{"id":"d42d34ca-46cc-4181-8584-5105f75e95bc","arxiv_id":"2506.06766","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence and uniqueness of strong solutions is established for abstract SPDEs with fully local monotonicity, then applied to fractional p-Laplace equations with arbitrary-order polynomial drift and superlinear transport noise.","lead":"This paper proves that a class of stochastic partial differential equations, including the fractional p-Laplace equation with arbitrary polynomial drift and superlinear transport noise, has unique solutions under general monotonicity conditions. It also repairs gaps in a recent well-posedness paper by using a topological version of the Skorokhod representation theorem and a weak-convergence argument for stochastic integrals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central argument survives scrutiny. Only minor clarifications (Simon's corollary statement, choice of Q_n basis) are warranted.","rationale":"The paper's main theorem is supported by a long but internally consistent argument. I examined the Galerkin approximation, the auxiliary Hilbert space construction, the tightness proof, the Skorokhod-Jakubowski application, and the monotonicity limit. The reader's weakest assumption about the auxiliary Hilbert space is standard and not the place where a serious gap would arise. The tightness argument in Lemma 2.6 relies on [31, Corollary 5]; the hypotheses appear to be met because the boundedness in L^q(0,T;V), the W^{σ,2}(0,T;H*) estimate, and the sup_H bound together imply the needed translation estimate, but the paper should state the exact corollary to remove any ambiguity. Similarly, the projection Q_n used to cut off the cylindrical Wiener process must be chosen from the singular basis of the Hilbert-Schmidt embedding U⊂U_0 to coincide with the U-orthogonal projection; with that standard choice, the term ∥h^*_kB(Q_n−I)∥→0 in (2.60) is justified. I also confirmed the conditions (H1)-(H5) are satisfied for the fractional p-Laplace applications, with the smallness conditions (3.19) and (3.34) matching the abstract condition (2.7). No step appears to invalidate the existence, uniqueness, or moment estimates of Theorem 2.2 or its applications. The verdict CONDITIONAL from the reader is therefore appropriate as a call for clarifications rather than for substantive revision, and I do not see a reason to change it.","tokens_in":38205,"tokens_out":52364,"duration_ms":492553,"concrete_test":"Verify, by quoting [31, Corollary 5] and checking its hypotheses for the set K in (2.45), that boundedness in L^q(0,T;V) ∩ W^{σ,2}(0,T;H*) plus sup_{[0,T]}||y||_H implies relative compactness in L^q(0,T;H) for q=min_j q_j; if the corollary requires σ>1/2, Lemma 2.6 would need a fractional-derivative estimate in W^{σ,r}(0,T;H*) with r>1, which is not currently proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After review, I find no load-bearing concern with the central claim. The auxiliary Hilbert space H⊆V from [5, Lemma C.1] is a standard spectral construction; the projection bounds (2.19)-(2.20) and the tightness set K in Lemma 2.6 are justified. The use of W^{σ,2}(0,T;H*) with σ∈(0,1/2) together with boundedness in L^q(0,T;V) and sup_H bound yields the required translation estimate for [31, Corollary 5]; the paper would benefit from quoting the corollary precisely. The passage to the limit in stochastic integrals via (2.62) and [8, Lemma 2.1] is sound, and the monotonicity argument in Section 2.4 correctly identifies the drift and diffusion limits. Minor issues: the topology T in (2.44) is not literally stronger than the norm topology of L^q(0,T;V), but a countable separating family for (Y,T) can be taken from (L^1(0,T;H))* since Y⊂L^1(0,T;H); and the projection Q_n on U_0 should be understood as built from the singular basis of the Hilbert-Schmidt embedding so that Q_n|_U equals the U-orthogonal projection. Neither point threatens the main theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an abstract well-posedness theorem for stochastic evolution equations in a variational setting under a fully local monotonicity condition, allowing both functions φ and ψ in the local monotonicity inequality (2.1) to be nonzero. The proof combines Galerkin approximations, uniform a priori estimates, tightness in a non-metrizable space built from L^q(0,T;H), weak-* L∞(0,T;H), and weak L^{q_j}(0,T;V_j), the Skorokhod-Jakubowski representation theorem, and a monotone limit argument. The abstract theorem is then applied to fractional stochastic p-Laplace equations with polynomial drift of arbitrary order and superlinear transport noise, under smallness conditions such as (3.19) or (3.34). A separate result covers the standard fractional Laplacian case p=2 with an additional gradient-type noise term.","tokens_in":38464,"tokens_out":13829,"duration_ms":139895,"significance":"If the proof is completed, the paper would remove several limitations of the recent work [32]: it avoids the use of the false strong Skorokhod representation theorem, repairs the tightness argument in L^r(0,T;H), and allows polynomial nonlinearities of arbitrary order under fully local monotonicity. The abstract framework is clean and the assumptions (H1)-(H5) are explicit, with no fitted parameters or normalization that encodes the answer. The applications to fractional p-Laplace equations with superlinear transport noise are substantial and go beyond existing results for linearly growing noise. The paper also gives quantitative moment estimates (2.15), which are valuable for further dynamical and ergodic questions. The main caveat is that the proof contains a real but reparable gap in the construction of the separating family for the Skorokhod-Jakubowski theorem, and a technical issue with the definition of the projections Q_n.","major_comments":[{"comment":"The assertion that the supremum topology T on Y is stronger than the norm topology of L^q(0,T;V) is false: convergence in T consists of convergence in L^q(0,T;H), weak-* convergence in L∞(0,T;H), and weak convergence in each L^{q_j}(0,T;V_j), none of which implies norm convergence in L^q(0,T;V). Consequently, a functional φ_j^* ∈ (L^q(0,T;V))^* need not be continuous on (Y,T), and the separating family constructed in the proof does not justify the application of Proposition 4.1. This is not fatal: since Y⊂L^1(0,T;H) and T is stronger than the L^1(0,T;H) norm topology, one can take a countable separating family from (L^1(0,T;H))^* ≅ L∞(0,T;H*), or from the duals of the spaces L^{q_j}(0,T;V_j) using their weak topologies. The repair should be written out explicitly, but it does not change the main result.","section":"Lemma 2.7, proof after (2.44)"},{"comment":"The projection Q_n is defined as the orthogonal projection of U0 onto span{u_1^0,...,u_n^0} for an arbitrary orthonormal basis {u_i^0} of U0. In equation (2.21) the stochastic integral contains P_nB(s,Z_n(s))Q_n dW(s), where W is a cylindrical Wiener process in U; for this integrand to be a Hilbert-Schmidt operator from U to H, Q_n must map U into U and act as an orthogonal projection on U. An arbitrary basis of U0 does not have this property. The construction should use the singular value decomposition of the Hilbert-Schmidt embedding U⊂U0 so that Q_n|_U is the U-orthogonal projection onto the first n singular vectors; this also makes the convergence estimates in (2.60)-(2.63) valid. This is a technical but necessary correction.","section":"Section 2.2, definition of Q_n"},{"comment":"The uniqueness part of Theorem 2.2 is not proved in the manuscript; the text says that 'by the standard argument (see, e.g., [32])' pathwise uniqueness holds and then invokes the Yamada-Watanabe theorem. Since Theorem 2.2 is the central abstract result and the fully local monotonicity condition (H2) is precisely the delicate part of the problem, the uniqueness proof should either be included or the exact statement in [32] that applies should be identified. Without this, the uniqueness assertion of the main theorem is not verified within the paper.","section":"Section 2.4, final paragraph"}],"minor_comments":[{"comment":"The citation to [31, Corollary 5] should be made precise: as usually stated, the corollary requires a translation estimate in L^p(0,T;H*) with the same integrability exponent p as the space L^p(0,T;V). Here the proof has L^q(0,T;V) and W^{σ,2}(0,T;H*); the missing step is that the L∞(0,T;H) bound converts the W^{σ,2} translation estimate into the required L^q translation estimate. This is a clarification, not a fatal gap.","section":"Lemma 2.6, use of [31, Corollary 5]"},{"comment":"The exponent q̃/(q̃-1) is typeset with a bar over q in several places, and the definition of q = min{2, q̃/(q̃-1)} in (2.39) should be double-checked for consistency with the use of q in the W^{σ,2} estimate.","section":"Lemma 2.5, equations (2.40)-(2.42)"},{"comment":"The derivation of (3.31) from (2.7) uses the convention that γ_{2,j}=0 gives an infinite ratio γ_{1,j}/γ_{2,j}; this convention should be stated explicitly, since otherwise the minimum in (2.7) is undefined when some γ_{2,j} vanish, as happens in the applications.","section":"Section 3, equation (3.31)"},{"comment":"The phrase 'the strong Skorokhod representation theorem is incorrect even in a complete separable metric space' is potentially confusing: the classical Skorokhod representation theorem for tight sequences in Polish spaces is correct, while the 'strong' version on the original probability space is what fails. The wording should distinguish these two statements.","section":"Abstract and Section 1"}],"recommendation":"major_revision","confidential_remarks":"The main mathematical strategy appears sound and the results are likely correct after a moderate revision. The most important point for the editor is that the gap in Lemma 2.7 concerning the separating family is real but local; the author should be asked to supply the corrected separating family and to fix the definition of Q_n. The omitted pathwise uniqueness argument should also be supplied or precisely referenced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is the real thing. It proves an abstract well-posedness theorem for SPDEs with fully local monotonicity (both phi and psi nonzero, polynomial drift of arbitrary order) and then applies it to the fractional stochastic p-Laplace equation with superlinear transport noise. The main advance over Röckner–Shang–Zhang [32] is that it handles both phi and psi without vanishing, removes the growth restriction on the polynomial, and patches two genuine gaps: the invalid use of the strong Skorokhod theorem and the incomplete tightness argument in L^r(0,T;H). The tightness fix via boundedness in W^{σ,2}(0,T;H*) plus Simon's compactness corollary is the right approach.\n\nThe abstract Theorem 2.2 is a reusable tool, not a one-off. The proof structure is standard but executed carefully: Galerkin approximation, uniform estimates, tightness in a non-metrizable space, Skorokhod–Jakubowski representation, weak convergence of stochastic integrals, and a monotone limit argument. I checked the key estimates and they hold. The assumptions (H1)–(H5) are explicit and the applications verify them with sharp constants; there is no circular fitting or hidden normalization.\n\nSoft spots are minor and repairable. First, in Lemma 2.7 the proof claims the topology T on Y is stronger than the topology of L^q(0,T;V); that's not literally true. A countable separating family for (Y,T) can instead be taken from (L^1(0,T;H))* since Y embeds in L^1(0,T;H), so the argument is fixable but needs a sentence. Second, there is a typo in (2.60) where Q_n is dropped in one line; harmless because Q_n is a contraction, but it should be corrected. Third, the choice of Q_n should be stated as built from the singular basis of the Hilbert–Schmidt embedding U⊂U_0 so that Q_n restricted to U is the U-orthogonal projection. Fourth, Lemma 2.6 invokes a specific corollary of Simon's compactness theorem without quoting it; a precise statement would help the reader.\n\nThese are presentation and clarity issues, not load-bearing flaws. The central argument survives scrutiny. This paper deserves a serious referee and, after minor revision, publication in a strong journal. For your own work on monotone SPDEs, it's worth citing: the abstract theorem will save you from redoing this tightness argument.","headline":"A serious, technically sound extension of the fully local monotonicity framework that fixes real gaps in the recent Annalen paper; only small clarifications are needed.","tokens_in":38997,"tokens_out":3196,"would_cite":true,"duration_ms":30825,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60H15","37L55","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that fractional p-Laplace stochastic equations with polynomial drift of arbitrary order and superlinear transport noise have unique solutions.","keywords":["fractional p-Laplace equation","transport noise","fully local monotonicity","Galerkin approximation","tightness","Skorokhod-Jakubowski theorem","well-posedness","superlinear noise"],"falsifier":"A concrete check: for $V=W^{s,p}(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ on a bounded domain, try to write down the auxiliary Hilbert space and basis promised by [5, Lemma C.1] and verify the identity $\\|v\\|_H^2=\\|P_nv\\|_H^2+\\|(I-P_n)v\\|_H^2$ for every $v\\in H$; a counterexample to this basis property would invalidate Lemma 2.6. Alternatively, test condition (2.9): find a diffusion $B$ satisfying (H2)-(H5) and a sequence $u_n\\to u$ in $L^1(0,T;H)$ for which $v^*B(\\cdot,u_n)$ fails to converge in $L^2(0,T;L^2(U,\\mathbb{R}))$ for some $v\\in V$, which would block the identification of the limit noise term.","tokens_in":37988,"feed_emoji":"","tokens_out":6721,"duration_ms":62422,"temperature":0.7,"pith_summary":"The paper proves an abstract existence-and-uniqueness theorem for stochastic evolution equations whose drift and diffusion satisfy a fully local monotonicity condition, then applies it to the fractional p-Laplace equation. The drift may be a polynomial of arbitrarily high order, and the transport noise may grow superlinearly and depend on spatial derivatives of the solution, a regime previous results could not cover. The proof uses Galerkin approximations, uniform moment estimates, tightness in a non-metrizable path space, and the Skorokhod-Jakubowski representation theorem. The main analytic difficulty is proving tightness and uniform integrability of the approximate solutions, which the paper handles with an auxiliary Hilbert space basis and a compactness criterion. If correct, the theorem gives unique solutions for a broad class of stochastic partial differential equations with genuinely superlinear noise.","feed_headline":"Unique solutions proved for fractional p-Laplace SPDEs","feed_subtitle":"Proof handles polynomial drift of any order plus superlinear transport noise.","key_machinery":"The argument rests on four interlocking devices. First, the Galerkin approximations are built from an auxiliary separable Hilbert space $\\mathcal{H}\\subseteq V$ whose orthonormal basis $\\{h_k\\}$ is also an orthogonal basis of the solution space $H$; this gives the projection identities (2.19)-(2.20) used throughout. Second, Itô's formula combined with the coercivity and growth assumptions (H3)-(H5) yields uniform moment bounds for the approximate solutions in $L^\\infty(0,T;H)$, $L^{q_j}(0,T;V_j)$, and $W^{\\sigma,2}(0,T;H^*)$. Third, tightness is proved in the non-metrizable space $L^q(0,T;H)\\cap L^\\infty_{w^*}(0,T;H)\\cap\\bigcap_j L^{q_j}_w(0,T;V_j)$ using the Simon compactness criterion, with the compact embedding $V\\subseteq H$ supplying the needed compactness in $L^q(0,T;H)$; this repairs an incomplete tightness argument in the earlier literature. Fourth, passage to the limit uses the Skorokhod-Jakubowski representation theorem in a topological space instead of the strong Skorokhod theorem, which is false, followed by a weak-convergence argument for stochastic integrals and the monotone trick to identify the weak limit of the drift as $A(\\cdot,Z)$.","core_discovery":"The central claim is Theorem 2.2: if the drift $A=\\sum_{j=1}^J A_j$ and diffusion $B$ satisfy hemicontinuity, full local monotonicity, coercivity, growth bounds, and the weak-noise-continuity condition (2.9), and if the embedding $V\\subseteq H$ is compact, then for every $x\\in H$ the abstract equation $dX(t)=A(t,X(t))dt+B(t,X(t))dW(t)$ has a unique solution in the sense of Definition 2.1, with uniform moment estimates. Full local monotonicity means the estimate $2\\sum_j(A_j(t,u)-A_j(t,v),u-v)+\\|B(t,u)-B(t,v)\\|^2_{L_2(U,H)}\\le(g(t)+\\varphi(u)+\\psi(v))\\|u-v\\|_H^2$ holds with both $\\varphi$ and $\\psi$ nonzero; earlier theorems only treated the case where one of them vanishes. The paper then verifies conditions (H1)-(H5) for the fractional $p$-Laplace operator with polynomial drift and superlinear transport noise, obtaining existence and uniqueness under explicit conditions such as (3.19), (3.34), or (3.60).","pith_inferences":["The auxiliary-Hilbert-space construction used for tightness suggests the method transfers to other Gelfand triples where a compactness criterion like Simon's applies, so the abstract theorem may cover intersections of fractional Sobolev spaces beyond $W^{s,p}\\cap L^q$.","Because the paper only needs the weak noise-continuity condition (2.9) rather than strong continuity of $B$, the same proof scheme may extend to noise coefficients that are merely weakly continuous in the state variable, a testable weakening for equations where strong continuity fails.","The repaired tightness argument, based on boundedness in $L^q(0,T;V)\\cap W^{\\sigma,2}(0,T;H^*)$, gives a general template for proving tightness in locally monotone SPDEs with multiple drift components, which could be reused in other settings.","One could numerically probe whether the smallness conditions such as $\\sum_i\\beta_i<\\delta_1$ in (3.34) are sharp, for instance by taking $f(u)=-|u|^{q-2}u$ and letting $\\sum_i\\beta_i$ approach the critical bound while watching for blow-up or non-uniqueness."],"forward_implications":["The fractional stochastic p-Laplace equation (1.1)-(1.3) with polynomial drift of any order $q\\ge2$ and superlinear transport noise has a unique solution for every $u_0\\in H$ under the conditions of Theorem 3.1 or Theorem 3.2.","For $p=2$, the standard fractional Laplacian equation with an additional noise term $G$ satisfying (3.42)-(3.43) also has a unique solution under condition (3.60), including multiplicative noise of the form $\\sum_i a_i g_i(-\\Delta)^{s/2}u$.","The solution satisfies the uniform moment estimates (2.15), so its $H$-norm and $V_j$-norms are controlled in expectation by the initial data for every $p$ in the range (2.14).","Pathwise uniqueness holds for the abstract equation, so existence on a new probability space upgrades to a unique strong probabilistic solution by the Yamada-Watanabe theorem.","The abstract theorem applies to a family of superlinear-noise SPDEs, including the 2D Navier-Stokes, Allen-Cahn, Cahn-Hilliard, and Allen-Cahn-Navier-Stokes equations with transport noise mentioned in the introduction."],"supporting_citations":[{"why":"Supplies the auxiliary Hilbert space with an orthonormal basis that is also an orthogonal basis of H, which underlies the Galerkin projections and projection bounds (2.19)-(2.20).","marker":"[5]"},{"why":"The prior well-posedness result for fully local monotone coefficients that this paper extends; it also provides the open problems and the gaps the paper repairs.","marker":"[32]"},{"why":"The counterexample to the strong Skorokhod representation theorem that forces the paper to use a weak-convergence argument for stochastic integrals.","marker":"[28]"},{"why":"Supplies the compactness criterion (Corollary 5) used in Lemma 2.6 to prove tightness of the approximate solutions in $L^q(0,T;H)$.","marker":"[31]"},{"why":"Provides the Skorokhod-Jakubowski representation theorem in non-metric spaces, which is the tool used to pass from tightness to almost-sure convergence on a new probability space.","marker":"[18]"},{"why":"Provides the existence theory for the finite-dimensional Galerkin equations and the Itô-formula and monotone-argument framework used throughout the proof.","marker":"[20]"},{"why":"Supplies Lemma 2.1 used to pass to the limit in stochastic integrals with respect to a sequence of Wiener processes.","marker":"[8]"}],"fun_headline_variants":["Fractional p-Laplace SPDEs: unique solutions under superlinear noise","Arbitrary polynomial drift: well-posedness for p-Laplace SPDEs","Full local monotonicity yields uniqueness for abstract SPDEs","New proof: fractional p-Laplace with transport noise well-posed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the auxiliary Hilbert space $\\mathcal{H}\\subseteq V$ from [5, Lemma C.1] really exists, with an orthonormal basis that is also an orthogonal basis of the solution space $H$; if that construction fails, the Galerkin projections, the bounds (2.19)-(2.20), and the tightness argument in Lemma 2.6 collapse.","fun_headline_variants_meta":{"raw":{"variants":["Fractional p-Laplace SPDEs: unique solutions under superlinear noise","Arbitrary polynomial drift: well-posedness for p-Laplace SPDEs","Full local monotonicity yields uniqueness for abstract SPDEs","New proof: fractional p-Laplace with transport noise well-posed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"total_tokens":1319,"prompt_tokens":959,"completion_tokens":360,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":280}},"tokens_in":575,"tokens_out":360,"duration_ms":3699,"temperature":1.0,"reasoning_tokens":280,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:51:06.847902+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: for $V=W^{s,p}(\\mathbb{R}^n)\\cap L^q(\\mathbb{R}^n)$ on a bounded domain, try to write down the auxiliary Hilbert space and basis promised by [5, Lemma C.1] and verify the identity $\\|v\\|_H^2=\\|P_nv\\|_H^2+\\|(I-P_n)v\\|_H^2$ for every $v\\in H$; a counterexample to this basis property would invalidate Lemma 2.6. Alternatively, test condition (2.9): find a diffusion $B$ satisfying (H2)-(H5) and a sequence $u_n\\to u$ in $L^1(0,T;H)$ for which $v^*B(\\cdot,u_n)$ fails to converge in $L^2(0,T;L^2(U,\\mathbb{R}))$ for some $v\\in V$, which would block the identification of the limit noise term.","supporting_citations":[{"cited_title":"Brze´ zniak and L","cited_arxiv_id":null,"evidence_quote":"Supplies the auxiliary Hilbert space with an orthonormal basis that is also an orthogonal basis of H, which underlies the Galerkin projections and projection bounds (2.19)-(2.20)."},{"cited_title":"Rockner, S","cited_arxiv_id":null,"evidence_quote":"The prior well-posedness result for fully local monotone coefficients that this paper extends; it also provides the open problems and the gaps the paper repairs."},{"cited_title":"Ondrejat and J","cited_arxiv_id":null,"evidence_quote":"The counterexample to the strong Skorokhod representation theorem that forces the paper to use a weak-convergence argument for stochastic integrals."},{"cited_title":"Simon, Compact sets in the spaceL p(0, T;B),Annali di Matematica Pura ed Applicata, 146(1987), 65-96","cited_arxiv_id":null,"evidence_quote":"Supplies the compactness criterion (Corollary 5) used in Lemma 2.6 to prove tightness of the approximate solutions in $L^q(0,T;H)$."},{"cited_title":"Jakubowski, The almost sure Skorokhod representation for subsequences in nonmetric spaces,Theory of Probability and Its Applications,42(1988), 167-175","cited_arxiv_id":null,"evidence_quote":"Provides the Skorokhod-Jakubowski representation theorem in non-metric spaces, which is the tool used to pass from tightness to almost-sure convergence on a new probability space."},{"cited_title":"Krylov and B.L","cited_arxiv_id":null,"evidence_quote":"Provides the existence theory for the finite-dimensional Galerkin equations and the Itô-formula and monotone-argument framework used throughout the proof."},{"cited_title":"Debussche, N","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.1 used to pass to the limit in stochastic integrals with respect to a sequence of Wiener processes."}],"review_version":1}