fully correct, this is the first work to construct computable approximations of invariant measures for stochastic non-Newtonian Stokes equations with transport noise, a genuinely new contribution to the numerical SPDE literature. The paper's strengths include the transparent tracking of all constants, the elegant diagonal-vanishing antisymmetric noise discretisation that preserves discrete energy cancellation, the fully written proofs of the main stability and uniqueness theorems, and the honest and explicit discussion of the remaining gaps (Remark 9, Remark 24, Section 7). The numerical experiments are reproducible in principle through the linked Firedrake-based code and directly probe the paper's central abstract condition. However, the advertised 'broad class' of gradient discretisations is not actually demonstrated: the uniform Sobolev-stability of the L2-projection onto discretely divergence-free subspaces (Assumption 10) is unverified for any family of discretisations, and the semigroup property -- essential for the invariant measure framework -- is only proved informally.","major_comments":[{"comment":"The uniform Sobolev-stability assumption on the L2-projection onto the discretely divergence-free subspace is the key structural condition that makes the time-global stability estimate (3.9) independent of the spatial discretisation. It enters the proof of Theorem 11 through the constant K = Gamma^3 ||sigma||_Linf ||(sigma . grad) g||_{W^{1,p'}} in the estimate of Term II in (5.6); without uniformity in the discretisation family, (3.9) carries a mesh-dependent constant, and the tightness argument in Lemma 21 loses the uniformity needed to produce a limit measure. The authors explicitly state in Remark 9 that stability for constrained subspaces such as E_{D,0} 'remains to be explored'. As it stands, no example of a Gradient Discretisation family satisfying Assumption 10 is provided, so the claim of long-term stability for a broad class of particular GDs is not supported by any concrete family. The manuscript should either supply such a family (for instance by verifying the assumption for Taylor--Hood elements under mesh refinement) or relax the assumption and show which parts of the theory survive without it.","section":"Section 3.1, Assumption 10, Definition 8, Eq. (3.5); proof of Theorem 11, Eq. (5.6)"},{"comment":"The semigroup property of P_{tau,D} is foundational: it justifies calling P_{tau,D} a discrete transition semigroup, is used in the definition of an invariant measure, and underpins the uniqueness theorem (Theorem 18) and the asymptotic invariance estimate (Lemma 19). However, the proof of Lemma 15 is only informal: it states 'We provide an informal argument only' and defers the rigorous justification to a limit passage over finite sums that is never carried out. In particular, the representation S(n,v_in) = S^{diamond n}_{CN}[K_D v_in, Delta_1 W, ..., Delta_n W] and the use of continuity in the random update (Lemma 31) require a careful treatment of the measurability and approximation steps. This gap must be closed before the invariant measure construction can be considered complete.","section":"Section 5.2.1, Lemma 15"},{"comment":"The numerical experiments are performed with a 'slightly modified' scheme that includes an external deterministic force F and, in EXP-2, non-solenoidal boundary data g, while the theoretical analysis of Theorem 11 and the energy identities are established for the scheme (3.8) under Assumption 1, which requires g to be solenoidal and uses the shift v = u - g. The modified scheme (6.1) is not shown to satisfy the structural properties used in the analysis (for example, the diagonal cancellation of the noise term after the shift, or the pathwise energy identity for g=0). Consequently, the numerical conclusion that condition (4.2) fails in general (Figure 1, Table 1) is not direct evidence about the analysed scheme. The authors should either prove that the modified scheme inherits the relevant structural properties or reformulate the numerical investigation so that it tests the analysed scheme.","section":"Section 6.3, Eq. (6.1); Section 4.2, Eq. (4.2), Figure 1, Table 1"},{"comment":"The only general-boundary result that yields existence of an invariant measure is the abstract condition (4.2), and the only example where it is verified is the trivial-boundary case g = 0 (Example 25), where the invariant measure is delta_0. For nontrivial g, no example satisfying (4.2) is given, and the numerical evidence suggests that the condition may fail for small time steps. The paper is honest about this, but the advertised 'recovering the existence of an invariant measure' is thus a conditional statement without a nontrivial instantiation. It would considerably strengthen the paper to provide a nontrivial class of data for which (4.2) can be verified, or at least a precise discussion of why such a class is expected to be empty or nonempty.","section":"Section 4.2, condition (4.2); Example 25"}],"minor_comments":[{"comment":"The definition of Y_{D,0} is self-referential: it reads 'Y_{D,0} = {q in Y_{D,0} : ...}'. The right-hand side should use the full space Y_D or a different symbol for the mean-free subspace.","section":"Definition 2(f)"},{"comment":"The labels in the bullet list appear swapped: the third bullet says 'space discrete case (D fixed, tau -> 0 and N -> infinity such that tau N >= 1)', which is a time-discrete case, while the fourth bullet describes the fully continuous limit. The terminology should be corrected.","section":"Remark 22"},{"comment":"The set D_R is defined as all L2 vector fields of the form Pi_D v_D with ||epsilon_D v_D||_{L^p} <= R, and is claimed to be compact. Strictly, D_R as defined may not be closed; for the tightness argument one can take its closure, which is compact because it is a bounded subset of the finite-dimensional space Pi_D X_{D,0}. This detail should be stated.","section":"Proof of Lemma 21"},{"comment":"The phrase 'the stochastic velocity vector points in all directions with positive probability' is a strong statement about the support of the stationary distribution at a fixed point; the numerical evidence is only based on a finite sample and finite-time observations, so this should be phrased as an observation rather than an established fact.","section":"Section 6.4, EXP-1"},{"comment":"In the estimate of R3, the martingale increments are centred and the quadratic variation is handled correctly, but the notation 'N0 ∋ M' is unusual; this is a minor stylistic point.","section":"Section 5.1, proof of Theorem 11"},{"comment":"The formal energy inequality (2.4) uses the Helmholtz--Leray projection P in the Itô--Stratonovich corrector, but the corrector term is not fully derived; a reference or a couple of additional lines would help the reader. Since this is only motivation, the informal derivation is acceptable.","section":"Section 2.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about its limitations, and the core algorithmic idea and stability analysis are promising. The main reservations are the unverified uniform Sobolev-stability assumption (Assumption 10) for any discretisation family and the informal proof of the semigroup property (Lemma 15). Both are load-bearing, and the numerical experiments test a modified scheme rather than the analysed one. In my view these are fixable within the scope of a major revision, not grounds for rejection. I would also ask the editor to ensure that the authors provide the missing example family for Assumption 10 or clearly restrict the claims, and that they either prove the modified scheme's structural properties or align the numerics with the analysed scheme."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuine content here is Theorem 11 (long-term stability with constants linear in tau N), the uniqueness of the discrete invariant measure (Theorem 18), and the two-sequence construction for approximate invariant measures. The proofs use standard tools — energy estimates, Burkholder-Davis-Gundy, tightness — and I found no circular reasoning or fitted constants. The paper is notably honest: it says plainly that condition (4.2) likely fails in general, and the numerics support that, so the advertised \"recovering the existence of an invariant measure\" is conditional. The homogeneous-boundary example is clean but collapses to delta_0.\n\nThe soft spots are real but proportionate. The most load-bearing is Assumption 10: uniform Sobolev-stability of the L2-projection onto the discretely divergence-free subspace. It enters Theorem 11 through the noise estimate, and without it the tightness argument for the second measure sequence loses uniformity. The authors flag in Remark 9 that stability for constrained subspaces is unexplored. For the Taylor-Hood element on a fixed mesh the constant is finite by finite-dimensionality, but no refined or unstructured family is shown to satisfy the assumption. That should be stated more prominently as an open condition, ideally with one verified family.\n\nLemma 15's semigroup proof is explicitly informal, and that is a genuine gap in a paper whose main object is the discrete semigroup and its invariant measure. It should be completed or clearly marked as a conditional step. The numerics use a 'slightly modified version' with an external force and a non-solenoidal lid in EXP-2, which puts them outside the paper's assumptions; they are illustrative but not a verification of the theory. The physical claims about mixing and vortex size are interesting but have no uncertainty quantification.\n\nWho this is for: numerical analysts working on stochastic PDEs, invariant measures, and GDM. It deserves a serious referee. The proven parts are original and independently checkable. A revision should tighten the abstract, complete or relegate Lemma 15, align the numerics with the assumptions or reclassify them as heuristic, and address the verifiability of Assumption 10.","headline":"Honest, technically strong paper whose real results are the time-uniform stability theorem and the discrete invariant-measure machinery; the general-boundary existence claim is conditional on a condition the authors expect to fail.","tokens_in":651,"tokens_out":914,"would_cite":true,"duration_ms":26892,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60H35","60J22","65C20","65C30","65C40","65M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Constructs a long-term stable, fully discrete scheme for the stochastic non-Newtonian Stokes equations and uses it to build approximate invariant measures, with the zero-boundary equilibrium shown to be the Dirac mass at zero.","keywords":["stochastic fluid equations","transport noise","non-Newtonian Stokes","invariant measure","long-term stability","Gradient Discretisation Method","power-law fluids","turbulence"],"falsifier":"Choose a sequence of increasingly refined mixed finite-element discretisations of the divergence-free Stokes problem and compute C^Sob_D(p)+C^Sob_D(p') on the discrete divergence-free subspace; if these constants grow without bound, the proof of Theorem 11 fails at the noise estimate Term II. Alternatively, fix τ and estimate the time-averaged mean squared velocity increment E[(1/N) Σ_ℓ ||Π_D $v^{{ℓ+1}}$_D − Π_D v^ℓ_D||^2] for large N; the numerical results suggest it tends to a positive τ-dependent constant rather than zero, which would violate condition (4.2) and block the invariance of the second measure sequence for fixed τ.","tokens_in":41235,"feed_emoji":"🌊","tokens_out":7863,"duration_ms":67469,"temperature":0.7,"pith_summary":"This paper proposes a fully discrete numerical scheme for the generalised Stokes equations driven by transport noise, a model of non-Newtonian power-law fluids with applications to turbulence. The authors show that the scheme is long-term stable: for every moment q>0, the discrete velocity and strain-rate accumulations stay bounded over N steps by C(||v_in||^{2q} + (τN)^q), with C independent of the time step, the horizon, and the spatial discretisation within a Γ-stable family. That linear growth is exactly what is needed to discuss equilibrium, so they build a discrete transition semigroup and two sequences of approximate invariant measures. The first sequence is asymptotically invariant but lacks a proved limit; the second has a limit measure but its invariance is not proved in general. For homogeneous boundary data the invariant measure is shown to exist, be unique, and equal the Dirac mass at zero, with both measure sequences converging to it.","feed_headline":"New scheme computes equilibrium of noisy non-Newtonian fluids","feed_subtitle":"A gradient-discretisation algorithm bounds error linearly in time, yielding approximate invariant measures for power-law fluids.","key_machinery":"The engine of the construction is the pair consisting of a Gradient Discretisation in space and a semi-implicit Crank-Nicolson step in time. A Gradient Discretisation packages a spatial scheme through reconstruction operators; the paper requires a Γ-stable family in which coercivity, inf-sup stability, and the Sobolev stability of the L2-projection onto the discrete divergence-free space are uniformly controlled. The transport noise is discretised through Temam's symmetrisation, B^σ_D(v,ξ)=1/2(σ^T∇_D v, Π_D ξ)−1/2(Π_D v, σ^T∇_D ξ), which is skew-symmetric and vanishes on the diagonal, so the noise term cancels in the discrete energy balance. The proof of Theorem 11 then sums local energy inequalities, converts dissipation into Lp control of the symmetric gradient via the relation S(A):(A−B) ≂ |V(A)−V(B)|^2, and absorbs stochastic terms using the Burkholder-Davis-Gundy inequality.","core_discovery":"The central discovery is that a Crank-Nicolson time discretisation combined with a Gradient Discretisation in space reproduces, on the discrete level, the energy dissipation mechanism that makes transport noise special: the noise term is discretised by a skew-symmetric bilinear form that vanishes on its diagonal, so it drops out of the discrete energy identity. This yields a pathwise energy equality for zero boundary data and an energy inequality with only linear growth in the horizon for general Dirichlet data. From that stability the paper derives a discrete Markov semigroup, proves it has at most one invariant measure, and identifies two constructor sequences: an ergodic-average sequence that is asymptotically invariant to order n/N, and a shifted-average sequence that is tight and therefore has a limit measure. A condition on the time-averaged squared velocity increment merges the two properties, and the authors show it holds in the trivial-boundary case, where the unique invariant measure is δ0 and both sequences converge to it.","pith_inferences":["If condition (4.2) indeed fails for fixed τ, as the numerical results suggest, then for a fixed discretisation one should not expect a stationary distribution at integer times; the authors' planned route through a time-continuous limit is then the natural way to obtain an invariant measure.","The pathwise energy equality for zero boundary data makes the scheme a candidate for unbiased sampling of the trivial equilibrium: because energy dissipates monotonically along each trajectory, burn-in can be monitored deterministically rather than statistically.","A direct test of the paper's main assumption would be to compute the Sobolev-stability constant of the L2-projection onto the discrete divergence-free space on a sequence of refined meshes; if it grows without bound, the uniform Γ-stability needed by Theorem 11 cannot be verified empirically for refined families.","The implemented algorithm already accommodates an external deterministic force and non-solenoidal boundary data; extending the invariant-measure analysis to that setting would give a nontrivial limit measure, unlike the δ0 obtained for homogeneous data."],"forward_implications":["For every moment q>0, discrete solutions satisfy a time-uniform bound growing only linearly in the horizon, so simulations can be run to arbitrarily long times without blow-up.","The discrete transition semigroup has at most one invariant measure, so any successfully constructed stationary distribution is the unique equilibrium of the discrete dynamics.","The empirical measure sequence (4.1a) is asymptotically invariant with an O(n/N) discrepancy, meaning any weak limit point of that sequence is invariant for the discrete semigroup.","The shifted sequence (4.1b) is tight and therefore has a weak limit measure for fixed discretisation parameters and also in various joint refinement regimes, provided the analytic horizon stays non-degenerate.","For homogeneous boundary data g=0, the unique invariant measure is δ0 and both measure sequences converge to it, characterising the equilibrium completely in that case."],"supporting_citations":[{"why":"Supplies the Gradient Discretisation framework, including the definition of GD, coercivity/inf-sup/inverse-estimate constants, and compactness notions used throughout.","marker":"[DEGGH18]"},{"why":"Introduces Gradient Schemes for the Stokes problem, the spatial-discretisation template adapted here to the stochastic non-Newtonian setting.","marker":"[DEF15]"},{"why":"Provides the classical Crank-Nicolson time-stepping idea on which the paper's semi-implicit algorithm is built.","marker":"[CN47]"},{"why":"Contributes the symmetrisation device (integration by parts before discretisation) used to define the skew-symmetric noise form B^σ_D.","marker":"[Tem68]"},{"why":"Presents the stochastic Navier-Stokes algorithm with transport noise whose first experiment (EXP-1) is reproduced, serving as the numerical benchmark for the new scheme.","marker":"[BMPW24]"},{"why":"Studied Sobolev and Lp-stability of the L2-projection; cited for the projection-stability property whose extension to constrained divergence-free subspaces remains open.","marker":"[DST21]"},{"why":"Supplies known H1-stability results for the L2-projection onto finite element spaces, the background for the uniform Sobolev-stability assumption.","marker":"[BY13]"},{"why":"Provides the only known strong-error convergence result for an approximation of the stochastic p-Stokes system, used to contextualise the open question of quantified convergence of the approximate measures.","marker":"[LW24]"}],"fun_headline_variants":["Stable scheme computes equilibrium of noisy non-Newtonian fluids","Invariant measures via gradient discretisation for stochastic Stokes","Long-term stability for transport-noise Stokes with power-law fluids","Algorithm for noisy non-Newtonian Stokes yields invariant measure","Transport noise enhances dissipation in power-law fluid scheme"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main stability theorem requires the L2-projection onto the discretely divergence-free velocity space to be uniformly Sobolev-stable, meaning bounded by a fixed constant across the whole family of discretisations; the paper itself notes that this property has not been verified for constrained divergence-free subspaces. If this constant blows up under mesh refinement, the time-uniform estimate, the tightness of the second measure sequence, and hence the invariant-measure construction would no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Stable scheme computes equilibrium of noisy non-Newtonian fluids","Invariant measures via gradient discretisation for stochastic Stokes","Long-term stability for transport-noise Stokes with power-law fluids","Algorithm for noisy non-Newtonian Stokes yields invariant measure","Transport noise enhances dissipation in power-law fluid scheme"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000213,"raw_usage":{"total_tokens":1414,"prompt_tokens":931,"completion_tokens":483,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":404}},"tokens_in":547,"tokens_out":483,"duration_ms":4917,"temperature":1.0,"reasoning_tokens":404,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:21:36.521145+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a sequence of increasingly refined mixed finite-element discretisations of the divergence-free Stokes problem and compute C^Sob_D(p)+C^Sob_D(p') on the discrete divergence-free subspace; if these constants grow without bound, the proof of Theorem 11 fails at the noise estimate Term II. Alternatively, fix τ and estimate the time-averaged mean squared velocity increment E[(1/N) Σ_ℓ ||Π_D $v^{{ℓ+1}}$_D − Π_D v^ℓ_D||^2] for large N; the numerical results suggest it tends to a positive τ-dependent constant rather than zero, which would violate condition (4.2) and block the invariance of the second measure sequence for fixed τ.","supporting_citations":[],"review_version":1}