{"id":"652c84a9-f42e-4edb-bd0c-41661ac9bc77","arxiv_id":"2502.09292","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A convex integration solution with lower action than the 1-D Riemann solution is constructed, so the least action admissibility principle fails to select the physically intuitive solution.","lead":"This paper shows that a recently proposed selection rule, the least action admissibility principle, rejects the physically intuitive solution for a special gas dynamics Riemann problem. The result adds another failed criterion to the search for a solution concept for the multi-dimensional Euler equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof hinges on an unproven finite-time convex-integration theorem with terminal trace; if Proposition 2.1 is not a valid consequence of the cited global-in-time results, the glued competitor is not a weak solution.","rationale":"The reader's weakest assumption identifies Proposition 2.1's finite-time version and the unshown 'easy verifications' as the main risk. My stress test confirms that the algebraic checks in §3.2 are sound: the chosen constants satisfy the Rankine-Hugoniot conditions, the subsolution inequalities hold exactly (the first factor C1/2+γ1 equals 1), and the final action inequality Kex-K1d<0 follows from the bounds ϱM<94 and σ<11/10. The genuinely load-bearing uncertainty is Proposition 2.1's terminal trace prescription. The paper cites global-in-time convex-integration theorems and asserts without proof that the method works on [0,T0] with the trace terms (2.2)-(2.4). Since the whole gluing argument in Corollary 2.4 depends on the momentum trace being ρ1u1 in the middle region while the interior has |u|2=C1, this is not a cosmetic gap: if the terminal trace cannot be imposed, the competitor is not a weak solution and the action comparison is moot. The proposed concrete test checks exactly whether the cited theorems already contain this trace statement or whether an independent proof is needed. I found no additional fatal flaw in the action computation, the two-shock fan after T0, or the final inequality, so the verdict remains CONDITIONAL as the reader stated.","tokens_in":11552,"tokens_out":32917,"duration_ms":361587,"concrete_test":"Inspect the statements of [2, Props 3.6 and 5.1], [3, Props 3.1 and 4.1], and [12, Thm. 7.3.4 and Prop. 7.3.5]; if none contains a finite-interval existence result with terminal traces as in (2.2)-(2.4), attempt to derive Proposition 2.1 from the cited global theorem by extending the subsolution beyond T0 and checking that the trace at T0 equals the prescribed piecewise-constant state. If the terminal momentum trace cannot be prescribed, the gluing in Corollary 2.4 fails and the action comparison is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 2.4 glues a convex-integration solution on [0,T0] to a classical solution on [T0,T]. For the glued function (2.9) to be a weak solution, the terminal trace of the first solution at t=T0 must equal the piecewise-constant initial data of the second solution, in particular the momentum ρ1u1 in the middle region. Proposition 2.1 asserts exactly this trace condition in (2.2)-(2.4), while also asserting |u|2=C1 almost everywhere in Γ1. This is a strong boundary-prescription property: the interior velocity oscillates with magnitude √C1, yet the weak momentum trace is forced to be the constant ρ1u1. The cited results [2,3,12] are global-in-time theorems, and Remark 2.2 only says the finite-time adaptation is 'easy to observe' without proof. If the quoted propositions do not deliver terminal traces of this form, the mass and momentum equations have an uncontrolled surface term at t=T0, so (ϱex,uex) is not an admissible weak solution and Lemma 3.1 does not refute the least action principle. I checked the algebraic constants and the final action inequality in §3; they are internally consistent, so the finite-time trace theorem is the weakest load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the least action admissibility principle proposed by Gimperlein, Grinfeld, Knops, and Slemrod as a selection criterion for admissible weak solutions of the two-dimensional barotropic compressible Euler equations. For the pressure law p(ρ)=ρ^2, the authors exhibit Riemann initial data with symmetric left and right states and, for every T>0, construct a competitor weak solution by gluing a convex-integration solution on [0,T/2] to an explicit two-shock solution on [T/2,T]. They compute the actions of both the self-similar 1-D solution and the competitor, obtaining the explicit constants K1d and Kex in Lemma 3.1, and show Kex<K1d using the bounds ρM<94 and σ<11/10. The conclusion is that the 1-D solution does not satisfy the least action principle, so the principle must either be rejected or the physical intuition reconsidered.","tokens_in":11766,"tokens_out":9946,"duration_ms":93841,"significance":"If the core existence input is valid, the theorem is a clean negative answer to a recently proposed admissibility criterion, and it complements earlier failures of the maximal dissipation criterion by the first author. The action comparison is explicit and self-contained, with concrete constants and no free parameters beyond the advertised ones. The main weakness is that the finite-time convex-integration Proposition 2.1 is quoted rather than proved; the result is conditional on that ingredient. The paper is a meaningful contribution to the ongoing assessment of selection criteria for multi-dimensional conservation laws.","major_comments":[{"comment":"Proposition 2.1 is the load-bearing existence theorem for the competitor on [0,T0], and in particular it prescribes terminal traces (2.2)-(2.4) so that the glued function (2.9) satisfies the weak formulation across t=T0. The proof is not given: the paper refers to [2, Props. 3.6 and 5.1], [3, Props. 3.1 and 4.1] and [12, Thm. 7.3.4 and Prop. 7.3.5], and Remark 2.2 asserts that the finite-interval version is easy to observe. The cited works are global-in-time theorems, and the trace condition at t=T0 is a stronger boundary-prescription property that is not obviously contained in them. Since the momentum trace in Γ1 is forced to the constant ρ1u1 while |u|^2=C1 a.e., this needs a proof or a precise reference with statement. Without it, the glued competitor is not known to be a weak solution and Lemma 3.1 does not refute the least action principle.","section":"Section 2, Proposition 2.1 and Remark 2.2"},{"comment":"The proof of Lemma 2.3 invokes Glimm's theorem [10] for arbitrary one-dimensional initial data (2.5). Glimm's classical theorem requires sufficiently small total variation, while the data here has jumps of size 2 from density 1 to 3; the stated general form is not a consequence of [10]. For the specific data chosen in Section 3.2 an explicit two-shock solution is available and is in fact described there, so the gap is fixable, but the proof as written does not establish the lemma in the needed generality.","section":"Section 2, Lemma 2.3"},{"comment":"Several load-bearing algebraic verifications are asserted without computation: the parameter values in (3.3) are said to satisfy all equations and inequalities of Proposition 2.1 (one can easily verify), the post-T0 solution is said to consist of two shock waves (one can easily verify), and in the proof of Theorem 1.4 the inequality Kex-K1d<0 is said to follow from (3.2) (which can easily be verified). These checks are central to the construction and to the final comparison; at least the explicit verification of the subsolution inequalities (2.1) and of the final numerical inequality should be included, for example in an appendix.","section":"Section 3.2-3.3"}],"minor_comments":[{"comment":"The word 'followng' is a typo for 'following'.","section":"Section 1, Definition 1.1"},{"comment":"The bounds ρM<94 and σ<11/10 in (3.2) are stated as simple to verify; since they are used in the proof of Theorem 1.4, include the short derivation.","section":"Section 3.1"},{"comment":"The statements (4.1) and (4.2) about the action for t<T0 and for t>T0 are presented without proof; as they are not needed for Theorem 1.4, label them as heuristic remarks or provide supporting references.","section":"Section 4"},{"comment":"In the proof of Lemma 3.1, the factor 4 in (3.4) and (3.5) is explained only implicitly; a sentence describing the symmetry in y and the x-periodicity would improve readability.","section":"Section 3.3, Lemma 3.1"},{"comment":"The notation u1 is used both for a two-dimensional velocity vector and for its horizontal component (u1,v1); an explicit notational remark would remove possible ambiguity for the reader.","section":"Section 2, Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends at its core on finite-time convex-integration existence with terminal traces (Proposition 2.1), which is quoted from the first author's own monograph and papers. Given the self-citation pattern, I would encourage the editor to ask for a detailed proof or an independent verification of the trace condition, because this is the only point at which the main theorem could fail. The algebraic computations in Section 3 are internally consistent and can be checked by the authors as an appendix. The paper is otherwise a solid, concise contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this note does what it claims. For a specific symmetric Riemann problem, it builds a convex-integration competitor whose action is lower than the 1-D solution's, so the least action admissibility principle would rule out the physically relevant solution. That is a genuine negative result against Gimperlein–Grinfeld–Knops–Slemrod's 2024 proposal, and it is the first direct one I know of.\n\nWhat I like: the structure is clear. Construct a wild solution with the authors' earlier machinery, glue it at T0 to a classical two-shock solution, compute both actions, and compare. Lemma 3.1 is explicit; I rechecked the signs and the final inequality Kex − K1d < 0 does follow from (3.2). The energy jump at T0 (Remark 2.5) is a neat mechanism: by arranging the kinetic energy to drop, you get the action below the 1-D curve. Self-citations for the existence theorem are not per se a problem; the cited statements exist and are published.\n\nThe soft spot, and it is load-bearing: Proposition 2.1 is quoted, not proved, and Remark 2.2 only says the finite-time adaptation is 'easy to observe'. But the terminal trace conditions (2.2)–(2.4) are exactly what make the glued function (2.9) a weak solution at t = T0. If the quoted global-in-time theorems do not deliver those traces on [0,T0], the competitor fails. That is a real gap and it needs a proof or a precise citation to a finite-interval statement. The 'one can easily verify' checks in Section 3.2—subsolution inequalities and the shock fan after T0—look right and a referee can check them, but they are also left as exercises. The constants are ugly but internally consistent.\n\nOverall: this deserves a serious referee. It is exactly the kind of paper that should be sent to review with a request to fill the finite-time gap. I would accept it if the authors close that hole, or point to a theorem that covers it. My own verdict is conditionally positive; I would not cite it as a definitive refutation until that happens.","headline":"A targeted counterexample to the least action principle; the algebra checks, but the finite-time convex-integration existence theorem is the weak spot.","tokens_in":12404,"tokens_out":2614,"would_cite":false,"duration_ms":26010,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76N10","35Q31","35D30","35L65"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every final time T, the least action admissibility principle rules out the standard one-dimensional Euler solution.","keywords":["compressible Euler equations","least action admissibility principle","Riemann problem","convex integration","admissible weak solutions","non-uniqueness","selection criteria","energy inequality"],"falsifier":"Compute the left-hand sides of the Rankine–Hugoniot and subsolution inequalities in Proposition 2.1 for the constants in (3.1) and Section 3.2, namely $\\varrho_1=3$, $u_1=0$, $v_1=0$, $C_1=\\frac{1121\\sqrt{1281}}{20}+\\frac{28037}{12}$, $\\gamma_1=-\\frac{1121\\sqrt{1281}}{40}-\\frac{28013}{24}$, $\\delta_1=0$, and $\\mu_0=-\\frac{57\\sqrt{35}}{20}-\\frac{59\\sqrt{915}}{60}$; if any required inequality is violated, the constructed competitor is not admissible. Alternatively, evaluate $K_{\\mathrm{ex}}-K_{1\\mathrm d}$ from Lemma 3.1 using $\\varrho_M<94$ and $\\sigma<11/10$; the theorem's claim is exactly that this number is negative.","tokens_in":11223,"feed_emoji":"📉","tokens_out":14122,"duration_ms":106088,"temperature":0.7,"pith_summary":"The paper tests the least action admissibility principle, a selection rule proposed for the multi-dimensional barotropic compressible Euler equations: among all admissible weak solutions with the same initial data, the physical one should minimize the action. The authors prove the principle fails. For initial data that are constant on each side of a horizontal line (a Riemann problem) and for every final time $T>0$, they exhibit another admissible weak solution, built by the solution-manufacturing technique of convex integration, whose action is strictly smaller than that of the standard one-dimensional shock solution. Thus the least action principle rules out the solution that is intuitively physical; the paper concludes that either intuition must be revised or the principle must be discarded.","feed_headline":"Least-action principle rejects the physical Euler solution","feed_subtitle":"This competitor beats the 1-D shock solution on action for every final time, so the criterion cannot stand.","key_machinery":"The load-bearing construction is Corollary 2.4's gluing recipe: take a convex integration solution on $[0,T_0]$ (with $\\varrho=\\varrho_1$ and $|u|^2=C_1$ on the wedge $\\Gamma_1$), then continue with the classical solution from the trace data at $t=T_0$. The algebraic conditions of Proposition 2.1—Rankine–Hugoniot relations on the two interfaces, the subsolution inequality $C_1-(u_1)^2-(v_1)^2>0$, and the two entropy/energy inequalities at the interfaces—guarantee that infinitely many such solutions exist. The identity that does the work is the energy jump $\\frac12\\varrho_1C_1\\mapsto\\frac12\\varrho_1|u_1|^2$ at $t=T_0$: because $C_1>|u_1|^2$, the jump is downward, which preserves the energy inequality and lowers the action density. The action functional here is $A[\\varrho,u]=\\int_0^T\\int(\\frac12\\varrho|u|^2-P(\\varrho))\\,dy\\,dx\\,dt$, and the comparison reduces to quadrature of the piecewise-linear action density, yielding the $T^2$ coefficients in Lemma 3.1.","core_discovery":"The central claim is Theorem 1.4: for each $T>0$, there exist Riemann initial data $(\\varrho_\\pm,u_\\pm)$ such that the one-dimensional (1-D) solution to (1.1) with pressure $p(\\varrho)=\\varrho^2$ does not fulfill the least action admissibility principle. The proof selects the explicit symmetric data (3.1), sets $T_0=T/2$, and uses convex integration on $[0,T_0]$ to form a competitor $(\\varrho_{\\mathrm{ex}},u_{\\mathrm{ex}})$ that is constant, $\\varrho_1=3$ with $|u|^2=C_1$, inside the wedge between two speeds $\\mu_0t<y<\\mu_1t$, then glues the classical two-shock 1-D solution on $[T_0,T]$. Lemma 3.1 gives the actions $A[\\varrho_{\\mathrm{ex}},u_{\\mathrm{ex}}]=K_{\\mathrm{ex}}T^2$ and $A[\\varrho_{1\\mathrm d},u_{1\\mathrm d}]=K_{1\\mathrm d}T^2$ with explicit constants, and since $K_{\\mathrm{ex}}-K_{1\\mathrm d}<0$, the competitor wins for every $T$. The competitor is an admissible weak solution in the sense of Definition 1.1; the key to admissibility is that the kinetic energy jumps downward, not upward, at the gluing time.","pith_inferences":["Because all inequalities in Proposition 2.1 are strict, the explicit numbers are unlikely to be isolated: small perturbations of the Riemann data should still satisfy the conditions, giving an open set of counterexamples.","The construction is written for $p(\\varrho)=\\varrho^2$, but the downward-jump gluing mechanism itself is not tied to that exponent; a natural test is whether the same recipe refutes the principle for other polytropic pressures or for the full Euler system.","The horizon dependence is a structural warning: any selection rule defined by minimizing a time-integrated functional on a fixed horizon $[0,T]$ can be gamed by inserting an energy-decreasing jump after the interval where the competitor is more expensive.","Proponents of the principle could try to rescue it by restricting admissible competitors, for example by demanding structural symmetry or a bound on oscillation; the paper does not explore such restrictions."],"forward_implications":["For every $T>0$ the 1-D solution fails the least action admissibility principle, so the principle does not single out the solution conventionally regarded as physical.","The failure is of the same type as for the global and local maximal dissipation criteria: in each case a convex-integration solution beats the 1-D solution on the criterion's own score.","Which solution is preferred depends on the final time: before $T_0$ the competitor is more expensive, and only after the downward kinetic-energy jump does its cumulative action fall below the 1-D solution's.","The downward energy jump at $t=T_0$ is the decisive mechanism; without it, the action comparison would go the other way.","Accepting the 1-D solution as physical forces one to discard the least action admissibility principle; retaining the principle forces one to revise what counts as the physical solution."],"supporting_citations":[{"why":"Proposes the least action admissibility principle that the paper sets out to refute; supplies Definition 1.3 and the comparison baseline.","marker":"[9]"},{"why":"Establishes the convex integration framework for the barotropic Euler equations and is the source quoted for Proposition 2.1.","marker":"[2]"},{"why":"Provides the energy-dissipation construction and the 1-D Riemann solution formulas on which the competitor and the action computation build.","marker":"[3]"},{"why":"Contains the convex integration existence theorems cited as the basis of Proposition 2.1.","marker":"[12]"},{"why":"Parallel failure of the local maximal dissipation criterion on the same type of example; the paper's strategy mirrors it.","marker":"[13]"},{"why":"Glimm's existence theorem is used in Lemma 2.3 to continue the solution by classical 1-D wave fans after the gluing time.","marker":"[10]"}],"fun_headline_variants":["Least action principle fails on Euler shocks","Least action picks wrong Euler solution","Least action admissibility contradicted by shock","Least action cannot select physical shock","Least action criterion rejects real Euler solution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hangs on the finite-time convex integration existence theorem (Proposition 2.1) holding with the stated data on the interval $[0,T_0]$, and on the 'easily verified' algebraic inequalities for the explicit constants being true; if either fails, the competitor is not an admissible weak solution and the action comparison proves nothing.","fun_headline_variants_meta":{"raw":{"variants":["Least action principle fails on Euler shocks","Least action picks wrong Euler solution","Least action admissibility contradicted by shock","Least action cannot select physical shock","Least action criterion rejects real Euler solution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000317,"raw_usage":{"total_tokens":1843,"prompt_tokens":1045,"completion_tokens":798,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":738}},"tokens_in":661,"tokens_out":798,"duration_ms":6935,"temperature":1.0,"reasoning_tokens":738,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:03:29.691162+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the left-hand sides of the Rankine–Hugoniot and subsolution inequalities in Proposition 2.1 for the constants in (3.1) and Section 3.2, namely $\\varrho_1=3$, $u_1=0$, $v_1=0$, $C_1=\\frac{1121\\sqrt{1281}}{20}+\\frac{28037}{12}$, $\\gamma_1=-\\frac{1121\\sqrt{1281}}{40}-\\frac{28013}{24}$, $\\delta_1=0$, and $\\mu_0=-\\frac{57\\sqrt{35}}{20}-\\frac{59\\sqrt{915}}{60}$; if any required inequality is violated, the constructed competitor is not admissible. Alternatively, evaluate $K_{\\mathrm{ex}}-K_{1\\mathrm d}$ from Lemma 3.1 using $\\varrho_M<94$ and $\\sigma<11/10$; the theorem's claim is exactly that this number is negative.","supporting_citations":[{"cited_title":"The Least Action Admissibility Principle","cited_arxiv_id":"2409.07191","evidence_quote":"Proposes the least action admissibility principle that the paper sets out to refute; supplies Definition 1.3 and the comparison baseline."},{"cited_title":"Global ill-posedness of the isentropic system of gas dynamics","cited_arxiv_id":null,"evidence_quote":"Establishes the convex integration framework for the barotropic Euler equations and is the source quoted for Proposition 2.1."},{"cited_title":"On the energy dissipation rate of solutions to the com- pressibleisentropicEulersystem","cited_arxiv_id":null,"evidence_quote":"Provides the energy-dissipation construction and the 1-D Riemann solution formulas on which the competitor and the action computation build."},{"cited_title":"Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations","cited_arxiv_id":null,"evidence_quote":"Contains the convex integration existence theorems cited as the basis of Proposition 2.1."},{"cited_title":"A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion","cited_arxiv_id":null,"evidence_quote":"Parallel failure of the local maximal dissipation criterion on the same type of example; the paper's strategy mirrors it."},{"cited_title":"Solutions in the large for nonlinear hyperbolic systems of equations","cited_arxiv_id":null,"evidence_quote":"Glimm's existence theorem is used in Lemma 2.3 to continue the solution by classical 1-D wave fans after the gluing time."}],"review_version":1}