{"id":"ad381b33-92b4-4def-ba6d-debee6d76b29","arxiv_id":"2506.06621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence, spacetime bounds, eventual regularity and uniqueness are established for MHD and damped viscoelastic Navier-Stokes solutions in Wiener amalgam spaces, extending the Navier-Stokes theory.","lead":"This paper proves existence of mild and global weak solutions for the 3D MHD equations and the damped viscoelastic Navier-Stokes equations in Wiener amalgam spaces, which allow non-decaying initial data. It extends a framework previously developed for the Navier-Stokes equations to these coupled fluid systems, including eventual regularity and small-large uniqueness.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The viscoelastic Navier-Stokes theorems are asserted without proofs; the paper's central claim for both systems is therefore not established as written.","rationale":"The reader's verdict was CONDITIONAL, flagging two issues: convergence of the pressure expansion in the q<2 MHD case, and the omitted vNSEd proofs. I agree that the omitted vNSEd proofs are a load-bearing concern because Theorem 1.14 (global local energy solution for vNSEd in E2_q) is one of the paper's headline claims, and it is not proved anywhere. The paper explicitly defers the proofs, so a rigorous reader cannot verify these results from the manuscript. The MHD part appears plausible, and the analogy to vNSEd is strong, but the analogy is not a substitute for proof. Supplying the missing arguments would either resolve the concern or reveal a subtle obstruction in the column-sum structure. Therefore the verdict should remain CONDITIONAL: the paper can be accepted only if the viscoelastic proofs are provided and the internal reference errors are fixed. I do not see a clear fatal flaw in the MHD arguments, so I am not moving the verdict to REJECT or UNVERDICTED.","tokens_in":71518,"tokens_out":46400,"duration_ms":464668,"concrete_test":"Supply the full proof of Theorem 1.14 by adapting Sections 3.3.1 and 3.3.2 to vNSEd, explicitly checking the pressure expansion (1.32) and the local energy inequality (1.33) for the weak limit. In particular, verify that the column-wise estimates for the terms -2 sum_n ∫ (v·f_n)(f_n·∇φ) and for the far-field pressure containing sum_n f_n⊗f_n hold with constants independent of the regularization epsilon and the radius R. If any step requires an extra assumption such as div F = 0 or a modified localized system, that assumption must be stated and proved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper advertises and abstracts existence of mild and local energy solutions in Wiener amalgam spaces for both MHD and the viscoelastic Navier-Stokes equations with damping. For MHD, Theorems 1.1-1.3 and 1.8-1.10 are proved in detail. However, the vNSEd counterparts are explicitly not proved: Section 2 states 'The proofs of Theorems 1.4, 1.5, and 1.6 for (vNSEd) are omitted for brevity', and Section 3 states 'The details of verification of Theorems 1.12, 1.13, 1.14 for (vNSEd) are left to the readers.' The vNSEd system is not a trivial relabeling of MHD: F is a tensor with three columns, the nonlinearities in (1.18) and the local energy inequality (1.33) have sums over columns, and the pressure expansion (1.32) contains sum_n f_n⊗f_n. The weak-limit construction for q<2 in Section 3.3.2 relies on MHD-specific estimates and compactness for (v,b); transferring these to (v,F) requires verifying that the regularized, localized equations for vNSEd satisfy the same a priori bounds (3.18)-(3.20) with the column-sum pressure, and that the local energy inequality survives the diagonal limit. Without this verification, the stated central claim of the paper for the viscoelastic system is unsupported, regardless of how natural the analogy appears.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the three-dimensional incompressible MHD equations and the incompressible viscoelastic Navier–Stokes equations with damping (vNSEd) in Wiener amalgam spaces E^p_q. For the MHD system it proves: local well-posedness of mild solutions in subcritical spaces (Theorem 1.1), in critical spaces with small data (Theorem 1.2), and in critical spaces with sufficiently decaying data (Theorem 1.3); and for local energy weak solutions it proves eventual regularity (Theorem 1.8), an explicit growth-rate estimate for the local energy (Theorem 1.9), and time-global existence of local energy solutions for divergence-free data in E^2_q, 1≤q<∞ (Theorem 1.10). Theorems 1.4–1.6 and 1.12–1.14 state the analogous results for the vNSEd system, but their proofs are explicitly omitted in Sections 2 and 3. The MHD proofs adapt the semigroup, interpolation, and epsilon-regularity machinery of Bradshaw–Lai–Tsai and Bradshaw–Tsai to the coupled (v,b) system, with a separate construction for q≥2 via a perturbed system and for 1≤q<2 via localized regularized equations and a diagonal subsequence argument.","tokens_in":71833,"tokens_out":15142,"duration_ms":142645,"significance":"If the claims are correct, the paper extends the Wiener amalgam framework from the Navier–Stokes equations to two coupled fluid models, giving mild solutions with spacetime integral bounds and time-global local energy solutions for data in E^2_q. The MHD results are presented in considerable detail and rely on the previously published estimates of [1,3,4], which are adapted rather than re-derived; this is a strength because the main technical work is explicit. A serious limitation is that the vNSEd theorems, which are advertised in the abstract as part of the paper's central contribution, are not proved at all. Since the vNSEd system is not a literal relabeling of MHD (the velocity equation contains a sum over columns and the local energy inequality and pressure expansion have column-sum structure), the analogy argument needs a written verification before the viscoelastic claims can be accepted. With the MHD part alone the paper is a substantial progress report; with the vNSEd part it is currently incomplete.","major_comments":[{"comment":"The proofs of all six viscoelastic main theorems are omitted. Section 2 states 'The proofs of Theorems 1.4, 1.5, and 1.6 for (vNSEd) are omitted for brevity', and Section 3 states 'The details of verification of Theorems 1.12, 1.13, 1.14 for (vNSEd) are left to the readers.' This is not a harmless bookkeeping omission: the vNSEd velocity equation in (1.18) contains the sum ∑_n f_n⊗f_n, the pressure expansion (1.32) contains ∑_n f_n⊗f_n, and the local energy inequality (1.33) contains the column-sum term −2∑_n (v·f_n)(f_n·∇φ). The MHD estimates and the weak-limit construction in Section 3.3.2 are written for a single magnetic field b; transferring them to the tensor F requires at minimum verifying the analogous a priori bounds, the pressure decomposition, and the survival of the local energy inequality under the regularized limits for the column-sum structure. Since the abstract advertises existence results for both systems, these theorems are load-bearing and their proofs must be supplied or the claims must be restricted to the MHD system.","section":"§2 and §3, Theorems 1.4–1.6 and 1.12–1.14"},{"comment":"The linearized term L_t^(2)(v_ǫ,b_ǫ) in the integral equation (3.32) does not match the second equation of the perturbed system (3.30) or (3.34). The displayed formula contains both '+ b_ǫ⊗(η_ǫ*u)' and '− b_ǫ⊗(η_ǫ*u)', which cancel, and it omits the term '− v_ǫ⊗(η_ǫ*a)' that is required by the linearization. As written, the right-hand side of the b-equation reduces to (η_ǫ*u)⊗b_ǫ − (η_ǫ*a)⊗v_ǫ, missing both the b_ǫ⊗(η_ǫ*u) and v_ǫ⊗(η_ǫ*a) contributions that appear in (3.30). Since Lemma 3.9 is used in the q≥2 global-existence construction of Section 3.3.1, this incorrect formula needs to be corrected and the estimates below it rechecked against the correct integral equation.","section":"Lemma 3.9, Eq. (3.32)"},{"comment":"The diagonal-limit argument asserts that the local pressure expansion and the local energy inequality are inherited by the limiting solution, but the verification is compressed into the sentence 'The local energy inequality follows from the local energy equality for (v_k,b_k) and π_k, and (3.67), (3.68), (3.69), and π̂_n(x,t)=π(x,t)−c_n(t)'. This is a nontrivial step: the pressure representation (1.27) is defined at every scale, yet the compactness statements (3.67)–(3.68) are obtained on expanding balls B_n, and the far-field pressure tail is handled by choosing M large after fixing the scale. For a complete proof, the author should explain explicitly why the scale-by-scale pressure convergence in (3.68) is uniform enough to pass to the limit in the local energy inequality for all cylinders, rather than only on the fixed diagonal sequence. As written, the global existence claim for 1≤q<2 depends on this unstated uniformity.","section":"Theorem 1.10, proof for 1≤q<2, §3.3.2"}],"minor_comments":[{"comment":"In the paragraph after Definition 1.11, the set of local energy solutions is denoted 'N_MHD(v0,F0)' but should be 'N_vNSEd(v0,F0)'.","section":"Definition 1.11"},{"comment":"Theorem 1.3 says 'Instead of (1.25), if we assume ...' but equation (1.25) belongs to Theorem 1.6 for the viscoelastic system; the reference should be to condition (1.16) in the same theorem. In addition, (1.16) states 'm>p′' while (1.17) uses 'm≥p′'; please clarify whether this difference is intentional.","section":"Theorem 1.3, conditions (1.16) and (1.17)"},{"comment":"The displayed criterion in Lemma 3.2 contains a duplicated term: it reads '|v|^3 + |v|^3 + |π|^{3/2}' but should be '|v|^3 + |b|^3 + |π|^{3/2}'.","section":"Lemma 3.2, epsilon-regularity criterion"},{"comment":"The symbol '/BD' appearing in both displayed estimates is a corrupted artifact; the intended conditional indicator for the case q≤s should be typeset properly so the statement is unambiguous.","section":"Theorems 1.2 and 1.5, Eq. (1.14) and (1.23)"},{"comment":"In the first sentence of the proof of Lemma 3.10, the text says 'where c0 is given in Lemma 3.11'; the reference should be to Lemma 3.10, which is the lemma whose constants are being used.","section":"Section 3.3.1, Lemma 3.10"}],"recommendation":"major_revision","confidential_remarks":"The paper is an extension of the author's earlier work with Bradshaw and Tsai, and the heavy reliance on [1,3,4] is transparent and appropriate. The main issue for the editor is scope: the abstract and introduction promise theorems for the viscoelastic system that are not proved. If the author either supplies the missing vNSEd proofs or revises the title/abstract to claim only the MHD results, the MHD portion appears to be a solid contribution that could be accepted after the technical corrections in Lemmas 3.9 and the 1≤q<2 diagonal limit are addressed. The self-citation pattern is not problematic; the prior results are cited as published theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper transfers the Wiener amalgam framework from Navier-Stokes to MHD and to the damped viscoelastic Navier-Stokes equations. The MHD part is done in detail. Theorems 1.1-1.3 and 1.8-1.10 are proved with the expected semigroup, bilinear, and epsilon-regularity estimates. The local energy construction with the q>=2 perturbed-system argument and the separate 1<=q<2 localized/regularized argument is substantive. If the MHD proofs are correct, that is a genuine extension of the Bradshaw-Lai-Tsai framework to a coupled system, and the paper is worth having for that alone.\n\nThe soft spot is not subtle. Theorems 1.4-1.6 and 1.12-1.14 for the viscoelastic system are stated as main results, and the text says the proofs are 'omitted for brevity' or 'left to the readers.' That is not a trivial relabeling. F is a tensor with three columns; the local energy inequality and pressure expansion have sums over columns. The weak-limit construction for q<2 in Section 3.3.2 is MHD-specific in its estimates, and transferring it to (v,F) requires checking that the regularized localized equations satisfy the same a priori bounds with the column-sum pressure, and that the local energy inequality survives the diagonal limit. None of that is written down. So the paper's advertised central claim for the viscoelastic system is unsupported.\n\nThere are also small internal errors: Theorem 1.3 refers to (1.25), which belongs to Theorem 1.6, and Definition 1.11 writes NMHD(v0,F0) for the viscoelastic solution set. Minor, but they contribute to the sense that the paper was assembled quickly.\n\nOn the MHD side, the proof of the pressure convergence at every scale for 1<=q<2 is sketched rather than fully written. It looks plausible, but a referee should check it line by line.\n\nBottom line: The MHD results are a real contribution and the paper deserves a serious referee. But as it stands, it is not ready for publication in a journal that expects the stated theorems to be proved. I would send it to peer review with the clear instruction that the viscoelastic proofs must be supplied or those theorems removed. If the author is not willing to do that, the paper should be rejected.","headline":"The MHD half is a real contribution; the viscoelastic half is unproved, and the paper as a whole is not ready as stated.","tokens_in":72353,"tokens_out":3855,"would_cite":false,"duration_ms":37496,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q30","35Q35","35D30","76D05","76W05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves global weak solutions for MHD and damped viscoelastic flows in Wiener amalgam spaces, extending the Navier–Stokes theory.","keywords":["Wiener amalgam spaces","MHD equations","viscoelastic Navier-Stokes equations with damping","local energy solutions","mild solutions","eventual regularity","global existence","weak-strong uniqueness"],"falsifier":"For some $1 \\le q < 2$ and a fixed scale $R$, exhibit a sequence of regularized solutions $(v_k,b_k,\\pi_k)$ to (3.31) whose pressure term $\\hat{\\pi}^k_{x_0,R}$ fails to converge to $\\hat{\\pi}_{x_0,R}$ in $L^{3/2}(B_{2R}(x_0)\\times(0,T))$; this would break the local pressure expansion and refute Theorem 1.10 for that range. Alternatively, find divergence-free initial data in $E^2_q$ whose diagonal limit violates the local energy inequality on a single cylinder.","tokens_in":71283,"feed_emoji":"🌊","tokens_out":9642,"duration_ms":84607,"temperature":0.7,"pith_summary":"This paper extends the Wiener amalgam space theory for the Navier–Stokes equations to two coupled three-dimensional systems: incompressible magnetohydrodynamics (MHD) and the incompressible viscoelastic Navier–Stokes equations with damping. It aims to prove the existence of mild solutions for subcritical and critical data in these spaces, and, more substantially, to construct global-in-time local energy weak solutions for divergence-free initial data in $E^2_q$ with $1 \\le q < \\infty$, satisfying finite spacetime integral bounds on every finite time interval. If the construction is correct, large or non-decaying initial data whose local $L^2$ mass decays only in an $\\ell^q$ sense still produce global weak solutions, with eventual regularity for $1 \\le q \\le 3$. The same conclusions are asserted for the damped viscoelastic system, with proofs omitted by analogy.","feed_headline":"Global weak solutions found for MHD and damped viscoelastic flows","feed_subtitle":"Large non-decaying initial data still yield time-global local energy solutions for both coupled systems.","key_machinery":"The carrying object is the Wiener amalgam space $E^p_q$, defined by the norm $\\|f\\|_{E^p_q} = \\| \\|f\\|_{L^p(B_1(k))} \\|_{\\ell^q(k \\in \\mathbb{Z}^3)}$, which blends local integrability with a global $\\ell^q$ decay pattern; $E^p_\\infty$ is $L^p_{\\mathrm{uloc}}$. Two spacetime norms, $L^s_T E^p_q$ and $E^{s,p}_{T,q}$, convert the local-in-space bounds into global-in-time integral estimates. The weak-solution half of the paper is carried by the notion of a local energy solution, whose definition includes a local pressure expansion of the form $\\pi = -\\Delta^{-1}\\mathrm{div}\\,\\mathrm{div}[(v\\otimes v - b\\otimes b)\\chi_{4R}] + \\text{(far-field term)} + c_{x_0,R}(t)$ and a local energy inequality; these are the structures that must survive the limiting arguments. For $q \\ge 2$, the proof mechanism is perturbation and restarting; for $1 \\le q < 2$, it is a diagonal subsequence of solutions to the localized-regularized equations (3.31), with the pressure representation required to converge at every scale.","core_discovery":"The central discovery is that the Wiener amalgam framework carries over from the Navier–Stokes equations to the MHD system and to the damped viscoelastic Navier–Stokes system. For divergence-free $v_0,b_0 \\in E^2_q$ with $1 \\le q < \\infty$, Theorem 1.10 asserts the existence of a time-global local energy solution $(v,b)$ with an associated pressure $\\pi$ such that $\\|(v,b)\\|_{LE_q(0,T)} < \\infty$ for every finite $T$, and in particular $(v,b) \\in L^\\infty(0,T;E^2_q \\times E^2_q)$. The proof treats $q \\ge 2$ by perturbing the MHD equations and restarting at regular times, and $1 \\le q < 2$ by taking a diagonal subsequence of solutions to localized, regularized equations on expanding balls. Along the way the paper establishes eventual and initial regularity, an explicit growth rate for the localized energy, and a uniqueness theorem for data small at high frequencies. Theorem 1.14 states the analogous global existence result for the viscoelastic system with damping.","pith_inferences":["The same amalgam machinery would likely apply to other coupled parabolic systems whose nonlinearity is a divergence of a quadratic form, such as Boussinesq or Oldroyd-B type models, provided an $\\epsilon$-regularity criterion is available.","The $1 \\le q < 2$ case rests on a delicate diagonal pressure argument; a uniform-in-scale pressure convergence lemma would replace the case-by-scale check and make the existence proof more transparent.","The viscoelastic system is handled entirely by analogy, so the claimed global existence is conditional on that analogy; an explicit verification of the local pressure expansion for the vNSEd system would remove the gap.","If the theorems hold for every $q < \\infty$, the boundary case $q = \\infty$ (locally square-integrable data with no global decay) remains the natural place to test whether the method can be pushed further."],"forward_implications":["For any divergence-free initial data in $E^2_q$ with $1 \\le q < \\infty$, the MHD equations admit a local energy solution that exists for all positive times and has finite $\\ell^q$ local energy on every finite interval.","For $1 \\le q \\le 3$, every such MHD solution becomes regular after a finite time, with a $t^{1/2}$ bound on the $L^\\infty$ norm at large times; the same is asserted for the damped viscoelastic system.","Small critical data in $E^3_q$ give unique mild solutions with the expected spacetime integral bounds, so data that do not decay at infinity are still well-posed in an appropriate sense.","Local energy solutions of MHD are unique when the initial data are small at high frequencies, and this implies short-time uniqueness for data in $E^3$."],"supporting_citations":[{"why":"supplies the Picard iteration, linear and bilinear estimates, and spacetime integral bounds for mild solutions in Wiener amalgam spaces that the paper adapts to MHD and vNSEd.","marker":"[1]"},{"why":"supplies the local energy solution framework in Wiener amalgam spaces, including the LE_q norm and the global existence strategy for Navier-Stokes that is extended here.","marker":"[4]"},{"why":"supplies eventual regularity, explicit growth rate, and uniqueness methods for Navier-Stokes local energy solutions that are adapted in Sections 3.1-3.3.","marker":"[3]"},{"why":"supplies the diagonal subsequence and pressure convergence technique used in the 1 <= q < 2 case.","marker":"[23]"},{"why":"supplies the epsilon-regularity criterion for suitable weak MHD solutions used to prove initial and eventual regularity.","marker":"[34]"},{"why":"supplies the epsilon-regularity criterion for the damped viscoelastic system that underlies the analogous regularity claims.","marker":"[17]"},{"why":"introduces the damped viscoelastic Navier-Stokes model with the damping term that the paper studies.","marker":"[29]"},{"why":"establishes the foundational global weak solutions for finite-energy MHD that the amalgam-space result extends.","marker":"[11]"}],"fun_headline_variants":["Large-data global solutions in Wiener amalgam spaces for MHD and viscoelastic","Wiener amalgam method yields global MHD and damped viscoelastic solutions","Global weak and mild solutions for MHD and viscoelastic in Wiener amalgam","MHD and viscoelastic flows solved globally in Wiener amalgam spaces","Wiener amalgam spaces deliver global solutions for MHD and viscoelastic flows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires that the local pressure formula and the local energy inequality pass unchanged to the weak limits built in Section 3.3 at every ball scale, and it assumes the viscoelastic system behaves exactly like MHD even though those proofs are omitted.","fun_headline_variants_meta":{"raw":{"variants":["Large-data global solutions in Wiener amalgam spaces for MHD and viscoelastic","Wiener amalgam method yields global MHD and damped viscoelastic solutions","Global weak and mild solutions for MHD and viscoelastic in Wiener amalgam","MHD and viscoelastic flows solved globally in Wiener amalgam spaces","Wiener amalgam spaces deliver global solutions for MHD and viscoelastic flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000239,"raw_usage":{"total_tokens":1518,"prompt_tokens":954,"completion_tokens":564,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":460}},"tokens_in":570,"tokens_out":564,"duration_ms":5355,"temperature":1.0,"reasoning_tokens":460,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:53:07.229947+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For some $1 \\le q < 2$ and a fixed scale $R$, exhibit a sequence of regularized solutions $(v_k,b_k,\\pi_k)$ to (3.31) whose pressure term $\\hat{\\pi}^k_{x_0,R}$ fails to converge to $\\hat{\\pi}_{x_0,R}$ in $L^{3/2}(B_{2R}(x_0)\\times(0,T))$; this would break the local pressure expansion and refute Theorem 1.10 for that range. Alternatively, find divergence-free initial data in $E^2_q$ whose diagonal limit violates the local energy inequality on a single cylinder.","supporting_citations":[{"cited_title":"Bradshaw, C.-C","cited_arxiv_id":null,"evidence_quote":"supplies the Picard iteration, linear and bilinear estimates, and spacetime integral bounds for mild solutions in Wiener amalgam spaces that the paper adapts to MHD and vNSEd."},{"cited_title":"Bradshaw and T.-P","cited_arxiv_id":null,"evidence_quote":"supplies the local energy solution framework in Wiener amalgam spaces, including the LE_q norm and the global existence strategy for Navier-Stokes that is extended here."},{"cited_title":"Bradshaw and T.-P","cited_arxiv_id":null,"evidence_quote":"supplies eventual regularity, explicit growth rate, and uniqueness methods for Navier-Stokes local energy solutions that are adapted in Sections 3.1-3.3."},{"cited_title":"Kwon and T.-P","cited_arxiv_id":null,"evidence_quote":"supplies the diagonal subsequence and pressure convergence technique used in the 1 <= q < 2 case."},{"cited_title":"Mahalov, B","cited_arxiv_id":null,"evidence_quote":"supplies the epsilon-regularity criterion for suitable weak MHD solutions used to prove initial and eventual regularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the epsilon-regularity criterion for the damped viscoelastic system that underlies the analogous regularity claims."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the damped viscoelastic Navier-Stokes model with the damping term that the paper studies."},{"cited_title":"Duvaut and J.-L","cited_arxiv_id":null,"evidence_quote":"establishes the foundational global weak solutions for finite-energy MHD that the amalgam-space result extends."}],"review_version":1}