{"id":"fe75e1cb-2fad-41dc-a5f8-36252e48abfe","arxiv_id":"1908.03539","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors establish existence of random attractors for locally monotone SPDE driven by additive Lévy noise, unifying and extending many previous case-specific results.","lead":"This paper proves that a large family of stochastic partial differential equations with locally monotone coefficients and additive Lévy noise generate random dynamical systems and random attractors. The unified framework covers fluid and phase-field models such as stochastic Burgers, 2D Navier-Stokes, the 3D Leray-alpha model, Cahn-Hilliard and Kuramoto-Sivashinsky type equations, replacing case-by-case proofs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Theorem 5.1's assumptions are explicit and the load-bearing stationary conjugation is proved and verified in the examples.","rationale":"The reader's weakest assumption identifies precisely the most delicate part of the argument: the existence and integrability of the stationary process u_t from Assumption (V), together with the structural condition (4.5). The paper does not hide this dependence; it states (V) and (4.5) as hypotheses, proves the stationary OU construction in Theorem 3.1, verifies (H1)-(H4) for the conjugated random PDE in Theorem 4.1, and checks the hypotheses in a long list of examples. I found no internal inconsistency in the main line of proof. The minor overstatement about all p-th moments in Theorem 3.1(vi)-(vii) is real but does not feed into Theorem 5.1, which only needs p=2 for sublinear growth of the conjugation and p=4 for the exponential integrability of the forcing term in Proposition 5.2. The delegation of the compactness proof to [35, Theorem 3.1] is a gap in self-containment rather than a demonstrated error; the variational assumptions are the standard ones under which such compactness is expected. Therefore the reader's ACCEPT verdict should stand unchanged.","tokens_in":33615,"tokens_out":40944,"duration_ms":431860,"concrete_test":"Re-check Theorem 3.1(vi)-(vii) against the cited [34, Lemma 5.2]: determine whether the claimed p-th moment bounds for all p rely on Gaussian or exponential integrability of the driving noise or only on the fourth-moment condition (N). If only fourth moments are assumed, restrict (vi)-(vii) to p<=4 and confirm that the proof of Theorem 5.1 uses only the p=2 and p=4 cases, so no correction to the main theorem is needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify a load-bearing flaw in the central claim. The main risk is exactly the one the reader names: the whole construction hinges on Assumption (V) and condition (4.5), which let Theorem 3.1 produce a stationary H-valued OU-type process u_t taking values in V, and make the conjugated random PDE A_omega(t,v)=A(v+u_t)-sigma M(u_t) satisfy (H1)-(H4). The paper states these hypotheses explicitly in Theorems 4.1 and 5.1, and the examples verify them in each application, so this is a conditional theorem with a clearly identified scope rather than a hidden assumption. One non-central overstatement: Theorem 3.1(vi)-(vii) assert Cesaro convergence and sublinear growth of ||u_t||_H^p for every p in N under Assumption (N), which only supplies Levy moments through order four; for p>4 these statements may fail and are not needed for Theorem 5.1. The compactness assertion in Theorem 5.1(i) is delegated to [35], so the proof is not fully self-contained, but the cited mechanism is standard for this variational framework. Neither observation undermines the attractor claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for the long-time behavior of locally monotone stochastic partial differential equations driven by additive trace-class Lévy noise. On a Gelfand triple V ⊂ H ⊂ V*, the authors impose structural conditions (A1)–(A5) on the drift A, assumption (V) that A has a strongly monotone part M, and condition (4.5) on the local-monotonicity functions η and ρ. Theorem 3.1 constructs a strictly stationary Ornstein–Uhlenbeck-type process u_t solving du_t = σM(u_t)dt + dN_t by taking limits as the initial time tends to -∞. This process is used in Section 4 to conjugate the original SPDE into a pathwise random PDE, for which the variational well-posedness theorem from Appendix A is verified. The resulting stochastic flow S is proved to be a continuous cocycle (Theorem 4.1), and Theorem 5.1 proves compactness and, under the additional condition K < γλ/4 when α = 2, existence of a random D-attractor. The abstract conditions are verified on a broad list of examples, including Burgers-type equations, 2D Navier–Stokes, the 3D Leray-α model, power-law fluids, the Ladyzhenskaya model, Cahn–Hilliard-type equations, Kuramoto–Sivashinsky-type equations, and previously studied monotone SPDE.","tokens_in":33761,"tokens_out":17374,"duration_ms":185315,"significance":"If the results are correct, this is a substantial contribution to the random-attractor literature. It unifies many case-by-case results by providing an abstract attractor theorem for locally monotone SPDE, going beyond the earlier monotone-operator frameworks of [34, 37] and covering fluid-dynamics-type nonlinearities. A particular strength is that the paper states all hypotheses explicitly and then checks them in detail for each application, including the nontrivial interpolation estimates in the fluid examples. The stationary Ornstein–Uhlenbeck-type construction under Lévy noise with only fourth-order moments is also of independent interest. The main theorem is conditional on the clearly stated hypotheses (V) and (4.5), which are verified in every example; this is a properly scoped conditional result rather than a hidden assumption. I did not identify a load-bearing flaw in the central derivation.","major_comments":[],"minor_comments":[{"comment":"Parts (vi) and (vii) assert Cesàro convergence and sublinear growth of ‖u_t‖_H^p for every p ∈ ℕ, but Assumption (N) only provides Lévy moments up to order 4 and the p-th-moment Itô estimates in the proof are only closed for p ≤ 4. For p > 4, E‖u_t‖_H^p need not be finite without additional moment assumptions. This is a local overstatement: the attractor argument uses only the p = 2 and p = 4 cases through estimate (3.4), so the unbounded range p > 4 is not needed. Please restrict (vi)–(vii) to 2 ≤ p ≤ 4 or strengthen (N) accordingly.","section":"Section 3, Theorem 3.1(vi)–(vii)"},{"comment":"The compactness of the cocycle S is asserted by reference to [35, Theorem 3.1] without further detail. Since compactness is a key step in the attractor proof, a short indication of why compactness transfers through the stationary conjugation T would improve self-containedness, even if the cited argument is standard in this variational framework.","section":"Section 5, proof of Theorem 5.1(i)"},{"comment":"The introductory theorem statement omits the extra condition K < γλ/4 for α = 2 that appears in the formal statement of Theorem 5.1. Please align the informal statement with the formal theorem.","section":"Section 1, informal statement of Theorem 5.1"},{"comment":"The displayed formula for the critical exponent p_c is typeset in a way that is hard to read; please clarify the expression so that the numerical range claimed in the example is unambiguous.","section":"Section 6.5, Example 6.9"}],"recommendation":"minor_revision","confidential_remarks":"None."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the real thing. It supplies a single abstract theorem that yields random attractors for locally monotone SPDE with additive trace-class Lévy noise, and the applications include models not covered before, notably the 3D Leray-alpha model and Cahn-Hilliard type equations. The paper does not fold the hard part into a citation: the stationary OU-type conjugation is constructed in Theorem 3.1, the random PDE estimates are in Section 4, and the examples verify the abstract hypotheses in detail. That is a lot of honest work.\n\nWhat is new: previous random attractor results for this variational framework were largely for monotone operators or for noise more regular than H-valued Lévy noise. Here the local monotonicity from Liu–Röckner and the Lévy well-posedness from Brzeźniak–Liu–Zhu are combined with the stationary conjugation idea of Gess. The structural assumption (V)—that the drift has a strongly monotone part M—is explicit, and each example names an M. The additional smallness condition on K in (A3) when α=2 is also stated clearly. Given how many examples are worked (Burgers, 2D NSE, Leray-alpha, power law fluids, Ladyzhenskaya, Cahn-Hilliard, Kuramoto-Sivashinsky, porous media, p-Laplace), this is a useful unification, not just a theorem for one equation.\n\nThe soft spots are genuine but minor. First, Theorem 3.1(vi)–(vii) assert Cesàro convergence and sublinear growth of ||u_t||_H^p for every p∈N, while Assumption (N) only supplies Lévy moments through order four. For p>4 those claims need moments that are not there; they are not used in the attractor proof, so this is an overstatement, not a load-bearing flaw. Second, in Theorem 5.1(ii), K(ω) is an image of a bounded set under a compact map; that is relatively compact, not necessarily closed. Taking the closure would make it a compact absorbing set. Minor fix. Third, compactness in Theorem 5.1(i) is delegated to [35, Theorem 3.1]; the paper is not fully self-contained at that point, but the cited mechanism is standard for this framework.\n\nThe citation pattern is heavy on the authors' own prior work, but those citations are used as tools—well-posedness for locally monotone SPDE, the conjugation idea—rather than as a way to avoid proving the new claim. The construction of u_t and the conjugated random PDE are in the paper.\n\nWho this is for: anybody working on long-time behavior of SPDE, especially fluid and phase-field models with jumps. It deserves a serious referee. I would send it out, expecting minor revision.","headline":"A genuinely general random attractor theorem for locally monotone SPDE with additive Lévy noise; the main argument holds up, and the flaws are minor.","tokens_in":34382,"tokens_out":2613,"would_cite":true,"duration_ms":28555,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37L55","60H15","35Q35","47H05","35G31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random attractors exist for a broad class of locally monotone SPDE driven by additive Lévy noise.","keywords":["random attractors","random dynamical systems","locally monotone SPDE","Lévy noise","Navier-Stokes equations","Burgers equation","Cahn-Hilliard equation","stochastic p-Laplace equations"],"falsifier":"Take a locally monotone equation with $\\alpha = 2$ and tune the linear term so that $K$ reaches $\\gamma\\lambda/4$, the boundary excluded in Theorem 5.1, and compute whether the pullback bound (5.2) still stays finite; if absorption still holds, the condition is not sharp, and if it fails, the threshold is necessary. Alternatively, for a candidate strongly monotone $M$, check whether the auxiliary stationary process $u_t$ really lies in $L^\\alpha_{\\mathrm{loc}}(\\mathbb{R}; V)$ and is exponentially integrable in the sense of Theorem 3.1; a Lévy noise with finite fourth moments that violates this would break the conjugation.","tokens_in":33337,"feed_emoji":"🌊","tokens_out":7600,"duration_ms":74460,"temperature":0.7,"pith_summary":"This paper proves that random attractors exist for a large class of stochastic partial differential equations whose drift is only locally monotone, not globally monotone, when the noise is additive and of trace-class Lévy type. Local monotonicity means the one-sided Lipschitz estimate may degrade by factors $\\eta(v_1)+\\rho(v_2)$ that are locally bounded in the solution norm, which is exactly the flexibility needed to include convection terms such as those in Burgers and 2D Navier-Stokes equations. The proof constructs a random dynamical system by subtracting a strictly stationary Ornstein-Uhlenbeck-type process built from a strongly monotone part of the drift, then uses compactness of the embedding $V \\subseteq H$ to upgrade bounded absorption to a compact absorbing set. The result turns many previously case-by-case attractor proofs into corollaries of one abstract theorem.","feed_headline":"Random attractors proven for locally monotone SPDE with Lévy noise","feed_subtitle":"The result covers Burgers, 2D Navier-Stokes, Cahn-Hilliard, p-Laplace and other stochastic PDEs.","key_machinery":"The load-bearing object is the stationary conjugation map $T(t,\\omega)y = y - u_t(\\omega)$, where $u_t$ is the strictly stationary solution of the strongly monotone auxiliary equation $du_t = \\sigma M(u_t)dt + dN_t$ built from a strongly monotone part $M$ of the drift $A$. Conjugating by $T$ turns the stochastic equation into a pathwise random PDE with coefficients $A_\\omega(t,v) = A(v+u_t) - \\sigma M(u_t)$; the assumptions (V) and (4.5) are exactly what make these random coefficients satisfy the local monotonicity, coercivity and growth hypotheses (H1)-(H4) of the deterministic well-posedness theory. Compactness of the embedding $V \\subseteq H$ then makes the cocycle compact, so after establishing a random bounded absorbing set the standard random attractor criterion applies.","core_discovery":"The central claim is Theorem 5.1: under assumptions (A1)-(A5), (V) and the structural inequality (4.5), the continuous cocycle $S$ generated by $dX_t = A(X_t)dt + dN_t$ is compact, and when $\\alpha = 2$ one additionally needs $K < \\gamma\\lambda/4$ in the coercivity estimate; then there is a random $\\mathcal{D}$-attractor. The conditions are satisfied by stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the 3D Leray-$\\alpha$ model, power law fluids, the Ladyzhenskaya model, Cahn-Hilliard type equations, Kuramoto-Sivashinsky type equations, porous media equations and $p$-Laplace equations, all driven by additive trace-class Lévy noise with finite fourth moments. In particular, the noise is only required to take values in the Hilbert space $H$, not in the domain of the drift operator, because the auxiliary stationary process supplies the missing spatial regularity.","pith_inferences":["The stationary conjugation scheme is not tied to additive Lévy noise in an essential way: any noise that can be absorbed into a strictly stationary $V$-valued process for a strongly monotone $M$ should fit the same template.","The structural condition (4.5) and the restriction $\\beta(\\alpha-1) \\le 2$ suggest the framework will not reach reaction terms of higher polynomial growth, such as the classical cubic double-well Cahn-Hilliard potential; a different route would be needed there.","The theorem leaves open whether the same attractor exists in the energy space $V$; the proof deliberately stops at compactness of the embedding, so regularity of the attractor is not addressed.","One testable extension is to let the Lévy measure have only finite second moments; the current proof uses moments up to order four, so weakening (N) would widen the class of admissible noises."],"forward_implications":["Stochastic Burgers, 2D Navier-Stokes, 3D Leray-$\\alpha$, power law fluid, Ladyzhenskaya, Cahn-Hilliard, Kuramoto-Sivashinsky, porous media and $p$-Laplace equations each admit a continuous random dynamical system and a random attractor under additive trace-class Lévy noise with finite fourth moments.","The noise only needs to take values in $H$, not in the domain of $A$, because the stationary process $u_t$ supplies the missing regularity.","Previously known random attractor results for monotone SPDE are recovered and extended to the locally monotone class.","For $\\alpha = 2$, the smallness condition $K < \\gamma\\lambda/4$ on the linear part is the price for bounded absorption; in examples where $K = 0$ the condition is automatic.","The compact-cocycle argument avoids higher regularity assumptions on the noise that earlier attractor proofs needed."],"supporting_citations":[{"why":"Supplies existence and uniqueness of variational solutions for locally monotone SPDE driven by Lévy noise.","marker":"[14]"},{"why":"Provides the stationary conjugation construction and Ornstein-Uhlenbeck-type process for Wiener noise that this paper generalizes.","marker":"[34]"},{"why":"Provides measurability and adaptedness arguments for random flows generated by additive noise.","marker":"[37]"},{"why":"Supplies the local monotonicity, coercivity and growth framework for SPDE with locally monotone coefficients.","marker":"[61]"},{"why":"Gives the generalized coercivity conditions and well-posedness result used for the random PDE in Theorem 4.1.","marker":"[62]"},{"why":"Provides the random attractor existence criterion used to conclude Theorem 5.1.","marker":"[22]"},{"why":"Supplies the compactness argument for the cocycle used in the proof of Theorem 5.1.","marker":"[35]"},{"why":"Provides the stochastic flow, cocycle and temperedness framework used throughout Appendix B.","marker":"[3]"}],"fun_headline_variants":["Random attractors proven for wide class of SPDEs with Lévy noise","Lévy noise in SPDEs: random attractors for Burgers to p-Laplace","One theorem, many SPDEs: random attractors under Lévy noise","Additive Lévy noise SPDEs: random attractors exist","From Burgers to Navier-Stokes: random attractors with Lévy noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the drift splits as a strongly monotone part $M$ plus a perturbation, with the strictly stationary process $u_t$ from $du_t = \\sigma M(u_t)dt + dN_t$ living in $V$ and obeying the growth bounds of (4.5); if no such $M$ exists, the conjugation that removes the noise fails and the whole attractor argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Random attractors proven for wide class of SPDEs with Lévy noise","Lévy noise in SPDEs: random attractors for Burgers to p-Laplace","One theorem, many SPDEs: random attractors under Lévy noise","Additive Lévy noise SPDEs: random attractors exist","From Burgers to Navier-Stokes: random attractors with Lévy noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000222,"raw_usage":{"total_tokens":1414,"prompt_tokens":865,"completion_tokens":549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":445}},"tokens_in":481,"tokens_out":549,"duration_ms":5445,"temperature":1.0,"reasoning_tokens":445,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:11:40.048233+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a locally monotone equation with $\\alpha = 2$ and tune the linear term so that $K$ reaches $\\gamma\\lambda/4$, the boundary excluded in Theorem 5.1, and compute whether the pullback bound (5.2) still stays finite; if absorption still holds, the condition is not sharp, and if it fails, the threshold is necessary. Alternatively, for a candidate strongly monotone $M$, check whether the auxiliary stationary process $u_t$ really lies in $L^\\alpha_{\\mathrm{loc}}(\\mathbb{R}; V)$ and is exponentially integrable in the sense of Theorem 3.1; a Lévy noise with finite fourth moments that violates this would break the conjugation.","supporting_citations":[{"cited_title":"Strong so lutions for SPDE with locally mono- tone coefficients driven by Levy noise","cited_arxiv_id":null,"evidence_quote":"Supplies existence and uniqueness of variational solutions for locally monotone SPDE driven by Lévy noise."},{"cited_title":"Random Attractors for Degenerate Stoch astic Partial Differential Equations","cited_arxiv_id":null,"evidence_quote":"Provides the stationary conjugation construction and Ornstein-Uhlenbeck-type process for Wiener noise that this paper generalizes."},{"cited_title":"Random at tractors for a class of stochastic partial differential equations driven by general additive n oise","cited_arxiv_id":null,"evidence_quote":"Provides measurability and adaptedness arguments for random flows generated by additive noise."},{"cited_title":"SPDE in Hilbert space with locally monotone coefficients","cited_arxiv_id":null,"evidence_quote":"Supplies the local monotonicity, coercivity and growth framework for SPDE with locally monotone coefficients."},{"cited_title":"Local and global well-pos edness of SPDE with generalized coercivity conditions","cited_arxiv_id":null,"evidence_quote":"Gives the generalized coercivity conditions and well-posedness result used for the random PDE in Theorem 4.1."},{"cited_title":"Attractors for random dynamical systems","cited_arxiv_id":null,"evidence_quote":"Provides the random attractor existence criterion used to conclude Theorem 5.1."},{"cited_title":"Random attractors for singular stochas tic evolution equations","cited_arxiv_id":null,"evidence_quote":"Supplies the compactness argument for the cocycle used in the proof of Theorem 5.1."}],"review_version":1}