{"id":"4570ebdb-0d52-481f-8763-2f20d04b35b9","arxiv_id":"2608.11194","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"A real scalar 1/2-Hölder coefficient arbitrarily close to 1 can destroy maximal L2-regularity for divergence-form parabolic equations, confirming the Auscher-Egert conjecture.","lead":"This paper finds a diffusion coefficient that is only 1/2-Hölder continuous in time yet still breaks the standard maximal-regularity guarantee for parabolic equations. The counterexample settles the sharp endpoint version of Lions' problem for scalar divergence-form operators.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's weakest assumption, the lacunary orthogonality of span{p_K,c_K} across factor-16 frequencies, is indeed the structural hinge that makes the divergent block contributions add instead of cancel. That assumption is explicitly proved in Lemma 2.1 by disjointness of the frequency sets {K-2,K+2} and {16K-2,16K+2}, so it is secure. I checked the other delicate points: the C^{0,1/2} estimate is valid even when h is so small that the cutoff frequency falls below K_{j,1}, since then K_{j,1}^{-1}=ell_j/16 bounds the tail and h^{1/2} >= ell_j/16; the forcing decomposition (5.4) has both R and epsilon L_beta v continuous in H with uniformly vanishing tails; and the L^2-lower-bound computation on J_j is rigorous, with the harmonic series divergence following from M_j ell_j = 1/(16(j+1)). The zero-extension and parabolic-localisation steps also respect the stated support and trace conditions. I therefore find no concrete defect and no reason to change the reader's ACCEPT verdict. A symbolic re-derivation of (5.2) is the most worthwhile sanity check, since the whole H-continuity of the forcing rests on that cancellation.","tokens_in":16687,"tokens_out":37386,"duration_ms":350588,"concrete_test":"Recompute the cancellation identity (5.2) symbolically for a generic lacunary frequency K: substitute y_{j,m} from (4.2) and the identities (2.5), (2.6) into (partial_t + A_{0,H})(y p_K) + epsilon L(eta K^{-1} sin theta cos(Kx))(rho phi). If the p_K coefficients do not cancel exactly, the claimed H-continuity of the forcing f would fail and the counterexample would collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central construction is internally consistent: Lemma 2.1 proves the lacunary orthogonality directly from disjoint Fourier frequencies; identities (2.5) and (2.6) combine with the amplitude choice (4.2) so that the leading p_K terms cancel exactly in (5.2), leaving H-norm remainders O(epsilon ell_j); the quadratic term epsilon L_beta v is bounded by O(epsilon^2/(j+1)); and the lower bound in Lemma 4.1 produces a divergent harmonic series. I traced the C^{0,1/2} bound through the geometric-frequency split and the cross-block argument and found no gap. The proof never needs cross-block orthogonality because the time supports are disjoint and each one-block estimate is uniform. I did not find an unverified estimate or a circular step in the main theorem; the stated limitations in Section 7 are comparisons after the fact, not inputs to the construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper gives a counterexample to Lions' maximal L^2-regularity problem for divergence-form operators on a bounded interval: for every δ>0 it constructs a real uniformly elliptic coefficient a∈C^{0,1/2}([0,1];L∞(0,π)) with ||a−1||_{C^{0,1/2}}<δ, and an H-valued continuous forcing f, such that the unique Lions variational solution has ∂_t u∉L^2(0,1;H). The construction superposes lacunary oscillatory modes p_K=sin(Kx)sin(2x) on shrinking time blocks, with temporal frequency K^2 and amplitude K^{-1}; a scalar corrector y_{j,m} cancels the leading p_K contributions so that the forcing is continuous in H, while the norm of ∂_t u on the blocks contains a divergent harmonic series. The authors then extend the construction by zero to the real line, tensorise it to R^d, and localise it by parabolic rescaling to arbitrary bounded domains, and they compare the coefficient with the known endpoint sufficient conditions in the literature.","tokens_in":16805,"tokens_out":24136,"duration_ms":200409,"significance":"If the construction is correct, it settles the endpoint question for Lions' maximal regularity in the divergence-form setting: C^{0,1/2} time regularity alone is insufficient, even for real scalar coefficients that are arbitrarily small perturbations of the identity. This confirms Auscher–Egert's conjecture and sharpens earlier abstract counterexamples by Fackler and Dier. The proof is explicit and self-contained; the key identities (Lemma 2.1), the cancellation (5.2), and the divergent block sum (Lemma 4.1) are verified with estimates that carry no fitted parameters. The passage to R^d and to bounded domains is a genuine extension of the interval construction, not a formal modification, and the comparison with sufficient hypotheses in Section 7 clarifies the precise position of the example.","major_comments":[],"minor_comments":[{"comment":"The proof of the C^1 regularity of v cites [2, Chap. V, §2, Thm. 2.8, p. 373] for the termwise differentiation of a uniformly convergent series of V'-valued functions; this reference appears to be an ordinary analysis textbook rather than a Bochner-space reference, and the pagination may not correspond to the cited edition. The argument itself is standard, but the citation should be checked and, if necessary, replaced by a standard reference on vector-valued differentiation.","section":"§4, Lemma 4.1"},{"comment":"In the proof of the 1/2-Hölder estimate, the step 'Ch+Ch^{1/2} ≤ Ch^{1/2}' uses h≤ℓ_j≤1 implicitly; making this explicit would improve readability, especially in the cross-block case where the same inequality is used with h>ℓ_j.","section":"§3, Proposition 3.1"},{"comment":"The exclusions in Corollary 7.1 are all proved by contradiction from known positive results, which is logically sufficient. A short direct indication of why the constructed coefficient fails the Auscher–Egert square condition (7.1), for instance by the same Fourier-coefficient argument as in Proposition 3.2, would make the comparison more transparent for readers who do not immediately see the divergence of the integral.","section":"§7, Corollary 7.1"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically convincing; the only issues I found are presentation-level, with the reference in Lemma 4.1 being the main item to correct. The unusual AI-use disclosure is handled openly; if the journal has a specific disclosure policy, the authors should make sure the current declaration conforms to it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper does what it says. It gives the first counterexample to maximal L2-regularity for real scalar divergence-form operators with C^{0,1/2} time dependence, in one dimension, with coefficient arbitrarily close to the heat coefficient. That settles the Auscher–Egert conjecture for scalar operators. The construction is explicit and the proof is self-contained.\n\nWhat is actually new: Fackler and Dier had abstract counterexamples, and Bechtel–Mooney–Veraar had divergence-form ones but with non-Hermitian coefficients, spatial discontinuities, and d≥2. This one is a real scalar coefficient on an interval, so it removes the hope that scalar 1/2-Hölder regularity is enough. The mechanism is transparent: lacunary oscillatory modes on shrinking time intervals, with a coefficient oscillation of size K^{-1} that produces an L2-sized leading term after the divergence-form derivative, and a scalar corrector that cancels it so the forcing stays continuous in H while the time derivative accumulates a divergent harmonic series.\n\nThe technical work is done properly. Proposition 3.1 proves the 1/2-Hölder bound via a clean geometric-frequency split; Proposition 3.2 shows sharpness within the Hölder scale; Lemma 4.1 identifies the divergent part; Proposition 5.1 shows the forcing is continuous in H. I traced the cancellation in (5.2) and it works: the leading p_K terms cancel exactly, leaving remainders of order ε/K. The zero-extension argument (Lemma 6.1) handles the endpoint Dirac-mass issue because both profile and modes vanish with their derivatives. The tensorisation and localisation in Corollary 1.2 are standard but carefully checked.\n\nSoft spots: none load-bearing. The paper is long and intricate; a few constants are left as 'C' but the dependencies are clear. The AI-use declaration is honest and does not affect the mathematics. The comparison section (Section 7) simply checks that the constructed coefficient fails the known sufficient hypotheses, which is a useful sanity check rather than a circular step. I agree with the reader that the weakest point is the reliance on lacunary orthogonality, but Lemma 2.1 proves it from disjoint Fourier frequencies, so it is solid. If there is a hidden error, it would be a sign or constant mistake in the many estimates, and I did not find one.\n\nThe paper is for anyone working on non-autonomous maximal regularity, Lions' problem, or divergence-form operators. It deserves a serious referee and, assuming no hidden error, acceptance. I would cite it if I were writing in this area.\n\nRecommendation: send to a strong journal, definitely peer-review rather than desk-reject. My own verdict: accept.","headline":"Genuine endpoint counterexample for scalar divergence-form operators with C^{0,1/2} coefficients; explicit, self-contained, and convincing.","tokens_in":17387,"tokens_out":2006,"would_cite":true,"duration_ms":17510,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K90","35B65","35R05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Even a 1/2-Hölder coefficient arbitrarily close to 1 can make the heat equation's unique solution lose square-integrable time regularity.","keywords":["non-autonomous maximal regularity","Lions' problem","divergence-form operators","1/2-Hölder endpoint","counterexample","variational solution","lacunary frequencies","bounded domains"],"falsifier":"For the explicitly constructed coefficient and forcing, compute or simulate with the first N blocks the integral $\\int_0^1 \\|\\partial_t u(t)\\|_{L^2(0,\\pi)}^2\\,dt$. The paper shows this is bounded below by a constant times $\\sum_{j=1}^N M_j\\ell_j$, which grows like the harmonic series; if a careful high-resolution computation found the partial sums bounded, the claimed divergence—and hence the theorem—would be false.","tokens_in":16437,"feed_emoji":"🔥","tokens_out":10205,"duration_ms":84697,"temperature":0.7,"pith_summary":"This paper settles Lions' maximal-regularity problem at the critical half-Hölder endpoint. It constructs, on the interval $(0,\\pi)$, a uniformly elliptic real scalar diffusion coefficient that is $1/2$-Hölder continuous in time with values in $L^\\infty$ and can be chosen arbitrarily close to the constant coefficient $1$. For zero initial data and a forcing term that is continuous in time and square-integrable in space, the unique variational solution has a time derivative that is not square-integrable in space-time. This proves that $C^{0,1/2}$ time regularity alone does not imply maximal $L^2$-regularity for divergence-form operators, confirming a conjecture discussed in the paper. Zero extension, tensorisation, and parabolic rescaling turn the interval counterexample into real symmetric isotropic counterexamples on $\\mathbb{R}^d$ and on every bounded domain, for all $d\\ge1$.","feed_headline":"Half-Hölder smoothness can break heat-equation regularity","feed_subtitle":"The time derivative escapes square integrability even for arbitrarily small perturbations of the heat equation.","key_machinery":"The engine is the coefficient–profile identity $L_{K^{-1}\\cos(Kx)}\\phi = \\sin(Kx)\\sin(2x) - 2K^{-1}\\cos(Kx)\\cos(2x)$, with $\\phi(x)=\\sin^2 x$. It says that a coefficient oscillation of amplitude $K^{-1}$ and spatial frequency $K$, differentiated once, produces a leading mode whose $L^2$ norm is independent of $K$; the leftover term carries an extra factor $K^{-1}$. The coefficient modulates this at temporal frequency $K^2$, which matches parabolic scaling and gives exactly the factor $K^{-1}\\min\\{2,K^2|t-s|\\}\\le\\sqrt2|t-s|^{1/2}$, so the coefficient is $1/2$-Hölder and no better. Lacunary spacing $K_{j,m+1}=16K_{j,m}$ makes the modes orthogonal across blocks, so the large contributions to $\\partial_t u$ add up instead of cancelling; scalar amplitudes $y_{j,m}$ satisfying $\\partial_t y_{j,m}+K_{j,m}^2 y_{j,m}$ are tuned to cancel the leading mode in the forcing, keeping $f$ in $C([0,1];H)$ while the time derivative accumulates a divergent harmonic series.","core_discovery":"The central claim is Theorem 1.1: for every $\\delta>0$ there exist a uniformly elliptic real coefficient $a\\in C^{0,1/2}([0,1];L^\\infty(0,\\pi))$ with $\\|a-1\\|_{C^{0,1/2}}<\\delta$ and a forcing $f\\in C([0,1];L^2)\\cap L^2$ such that the unique Lions variational solution $u$ with $u(0)=0$ satisfies $u(t)\\in D(A_H(t))$ for every $t$, yet $\\partial_t u\\notin L^2(0,1;H)$ and $A_H(\\cdot)u(\\cdot)\\notin L^2(0,1;H)$, where $H=L^2(0,\\pi)$. The coefficient is formed by superposing oscillations at spatial frequency $K$ and temporal frequency $K^2$ on disjoint time blocks of lengths $\\ell_j\\simeq 1/(16j(j+1))$, with $j$ modes on the $j$-th block. On block $j$ the squared $L^2$ norm of $\\partial_t u$ is comparable to $\\varepsilon^2/(16(j+1))$, so the total diverges like a harmonic series even though the forcing is continuous with values in $H$.","pith_inferences":["A testable extension is whether a similar lacunary construction can be built for quasilinear or higher-order parabolic problems; the load-bearing identity is specific to second-order divergence form, so the failure mechanism would need to be re-derived there.","The construction suggests that any sufficient condition for endpoint maximal regularity must be non-local in time and must couple spatial structure, because pointwise $1/2$-Hölder continuity with values in $L^\\infty$ is not enough even at arbitrarily small amplitude.","A quantitative version may be possible: truncating the construction at $N$ blocks should make the squared $L^2$ norm of $\\partial_t u$ grow like $\\log N$, giving a concrete rate at which regularity degrades as the counterexample is approximated.","Because the flux vanishes at the endpoints, the zero-extension trick is robust; a similar profile with vanishing first derivatives could be sought for other boundary conditions, though such an extension is not part of the paper."],"forward_implications":["At the critical Hölder exponent $1/2$, no general maximal $L^2$-regularity theorem can hold for scalar divergence-form operators on intervals, because the coefficient here is real and uniformly elliptic and can be taken arbitrarily close to $1$.","The counterexample transfers to the full space $\\mathbb{R}^d$ and to every bounded domain $\\Omega\\subset\\mathbb{R}^d$ for all $d\\ge1$ with a real symmetric isotropic coefficient matrix, so the failure is neither a boundary artefact nor a low-dimensional phenomenon.","The construction confirms that the sufficient endpoint conditions in the paper's cited positive results—bounded variation, Dini-type moduli, piecewise $H^{1/2}$, and the scale-invariant square condition—are all genuinely needed; the constructed coefficient manages to violate each of them.","Maximal regularity fails despite the solution being well-behaved pointwise: $u(t)$ lies in the operator domain $D(A_H(t))$ for every $t$ and $A_H(\\cdot)u(\\cdot)$ belongs to $L^1$, but the square-integrable regularity of the time derivative is exactly what is lost.","The same construction gives a coefficient that is $1/2$-Hölder with arbitrarily small norm but not $C^{0,\\alpha}$ for any $\\alpha>1/2$, so the endpoint scaling is sharp."],"supporting_citations":[{"why":"Supplies the Lions variational uniqueness theorem used to identify the constructed u as the unique solution and to guarantee well-posedness.","marker":"[13]"},{"why":"Contains the conjecture that C^{0,1/2}([0,T];L^∞) regularity is insufficient for divergence-form operators and the square condition (8) that the main theorem confirms must fail.","marker":"[4]"},{"why":"Shows failure for abstract symmetric forms with C^{0,1/2} time dependence; the present paper transfers the obstruction to concrete differential operators.","marker":"[10]"},{"why":"Gives maximal regularity for C^{0,α}, α>1/2, marking the threshold at which the new counterexample lives.","marker":"[14]"},{"why":"Positive maximal-regularity result for forms of bounded variation that the constructed coefficient is shown to violate.","marker":"[8]"},{"why":"Positive result under a Dini-type modulus that the construction is shown not to satisfy.","marker":"[11]"},{"why":"Provides a positive endpoint result under piecewise H^{1/2} regularity with a local smallness condition; the constructed path is shown to fail piecewise H^{1/2} regularity.","marker":"[1]"}],"fun_headline_variants":["Half-Hölder coefficients can break maximal L2 regularity","Tiny 1/2-Hölder perturbation of heat equation loses regularity","1/2-Hölder smoothness not enough for heat-equation control","Small 1/2-Hölder diffusion coefficients defeat maximal regularity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The counterexample depends on the lacunary separation of the oscillatory modes: consecutive spatial frequencies differ by a factor of 16, which makes the mode families orthogonal and prevents the bad time-derivative contributions from cancelling across blocks; if that separation were removed, cross-mode interaction could restore square integrability.","fun_headline_variants_meta":{"raw":{"variants":["Half-Hölder coefficients can break maximal L2 regularity","Tiny 1/2-Hölder perturbation of heat equation loses regularity","1/2-Hölder smoothness not enough for heat-equation control","Small 1/2-Hölder diffusion coefficients defeat maximal regularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1642,"prompt_tokens":1079,"completion_tokens":563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":486}},"tokens_in":695,"tokens_out":563,"duration_ms":4746,"temperature":1.0,"reasoning_tokens":486,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:17:24.676944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicitly constructed coefficient and forcing, compute or simulate with the first N blocks the integral $\\int_0^1 \\|\\partial_t u(t)\\|_{L^2(0,\\pi)}^2\\,dt$. The paper shows this is bounded below by a constant times $\\sum_{j=1}^N M_j\\ell_j$, which grows like the harmonic series; if a careful high-resolution computation found the partial sums bounded, the claimed divergence—and hence the theorem—would be false.","supporting_citations":[{"cited_title":"Auscher and M","cited_arxiv_id":null,"evidence_quote":"Contains the conjecture that C^{0,1/2}([0,T];L^∞) regularity is insufficient for divergence-form operators and the square condition (8) that the main theorem confirms must fail."},{"cited_title":"Fackler,J.-L","cited_arxiv_id":null,"evidence_quote":"Shows failure for abstract symmetric forms with C^{0,1/2} time dependence; the present paper transfers the obstruction to concrete differential operators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives maximal regularity for C^{0,α}, α>1/2, marking the threshold at which the new counterexample lives."},{"cited_title":"Dier,Non-autonomous maximal regularity for forms of bounded variation, J","cited_arxiv_id":null,"evidence_quote":"Positive maximal-regularity result for forms of bounded variation that the constructed coefficient is shown to violate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Positive result under a Dini-type modulus that the construction is shown not to satisfy."},{"cited_title":"Achache and E","cited_arxiv_id":null,"evidence_quote":"Provides a positive endpoint result under piecewise H^{1/2} regularity with a local smallness condition; the constructed path is shown to fail piecewise H^{1/2} regularity."}],"review_version":1}