{"id":"d2678255-0c53-4f2f-89d4-c1a71ee17b32","arxiv_id":"2507.02338","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For large tailored forces, the forced two-dimensional Navier-Stokes equations in the half space admit at least two distinct mild solutions starting from zero, one of which is a self-similar flow pinned to the boundary.","lead":"This paper proves that the two-dimensional forced Navier-Stokes equations in a half space with a no-slip wall can have more than one solution for the same force and zero initial data. The second solution is produced by a self-similar flow that concentrates at the wall as time starts, which forces a boundary-layer analysis that earlier constructions could avoid.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The alpha^(-1/4) boundary-corrector estimate (58) in Prop. 4.1 is the load-bearing premise for Theorem 1.3; its proof omits the explicit alpha-scaling of the Helmholtz-projection terms, so a direct verification is needed.","rationale":"The reader correctly identifies inequality (58) as the load-bearing premise for Theorem 1.3, and I agree that the boundary-layer corrector is the part of the argument where a hidden scaling error would break the resolvent convergence. However, a spot check of the scaling suggests the estimate may be correct: the ansatz gives ||grad^2 J||_L2 ~ alpha^(3/4), so alpha^(-1)||Delta J||_L2 ~ alpha^(-1/4), and since P_sigma is L2-bounded, ||P_sigma Delta J||_L2 inherits the same scaling. The nonlocal pressure removed by the projection is not part of P_sigma Delta J, so I did not find a definite contradiction. The concern is that this critical scaling is asserted rather than demonstrated, and the uniform-in-lambda and uniform-in-alpha constants in (66), (70), and (72) are not fully expanded. Because the theorem's central claim depends on this estimate, the paper deserves a conditional verdict with a verification step, rather than outright acceptance. The reader's CONDITIONAL verdict is therefore retained unchanged.","tokens_in":30726,"tokens_out":54643,"duration_ms":601340,"concrete_test":"Take the boundary-layer ansatz (62) with a Schwartz function h and compute explicitly the alpha-scaling of ||f||_L2 = || -lambda J[h] + alpha^(-1)(P_sigma Delta + xi/2 . grad + 1/2)J[h] ||_L2 for alpha = 2^k, k large, uniformly in lambda on the contour partial B_epsilon(lambda_E). In particular, evaluate the nonlocal pressure contribution to P_sigma Delta J[h] via the Fourier representation of the Helmholtz projection in the half space and check whether ||P_sigma Delta J[h]||_L2 is O(alpha^(3/4))||h||_Z. If the scaling is instead O(alpha), then inequality (66) fails, the alpha^(-1/4) bound in (58) is false, and the resolvent convergence in Theorem 1.3 must be reconsidered.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.3 reduces non-uniqueness to resolvent convergence of the no-slip linearized operator to the linearized Euler operator as alpha tends to infinity. The decisive step is Proposition 4.1, inequality (58): the boundary-layer corrector v_bc has L2 norm O(alpha^(-1/4))||g||. In the proof, v_bc = v_app[(I+Psi)^(-1)h], where v_app = J[h] + R[h] + v_slip^(2)[h]. The bound (66) for R[h] is obtained by applying Lemma 2.2 to f = -lambda J[h] + alpha^(-1)(P_sigma Delta + xi/2 . grad + 1/2)J[h]. The key hidden point is that J[h] has nonzero tangential trace at the wall, so P_sigma Delta J[h] is not a local differential expression: the Helmholtz projection removes a harmonic pressure whose boundary data comes from the normal component of Delta J[h]. The paper controls this only by saying 'where (64) is used', but (64) is a pointwise boundary-layer bound for J and its derivatives; it does not explicitly track the nonlocal pressure contribution. If the pressure term contributes at order O(alpha) in L2 (rather than O(alpha^(3/4)) after projection), then ||f||_L2 would be O(1) rather than O(alpha^(-1/4)), and the decay in (58) would fail. Uniform-in-lambda bounds for all terms are also asserted without full expansion. This is therefore the single most load-bearing estimate in the paper: every later convergence statement in (74)-(75) and the contradiction argument of Theorem 1.3 depends on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves non-uniqueness of mild solutions to the forced two-dimensional Navier-Stokes equations in the half space with no-slip boundary condition, following the Albritton–Brué–Colombo program. The force is chosen as F_alpha = alpha (∂_t u_E - Δ u_E), where u_E is the self-similar scaling of a compactly supported, linearly unstable stationary Euler flow U_E located away from the boundary. One solution is the explicit self-similar flow alpha u_E; the other is constructed from an unstable eigenmode of the linearized operator L_alpha around alpha U_E. The key analytical step is Theorem 1.3, which shows that L_alpha inherits an unstable eigenvalue from the Euler linearization as alpha → ∞. This is obtained by a resolvent comparison: the no-slip linearized Navier-Stokes resolvent is shown to converge to the linearized Euler resolvent, with the discrepancy corrected by a boundary layer whose L2 norm is O(alpha^{-1/4}). A concrete example of an unstable U_E is constructed in Section 6 by reflecting a Vishik-type unstable vortex across the boundary. The paper is clearly written, follows a well-established strategy, and makes a serious attempt to handle the genuinely new difficulty caused by the boundary.","tokens_in":31042,"tokens_out":24770,"duration_ms":282382,"significance":"If the proof is completed, the result would be a meaningful extension of the non-uniqueness program to a setting where the self-similar profile concentrates at the boundary at the initial time, forcing genuine boundary-layer effects. The paper contains several strong elements: an explicit boundary-layer ansatz, resolvent estimates with uniform-in-alpha constants, and a clean Riesz-projection contradiction argument. The borrowing of the unstable Euler flow from Vishik and ABC is honest and clearly attributed. The main unresolved issue is a missing verification in the boundary-layer estimate; it appears fixable and does not undermine the overall strategy. The result is significant for the theory of non-uniqueness of the forced Navier-Stokes equations in domains with boundary.","major_comments":[{"comment":"The estimate (66) is the only justification for the boundary-layer bound (58), but it is not derived in the text: applying Lemma 2.2 to f = -λ J[h] + α^{-1}(P_σ Δ + ξ/2·∇ + 1/2)J[h] requires an estimate for P_σ Δ J[h], and the displayed use of (64) controls only Δ J[h] and its derivatives. Since J[h] is divergence-free, P_σ Δ J[h] = Δ J[h] - ∇p, where p is the harmonic function with ∂_2 p = (Δ J[h])_2|_{ξ_2=0}. For the ansatz (62) this boundary datum is of size 2 α^{1/2} ∂_1 h, so a direct computation gives ∥∇p∥_{L^2} ≲ α^{1/2} ∥h∥_Z. Thus α^{-1}∥P_σ Δ J[h]∥_{L^2} is indeed O(α^{-1/4}), but this verification is absent and does not follow from (64) as written. Because (58) is the load-bearing input for the convergence in (75) and the contradiction in Theorem 1.3, the proof must supply this pressure estimate explicitly.","section":"§4, Prop. 4.1, Eq. (66)"},{"comment":"The sentence 'Since possible spectrum of L_{α_n} in B_{ε_0}(λ_E) consists only of isolated eigenvalues' is asserted before any spectral analysis of L_α is available in the paper. The essential-spectrum and compactness argument that would justify it appears only later, in Proposition 4.2, where e^{τ α L_α} is shown to be a compact perturbation of e^{τ H}. The proof of Theorem 1.3 should be reordered so that this discreteness statement is established before the Riesz-projection argument, or a short lemma should be added before Eq. (74).","section":"§4, proof of Theorem 1.3, Eq. (74)"},{"comment":"The construction of the unstable Euler profile in the half space is not quite complete. Proposition 6.2 asserts without proof that λ_∞ remains isolated in L^2(B_{R_0}(0)), and Proposition 6.3 only proves that the spectral projection P_R^r is nonzero. A nonzero spectral projection implies that there is spectrum inside the contour, but not that it is an isolated eigenvalue; Assumption 1.1 requires an isolated eigenvalue of Λ_E. The authors should either show directly that the spectral subspace is finite-dimensional (for example by a Fredholm or meromorphic resolvent argument using the smallness of U_R) or construct the eigenfunction by a perturbation argument around the known eigenfunction of Λ̃.","section":"§6, Props. 6.2 and 6.3"}],"minor_comments":[{"comment":"The boundary-layer ansatz is printed inconsistently: the first line uses e^{-Ξ_2} while the integral uses e^{-η^2}. As printed, the identity ∫_0^∞ (1-η^2)e^{-η^2} dη = 0 is false, and the stated divergence-free property of J[h] does not follow. Please state the profile consistently; the standard choice (1-η)e^{-η} works.","section":"§4, Eqs. (62)-(63)"},{"comment":"There are several typographical errors: 'Lerey-Hopf' in the Introduction, 'unqiue' in Section 5, 'intorudce' in Proposition 4.1, 'formua' in the proof of Theorem 3.1, and 'complepte' in Appendix A.","section":"Throughout"},{"comment":"The force F_α(t) behaves like α t^{-1} in L^2 near t=0, so it is not integrable up to t=0; the mild formulation in Definition 1.1 only uses the equation for 0 < s < t, so this is consistent, but the paper should state explicitly that the forcing is considered on (0,∞) and need not be integrable at t=0.","section":"§5"},{"comment":"The proof of Proposition 4.2 uses boundedness of the Helmholtz projection on the weighted space L^2(⟨ξ⟩^{1/4}), citing [9, Theorem 2.4]; please verify that the cited theorem covers the half-space with this weight and the full Stokes semigroup setting, since the domain is unbounded.","section":"§4, Prop. 4.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is well within scope and the overall strategy is sound: the missing pressure estimate in Proposition 4.1 is local and appears routine to fill, and the spectral ordering issue can be repaired by reorganization. The Section 6 construction also needs a small but genuine completion. I would not recommend rejection; these are fixable load-bearing gaps."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is solid and worth engaging with. What is actually new is the half-space geometry with no-slip boundary and a self-similar flow whose support collapses onto the wall at t=0. That forces a genuine boundary-layer analysis, which is the technical core here: the explicit corrector (62), the alpha^{-1/4} decay estimate in Proposition 4.1, and the resolvent-convergence contradiction in Theorem 1.3. This is a real step beyond the earlier ABC constructions, where the unstable vortex stayed away from the boundary and the wall was a spectator.\n\nThe paper is also honest about what it borrows. The force is chosen exactly to make the self-similar flow an explicit solution, and the instability input comes from Vishik/ABC; the authors say so. The construction of the unstable Euler profile in Section 6 is detailed and self-contained enough to check, including the odd reflection and the convergence of the remainder operator U_R.\n\nI looked specifically at the stress-test worry about the Helmholtz projection in inequality (66). That concern does not land. For the ansatz (62), J is divergence-free, J_2 vanishes at the wall, and a direct computation gives ∂_2^2 J_2 = -α^{1/2} F''(Ξ)∂_1 h with F''(0)=0; so ΔJ_2 also vanishes at the wall. Since ΔJ is divergence-free and has zero normal trace, P_σ ΔJ = ΔJ with no harmonic pressure correction. The estimate (66) then follows from the pointwise boundary-layer bounds in (64). The alpha^{-1/4} rate is doing real work, but the pressure term is not hiding an extra alpha.\n\nThe soft spots are real but not fatal. Theorem 1.2 is not proved in the paper: the nonlinear construction is dismissed as \"almost parallel\" to ABC Section 4. That is probably true, but given the audience and the importance of the claim, the authors should either write out the fixed-point argument in the self-similar variables or relegate a complete version to an appendix. There are also scattered typos and a few estimates in Appendix B that are sketched rather than fully expanded. Minor stuff.\n\nI would send this to a serious referee. The main theorem is significant if true, the linear analysis is checkable, and the one place I was suspicious turned out to be fine. The referee should focus on the deferred nonlinear argument and on the uniformity in lambda in Proposition 4.1, not on the boundary-layer pressure term.","headline":"A serious, carefully built extension of the ABC non-uniqueness program to the half-space with the singular vortex pinned at the boundary; the load-bearing boundary-layer estimate looks right after checking the pressure term, but the nonlinear half is still deferred.","tokens_in":31660,"tokens_out":5270,"would_cite":true,"duration_ms":57994,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A02","35B30","35Q30","35Q35","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A forced half-space flow has two solutions from one zero initial state","keywords":["non-uniqueness","Navier-Stokes equations","half-space","mild solutions","self-similar vorticity","no-slip boundary condition","boundary layer","linear instability"],"falsifier":"A numerical check would settle the spectral transfer: for a concrete truncated unstable profile $U_E$, compare the exact resolvent solution $(\\lambda I-L_\\alpha)^{-1}K[g]$ with the slip solution plus the boundary-layer formula (62), and compute the eigenvalues of a discretized $L_\\alpha$ for growing $\\alpha$; if $\\|v_{\\mathrm{bc}}\\|$ does not decay like $\\alpha^{-1/4}$ or no eigenvalue converges to a positive-real-part point, the mechanism fails.","tokens_in":30447,"feed_emoji":"🌀","tokens_out":12051,"duration_ms":129378,"temperature":0.7,"pith_summary":"The paper proves that the two-dimensional forced Navier-Stokes equations in a half-space with the no-slip boundary condition (the fluid velocity is zero at the wall) and zero initial data can have more than one mild solution, a velocity that satisfies the integrated form of the equations and tends weakly to zero at the initial time. The force is chosen as a large multiple $\\alpha$ of the linearized Euler flow, large enough to sit above the scale-critical uniqueness threshold. One solution is the explicit self-similar flow $\\alpha u_E$; the other is built from an unstable eigenmode of the linearized operator around $\\alpha u_E$, a mode that grows in self-similar time while the whole velocity still tends weakly to zero at $t=0$. The new difficulty is that the self-similar vorticity shrinks toward the boundary as $t\\to 0$, so restoring the no-slip condition requires a boundary-layer correction, and the central technical step is to show that this correction is small when $\\alpha$ is large. If the proof is right, non-uniqueness survives in the flat half-space with the boundary at the center of the singularity, rather than only for vortex locations kept at finite distance from the wall.","feed_headline":"Forced half-space flow: two solutions from one zero initial state","feed_subtitle":"A near-boundary self-similar vortex turns unstable, giving two different flows from the same zero start.","key_machinery":"The load-bearing mechanism is the resolvent problem for the linearized operator $L_\\alpha$ in self-similar variables, together with a boundary-layer corrector that reconciles the slip-type linearized Euler flow with the no-slip condition. The perfect-slip part is $v_{\\mathrm{slip}}=K[(\\lambda I-M_\\alpha)^{-1}g]$, built from the linearized vorticity operator $M_\\alpha$ with $\\omega|_{\\xi_2=0}=0$. The no-slip part is $v_{\\mathrm{bc}}$ obtained from the explicit boundary-layer ansatz (62), and its size is controlled by the key estimate (58), $\\|v_{\\mathrm{bc}}[g;\\lambda,\\alpha]\\|_{L^2}\\le C\\alpha^{-1/4}\\|g\\|_{L^2}$. Because this corrector vanishes as $\\alpha\\to\\infty$, the resolvent $(\\lambda I-L_\\alpha)^{-1}K[g]$ converges to the Euler resolvent $K[(\\lambda I-\\Lambda_E)^{-1}g]$; a spectral-projection contradiction then forces the viscous operator to inherit the unstable eigenvalue.","core_discovery":"The central claim is Theorem 1.2: take a compactly supported stationary Euler flow $U_E$ in the half-space whose linearized vorticity operator $\\Lambda_E$ has an isolated eigenvalue $\\lambda_E$ with $\\Re(\\lambda_E)>0$, set $u_E(t,x)=t^{-1/2}U_E(x/t^{1/2})$, and force the system with $F_\\alpha=\\alpha(\\partial_t u_E-\\Delta u_E)$. Then there is $\\alpha_E\\ge 1$ such that every $\\alpha\\ge\\alpha_E$ yields at least two distinct mild solutions. The explicit one is $\\alpha u_E$ itself. The second is produced in self-similar variables as a perturbation along the unstable mode $e^{\\alpha\\lambda_\\alpha\\tau}V_{\\mathrm{unst}}$, with a remainder $w$ found by a contraction map on a weighted space. The proof rests on Theorem 1.3, which transfers the Euler instability to the viscous problem: the linearized no-slip operator $L_\\alpha$ has an isolated eigenvalue $\\lambda_\\alpha$ arbitrarily close to $\\lambda_E$ once $\\alpha$ is large.","pith_inferences":["Beyond the paper, the same resolvent-plus-boundary-layer strategy suggests the non-uniqueness should be stable under small perturbations of the flat boundary, at least as long as the profile's scaled distance to the wall vanishes in a controlled way.","A natural testable extension is to probe whether the $\\alpha^{-1/4}$ rate is sharp; improving the boundary-layer ansatz might transfer instability under weaker separation between vortex and wall, while a slower rate would mark the limit of this construction.","The paper leaves open whether a similar mechanism can produce non-uniqueness without an external force, since the force here is what plants the unstable self-similar profile; showing the same boundary-layer instability can be triggered internally would be a genuinely different result."],"forward_implications":["For every $\\alpha\\ge\\alpha_E$, the half-space problem with $F=F_\\alpha$ and zero initial data has at least two distinct mild solutions, so the small-data uniqueness theorem is false for large forces in this geometry.","The second solution is not an artifact of the boundary condition: it is driven by an unstable mode of the linearized operator around $\\alpha u_E$, and it differs from $\\alpha u_E$ at every later time while sharing the same weak zero limit at $t=0$.","The transfer of instability works even though the vortex's distance to the wall is $O(\\sqrt{t})$ at time $t$, because the boundary-layer part of the resolvent decays as $\\alpha^{-1/4}$.","Theorem 1.1 provides concrete profiles satisfying Assumption 1.1, so the main theorem is realized by an explicit family of unstable flows rather than a conditional statement.","For small $\\alpha$, the scale-critical norms are small and the solution is unique, so the threshold $\\alpha_E$ is a genuine point where uniqueness is lost as the force amplitude grows."],"supporting_citations":[{"why":"Supplies the non-uniqueness program, the truncation of an unstable vortex, and the nonlinear contraction argument that Theorem 1.2 adapts to the half-space.","marker":"[1]"},{"why":"Introduces the strategy of obtaining non-uniqueness from linear instability of a self-similar profile.","marker":"[8]"},{"why":"Gives the boundary-layer separation principle for initial vorticity away from the wall that motivates the $\\alpha^{-1/4}$ correction in Proposition 4.1.","marker":"[12]"},{"why":"Provides the unstable Euler vortex construction whose truncation gives the half-space profile $U_E$.","marker":"[17]"},{"why":"Provides the companion instability result for the Euler vortices used in Section 6.","marker":"[18]"},{"why":"Supplies the semigroup spectral theory used to equate the spectral bound with the growth bound in Proposition 4.2.","marker":"[5]"},{"why":"Supplies the weighted harmonic-oscillator estimates used throughout the resolvent and energy estimates.","marker":"[6]"},{"why":"Supplies weighted Stokes semigroup estimates used for the semigroup bounds and compactness in Section 4.","marker":"[9]"}],"fun_headline_variants":["Unstable boundary vortex spawns twin Navier-Stokes flows","Two distinct flows from same zero start in half-space","Near-boundary vortex breaks uniqueness in forced Navier-Stokes","High Reynolds boundary layer yields multiple solutions","Self-similar vortex near wall causes non-unique flows"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the boundary-layer correction restores the no-slip condition with an $L^2$ norm no larger than a constant times $\\alpha^{-1/4}$, and the derivation of this rate uses the profile's support staying a positive distance away from the wall; if the wall correction decays more slowly, the Euler instability may not transfer to the viscous operator.","fun_headline_variants_meta":{"raw":{"variants":["Unstable boundary vortex spawns twin Navier-Stokes flows","Two distinct flows from same zero start in half-space","Near-boundary vortex breaks uniqueness in forced Navier-Stokes","High Reynolds boundary layer yields multiple solutions","Self-similar vortex near wall causes non-unique flows"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000423,"raw_usage":{"total_tokens":2153,"prompt_tokens":905,"completion_tokens":1248,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":1170}},"tokens_in":521,"tokens_out":1248,"duration_ms":10025,"temperature":1.0,"reasoning_tokens":1170,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:33:27.542332+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A numerical check would settle the spectral transfer: for a concrete truncated unstable profile $U_E$, compare the exact resolvent solution $(\\lambda I-L_\\alpha)^{-1}K[g]$ with the slip solution plus the boundary-layer formula (62), and compute the eigenvalues of a discretized $L_\\alpha$ for growing $\\alpha$; if $\\|v_{\\mathrm{bc}}\\|$ does not decay like $\\alpha^{-1/4}$ or no eigenvalue converges to a positive-real-part point, the mechanism fails.","supporting_citations":[{"cited_title":"Albritton, E","cited_arxiv_id":null,"evidence_quote":"Supplies the non-uniqueness program, the truncation of an unstable vortex, and the nonlinear contraction argument that Theorem 1.2 adapts to the half-space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the strategy of obtaining non-uniqueness from linear instability of a self-similar profile."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the boundary-layer separation principle for initial vorticity away from the wall that motivates the $\\alpha^{-1/4}$ correction in Proposition 4.1."},{"cited_title":"Engel, R","cited_arxiv_id":null,"evidence_quote":"Supplies the semigroup spectral theory used to equate the spectral bound with the growth bound in Proposition 4.2."},{"cited_title":"Gallay, C","cited_arxiv_id":null,"evidence_quote":"Supplies the weighted harmonic-oscillator estimates used throughout the resolvent and energy estimates."},{"cited_title":"Kobayashi, T","cited_arxiv_id":null,"evidence_quote":"Supplies weighted Stokes semigroup estimates used for the semigroup bounds and compactness in Section 4."}],"review_version":1}