{"id":"7bfc1569-30c3-4f16-923c-d03eba374679","arxiv_id":"2601.19442","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global dissipative solutions and weak-strong uniqueness hold for a compressible non-Newtonian Navier–Stokes–Korteweg system with density-dependent viscosity on the torus in 2D and 3D.","lead":"Compressible fluids whose viscosity depends on density and whose stress responds nonlinearly to strain are proved to have global 'dissipative solutions'—weaker solutions that still obey a relative-entropy inequality and a weak-strong uniqueness property. The proof extends a known Newtonian result to non-Newtonian stresses, with the capillary term supplying the decisive density-gradient control.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumptions (1.4)-(1.5) omit S(0)=0; for p=1 the energy estimates can fail because S(Du):Du need not be nonnegative, so Theorem 1.2 is not established for the full stated class.","rationale":"I read the full manuscript. The overall architecture - Galerkin approximation, monotonicity method, relative entropy inequality, and limit passage - is coherent and largely self-contained. The reader's weakest assumption (the special Korteweg form K=1/ρ) is a genuine scope limitation but is explicitly stated and does not undermine Theorem 1.2 as the theorem is formulated. The load-bearing issue I found is different: the stress-tensor assumptions (1.4)-(1.5) do not guarantee that S(Du):Du is nonnegative when p=1, because S(0) is not constrained. The energy estimates, the weak lower-semicontinuity argument, and the final limiting procedure all require the dissipation terms to have a definite sign. A concrete counterexample to the implication '(1.4)-(1.5) ⇒ S(A):A≥0' exists for p=1, so the theorem is not fully established for the stated class. Since all physical examples in the paper satisfy S(0)=0, the fix is a modest, standard additional assumption; hence I recommend CONDITIONAL acceptance rather than rejection. The reader's weakest assumption is not the same as this concern, so I mark agreement as disagree.","tokens_in":30279,"tokens_out":52079,"duration_ms":538331,"concrete_test":"Construct explicitly a smooth monotone S:R→R with p=1, c=c1=1, S(A)=-1 for A≤-1, S(A)=1 for A≥1, S(0)=1/2, and S'(A)≥0. Verify that this S satisfies (1.4)-(1.5). Then check the Galerkin energy identity (3.8) with Du_N chosen to be -0.5·I on a set of positive measure and zero elsewhere on a suitable approximate state: the term ∫ϱS(Du_N):Du_N is strictly negative, so the step from (3.8) to (3.9) cannot be made with a nonnegative dissipation term. If such a state is admissible, Theorem 1.2 is not proven under (1.4)-(1.5) as written; adding S(0)=0 to the assumptions removes the obstruction.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.2 relies at every a priori stage on the energy identity/inequality (3.9), (3.17), (3.22), where the term ∫ϱS(Du):Du is treated as a nonnegative dissipation and dropped when bounding E. Assumptions (1.4)-(1.5) do not ensure this. For p>1 the growth bound |S(A)|≤C|A|^{p-1} forces S(0)=0, but for p=1 it does not. It is possible to choose a smooth, bounded, monotone S with S(0)>0, S(A)≤-c for A<-c1, S(A)≥c for A>c1, and S(A)·A<0 for some small A (e.g., a smoothed monotone function with S=-1 on (-∞,-1), S=1 on (1,∞), S(0)=1/2). Such S satisfies (1.4)-(1.5) but makes ϱS(Du):Du negative on a set where Du is small and negative. Then (3.8) has a possible source term, the uniform bounds derived from (3.9) are not justified, and the lower-semicontinuity passage to (3.17)-(3.22) does not follow. The intended examples in Remark 1.1 all have S(0)=0, so adding the explicit normalization S(0)=0 (or S(A):A≥0 for all A) repairs the argument, but as stated the theorem is broader than what the proof supports.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a dissipative-solution framework for the periodic compressible Navier–Stokes–Korteweg system (1.1) with density-dependent non-Newtonian viscous stress. The main result, Theorem 1.2, asserts global-in-time existence of dissipative solutions for initial data satisfying (1.3), under growth/coercivity assumptions (1.4) and monotonicity (1.5) on S and pressure assumptions (1.6)–(1.7). The proof combines a Galerkin approximation of a regularized system with artificial viscosity and parabolic density diffusion (Section 3), a monotonicity passage for the nonlinear stress, a relative entropy inequality for the approximate system (Section 4), and a final limit ε,ν→0. A weak-strong uniqueness statement for dissipative solutions is also given (Corollary 2.5). The paper is careful and detailed, but the main theorem is stated more broadly than the proof supports.","tokens_in":30558,"tokens_out":14624,"duration_ms":159469,"significance":"If the stated result is repaired as suggested below, the paper would be a valuable extension of the dissipative-solution theory for compressible Korteweg flows from the Newtonian case of Bresch–Gisclon–Lacroix-Violet to non-Newtonian, density-dependent viscous stresses. The relative entropy inequality (2.4), the weak-strong uniqueness corollary, and the explicit multi-parameter approximation scheme are concrete and useful. The proof is largely self-contained, and the monotonicity/Galerkin construction for the approximate system is credible. The main value is conceptual: it identifies a solution concept with weak-strong uniqueness for a class for which Leray–Hoff weak solutions remain open.","major_comments":[{"comment":"The proof treats ∫ϱS(Du):Du as a nonnegative dissipation at every a priori stage, but (1.4)–(1.5) do not imply S(A):A ≥ 0 when p=1. The bound |S(A)|≤C|A|^{p-1} forces S(0)=0 only for p>1; for p=1 it does not. A monotone, bounded S with S(0)>0 and S(A)A<0 for small negative A satisfies (1.4)–(1.5) but makes the dissipation term signed, so the uniform bounds derived from (3.9) are not justified for the full stated class. Since the intended examples satisfy S(0)=0, the fix is local: add S(0)=0 (or S(A):A≥0 for all A) to the hypotheses. Without this, Theorem 1.2 is not established.","section":"§1, assumptions (1.4)–(1.5); §3, energy identities (3.9), (3.17), (3.22)"},{"comment":"The theorem only assumes (1.3), i.e. √ϱ0∈H1 and √ϱ0u0∈L2. However the proof in Section 3 chooses √ϱ0,N→√ϱ0 in H1 and ϱ0,N→ϱ0 in L^γ, and E_N(0) is uniformly bounded only if ∫H(ϱ0) < ∞. Under (1.7), H(ϱ) is comparable to ϱ^γ, so one needs ϱ0∈L^γ. For γ>3, √ϱ0∈H1 only gives ϱ0∈L3, which is insufficient. Thus the statement of Theorem 1.2 (and the definition of the initial relative entropy in (2.4)) requires the additional hypothesis ϱ0∈L^γ, or equivalently E(0)<∞. As written, the theorem covers data for which the energy and the dissipative-solution inequality are not even defined.","section":"Theorem 1.2 and §3, Step 1 (approximation of initial data)"}],"minor_comments":[{"comment":"The weak-limit notation is not fully defined: it is not immediately clear where the overline denotes the limit of S(Du_N) and where it denotes the limit of S(Du_N):Du_N. The monotonicity step leading to the nonnegativity of the S-term is terse; please spell out the standard Minty-type argument.","section":"§3, Step 3"},{"comment":"The passage ε→0 in the relative entropy inequality is compressed. In particular, convergence of the terms in b(t) involving ∇lnϱε and the A1 term is only asserted; a short justification using the stated strong/weak convergences would improve readability.","section":"§4.1"},{"comment":"In the display before the final inequality, 'ϱ(∇lnϱ−lnr)' should read 'ϱ(∇lnϱ−∇lnr)'.","section":"Proposition 2.2"},{"comment":"The heading contains a typo: 'Poncaré' should be 'Poincaré'.","section":"Appendix A.1"}],"recommendation":"major_revision","confidential_remarks":"The two hypothesis gaps are local and easily fixed, and the main construction appears sound once those hypotheses are added. I recommend major revision rather than rejection because the central idea and proof architecture are credible, but Theorem 1.2 as stated is strictly stronger than what the proof establishes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real extension, not a repackaging, and the proof is mostly self-contained and careful. But the theorem as stated is broader than the proof supports. Assumptions (1.4)-(1.5) do not imply S(A):A >= 0 for small A when p=1, so the energy estimates (3.9), (3.17), (3.22) can fail. Adding S(0)=0 (or S(A):A >= 0 for all A) repairs the a priori bounds, and the intended examples in Remark 1.1 all satisfy that, but the paper should say so explicitly and prove the p=1 case with that normalization. As written, Theorem 1.2 claims more than the argument delivers.\n\nWhat is new and what works: previous dissipative-solution results for Korteweg systems were either Newtonian ([13]) or had density-independent viscosity ([1]). Here the combination of a density-dependent factor rho*S(Du), a monotone non-Newtonian stress, and the capillarity term for K(rho)=1/rho is handled by a two-level approximation: Galerkin for the epsilon,nu-regularized system, a monotonicity argument to identify the nonlinear stress, and a relative entropy inequality to pass epsilon,nu to zero. The technical appendices (density-weighted Poincare, p-Laplacian monotonicity, pressure-relative-entropy bound, Bohm identity) are real and mostly proved in the text. The relative entropy inequality (4.1) and the weak-strong uniqueness corollary are clean. The q-Laplacian regularization is doing genuine work, not just cosmetic.\n\nSoft spots: the p=1 gap is the main one; it is local and fixable, not a collapse of the whole strategy. Some convergence claims in Step 3 of Section 3 and in the epsilon->0 passage are compressed, but they look routine and I did not find a hidden error. The restriction to K(rho)=1/rho is acknowledged and is genuinely needed for the H1 bound on sqrt(rho); extending to general K would require substantially new ideas.\n\nBottom line: this deserves a serious referee. It is a solid step in a subfield where Leray-type weak solutions are still open. I would ask the authors to tighten the assumptions, treat p=1 explicitly with the normalization, and expand the compressed convergence passages. After that revision I would be comfortable citing it.","headline":"Genuine extension of the Newtonian Korteweg dissipative-solution theory to non-Newtonian stresses, but the stated assumptions have a repairable gap for p=1 that the energy estimates do not cover.","tokens_in":31146,"tokens_out":2682,"would_cite":true,"duration_ms":30777,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76N10","35D30","76A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that adding a special capillary (Korteweg) force yields global-in-time dissipative solutions for a compressible non-Newtonian fluid with density-dependent viscosity, and that these solutions inherit weak-strong unique","keywords":["compressible Navier–Stokes–Korteweg","non-Newtonian fluids","dissipative solutions","relative entropy","weak-strong uniqueness","density-dependent viscosity","capillarity","Galerkin approximation"],"falsifier":"Solve the regularized system (3.1) on the 3D torus with ν=ε for two independent sequences ε→0, starting from the same smooth data, with S(Du)=|Du|^{p−2}Du for p=3 and γ=2; if the two limits differ while a classical strong solution exists for that data, the weak-strong uniqueness claim (Corollary 2.5) is false.","tokens_in":30073,"feed_emoji":"🌊","tokens_out":4853,"duration_ms":53224,"temperature":0.7,"pith_summary":"The paper proves that a compressible non-Newtonian Navier–Stokes–Korteweg system with density-dependent viscous stress has global-in-time dissipative solutions in two and three space dimensions, under monotone growth assumptions on the stress and power-law pressure. Dissipative solutions are a weak notion built from a relative entropy inequality, and the payoff is weak-strong uniqueness: any dissipative solution starting from the same data as a smooth strong solution must coincide with it. The construction works because the particular capillary term κ div(ϱ∇²lnϱ) lets the energy control the H¹ norm of √ϱ, giving the compactness that the non-Newtonian stress alone does not provide. The proof passes through a regularized system with artificial diffusion and a p-Laplacian-type viscosity, then removes the regularizers by monotonicity arguments.","feed_headline":"Capillarity yields global dissipative solutions for non-Newtonian flows","feed_subtitle":"Add a Korteweg capillary force and a rough compressible model gains global solutions with weak-strong uniqueness.","key_machinery":"The load-bearing object is the relative entropy functional E(t) in (2.1), combining kinetic energy relative to a test velocity, a capillary term κϱ|∇lnϱ−∇lnr|², and the convex pressure-entropy H(ϱ|r). The capillary term is special: by the Böhm identity div(ϱ∇²lnϱ)=2ϱ∇(Δ√ϱ/√ϱ), the energy controls 4κ|∇√ϱ|², which provides the H¹ compactness of √ϱ that the non-Newtonian stress alone cannot. The monotonicity of S and of the p-Laplacian regularizer is what allows passage to the limit in the nonlinear stress terms.","core_discovery":"The central claim, Theorem 1.2, is that under assumptions (1.4)–(1.5) on the stress tensor S and (1.6)–(1.7) on the pressure p, the system (1.1) admits a global-in-time dissipative solution for initial data satisfying √ϱ₀∈H¹, √ϱ₀u₀∈L², on the torus in dimensions 2 and 3. A dissipative solution here is defined through the relative entropy inequality (2.4) with admissible test functions (r,v); this notion is strong enough to imply weak-strong uniqueness (Corollary 2.5). The proof constructs weak solutions to an approximate system with ε∆ϱ and ν div(|Du|^{q−2}Du), derives the relative entropy inequality at the approximate level, and then passes ε,ν→0, with the approximation errors vanishing tha","pith_inferences":["The proof depends on the precise capillary form K(ϱ)=1/ϱ; a testable extension is whether the same relative-entropy scheme works for K(ϱ)=ϱ^β with β in some range, or whether the H¹ control of √ϱ is genuinely necessary.","The weak-strong uniqueness suggests a numerical selection principle: any stable numerical method that converges to a dissipative solution will converge to the strong solution in the smooth regime, so the relative entropy could be used as an a posteriori error indicator.","The definition via a concrete relative entropy inequality, rather than a measure-valued formulation, may allow a maximal-dissipation selection criterion; one could test whether the entropy-production term ϱS(Du):Du is maximal among all admissible limits."],"forward_implications":["Global dissipative solutions exist for a broad class of non-Newtonian compressible models (including power-law and regularized Bingham-type stresses) with density-dependent viscosity, where previously only local strong solutions or one-dimensional results were available.","Any dissipative solution sharing initial data with a smooth strong solution must equal it, so the solution concept is unambiguous on the smooth regime.","The relative entropy framework supplies a stability tool: dissipative solutions can be compared against any smooth approximate solution, opening the way to inviscid and large-viscosity limits, including the Euler–Korteweg system.","Adding the special 1/ϱ capillarity is a physically motivated regularizer that breaks the obstruction to global existence for compressible non-Newtonian systems."],"fun_headline_variants":["Capillarity secures global dissipative solutions for non-Newtonian flows","Korteweg capillarity unlocks existence for viscoplastic compressible flows","Non-Newtonian Korteweg system: global dissipative solutions proven","A pinch of capillarity yields global solutions for tough flows","Korteweg term ensures dissipative solutions and weak-strong uniqueness"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Everything hinges on the special capillary term κ div(ϱ∇²lnϱ): the energy controls H¹ of √ϱ only for this (or equivalent) structure, and for a general Korteweg stress K(ϱ)≠1/ϱ the compactness argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Capillarity secures global dissipative solutions for non-Newtonian flows","Korteweg capillarity unlocks existence for viscoplastic compressible flows","Non-Newtonian Korteweg system: global dissipative solutions proven","A pinch of capillarity yields global solutions for tough flows","Korteweg term ensures dissipative solutions and weak-strong uniqueness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00042,"raw_usage":{"total_tokens":1991,"prompt_tokens":734,"completion_tokens":1257,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":478,"completion_tokens_details":{"reasoning_tokens":1161}},"tokens_in":478,"tokens_out":1257,"duration_ms":11398,"temperature":1.0,"reasoning_tokens":1161,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T07:41:00.238081+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the regularized system (3.1) on the 3D torus with ν=ε for two independent sequences ε→0, starting from the same smooth data, with S(Du)=|Du|^{p−2}Du for p=3 and γ=2; if the two limits differ while a classical strong solution exists for that data, the weak-strong uniqueness claim (Corollary 2.5) is false.","supporting_citations":[],"review_version":1}