{"id":"1e6f5f85-9a67-438a-823f-ed63b86018b7","arxiv_id":"2606.01321","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves local and global existence of weak solutions plus well-posedness for time-dependent fractional Kohn-Sham equations in 3D.","lead":"The paper proves local existence of weak solutions in H^s for time-dependent fractional Kohn-Sham equations in 3D with dispersion (1-Δ)^s for s in (0, 3/2) and various nonlinear interactions, plus global extension under an energy control assumption and well-posedness for s in [1, 3/2) via Strichartz estimates. A smart generalist might read it to understand the mathematical conditions under which fractional quantum models are guaranteed to have solutions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Global existence hinges on unverified control of interaction energies by kinetic energy","rationale":"The reader's weakest_assumption exactly matches the conditional step required for the global-existence portion of the central claim. Because the paper presents the result as conditional on this control, the load-bearing character of the assumption is already correctly flagged; no stronger internal inconsistency appears in the abstract-level structure.","tokens_in":1658,"tokens_out":334,"duration_ms":16877,"concrete_test":"Extract the precise inequality used for 'interaction energies controlled by kinetic energy' (likely in the global-existence section after the local-existence theorem); substitute the explicit Hartree and |u|^{p} terms with p < 2s*/(3-2s) and check whether the constant remains finite and independent of the solution norm for s=0.5 and s=1.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Local existence is obtained via regularization of nonlinearities, yielding weak solutions in H^s. Global extension then invokes the assumption that interaction energies (Hartree + subcritical powers) are controlled by the kinetic energy term to close a priori bounds from the conserved energy. The abstract states this assumption explicitly but provides no derivation or verification that it holds uniformly for the stated class in 3D with fractional s ∈ (0,3/2). If the control constant blows up or fails for any admissible interaction, the energy estimate cannot prevent finite-time blow-up and the global claim collapses. The Strichartz well-posedness for s ≥ 1 is independent of this step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves local existence of weak solutions in H^s to the 3D time-dependent fractional Kohn-Sham equations for s ∈ (0, 3/2) via regularization of the nonlinearities. Under the assumption that interaction energies (Hartree and subcritical powers) are controlled by the kinetic energy, global solutions follow from energy estimates. For s ∈ [1, 3/2), well-posedness is additionally obtained via Strichartz estimates.","tokens_in":1786,"tokens_out":424,"duration_ms":17510,"significance":"If the stated energy-control assumption holds, the results extend local and global existence theory for fractional nonlinear Schrödinger equations to the Kohn-Sham setting, with the Strichartz well-posedness providing a useful regularity upgrade for s ≥ 1.","major_comments":[{"comment":"Abstract: the global-existence statement is explicitly conditional on the assumption that 'interaction energies can be controlled by the kinetic energy,' yet the manuscript supplies neither a derivation of this control nor uniform bounds that hold for the full class of admissible interactions (external potentials, Hartree, subcritical powers) when s ∈ (0, 3/2).","section":"Abstract"},{"comment":"Global existence argument: the a priori energy estimates used to prevent finite-time blow-up close only if the interaction term is absorbed by the kinetic energy with a constant independent of the solution; without an explicit verification or a list of sufficient conditions on the potentials, the extension from local to global solutions remains formally conditional and cannot be applied directly to the stated problem class.","section":"Global existence section"}],"minor_comments":[{"comment":"Abstract: the interval for the Strichartz result is written s ∈ [1, 3/2); confirm whether the endpoint s = 3/2 is included and whether the fractional Laplacian is well-defined at that value.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive feedback. We agree that the conditional nature of the global existence result requires additional clarification with sufficient conditions on the interactions. We will revise the manuscript accordingly.","responses":[{"response":"We agree that the global existence is conditional on the stated assumption and that the manuscript does not derive uniform bounds for the entire class of interactions. In the revised version we will add a dedicated remark (or short subsection) listing sufficient conditions on external potentials, Hartree terms and subcritical powers that guarantee the interaction energies are controlled by the kinetic energy for s ∈ (0, 3/2). These conditions will be based on standard Sobolev and Gagliardo-Nirenberg inequalities and will cover the interactions commonly used in Kohn-Sham models.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the global-existence statement is explicitly conditional on the assumption that 'interaction energies can be controlled by the kinetic energy,' yet the manuscript supplies neither a derivation of this control nor uniform bounds that hold for the full class of admissible interactions (external potentials, Hartree, subcritical powers) when s ∈ (0, 3/2)."},{"response":"The a priori estimates close under the energy-control assumption precisely because the interaction term is absorbed with a solution-independent constant. We acknowledge that without explicit sufficient conditions the result cannot be applied directly to arbitrary admissible interactions. The revision will therefore include a list of concrete sufficient conditions on the potentials and nonlinearities, allowing the global existence statement to be used for standard choices arising in density-functional theory while keeping the general statement conditional as originally intended.","revision_made":"yes","referee_comment":"[Global existence section] Global existence argument: the a priori energy estimates used to prevent finite-time blow-up close only if the interaction term is absorbed by the kinetic energy with a constant independent of the solution; without an explicit verification or a list of sufficient conditions on the potentials, the extension from local to global solutions remains formally conditional and cannot be applied directly to the stated problem class."}],"tokens_in":1255,"tokens_out":451,"duration_ms":21880,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper proves local existence of weak solutions in H^s for the 3D time-dependent fractional Kohn-Sham equations with s in (0, 3/2), using an approximation that regularizes the nonlinearities. For s in [1, 3/2) it adds well-posedness via Strichartz estimates. Global extension follows from energy conservation provided the interaction energies (Hartree plus subcritical powers) stay controlled by the kinetic term.\n\nThe new part is carrying these standard tools over to the fractional dispersion operator paired with the listed interaction class. That combination had not been treated before in the integer-Laplacian literature the abstract cites, so the local result and the Strichartz range fill a gap.\n\nThe approximation procedure and the separation between weak local solutions and conditional global ones are handled cleanly. The Strichartz step for s >=1 looks routine once the fractional estimates are in place.\n\nThe soft spot is the control assumption itself. The abstract states it outright but gives no derivation or uniform bound for the 3D interactions at the stated s values. If the full paper only invokes the assumption without proving the constant stays finite, the global claim stays conditional and could fail for some admissible potentials. The local and Strichartz parts do not depend on it, so they stand on their own.\n\nThis is for people working on fractional dispersive equations or mathematical aspects of density-functional models. A reader already comfortable with Strichartz and approximation arguments will get concrete statements and ranges to compare against their own work.\n\nIt deserves a serious referee. The methods are appropriate and the statements are precise; the assumption just needs checking in review.","headline":"Local weak solutions for fractional Kohn-Sham via regularization, conditional global via energy control assumption, and Strichartz well-posedness for s >=1; the assumption is the main caveat.","tokens_in":2277,"tokens_out":422,"would_cite":false,"duration_ms":19037,"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":"Time-dependent fractional Kohn-Sham equations admit local weak solutions in H^s for s in (0, 3/2), with global extension when interaction energies are controlled by kinetic energy.","keywords":["fractional Kohn-Sham equations","time-dependent equations","weak solutions","local existence","global existence","Strichartz estimates","fractional Laplacian","nonlinear Schrödinger equations"],"falsifier":"An explicit interaction potential for which the interaction energy exceeds any multiple of the kinetic energy and produces a solution that blows up in finite time in the H^s norm.","tokens_in":2567,"feed_emoji":"","tokens_out":703,"duration_ms":17011,"temperature":0.7,"pith_summary":"The paper proves local existence of weak solutions to the time-dependent Kohn-Sham system in three dimensions when the kinetic term is replaced by the fractional operator (1-Δ)^s. An approximation scheme regularizes the nonlinear interaction terms to obtain these solutions in the Sobolev space H^s. When the interaction energies satisfy a bound relative to the kinetic energy, energy estimates prevent blow-up and yield global-in-time solutions. For the range s in [1, 3/2), Strichartz estimates further establish uniqueness and continuous dependence, giving well-posedness. These results address the well-posedness of fractional quantum evolution equations arising in density-functional models.","feed_headline":"Fractional Kohn-Sham equations have local weak solutions in H^s","feed_subtitle":"Local existence holds via regularization for s below 3/2; global solutions follow when interaction energy is controlled by kinetic energy.","key_machinery":"Regularization approximation of the nonlinearities to construct local weak solutions in H^s, combined with kinetic-energy control for global extension and Strichartz estimates for well-posedness when s ≥ 1.","core_discovery":"We prove the local existence of weak solutions in H^s using an approximation procedure regularizing the non-linearities. Assuming that the interaction energies can be controlled by the kinetic energy, we show that the solutions can be extended to global solutions using energy estimates. If s∈[1,3/2), we establish in addition the well-posedness of the time-dependent Kohn-Sham equations using Strichartz estimates.","pith_inferences":["The same control condition on interaction versus kinetic energy may be checkable for concrete Hartree or exchange potentials arising in atomic physics.","Well-posedness for s ≥ 1 opens the door to rigorous justification of time-dependent density-functional approximations that employ fractional dispersion.","The Strichartz-based uniqueness argument may extend to other dispersive regimes once suitable Strichartz estimates for the fractional operator are available."],"forward_implications":["Local-in-time weak solutions exist for every s in (0, 3/2) and every admissible interaction class.","Global solutions exist whenever the interaction energy is dominated by the kinetic energy.","For s in [1, 3/2) the initial-value problem is well-posed in H^s.","The same regularization-plus-energy-estimate strategy applies directly to related fractional nonlinear Schrödinger systems."],"fun_headline_variants":["Local H^s weak solutions for fractional time-dependent Kohn-Sham","Global extension of solutions when interaction energy controlled by kinetic","Well-posedness via Strichartz estimates for s in [1, 3/2)","Energy estimates extend fractional Kohn-Sham solutions to global time"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Interaction energies remain controlled by the kinetic energy so that energy estimates can prevent finite-time blow-up.","fun_headline_variants_meta":{"raw":{"variants":["Local H^s weak solutions for fractional time-dependent Kohn-Sham","Global extension of solutions when interaction energy controlled by kinetic","Well-posedness via Strichartz estimates for s in [1, 3/2)","Energy estimates extend fractional Kohn-Sham solutions to global time"]},"model":"grok-4.3","cost_usd":0.008174,"raw_usage":{"total_tokens":3593,"prompt_tokens":593,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":81740500,"prompt_tokens_details":{"text_tokens":593,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2927,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":593,"tokens_out":73,"duration_ms":18269,"temperature":1.0,"reasoning_tokens":2927,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T16:46:25.614268+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit interaction potential for which the interaction energy exceeds any multiple of the kinetic energy and produces a solution that blows up in finite time in the H^s norm.","supporting_citations":[],"review_version":1}