{"id":"1cb5a6fb-ade7-4269-b35c-41cbef1150f5","arxiv_id":"2608.00126","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For a dense set of initial data, the full Euler system admits infinitely many entropy-admissible weak solutions that take only finitely many constant states, with increasing entropy and prescribed terminal entropy.","lead":"This paper constructs infinitely many entropy-admissible weak solutions of the multidimensional Euler equations that jump between finitely many constant states, can increase their entropy over time, and can converge to a prescribed terminal entropy profile. It extends a known convex-integration construction from simplified isentropic gas dynamics to the full Euler system.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central construction rests entirely on an imported convex-integration proposition ([17, Prop. 1]) whose hypotheses are not checked for the present setting: arbitrary measurable subdomains, no boundary regularity, and test functions in (3.4)–(3.5) that do not vanish on the boundary. If that propo","rationale":"The reader identified the same weakest point: Proposition 4.1 is imported from [17] without checking its hypotheses, and the authors' application demands more generality than a typical convex-integration black box may provide. My reading of the manuscript confirms that every main theorem — discrete solutions, prescribed kinetic energy, terminal entropy profile — is built on this single imported local-existence result. The rest of the proof is a sequence of concatenations and energy-matching identities (Sections 4–5) that appear internally coherent once Proposition 4.1 is granted. I do not find a separate, independent foundational gap that would require a different verdict. The concern is real but not a demonstrated contradiction: it is a missing verification issue, which is exactly what a conditional verdict should reflect. Therefore I keep the reader's CONDITIONAL verdict unchanged and agree that the weakest assumption is the unverified applicability of [17, Proposition 1] to the boundary-free, non-regular subdomains used here. The concrete test I propose would settle the issue by inspecting the source proposition and, if needed, attempting the cube case with the exact test class; this is the most direct way to determine whether the central construction has a solid foundation.","tokens_in":15699,"tokens_out":25551,"duration_ms":271198,"concrete_test":"Obtain [17, Proposition 1] and verify three concrete hypotheses against the use made here: (i) What spatial regularity is required of Ω_n — smooth, Lipschitz, or merely open bounded, and can Ω_n be an arbitrary member of a finite measurable partition without regularity? (ii) Does the conclusion of [17, Prop. 1] provide (3.4)–(3.5) for all φ∈C^1([0,T]×Ω_n), i.e. does it enforce m_n·ν=0 on ∂Ω_n in the weak sense and m_n(0)=m_n(T)=0, or only for φ compactly supported in Ω_n×(0,T)? (iii) Does it allow the boundary-free test class needed in the gluing step (4.5)? If not, try to prove Proposition 4.1 for a single cube with the exact test class C^1([0,T]×(0,1)^d); if the proof fails, Theorems 2.3, 2.6 and 2.9 do not follow from the cited result.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof reduces the full Euler system to the local problem (3.4)–(3.6) on subdomains Ω_n of an arbitrary finite partition (3.1), with no regularity of ∂Ω_n assumed. The only source of existence of momenta m_n solving this local problem is Proposition 4.1, imported unchanged from Luo–Xie–Xin [17, Proposition 1]. The authors do not verify the hypotheses of [17] for Ω_n here, nor do they reproduce the proof. In particular, the test functions in (3.4)–(3.5) are all of C^1([0,T]×Ω_n), not compactly supported in Ω_n×(0,T); this is exactly what the gluing step (4.5) needs, since it allows summing over n without boundary contributions. If [17, Prop. 1] only supplies solutions for test functions vanishing on the boundary or requires Ω_n to be a torus/smooth bounded domain, then the asserted local solutions may not exist, and the derivation of (4.5)–(4.9) — and hence the construction of admissible weak solutions in Proposition 4.3 and the subsequent entropy-concatenation arguments — has no foundation. The manuscript itself flags no limitation here, but the dependence is explicit: Remark 4.2 identifies the parameters, but parameter identification is not hypothesis verification. The typographical slip in the vector count (d(d+3)/d versus d(d+3)/2) further suggests the citation has not been re-checked in detail. This is the single most load-bearing point because Proposition 4.1 is the only non-elementary ingredient; everything else is a concatenation/energy-matching argument built on top of it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs \"discrete\" entropy-admissible weak solutions to the full compressible Euler system (1.1)-(1.8) in bounded domains Ω⊂R^d, d=2,3, for polytropic gases with γ≤5/3. The authors fix a finite partition Ω=∪Ω_n into arbitrary measurable sets and piecewise-constant initial density and entropy. On each cell they solve a local 'incompressible-type' problem (3.4)-(3.6) for the momentum using convex integration imported from Luo-Xie-Xin [17], then piece the cells together. Because the local momentum takes values in a finite set of vectors and the densities/entropies are constant, the resulting weak solutions attain only finitely many constant states. By concatenating such solutions on consecutive time intervals and increasing the entropy at the interfaces while preserving the total energy through condition (5.4)-(5.5), they obtain infinitely many admissible solutions from the same initial data with strictly increasing total entropy (Theorem 2.3), and similar results with prescribed initial kinetic energy (Theorem 2.6) or total energy (Corollary 2.7). In the special case d=3, γ=5/3, an infinite concatenation with refining spatial partitions yields solutions whose entropy converges to an arbitrary prescribed Riemann-integrable profile at the terminal time (Theorem 2.9). The final sections discuss consequences for maximality criteria and long-time behaviour, including time-periodic variants and convergence to equilibria.","tokens_in":16167,"tokens_out":14965,"duration_ms":170844,"significance":"If correct, the paper would considerably strengthen the known non-uniqueness picture for the complete Euler system: the entropy inequality would impose almost no restriction on the structure or long-time behaviour of weak solutions, and the initial kinetic energy need not be large. The construction is conceptually clean and the entropy-concatenation mechanism is transparent. The paper also explicitly acknowledges that the resulting solutions are not maximal in DiPerna's or Dafermos' sense, which is an honest and useful observation. The main barrier to accepting the result is that the sole non-elementary ingredient, Proposition 4.1, is imported without a verification of its hypotheses; the paper's entire edifice rests on that black box.","major_comments":[{"comment":"Proposition 4.1 is the only non-elementary input, but its hypotheses are not checked for the present setting. The subdomains Ω_n in (3.1) are arbitrary measurable sets, no regularity of ∂Ω_n is assumed, and the test functions in (3.4)–(3.5) are C^1([0,T]×Ω_n) and do not vanish on ∂Ω_n. In addition, (3.9) asserts that the solutions satisfy m_n∈C_weak([0,T];L^q) and m_n(0)=m_n(T)=0; this endpoint condition is essential for the concatenation in Section 5. The paper only identifies parameters in Remark 4.2; it does not state the exact hypotheses of [17, Prop. 1] nor verify that they hold for arbitrary measurable partitions and non-compact test functions. If [17] requires, say, a smooth or periodic domain, or test functions with zero boundary values, then the local problem (3.4)–(3.6) may have no solution and the derivation of (4.5)–(4.9), Proposition 4.3, and all main theorems collapses. Ple","section":"Section 4, Proposition 4.1 and Remark 4.2"},{"comment":"The passage from a solution with m(0)=m(T)=0 to an admissible weak solution with a prescribed nonzero initial momentum m0 is described as 'a simple time shift,' but the construction is not given. The weak formulation requires the initial datum m0 in (2.5); if the constructed trajectory has zero trace at t=0, the initial momentum cannot be arbitrary. Please explain explicitly how m0 is selected (e.g., as a trace m(τ) for a suitable τ), prove that the shifted solution satisfies Definition 2.1, and ensure that the energy balance (2.6) holds at the initial time.","section":"Section 4, paragraph after (4.9) and Proposition 4.3"},{"comment":"The number of constant vectors is inconsistent: the text and Eqs. (4.1)–(4.2) contain 'd(d+3)/d' while Proposition 4.1 in (4.3) reads 'd(d+3)/2'. This is not merely cosmetic, since the cardinality of the finite set in (4.3) is used in the discreteness statement. Please correct the count and confirm that it matches [17, Lemma 3].","section":"Section 4, Eq. (4.1)–(4.3)"},{"comment":"In the proof of Theorems 2.3/2.6, the boundary case 1 - d/2(γ-1)=0 (i.e., d=2,γ=2, which is allowed by the assumptions) is not covered. Alternative 1 is stated only for d=3,γ=5/3, and alternative 2 with λ>0 requires the coefficient to be positive. For d=2,γ=2, the same argument as in alternative 1 (bΛ=Λ, bs=s+λ) applies. Please extend alternative 1 to γ=1+2/d, or otherwise handle this case.","section":"Section 5.1, alternatives after Eq. (5.5)"}],"minor_comments":[{"comment":"The phrase 'there exist d(d+3) d vectors' is garbled and should be corrected to 'd(d+3)/2 vectors' (or whatever the correct count is).","section":"Section 4, first paragraph"},{"comment":"In the displayed convex-hull condition, the kinetic term appears as '1/d |m^ℓ_n|/ϱ_{0,n}' but should be '1/d |m^ℓ_n|^2/ϱ_{0,n}', consistent with (3.5).","section":"Eq. (4.2)"},{"comment":"The sentence 'the density remains continuous' is inaccurate; the density is piecewise constant on the original partition. Also, in Remark 2.11, '˜ϱ' should presumably be 'es'.","section":"Section 5.2, proof of Theorem 2.9"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and the structural idea is appealing, but the unverified import of [17, Prop. 1] is a serious correctness risk. If the authors can supply a full proof or a precise verification of the required hypotheses, the result would be very interesting. I recommend major revision rather than rejection, as the gap may be fixable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nRead the paper. The new thing here is that the authors push convex integration to produce \"discrete\" weak solutions for the full Euler system: solutions taking only finitely many constant states, with total entropy increasing along a chosen family, and in the monoatomic 3D case, converging to a prescribed terminal entropy profile. If it holds, this is a nice strengthening of the known non-uniqueness — the entropy inequality really does almost nothing to restrict long-time behavior. The proof is structured well: piecewise-constant ansatz, reduction to local incompressible-type equations, time concatenation, energy matching. The entropy ordering and terminal profile constructions are explicit and reasonably transparent.\n\nThe soft spot is exactly what the stress-test note says: Proposition 4.1, the only non-elementary ingredient, is imported verbatim from Luo–Xie–Xin [17] without checking its hypotheses for the present setting. The local problem (3.4)–(3.6) is posed on arbitrary measurable subdomains Ω_n with test functions that do not vanish on ∂Ω_n — that's exactly what the gluing step needs. The paper simply asserts [17, Proposition 1] applies, with a parameter identification in Remark 4.2 but no verification of functional setup. If the original proposition actually requires a torus or smooth domain, or test functions vanishing at the boundary, then (4.5)–(4.9) have no foundation and the whole construction collapses. I can't tell from the text whether the gap is real or easily repairable — the original paper isn't reproduced — but the authors need to settle it. The typo in the vector count (d(d+3)/d instead of d(d+3)/2) is minor but suggests the citation wasn't re-checked carefully.\n\nThe rest of the proof is coherent. The energy-conservation matching (5.4)–(5.5) for the two cases is sensible. The terminal-entropy proof uses a monotone increasing staircase of piecewise-constant initial entropies on refinements; looks right for Riemann integrable targets. Section 6's discussion of admissibility is honest — they acknowledge their solutions are not maximal and likely unphysical.\n\nRecommendation: send to a serious referee. The result is important enough, and the gap is specific and checkable. The authors should be required to either reproduce Proposition 4.1 in the needed generality or verify the hypotheses of [17] in detail, and fix the small typos. I wouldn't cite it as established until that's done, but it's worth a reading group discussion.","headline":"A well-put convex-integration construction for discrete full-Euler solutions that hinges on an unverified imported proposition; referee it, but make the authors check the black box.","tokens_in":16597,"tokens_out":5401,"would_cite":false,"duration_ms":57270,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q31","35D30","35L65","76N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a dense set of initial density and entropy data, the Euler system admits infinitely many discrete, entropy-admissible weak solutions with increasing total entropy, even converging to a prescribed terminal entropy profile in the monoatom","keywords":["Euler system of gas dynamics","discrete solutions","convex integration","entropy admissibility","ill-posedness","long-time behaviour","weak solutions","terminal entropy profile"],"falsifier":"The decisive check is to test the local construction on a single measurable subdomain with no boundary regularity and with test functions that do not vanish on the boundary: exhibit one such subdomain for which the overdetermined system (3.4)–(3.6) has no bounded finite-valued momentum field. If such a subdomain exists, the imported lemma fails at the level of generality used and the dense-set theorems collapse.","tokens_in":1606,"feed_emoji":"💨","tokens_out":4653,"duration_ms":121153,"temperature":0.7,"pith_summary":"The paper sets out to show that the entropy inequality is almost powerless as a selection criterion for the full Euler system of gas dynamics. It constructs, for a dense set of initial densities and entropies, an initial momentum such that infinitely many weak solutions coexist. These solutions are discrete, meaning each one attains only finitely many constant states, so the non-uniqueness is not merely an artifact of wild oscillations. The total entropy can be made to increase from one solution to the next, and in the three-dimensional monoatomic case the entropy can also be steered toward any prescribed Riemann-integrable terminal profile as time approaches the horizon. If the argument holds, physically relevant admissibility criteria based on entropy growth alone cannot single out a unique solution.","feed_headline":"Dense set of gas-dynamics data yields infinitely many solutions","feed_subtitle":"Each solution assumes only finitely many constant states, yet total entropy can climb or converge to any prescribed profile.","key_machinery":"The load-bearing object is an imported convex-integration lemma (Proposition 4.1) asserting that on each subdomain of a finite partition, the overdetermined local problem (3.4)–(3.6)—weakly divergence-free momentum with test functions not vanishing on the boundary, plus the pointwise kinetic-energy constraint |m_n|^2/(2ϱ0,n)=Λ−(d/2)p(ϱ0,n,s0,n)—admits infinitely many solutions taking values in a finite set of vectors on a sphere. Around this, the paper assembles global solutions by time-splicing: concatenating solutions on consecutive intervals while augmenting entropy at each junction and adjusting Λ to keep total energy continuous; for terminal profiles, it uses a nested refinement of subd","core_discovery":"The central claim is Theorem 2.3 together with Theorem 2.9: there is a set of initial pairs (density, entropy), dense in the L^q topology, such that for each pair one can choose a bounded initial momentum for which the complete Euler system has infinitely many admissible weak solutions, indexed by λ≥0. Each solution takes values in finitely many constant states (ϱ_k, m_k, s_k). The solutions are ordered by total entropy: for λ1<λ2, the integral of ϱλ1 sλ1 is pointwise no larger than that of ϱλ2 sλ2, and they differ on a set of positive measure. In the monoatomic case d=3, γ=5/3, the same initial data also admit solutions whose entropy converges strongly in L^q to an arbitrary Riemann-integra","pith_inferences":["The paper leaves implicit that if the imported convex-integration lemma were shown to hold on all measurable subdomains without boundary regularity, the same time-splicing argument would likely produce discrete wild solutions for a broader class of equations of state, including the radiation-pressure models mentioned in the text.","A natural testable extension is to ask whether the discrete solutions can be forced to satisfy additional admissibility conditions, such as Clausius–Duhem-type inequalities or maximal entropy production; the paper's own discussion suggests that maximal entropy production would remove its solutions, but this is not proved.","Because the initial momentum can be chosen independently of the terminal entropy profile, the result suggests that the reachable set of entropy profiles from a fixed initial state is extremely large; one could try to determine whether any monotone path of entropy profiles can be realized in the same way.","In numerical terms, the terminal-entropy theorem provides a benchmark: any numerical scheme that provably converges to a unique entropy-admissible solution for these data would contradict the paper's conclusion, so the theorem delineates what numerical entropy dissipation can and cannot guarantee."],"forward_implications":["If the theorem is correct, the entropy inequality alone cannot select a unique weak solution: a dense set of initial density/entropy profiles is accompanied by infinitely many entropy-admissible solutions.","The constructed solutions are discrete, so non-uniqueness survives even when solutions are required to attain only finitely many constant states, not just among highly oscillatory fields.","The family of solutions is ordered by strictly increasing total entropy, so none of them is maximal with respect to the standard entropy-rate orderings considered in the literature; those criteria therefore do not, by themselves, eliminate such solutions.","In the 3D monoatomic case, the same initial data produce solutions with any prescribed Riemann-integrable terminal entropy profile, and the required initial momentum does not depend on the profile; entropy can thus be made to approach equilibrium at arbitrarily fast or slow rates.","The same machinery yields time-periodic entropy-admissible solutions and solutions whose kinetic energy decays to zero along a prescribed envelope, further illustrating the flexibility of weak solutions."],"fun_headline_variants":["Dense gas-data set yields infinitely many discrete solutions","Infinite discrete Euler solutions from dense initial data","Gas dynamics: dense data spawn infinite entropy-growing solutions","One dense set, infinite solutions: discrete Euler gas states","Dense data give infinite discrete solutions with rising entropy"],"cache_read_input_tokens":17792,"weakest_assumption_plain":"The entire construction rests on an imported convex-integration proposition that is assumed to supply infinitely many finite-valued momenta solving the local problem (3.4)–(3.6) on arbitrary measurable subdomains with no boundary regularity and non-vanishing boundary test functions; the paper does not verify the hypotheses of that proposition in this setting.","fun_headline_variants_meta":{"raw":{"variants":["Dense gas-data set yields infinitely many discrete solutions","Infinite discrete Euler solutions from dense initial data","Gas dynamics: dense data spawn infinite entropy-growing solutions","One dense set, infinite solutions: discrete Euler gas states","Dense data give infinite discrete solutions with rising entropy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1286,"prompt_tokens":662,"completion_tokens":624,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":406,"completion_tokens_details":{"reasoning_tokens":548}},"tokens_in":406,"tokens_out":624,"duration_ms":7813,"temperature":1.0,"reasoning_tokens":548,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T01:13:51.643405+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is to test the local construction on a single measurable subdomain with no boundary regularity and with test functions that do not vanish on the boundary: exhibit one such subdomain for which the overdetermined system (3.4)–(3.6) has no bounded finite-valued momentum field. If such a subdomain exists, the imported lemma fails at the level of generality used and the dense-set theorems collapse.","supporting_citations":[],"review_version":1}