{"id":"c4765750-f048-4e38-86ad-e5ca936f69a7","arxiv_id":"2607.27925","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Rotating Riesz star solutions exist and are nonlinearly stable in the mass-subcritical regime, and exist but are unstable in the mass-supercritical regime, with the effect of rotation depending on the Riesz singularity α.","lead":"This mathematics paper proves existence, stability, and instability of rotating steady-state “Riesz star” solutions for the compressible Euler–Riesz equations, which model self-gravitating fluids and plasmas with non-local interactions. It shows that rotation can stabilize or destabilize these stars depending on the singularity strength of the interaction, and it extends existence beyond the small-rotation regime.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stability/instability theorems are conditional on global solutions for rotating axisymmetric data that are not known to exist; without such solutions, the central claims are vacuous.","rationale":"The reader's weakest assumption identified essentially this concern: global finite-energy weak solutions with the angular-momentum structure are assumed but not proved, and known finite-time singularity results make the premise fragile. My stress-test confirms that this is the single most load-bearing issue. The variational existence results appear internally consistent and are significant, but the dynamical stability/instability claims are conditional on unproven global-existence and structural-conservation hypotheses. I do not see a more fundamental flaw in the variational compactness arguments themselves, so the appropriate verdict remains CONDITIONAL with moderate confidence. The concrete test proposed—checking whether the [13] existence theory extends to rotating axisymmetric data—would settle whether the theorems have any known instances beyond the non-rotating case. If it does not extend, the stability theorem is vacuous for rotating stars; if it does, the concern is resolved.","tokens_in":51831,"tokens_out":48542,"duration_ms":395505,"concrete_test":"Check whether the global-existence construction of [13] can be extended to axisymmetric initial data with nonzero angular momentum satisfying (I1)-(I3). Concretely: attempt to reproduce the a priori estimates of [13, Theorem 2.4] with the additional rotational kinetic energy term ∫ρL(mρ)/r^2 and with the angular-momentum structure (2.9). If the proof relies essentially on spherical symmetry to control the Riesz potential or to propagate the angular-momentum conservation (A2)-(A3), then Theorem 2.5's hypothesis has no known non-vacuous instance, and the advertised stability of rotating Riesz stars remains unsubstantiated for genuinely rotating data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central stability result (Theorem 2.5) is an implication whose premise requires a global finite-energy weak solution of the CEREs satisfying (A1)-(A4), including conservation of angular momentum and mass along particle paths. The paper does not prove such solutions exist for rotating axisymmetric data with nonzero angular momentum. The cited global existence result [13] is restricted to spherically symmetric data, and Choi–Jeong [21] establish finite-time singularity formation for classical solutions in the relevant attractive regime. Similarly, the instability theorem (Theorem 6.4) assumes global classical solutions, which are not known to exist and may be ruled out by the same singularity results. Thus, as stated, the theorems do not establish that rotating Riesz stars are actually stable or unstable solutions of the Euler–Riesz dynamics; they only show conditional statements about hypothetical global solutions. Since the abstract and introduction present nonlinear stability/instability as the main results, this missing existence premise is the most load-bearing weakness of the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies rotating steady states of the attractive three-dimensional compressible Euler–Riesz equations (CEREs), which it calls rotating Riesz stars. In the mass-subcritical regime (general pressure laws, polytropically γ > (3+α)/3), it proves existence of minimisers of the free-energy functional G over axisymmetric densities of fixed mass under subhomogeneity conditions on the angular momentum profile L, and shows these minimisers solve the rotating-star equations (2.4) (Corollary 3.10, Theorem 3.11). It then proves a conditional nonlinear stability theorem (Theorem 2.5) for a unique minimiser, assuming the existence of global finite-energy weak solutions satisfying structural assumptions (A1)–(A4). In the polytropic mass-supercritical regime (6/(6−α) < γ < (3+α)/3), it introduces a modified admissible class K and proves existence of minimisers of S_μ (Theorem 2.8) under superhomogeneity conditions, and shows these are compactly supported rotating Riesz stars. Finally, under an additional growth condition on L and assuming global classical solutions, it proves growth of the support for solutions starting near the star (Theorem 6.4), yielding a conditional instability result. The main technical novelty is a concentration-compactness argument adapted to axisymmetry that excludes loss of mass along rings with diverging radii.","tokens_in":52043,"tokens_out":13397,"duration_ms":112937,"significance":"Provided the conditional premises are accepted, the paper makes a significant contribution: it extends the classical Auchmuty–Beals/Luo–Smoller rotating-star theory from Newtonian to Riesz interactions, introduces a genuinely new compactness mechanism for axisymmetric sequences (control of ring concentration), and gives the first mass-supercritical existence result beyond small angular velocity. The variational Euler–Lagrange computations are coherent and the scaling analysis is internally consistent. However, the dynamical conclusions are conditional on global existence and uniqueness hypotheses that are not proved here; the cited global existence result is for spherically symmetric data, and known singularity results for classical solutions make the instability theorem's premise potentially empty. The paper's strengths include a detailed variational framework and explicit scaling analysis; its main weakness is the gap between the stated 'nonlinear stability/instability' and the conditional nature of the theorems.","major_comments":[{"comment":"The central dynamical theorems are conditional on global solutions that are not shown to exist. Theorem 2.5 assumes a global finite-energy weak solution satisfying (A1)–(A4); Theorem 6.4 assumes a global classical solution. The cited global existence result [13] is restricted to spherically symmetric initial data, and Choi–Jeong [21] prove finite-time singularity for classical solutions in the attractive regime. Consequently, the paper does not establish that rotating Riesz stars are actually stable or unstable solutions of the CEREs; it proves conditional statements about hypothetical global solutions. Since the abstract and introduction present nonlinear stability/instability as the main results, this missing existence premise is load-bearing. The author should either prove global existence for rotating axisymmetric data (or at least provide a local existence/continuation framework), o","section":"Theorems 2.5 and 6.4"},{"comment":"The stability theorem assumes ¯ρ is the unique minimiser of G over X_M up to vertical translation, but Corollary 3.10 only establishes existence of a minimiser. No uniqueness proof or example is provided. Without uniqueness, Theorem 3.2 only gives convergence of a minimising sequence to some minimiser; the contradiction argument in §4 cannot identify the limit with the reference state ¯ρ. Remark 4.2 refers to the non-unique case but gives no proof. The theorem should either prove uniqueness under (2.5)–(2.7), or be reformulated as stability of the set of minimisers, or the uniqueness hypothesis should be stated prominently in the abstract.","section":"Theorem 2.5"},{"comment":"The exclusion of concentration along rings with diverging radii is the key new compactness statement, but the proof is sketched. In particular, estimate (3.17) is asserted without proof, and the covering argument with Cov_k(R) is terse. Since this is the central new ingredient separating the paper from prior non-rotating compactness arguments, the author should provide a complete proof of (3.17), including the precise use of axisymmetry, and justify the covering estimate in detail.","section":"Theorem 3.2, Step 4"}],"minor_comments":[{"comment":"The two displayed inequalities are not clearly labelled; the first involves a different sign of the potential term and is later used as an energy upper bound. Please clarify which quantity is the physical energy and which is an auxiliary bound.","section":"Definition 2.1(b)"},{"comment":"The uniform convergence of m^S_{ϱ_k}(r) on [0,R] is justified by a 'variation of Dini's Theorem' from [61, p.167]; monotonicity in k is not shown, so please spell out the argument or replace it with a direct proof.","section":"Lemma 3.8"},{"comment":"The phrase 'main text of the thesis' should be 'main text of the paper'.","section":"Appendix A"},{"comment":"The relative energy is defined with a subtracted rotational kinetic term; please explain the physical meaning of this subtraction in the text.","section":"Equation (2.8)"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's own works [12] and [13] for the variational framework and for global weak-solution existence. The rotating results are new relative to those references, but the heavy self-citation should be kept in mind. The main dynamical theorems are conditional on global existence and uniqueness premises that are not established in the manuscript. The editor may wish to consider whether the journal is willing to publish conditional stability/instability results with open existence premises, or whether the scope should be narrowed to the variational existence results with the dynamical claims clearly flagged as conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing: this is a real paper. The existence theory for rotating Riesz stars — mass-subcritical minimizers and mass-supercritical polytropic minimizers — is new, and the axisymmetric concentration-compactness argument that rules out mass escaping along rings whose radii go to infinity is a genuine technical contribution. In the supercritical regime, the paper goes beyond the small-angular-velocity barrier that has limited the Euler–Poisson literature, and the κ(ϱ) construction for α∈(0,2) is a nice workaround for the non-uniqueness of critical scalings. If the variational results are correct, they are worth having.\n\nNow the soft spots. The stability theorem (Thm 2.5) is conditional on global finite-energy weak solutions that conserve angular momentum and mass along particle paths (A1–A4). The paper does not prove such solutions exist for rotating axisymmetric data with nonzero angular momentum. The cited global existence result [13] is spherically symmetric, and Choi–Jeong's finite-time singularity results for the attractive super-Newtonian range make the existence of global classical solutions doubtful in general. The instability theorem (Thm 6.4) assumes global classical solutions. So as stated, the stability and instability claims are implications about hypothetical solutions, not unconditional statements about the dynamics. The abstract and introduction lean on the unconditional reading; I would want the conditional language up front.\n\nThere are also smaller gaps. Theorem 2.5 assumes uniqueness of the minimizer up to vertical translation; only existence is proved. Remark 4.2 says the uniqueness assumption can be removed, but the argument is not included. A few technical estimates — the covering argument in Theorem 3.2, Step 4, and parts of Lemma 5.7 — are sketched and need careful referee work.\n\nNone of this sinks the paper. Conditional stability theorems are normal in this area, and the variational core is self-contained. The paper's own claims are mostly honest if you read the hypotheses. I would send it to peer review with a request to state the conditional nature of the stability and instability results explicitly, provide or cite global existence for the rotating axisymmetric class, and either prove the uniqueness claim or make the non-unique version precise.\n\nUseful to people who work on Euler–Poisson, Euler–Riesz, and variational methods for stellar structure. Not for the casual reader. I would cite it if I wrote on rotating Riesz stars, and I might bring it to reading group.","headline":"Genuine variational progress on rotating Riesz stars, but the stability and instability theorems are conditional on global solutions whose existence for rotating data is not established.","tokens_in":52539,"tokens_out":3253,"would_cite":true,"duration_ms":31237,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","35Q31","35B35","35A15","35R09","35L65","76N10","35B38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that rotating Riesz star solutions of the compressible Euler–Riesz equations exist as energy minimizers, are nonlinearly stable below a critical mass, and become unstable above it.","keywords":["compressible Euler-Riesz equations","rotating Riesz stars","nonlinear stability","nonlinear instability","concentration compactness","mass-supercritical regime","Riesz potential","axisymmetric steady states"],"falsifier":"Take α = 1, polytropic exponent γ = 6/5 (supercritical), and L(m) = m^ω with ω chosen slightly below the predicted threshold ω* + ω̄. If numerical solutions starting in the invariant set fail to grow support, the instability condition is falsified; more directly, constructing a global weak solution for rotating axisymmetric data that violates (A2) would falsify the stability theorem's premise.","tokens_in":51645,"feed_emoji":"⭐","tokens_out":3960,"duration_ms":33084,"temperature":0.7,"pith_summary":"The paper establishes a conditional dichotomy for rotating Riesz stars, axisymmetric steady states of the attractive compressible Euler–Riesz equations that rotate with angular velocity J(m(r))/r. In the mass-subcritical regime the stars exist as minimizers of a free-energy functional and are nonlinearly stable, provided global finite-energy weak solutions retain the assumed angular-momentum structure along particle paths. In the mass-supercritical polytropic regime the stars still exist as minimizers but are nonlinearly unstable: arbitrarily close initial data lead to solutions whose support grows without bound. The dividing line is the mass-critical exponent, and the proof hinges on new compactness for axisymmetric minimizing sequences, ruling out mass escaping along rings of diverging radius.","feed_headline":"Rotating Riesz stars flip from stable to unstable at critical mass","feed_subtitle":"New proof shows rotation can stabilize or destabilize gaseous stars depending on the Riesz interaction's singularity.","key_machinery":"The central machinery is the pair of free-energy functionals G (subcritical) and S_μ (supercritical) defined over axisymmetric densities with angular momentum squared L, together with the mass-preserving scaling ϱ_λ(x) = λ^{3/2} ϱ(λ^{1/2} x) and the quantity κ(ϱ) that separates the two possible critical scalings when α ∈ (0,2). These tools convert compactness of minimizers into existence of stars, and concavity of S_μ under scaling into growth of support, giving instability.","core_discovery":"For subcritical masses, rotating Riesz stars are variational minimizers and nonlinearly stable; for supercritical masses, they exist as minimizers but are unstable through growth of support. The steady state satisfies the profile equation (ρ e(ρ))_ρ + ∫_r^∞ L(m_ρ(s))s^{-3} ds + Φα * ρ = -μ on its support, with rotational velocity J(m_ρ(r))/r e_θ. The central compactness theorem shows that an axisymmetric minimizing sequence cannot lose mass along rings whose radii diverge, because the potential energy would then vanish and force the infimum to become non-negative, contradicting strict negativity. The instability proof uses concavity of the rescaled free energy and a virial identity to force","pith_inferences":["The virial-based instability argument likely transfers to finite-energy weak solutions provided the structural assumptions (A1)–(A4) hold, since it only uses the weak form of the equations.","The superhomogeneity threshold ω* + ω̄ may be sharp; testing with power-law angular momentum L(m) = m^ω near the threshold would provide a concrete numerical check of the existence and instability conditions.","The ring-drifting compactness result is a general template for axisymmetric variational problems with rotational terms: if minimizing mass drifts to infinite radius, potential energy vanishes and forces the energy to be non-negative.","Rotation appears to stabilize in the regime α ∈ (0,2) by slowing ring drift, but destabilizes for α ∈ [2,3) in the supercritical regime, indicating that the effect of rotation depends on the singularity of the Riesz kernel."],"forward_implications":["If the stability theorem holds, then for polytropic gases with γ > (3+α)/3, solutions starting close to a rotating Riesz star remain close in the relative-energy distance for all time, up to vertical translation.","In the supercritical range, any solution starting in the invariant set I_μ must have support growing to infinity, so the star is not even nonlinearly stable in a weak sense.","The stability result requires conservation of angular momentum and mass along particle paths; if these structural assumptions fail, the dichotomy may collapse.","The mass bounds (2.5)–(2.6) play the role of a critical mass; crossing it flips stability to instability for rotating Riesz stars.","The results generalize the classical rotating-star theory of the Euler–Poisson equations to the full Riesz range α ∈ (0,3)."],"fun_headline_variants":["Rotation stabilizes or destabilizes Riesz stars by mass","Critical mass decides stability of rotating Riesz stars","Subcritical stable, supercritical unstable: rotating Riesz stars","Rotating Riesz stars flip stability at a mass threshold","Rotation's dual role in Riesz star stability proven"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The theorems assume that global finite-energy weak or classical solutions of the Euler–Riesz equations exist and conserve angular momentum and mass along particle paths; the paper does not prove global existence for rotating axisymmetric data, and known finite-time singularity results make this premise fragile.","fun_headline_variants_meta":{"raw":{"variants":["Rotation stabilizes or destabilizes Riesz stars by mass","Critical mass decides stability of rotating Riesz stars","Subcritical stable, supercritical unstable: rotating Riesz stars","Rotating Riesz stars flip stability at a mass threshold","Rotation's dual role in Riesz star stability proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1333,"prompt_tokens":799,"completion_tokens":534,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":543,"completion_tokens_details":{"reasoning_tokens":455}},"tokens_in":543,"tokens_out":534,"duration_ms":4748,"temperature":1.0,"reasoning_tokens":455,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T22:59:25.176725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take α = 1, polytropic exponent γ = 6/5 (supercritical), and L(m) = m^ω with ω chosen slightly below the predicted threshold ω* + ω̄. If numerical solutions starting in the invariant set fail to grow support, the instability condition is falsified; more directly, constructing a global weak solution for rotating axisymmetric data that violates (A2) would falsify the stability theorem's premise.","supporting_citations":[],"review_version":1}