{"id":"611dc637-6ef1-48a0-9f14-b2050e36a3a8","arxiv_id":"2507.22449","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of time-periodic pulsatile flows is proved for variable-exponent power-law fluids in pipes, with explicit steady benchmark solutions and a constructive finite-element scheme.","lead":"This paper proves that time-periodic pipe flows exist for a class of non-Newtonian 'smart' fluids whose power-law index varies across the cross-section of the pipe. It also writes down explicit steady benchmark solutions and tests a convergent numerical scheme against them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Convergence proof requires κ2=0 (Lemma 4.13, Theorem 4.14), but the abstract's flagship example (δ+|a|)^{p-2}a with δ>0 and p<2 violates this; existence for the general (s.1)-(s.4) class is therefore unproven.","rationale":"The reader's weakest assumption pinpoints exactly the load-bearing issue: Theorem 4.14 and its supporting Lemma 4.13 require κ2=0, while the abstract and Section 1 advertise the general class (s.1)-(s.4), including the regularized stress (1.4) with δ>0 and arbitrary p∈(1,+∞). My independent reading confirms this: the strong stability estimates (4.48), which are the backbone of the weak compactness and Minty identification in Theorem 4.14, are derived using Lemma 3.3(iii), whose lower bound on V requires κ2=0. For p<2 and δ>0, the regularized model fails that condition, so the flagship example falls outside the proved convergence theorem. This is a real scope mismatch, not a manufactured objection. I also flag that Theorem 5.5 is asserted without proof; this is a missing support for the pressure-drop part of the abstract, though it is a smaller issue than the κ2 gap since the direct problem is simpler. The paper retains substantial value: the constructive fixed-point analysis in Lemma 4.9, the δ=0 case, the exact solutions in Section 6, and the numerical experiments in Section 7 all support the conditional acceptance. No ad hominem concerns; the critique is about the distance between the advertised theorem and the proved hypotheses. Because the reader's conditional verdict already reflects this gap, no verdict adjustment is needed.","tokens_in":45275,"tokens_out":5827,"duration_ms":71827,"concrete_test":"Analytically verify whether the regularized model satisfies (s.2) with κ2=0 when p<2. For s(x,a)=(δ+|a|)^{p-2}a, δ>0, and p<2, compute inf_{a≠0} (δ+|a|)^{p-2}|a|^{2-p}; this tends to 0 as |a|→0, so no κ1>0 can make s(x,a)·a ≥ κ1|a|^p pointwise. Hence the hypotheses of Lemma 4.13/Theorem 4.14 fail for the abstract's prototypical example on any region where p(x)<2. To settle the concern fully, either extend Lemma 4.13 to κ2>0 by a different compactness argument, or explicitly restrict the abstract/convergence claims to κ2=0, δ=0, or p≥2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central advertised claim is existence for the full class of stress tensors satisfying (s.1)-(s.4), with p(x)∈(1,+∞) and the prototypical model (δ+|a|)^{p(x)-2}a for δ≥0 at the center of the exposition. Theorem 4.14, however, explicitly assumes κ2=0 in (s.2), and Lemma 4.13's proof needs this assumption: the lower bound on the potential V in Lemma 3.3(iii) is used to convert the convexity identity (4.51) into the strong stability estimates (4.48a)-(4.48d), which are then the source of the compactness (4.55) and the Minty-type identification (4.65). For κ2>0 this chain breaks: Lemma 4.9 still gives an L^p(I×Σ) bound on the gradient, but not the uniform-in-time modular bound or the L^2 bound on d_τ v^τ_h that the weak compactness argument needs. The gap is not merely technical: for s(x,a)=(δ+|a|)^{p(x)-2}a with δ>0 and p(x)<2 on a set of positive measure, (s.2) cannot hold with κ2=0, because ((δ+|a|)^{p-2}|a|^2)/|a|^p = (δ+|a|)^{p-2}|a|^{2-p} → 0 as |a|→0. Thus the theorem's hypotheses exclude the study's own motivating example whenever the variable exponent dips below 2, while the abstract claims all p∈(1,+∞). A secondary, related support gap: Theorem 5.5 for assigned pressure drop is stated with its proof 'left to the interested reader' at the end of Section 5, so the pressure-drop half of the abstract is not established within the preprint. These are proof-scope gaps rather than observed contradictions, but they make the advertised scope of the central claim narrower than what is actually demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fully-developed, time-periodic, shear-dependent flows of p(x)-power-law fluids in an infinite pipe, for both prescribed flow rate and prescribed pressure drop. The main theoretical contribution is a fully-constructive existence proof for variational solutions via a fully-discrete finite-difference/finite-element scheme, together with explicit time-independent benchmark solutions for piecewise-constant variable exponents and numerical experiments comparing the reduced 1D model with a direct 2D approximation.","tokens_in":45735,"tokens_out":4646,"duration_ms":52169,"significance":"If the results are established at the claimed level of generality, the paper would extend the Womersley/Leray flow-rate and pressure-drop existence theory from constant p and from Navier-Stokes to variable-exponent p(x)-fluids for all p(x) in (1,∞), without an auxiliary Newtonian term. The fully-discrete constructive proof, the explicit benchmark solutions, and the numerical comparison between the reduced and full models are all valuable and largely reproducible contributions. However, as detailed below, the generality actually proved in the manuscript is narrower than the abstract's claim, and two load-bearing parts of the proof are either conditional on κ2=0 or omitted.","major_comments":[{"comment":"The convergence and existence theorem for the flow-rate problem is proved only under the assumption κ2=0 a.e. in Σ, as stated at the beginning of Lemma 4.13 and in Theorem 4.14. This excludes the paper's own motivating example, the (p(·),δ)-structure s(x,a)=(δ+|a|)^{p(x)-2}a with δ>0, whenever p(x)<2 on a set of positive measure, because for such x the quotient s(x,a)·a/|a|^{p(x)} = (δ+|a|)^{p(x)-2}|a|^{2-p(x)} tends to 0 as |a|→0, so (s.2) cannot hold with κ2=0. The proof of Lemma 4.13 uses κ2=0 precisely where the lower bound of the potential V in Lemma 3.3(iii) is needed to obtain the strong stability estimates (4.48), which in turn provide the compactness and the Minty-type identification in Theorem 4.14. The abstract and Section 1, however, advertise existence for the full class (S.1)-(S.4) and for the prototype (1.4) with δ≥0. This is a load-bearing scope mismatch: the main theorem should either be extended to κ2>0 or the claims in the abstract and introduction should be restricted accordingly.","section":"§4, Lemma 4.13 and Theorem 4.14; §2.2 (S.2)/§3 (s.2)"},{"comment":"The pressure-drop half of the central claim is not established within the manuscript: Lemma 5.4 (strong stability) and Theorem 5.5 (weak convergence and existence) are stated, but their proofs are left to the interested reader. The abstract and the introduction promise existence of time-periodic solutions with assigned pressure drop, so this is not a peripheral remark. At minimum, the authors need to provide the proofs or state the pressure-drop result as conditional/deferred with a precise reference to a future work.","section":"§5, Lemma 5.4 and Theorem 5.5"},{"comment":"Equation (4.38) contains a sign inconsistency in the application of the discrete Gronwall lemma. The text applies Lemma 4.11 with λ = -κ1/(cP 2^{p+}) in (4.37), but the resulting bound in (4.38) is written with (1+λτ)^m and c1/(-λτ). For λ<0, the factor 1+λτ is smaller than 1, and the second term {1 - 1/(1+λτ)^m} c1/(-λτ) is negative, which cannot serve as an upper bound. The correct expression from Lemma 4.11 would involve (1-λτ)^m = (1+|λ|τ)^m and a positive second term. This estimate is used to prove that the fixed-point operator maps the ball B^τ_h into itself, so the error affects the constructive existence argument in Lemma 4.9. The issue appears fixable by correcting the sign convention, but as written it is a load-bearing inconsistency in the discrete existence proof.","section":"§4, Eq. (4.38) and Lemma 4.11"}],"minor_comments":[{"comment":"In Lemma 4.8 the phrase 'in the sense of Definition 4.4' appears twice; the flux-free discrete formulation is Definition 4.7, not Definition 4.4.","section":"§4, Lemma 4.8"},{"comment":"The convergence stated as '(τ → +∞)' should read '(τ → 0+)'.","section":"§4, Eq. (4.57c)"},{"comment":"The sign assumption on ∂xv in (6.10) is imposed a priori. Since the goal is to construct explicit solutions, this is acceptable, but it would help to state explicitly that this sign condition is an ansatz used to select a branch of the solution, not a consequence of the boundary-value problem.","section":"§6, Eq. (6.10)"},{"comment":"The formula for the reduced stress vector in Section 7.3, displayed as '2^{2+p(x)/2}', is typographically ambiguous and the exponent appears algebraically wrong; the factor should be checked and stated without ambiguity, because the numerical comparison uses this scaling.","section":"§7.3, stress-tensor scaling"},{"comment":"The paper repeatedly cites [6,7,27] for technical lemmas and for the convergence-rate discussion in Remark 4.12. The citations are used appropriately, but a sentence in Remark 4.12 clarifying which results are taken from [6] and which are new to this paper would improve readability.","section":"§1 and §2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a substantial and mostly sound constructive scheme, and the numerical experiments are informative. The main concern is not correctness of the core fixed-point idea but the mismatch between the advertised scope and the proved scope: the flow-rate theorem is conditional on κ2=0, which excludes the paper's flagship example for p(x)<2 with δ>0, and the pressure-drop theorem is not proved in the preprint. The sign issue in (4.38) must also be corrected, as it affects the self-mapping argument. I recommend major revision, asking the authors to either extend the analysis to κ2>0 (or to a clearly specified subclass) and to supply the missing proofs, or to restate the main theorems and the abstract at the level that is actually established. There is no indication of improper citation; the self-citations are used for technical lemmas and previous numerical results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look, but with the advertised scope trimmed. The paper genuinely extends Galdi–Grisanti from constant p to variable p(·) for the flow-rate (inverse) problem, and gives a mostly self-contained fully-discrete existence proof via Edelstein's fixed point theorem. That is a real step for the non-Newtonian analysis community, and the explicit piecewise-constant-p steady solutions in Section 6 are useful benchmarks. The numerical experiments are serious: they check convergence rates, include a non-trivial time-periodic manufactured solution, and compare the reduced 1D model against a full 2D approximation. The authors also cite the relevant literature, including their own prior work where it is actually used for technical lemmas.\n\nNow the soft spots, in proportion. The biggest is that the flagship convergence theorem (Theorem 4.14) assumes κ2=0 in (s.2). For the prototypical (δ+|a|)^{p(x)-2}a with δ>0 and p(x)<2, that assumption fails, and the abstract's claim of existence for the full (s.1)–(s.4) class with p∈(1,∞) is not what is proved. The stress-test note is right: this is a proof-scope gap, not an observed contradiction, but it is central to how the paper sells itself. Second, Lemma 5.4 and Theorem 5.5 for the pressure-drop problem are stated with proofs 'left to the interested reader', so the pressure-drop half of the abstract is simply not established in this preprint. Third, the discrete Gronwall application in the proof of Lemma 4.9 looks off: (4.38) has a sign/format inconsistency (the bracket {1−1/(1+λτ)^m} times c1/(−λτ) is negative when λ is negative), and the text mentions 'λ=1' earlier without clear justification. This is likely fixable — the contraction property of the fixed point map is plausible — but the argument as written does not close cleanly. Minor: no code shipped (FEniCS is named, but no repository), and a few typos like (4.57c) say τ→+∞ instead of τ→0+.\n\nWho is this for? Researchers in non-Newtonian fluid analysis and numerical approximation, especially those working on Leray-type problems and variable-exponent models. The flow-rate existence result with κ2=0 is probably correct and the explicit solutions are genuinely citable. The paper deserves a serious referee, but it needs revision: either prove the general case or restrict the abstract accordingly, provide the missing pressure-drop proofs, and fix the Gronwall step. I would send it to review, not desk reject.","headline":"A real variable-exponent extension of the Womersley/Leray line, but the abstract oversells the generality: the convergence theorem needs κ2=0 and the pressure-drop half is left unproved.","tokens_in":869,"tokens_out":977,"would_cite":true,"duration_ms":49655,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76A05","76D05","65M60","35D30","35K55","35B10","76M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that pulsatile pipe flows exist for fluids with a position-dependent power-law index, for prescribed flow rate or pressure drop, and that a fully-discrete scheme converges to them.","keywords":["incompressible non-Newtonian fluids","fully-developed pulsatile flows","variable exponent rheology","exact solutions for smart fluids","electro-rheological fluids","time-periodic solutions","finite element discretization","numerical experiments"],"falsifier":"Take the $\\delta$-regularized stress $s(x,a)=(\\delta+|a|)^{p(x)-2}a$ with $\\delta>0$ and a cross-section where $p$ takes values on both sides of $2$ (so that (s.2) forces $\\kappa_2>0$), and compute the discrete solutions of (4.15)–(4.16b) on refining meshes with a genuinely time-dependent flow rate: if the bound on $\\|d_\\tau v^\\tau_h\\|_{I\\times\\Sigma}$ deteriorates as $\\tau,h\\to 0^+$ or the scheme fails to converge, the stated scope of the theorem is exactly the $\\kappa_2=0$ class, whereas if the bound holds, $\\kappa_2=0$ is not needed.","tokens_in":45086,"feed_emoji":"🌀","tokens_out":15672,"duration_ms":157627,"temperature":0.7,"pith_summary":"This paper aims to establish that pulsatile—time-periodic, fully-developed—pipe flows exist for a class of 'smart fluids': shear-dependent non-Newtonian fluids whose power-law exponent varies from point to point in the cross-section, as in electro-rheological fluids. It treats both ways of driving the flow, an assigned time-periodic flow rate (the 'inverse' problem) and an assigned time-periodic pressure drop (the Womersley-type 'direct' problem), and it proves the result constructively: a backward-Euler/finite-element discretization, a fixed-point argument on the initial data, stability bounds, and a monotonicity limit passage yield a unique variational solution, with the discrete solutions converging weakly to it. The same geometric setting yields explicit closed-form solutions for steady flows with piecewise-constant exponent—variable-exponent analogues of Hagen–Poiseuille flow—which the paper puts forward as benchmark tests. If the results stand, the periodic-flow existence theory known for Navier–Stokes and constant-$p$ fluids extends to variable-exponent fluids, with a computable algorithm attached.","feed_headline":"Time-periodic flows exist for smart fluids with variable power-law","feed_subtitle":"A provably convergent discretization also yields explicit Hagen–Poiseuille-style benchmarks.","key_machinery":"The central objects are the reduced scalar problem (3.2) obtained from the fully-developed ansatz, whose stress vector $s(x,a)=\\nu(x,|a|^2/2)\\,a/2$ inherits coercivity, growth, and strict monotonicity from the original tensor, and the flux-free shift $u=v-\\alpha\\chi$: an auxiliary function $\\chi$ with unit mean over the cross-section absorbs the prescribed flow rate $\\alpha$, converting the constrained problem into an equation for $u$ on the zero-mean subspace, with the pressure drop later recovered by $\\Gamma=-(\\partial_t v,\\chi)_\\Sigma-(s(\\cdot,\\nabla v),\\nabla\\chi)_\\Sigma$. Discrete solvability comes from the Edelstein fixed-point theorem (a contraction on a compact ball in the finite-element space) applied to the initial-to-final value map, with the discrete Gronwall lemma in difference form providing the invariant ball. The passage to the continuous problem rests on the strong stability bounds (4.48), which bound the time differences and gradient modulars and hold only when $\\kappa_2=0$, enabling a Minty-type monotonicity identification of the weak limit of the stress. The explicit solutions of Section 6 arise by integrating $-\\partial_x(|\\partial_x v|^{p(x)-2}\\partial_x v)=1$ on each interval where $p$ is constant and patching the pieces by continuity of $v$ and of the shear stress.","core_discovery":"Starting from the fully-developed ansatz $v(t,x)=v(t,x)e_1$, $\\pi(t,x)=\\Gamma(t)x_1$, the paper reduces the infinite-pipe problem to a $(d-1)$-dimensional scalar problem with stress vector $s(x,a)$ satisfying coercivity, growth, and strict monotonicity assumptions (s.1)–(s.4). Under the additional hypothesis that the coercivity offset $\\kappa_2$ in (s.2) vanishes identically on the cross-section, Theorem 4.14 establishes a unique variational solution $(v,\\Gamma)$ in $W^{1,2}(I;L^2(\\Sigma))\\cap L^\\infty(I;W_0^{1,p(\\cdot)}(\\Sigma))\\times L^2(I)$ for the flow-rate problem, and the fully-discrete solutions converge weakly to it as time step and mesh size go to zero; Theorem 5.5 makes the analogous claim for the pressure-drop problem, through the same strong-stability machinery. The discrete existence proof is constructive: the Edelstein fixed-point theorem applied to the map sending an initial datum to the final value of the corresponding initial-value problem gives a unique discrete solution and doubles as a Picard algorithm. For a strip with piecewise-constant exponent, the steady problem is solved in closed form, producing Lipschitz velocities patched from power functions, for both even and one-jump non-even exponent profiles; the numerical experiments report the predicted error decay rates and show the reduced one-dimensional model reproducing the interior of a full two-dimensional simulation.","pith_inferences":["The abstract claims existence for the whole (s.1)–(s.4) class, but the proof as written delivers it only for $\\kappa_2=0$; whether the $\\delta$-regularized model admits time-periodic solutions is therefore still open on the evidence of this paper.","Because $\\delta$-regularization is the standard device in numerical implementations, a stability estimate that absorbs $\\kappa_2$ (or a Leray–Schauder-type argument) would close the gap; the paper's own variable-exponent numerics use the unregularized stress, so they do not exercise the gap.","A testable extension suggested by Section 6: for rectangular ducts with $p$ depending on a single coordinate, the same one-dimensional patched solutions should carry over (as the paper hints in Remark 6.1), which comparison numerics could verify."],"forward_implications":["Time-periodic fully-developed flows with prescribed flow rate or prescribed pressure drop exist and are unique for variable-exponent fluids in the $\\kappa_2=0$ class, in particular for the pure $p(x)$-Laplacian-type stress $s(x,a)=|a|^{p(x)-2}a$.","Because the existence proof is constructive, the same discrete scheme—backward differences, finite elements, Picard iteration on the initial datum—can be used directly as a solver, with weak convergence guaranteed.","The closed-form steady solutions give variable-exponent analogues of Hagen–Poiseuille flow, including asymmetric exponent profiles, usable as benchmarks in CFD codes.","All $p(x)\\in(1,+\\infty)$ are treated uniformly and no auxiliary Newtonian term is needed, yielding an alternative proof of the constant-exponent results and fresh insight into the Newtonian case.","The reduced $(d-1)$-dimensional model matches the interior of the full $d$-dimensional simulation in the reported experiments, supporting the fully-developed reduction as a reliable computational shortcut."],"supporting_citations":[{"why":"the constant-power Womersley-type existence result for generalized Newtonian liquids that this paper extends to variable exponents.","marker":"[24]"},{"why":"the time-periodic Navier–Stokes (Leray problem) results whose flow-rate and pressure-drop settings are generalized here.","marker":"[37]"},{"why":"Womersley's exact pulsatile solution, the classical prototype and benchmark for the pressure-drop problem.","marker":"[40]"},{"why":"the Edelstein fixed-point theorem used to prove unique solvability of the fully-discrete problems.","marker":"[18]"},{"why":"the discrete Gronwall lemma in difference form that provides the invariant balls for the fixed-point argument.","marker":"[19]"},{"why":"the $(p(\\cdot),\\delta)$-structure inequalities used to quantify contraction of the Picard iteration.","marker":"[6]"},{"why":"the variable-exponent $\\varepsilon$-Young inequality underlying the energy estimates.","marker":"[27]"},{"why":"finite-element approximation properties of the $p(\\cdot)$-Laplacian used for the projection operator $\\Pi_h$.","marker":"[11]"},{"why":"the variable Lebesgue and Sobolev space framework supplying the norms and embedding theorems.","marker":"[16]"},{"why":"the electro-rheological fluid model that motivates the variable-exponent stress tensor.","marker":"[33]"}],"fun_headline_variants":["Time-periodic flows proven for variable power-law smart fluids","Constructive proof yields periodic flows in variable-exponent fluids","Closed-form benchmarks for periodic smart-fluid flows","Existence and numerics for pulsatile variable-power-law flows","Smart fluids: periodic flows, constructive proof, benchmarks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the coercivity offset $\\kappa_2$ in assumption (s.2) is identically zero on the cross-section: the strong stability bound (4.48) that drives the limit passage is proved only in that case, so for stresses such as the $\\delta$-regularized model $(\\delta+|a|)^{p(x)-2}a$ with $\\delta>0$, where $\\kappa_2>0$ in general, the paper's existence proof does not apply even though the abstract advertises the full (s.1)–(s.4) class.","fun_headline_variants_meta":{"raw":{"variants":["Time-periodic flows proven for variable power-law smart fluids","Constructive proof yields periodic flows in variable-exponent fluids","Closed-form benchmarks for periodic smart-fluid flows","Existence and numerics for pulsatile variable-power-law flows","Smart fluids: periodic flows, constructive proof, benchmarks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001239,"raw_usage":{"total_tokens":5182,"prompt_tokens":1135,"completion_tokens":4047,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":751,"completion_tokens_details":{"reasoning_tokens":3969}},"tokens_in":751,"tokens_out":4047,"duration_ms":34738,"temperature":1.0,"reasoning_tokens":3969,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:41:14.384856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $\\delta$-regularized stress $s(x,a)=(\\delta+|a|)^{p(x)-2}a$ with $\\delta>0$ and a cross-section where $p$ takes values on both sides of $2$ (so that (s.2) forces $\\kappa_2>0$), and compute the discrete solutions of (4.15)–(4.16b) on refining meshes with a genuinely time-dependent flow rate: if the bound on $\\|d_\\tau v^\\tau_h\\|_{I\\times\\Sigma}$ deteriorates as $\\tau,h\\to 0^+$ or the scheme fails to converge, the stated scope of the theorem is exactly the $\\kappa_2=0$ class, whereas if the bound holds, $\\kappa_2=0$ is not needed.","supporting_citations":[{"cited_title":"Kaltenbach, Pseudo-monotone operator theory for unsteady problems with variable exponents , Lect","cited_arxiv_id":null,"evidence_quote":"the variable-exponent $\\varepsilon$-Young inequality underlying the energy estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the constant-power Womersley-type existence result for generalized Newtonian liquids that this paper extends to variable exponents."},{"cited_title":"Beir ˜ao da Veiga , Time-periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains-Leray’s problem for periodic flows, Arch","cited_arxiv_id":null,"evidence_quote":"the time-periodic Navier–Stokes (Leray problem) results whose flow-rate and pressure-drop settings are generalized here."},{"cited_title":"Edelstein, On fixed and periodic points under contractive mappings, J","cited_arxiv_id":null,"evidence_quote":"the Edelstein fixed-point theorem used to prove unique solvability of the fully-discrete problems."},{"cited_title":"Emmrich, Discrete versions of Gronwall’s lemma and their application to the numerical analysis of parabolic problems, Tech","cited_arxiv_id":null,"evidence_quote":"the discrete Gronwall lemma in difference form that provides the invariant balls for the fixed-point argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the $(p(\\cdot),\\delta)$-structure inequalities used to quantify contraction of the Picard iteration."},{"cited_title":"Breit, L","cited_arxiv_id":null,"evidence_quote":"finite-element approximation properties of the $p(\\cdot)$-Laplacian used for the projection operator $\\Pi_h$."},{"cited_title":"R˚uˇziˇcka, Electrorheological fluids: modeling and mathematical theory , Lect","cited_arxiv_id":null,"evidence_quote":"the electro-rheological fluid model that motivates the variable-exponent stress tensor."}],"review_version":1}