{"id":"771717b4-c6aa-49f8-bfae-1128ea4cb6cf","arxiv_id":"2606.13065","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Uniqueness holds for dissipative SQG solutions in critical spaces without time-continuity or smallness via energy method with fractional heat semigroup smoothing and iteration.","lead":"This paper proves uniqueness of solutions to the dissipative surface quasi-geostrophic equation in scale-critical Lebesgue and non-homogeneous Besov spaces, without assuming time-continuity or smallness. Mathematicians studying PDE uniqueness for fluid models may read it to see an adaptation of energy methods that removes common technical restrictions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Justification of energy inequality for non-time-continuous solutions via semigroup smoothing + iteration may fail to close in critical spaces","rationale":"The reader's identification of the energy-inequality justification as the weakest assumption aligns exactly with the key technical step described in the abstract. No other internal inconsistency is detectable from the given information, and the approach follows a known Lions-Masmoudi template, so the verdict remains provisional pending verification of that step.","tokens_in":1571,"tokens_out":353,"duration_ms":12757,"concrete_test":"Extract the precise iteration scheme from the proof of the energy inequality (likely §3 or §4); recompute the first two iterates explicitly for a pair of solutions in the critical Lebesgue space with a jump discontinuity in time; check whether the L^1_t norm of the difference remains controlled as the semigroup parameter →0.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The uniqueness claim rests on obtaining an energy inequality for the difference of two solutions in scale-critical spaces (L^{2/α} or non-homogeneous Besov B^{2α-1}_{p,∞} etc.) without assuming any time continuity. The abstract states this is achieved by combining the smoothing effect of the fractional heat semigroup with an iteration scheme derived from the mild form of the integral equation. In the absence of time continuity, the passage from the mollified energy equality to the inequality requires controlling the nonlinear term at each iteration step; if the iteration does not produce a uniform bound independent of the mollification parameter (or if the critical-space embedding does not absorb the commutator errors), the inequality does not follow and the subsequent Gronwall-type argument collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to establish uniqueness of solutions to the dissipative surface quasi-geostrophic (SQG) equation in the scale-critical Lebesgue spaces L^{2/α} and non-homogeneous Besov spaces B^{2α-1}_{p,∞} (and related spaces), without any time-continuity assumption on the solutions and without smallness conditions. The argument adapts the energy method of Lions-Masmoudi (2001) for the Navier-Stokes equations, with the key step being the derivation of an energy inequality for the difference of two solutions via the smoothing properties of the fractional heat semigroup combined with an iteration scheme applied to the mild integral formulation.","tokens_in":1719,"tokens_out":526,"duration_ms":14927,"significance":"If the central argument closes, the result would be a meaningful extension of uniqueness theory for dissipative SQG, removing the time-continuity hypothesis that is often imposed in critical-space settings. It would also supply a template for handling weak solutions in other active-scalar equations where time regularity is unavailable. The paper explicitly credits the Lions-Masmoudi framework and the semigroup smoothing as independent ingredients.","major_comments":[{"comment":"The justification of the energy inequality (central to the uniqueness argument) proceeds by mollification followed by an iteration scheme on the mild form; however, the abstract and the described method do not specify how the commutator terms arising from the nonlinear advection are controlled uniformly in the mollification parameter when working in the scale-critical spaces L^{2/α} or B^{2α-1}_{p,∞}. Without a uniform bound independent of the regularization, passage to the limit fails and the subsequent Gronwall-type estimate for the difference of solutions cannot be obtained.","section":"Abstract (and the section containing the iteration scheme)"},{"comment":"In the critical Besov spaces B^{2α-1}_{p,∞}, the embedding and product estimates used to absorb the nonlinear term after each iteration step must be verified explicitly; the manuscript does not indicate whether the iteration produces a bound that remains controlled as the number of iterations tends to infinity while the mollification parameter tends to zero simultaneously.","section":"the energy-inequality justification"}],"minor_comments":[{"comment":"Notation for the fractional dissipation parameter α and the precise range of p should be stated uniformly from the outset rather than introduced piecemeal.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript to improve the explicitness of the estimates.","responses":[{"response":"We agree that the control of commutator terms arising from mollification requires more explicit justification in the critical spaces. The fractional heat semigroup smoothing is used to obtain the necessary regularity that yields bounds on the commutators independent of the mollification parameter, allowing passage to the limit before applying the iteration scheme on the mild formulation. We will add a dedicated paragraph or lemma detailing these commutator estimates in the revised version.","revision_made":"yes","referee_comment":"[Abstract (and the section containing the iteration scheme)] The justification of the energy inequality (central to the uniqueness argument) proceeds by mollification followed by an iteration scheme on the mild form; however, the abstract and the described method do not specify how the commutator terms arising from the nonlinear advection are controlled uniformly in the mollification parameter when working in the scale-critical spaces L^{2/α} or B^{2α-1}_{p,∞}. Without a uniform bound independent of the regularization, passage to the limit fails and the subsequent Gronwall-type estimate for the difference of solutions cannot be obtained."},{"response":"We acknowledge the need for explicit verification of the embedding and product estimates in B^{2α-1}_{p,∞}. The iteration is constructed so that each step absorbs the nonlinear contribution via the critical-space product laws, producing a bound independent of both the iteration index and the mollification parameter; the double limit is then justified by a diagonal argument. We will include an explicit verification of these estimates (including the relevant embeddings) in a revised subsection.","revision_made":"yes","referee_comment":"[the energy-inequality justification] In the critical Besov spaces B^{2α-1}_{p,∞}, the embedding and product estimates used to absorb the nonlinear term after each iteration step must be verified explicitly; the manuscript does not indicate whether the iteration produces a bound that remains controlled as the number of iterations tends to infinity while the mollification parameter tends to zero simultaneously."}],"tokens_in":1346,"tokens_out":477,"duration_ms":19685,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main claim is that uniqueness holds for the dissipative SQG in scale-critical spaces like L^{2/α} and certain non-homogeneous Besov spaces, even when solutions lack time continuity and without any smallness restriction. The argument follows the 2001 Lions-Masmoudi energy method but replaces the usual continuity assumption with a justification of the energy inequality that uses the smoothing of the fractional heat semigroup plus an iteration scheme on the mild integral equation.\n\nThis is a direct, incremental extension rather than a new method. Removing the time-continuity hypothesis is the concrete gain, since many existence theorems only deliver solutions that are integrable in time but not necessarily continuous. The iteration approach is a reasonable way to handle the lack of continuity while still recovering an energy inequality that feeds into Gronwall.\n\nThe soft spot is the passage from mollified equality to the inequality in the critical norm. The abstract indicates that semigroup smoothing controls the linear part and the iteration handles the nonlinear term, but critical spaces leave little room for commutator errors or approximation losses. If the iteration does not produce a bound uniform in the mollification parameter, the inequality fails and the uniqueness argument collapses. The paper needs to make that step explicit and check the constants carefully.\n\nThis is for people working on uniqueness questions in dissipative fluid equations. It is a clean technical improvement on a known technique, so it deserves a serious referee to verify the iteration details rather than a desk rejection.","headline":"The paper adapts Lions-Masmoudi to get uniqueness for dissipative SQG in critical Lebesgue and Besov spaces without time continuity or smallness.","tokens_in":2191,"tokens_out":369,"would_cite":false,"duration_ms":16298,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Uniqueness of dissipative SQG solutions holds in scale-critical Lebesgue and Besov spaces without time continuity.","keywords":["surface quasi-geostrophic equation","dissipative SQG","uniqueness","scale-critical Lebesgue spaces","Besov spaces","energy inequality","fractional heat semigroup"],"falsifier":"Exhibiting two distinct functions in a scale-critical Lebesgue space that both satisfy the integral form of the dissipative SQG equation but are not equal.","tokens_in":2462,"feed_emoji":"","tokens_out":596,"duration_ms":21818,"temperature":0.7,"pith_summary":"This paper proves uniqueness for solutions of the dissipative surface quasi-geostrophic equation in scale-critical Lebesgue spaces and non-homogeneous Besov spaces. The result requires neither time continuity of the solutions nor smallness assumptions. The argument adapts an energy method from the study of Navier-Stokes equations. Justification of the energy inequality comes from the smoothing effect of the fractional heat semigroup applied within an iteration scheme on the integral equation. Readers interested in partial differential equations for fluid dynamics would care because this broadens the class of solutions for which uniqueness is guaranteed.","feed_headline":"Dissipative SQG unique in critical spaces without time continuity","feed_subtitle":"Solutions remain unique in scale-critical Lebesgue and non-homogeneous Besov spaces even without time continuity or smallness assumptions.","key_machinery":"Energy method with justification of the energy inequality using the smoothing effect of the fractional heat semigroup and iteration on the integral equation structure.","core_discovery":"We show that the uniqueness holds in the scale-critical Lebesgue spaces and non-homogeneous Besov spaces. The proof is based on the energy method, inspired by the approach introduced by Lions and Masmoudi in the study of uniqueness for the Navier-Stokes equations. A key ingredient of the argument is the justification of the energy inequality via the smoothing effect of the fractional heat semigroup together with an iteration scheme based on the structure of the integral equation.","pith_inferences":["If similar justification techniques apply, uniqueness might hold for other fractional dissipation equations.","The removal of time-continuity could enable analysis of more irregular weak solutions in related models.","Extensions to inhomogeneous spaces suggest broader applicability in critical regularity regimes."],"forward_implications":["Unique solutions exist in scale-critical Lebesgue spaces for the dissipative SQG.","Unique solutions exist in non-homogeneous Besov spaces for the dissipative SQG.","The energy method applies without requiring time-continuity of solutions.","The approach works for solutions that are not necessarily small."],"fun_headline_variants":["Dissipative SQG unique in critical Lebesgue spaces without time continuity","SQG uniqueness holds in scale-critical Lebesgue and Besov spaces","Dissipative SQG solutions unique without time continuity in critical spaces","Uniqueness of dissipative SQG proven in critical spaces no time continuity"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The energy inequality holds after applying the smoothing effect of the fractional heat semigroup and iterating on the integral equation.","fun_headline_variants_meta":{"raw":{"variants":["Dissipative SQG unique in critical Lebesgue spaces without time continuity","SQG uniqueness holds in scale-critical Lebesgue and Besov spaces","Dissipative SQG solutions unique without time continuity in critical spaces","Uniqueness of dissipative SQG proven in critical spaces no time continuity"]},"model":"grok-4.3","cost_usd":0.005347,"raw_usage":{"total_tokens":2528,"prompt_tokens":563,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":53474500,"prompt_tokens_details":{"text_tokens":563,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1891,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":563,"tokens_out":74,"duration_ms":12423,"temperature":1.0,"reasoning_tokens":1891,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T06:11:55.236673+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibiting two distinct functions in a scale-critical Lebesgue space that both satisfy the integral form of the dissipative SQG equation but are not equal.","supporting_citations":[],"review_version":1}