{"id":"5e193791-659b-40cd-bfca-871311e84504","arxiv_id":"2504.15735","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bounded local weak solutions of nonlocal porous medium and fast diffusion equations with general measurable kernels are locally Hölder continuous.","lead":"This paper proves that solutions to a broad family of nonlinear diffusion equations with long-range interactions are locally Hölder continuous. The proof develops a De Giorgi-Nash-Moser iteration adapted to the equation's degeneracy and nonlocality, and applies to rough, merely measurable kernels.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Away-from-zero phase cites linear theory without verifying it applies to the W-equation, whose time coefficient depends on W and makes the absorbed kernel non-symmetric.","rationale":"The reader's weakest assumption correctly identifies Section 6.1.2/6.2.2 as the point where the proof switches from the nonlinear problem to cited linear theory. My stress-test sharpens the concern: the coefficient multiplying the time derivative is not merely time-dependent, it is solution-dependent, and absorbing it into the kernel destroys symmetry unless the coefficient is spatially constant. This makes the cited theorems inapplicable on their face, so the oscillation decay in the away-from-zero case is not established. The paper has substantial independent structure—energy estimates, De Giorgi lemmas, and careful tail controls—and the near-zero analysis is detailed. However, the away-from-zero step is a one-paragraph citation with no verification of the linear hypotheses. I also note that assumption (1.3) lacks an explicit symmetry condition even though Section 3's Caccioppoli estimates use symmetry; this is a related missing hypothesis worth checking. The reader's verdict of CONDITIONAL remains appropriate: the main theorem is plausible and likely repairable, but the current manuscript does not supply the required proof or precise citation for the linear reduction.","tokens_in":42812,"tokens_out":20558,"duration_ms":202856,"concrete_test":"Independently re-derive the equation satisfied by W in Section 6.1.2 and verify the hypotheses of the cited linear results: write the reduced equation as d_t W - a(x,t)L'W = 0 with a = phi'(U), and check whether the Caccioppoli step for the absorbed kernel a(x,t)K'(x,y) remains coercive. Concretely, compute the quadratic form Q = integral (W(x)-W(y))(W(x)xi^2(x)-W(y)xi^2(y)) a(x) K'(x,y) dx dy for a smooth nonnegative W and xi where a varies; the antisymmetric part (a(x)-a(y))K' should be shown to be controllable. If [Par23] and [CCV11] do not cover solution-dependent coefficients or non-symmetric absorbed kernels, then the away-from-zero phase lacks proof. Also check whether (1.3) is intended to include symmetry K(x,y;t)=K(y,x;t), since Lemma 3.2 relies on 'by symmetry'.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central gap is in Section 6.1.2 (and its fast-diffusion analogue Section 6.2.2). After rescaling and setting W = phi(U), the equation is d_t beta(W) - L'W = 0 with beta = phi^{-1} and |U| in [1/2, 2]. Equivalently, d_t W - a(x,t) L' W = 0, where a(x,t) = phi'(U(x,t)) = 1/beta'(W(x,t)) is bounded between positive constants but depends on the solution and varies in space and time. The paper asserts 'It is well known how to handle elliptic and more general degenerate coefficients in time' and cites [Par23, Par15, CCV11], but no theorem statement or hypothesis check is given. If a is absorbed into the kernel, K_a(x,y) = a(x,t)K'(x,y), the kernel is not symmetric unless a is spatially constant; the standard Caccioppoli/De Giorgi machinery for nonlocal operators requires symmetry or a separate treatment of the antisymmetric part. Paronetto's results cover coefficients depending on time, and CCV11 covers a fixed bounded kernel, but neither is shown to apply to d_t W - a(W)L W = 0. This step provides the oscillation decay in the away-from-zero case that is glued to the near-zero analysis to prove Theorem 6.1 and hence Theorem 1.1. Without a proof or a precise citation, the main theorem is not established. Relatedly, assumption (1.3) does not state K(x,y;t) = K(y,x;t), although the energy estimates in Section 3 use symmetry; this strengthens the need for a careful check of the linear reduction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies interior Hölder regularity for local weak solutions of the nonlocal nonlinear diffusion equation ∂_t u - Lφ(u)=0, where φ(u)=|u|^{m-1}u for m>0 and L is a nonlocal operator with a bounded measurable kernel K satisfying the two-sided bound (1.3). The main result, Theorem 1.1, asserts that locally bounded local weak solutions are locally C^{0,α}, with explicit oscillation estimates stated in Theorem 6.1. The proof splits into two phases: near the set {u≈0}, where the singular/degenerate nature of φ is handled through intrinsic space-time scaling and De Giorgi-type lemmas (Sections 3-5), and away from zero, where a rescaling is claimed to reduce the equation to a linear nonlocal parabolic equation to which known regularity theory is applied (Sections 6.1.2 and 6.2.2). Sections 3-5 contain detailed Caccioppoli estimates, a fractional isoperimetric inequality, tail alternatives, and two De Giorgi lemmas with explicit constants.","tokens_in":43201,"tokens_out":13143,"duration_ms":120078,"significance":"If the proof is completed, the result is a substantial contribution: it provides a De Giorgi-Nash-Moser type interior regularity theory for a general class of nonlocal porous medium and fast diffusion equations with measurable kernels, and it is new even for positive solutions of the constant-coefficient fractional equation. The paper contains genuine work in Sections 3-5: the energy estimates, the intrinsic scaling near {u≈0}, and the tail-control argument are carefully written and internally consistent. The use of [APT] is substantial but not circular, since the cited results are independent theorems. The main weakness is the final linear reduction in the away-from-zero phase, which is asserted rather than proved; this currently prevents the main theorem from being fully established.","major_comments":[{"comment":"The oscillation decay in the away-from-zero phase is not established. After the rescaling and the change of variables W=φ(U), the equation is written as ∂_t β(W)-L'W=0, equivalently ∂_t W - a(x,t)L'W=0 with a(x,t)=φ'(U(x,t)) bounded between positive constants but depending on the solution and only measurable in general. The text says 'It is well known how to handle elliptic and more general degenerate coefficients in time' and cites [Par23, Par15, CCV11], but no theorem statement or hypothesis check is given. The absorbed kernel a(x,t)K'(x,y) is time-dependent and, unless a is spatially constant, non-symmetric. Paronetto's results concern local parabolic equations with time-dependent coefficients, and CCV11 concerns a fixed bounded kernel; neither is shown to apply to ∂_t W - a(W)L'W=0. This step supplies the exponent α_1 in the final oscillation estimate of Theorem 6.1, so without a proof or a precise citation with verified hypotheses, the main theorem is not fully proved.","section":"§6.1.2 and §6.2.2 (Eqs. (6.10)-(6.11))"},{"comment":"Assumption (1.3) does not state that K(x,y;t)=K(y,x;t), but the energy estimates in Section 3 use symmetry. In Lemma 3.2 the full integral over K×R^n is split as A+2B 'by symmetry', and Lemma 3.4 uses a similar reduction; these steps require the kernel to be symmetric. If symmetry is not intended as a standing assumption, the estimates need to be revisited; if it is intended, it should be stated in (1.3). This also matters for the linear reduction in §6.1.2, because even for symmetric K', the kernel a(x,t)K'(x,y) with a depending on x is non-symmetric.","section":"§1.1, Eq. (1.3); §3, Lemma 3.2"}],"minor_comments":[{"comment":"The disjunction defining the first 'away from zero' index l writes |{u≥M_l}∩Q_l|≤ν|Q_l|; from the close-to-zero conditions it should be |{u≥-M_l}∩Q_l|≤ν|Q_l| in order to apply Lemma 4.2 or Lemma 4.3 and conclude |u|≥M_l/2.","section":"§6.1.2 and §6.2.2"},{"comment":"The notation Q_{R0} in the statement 'with Q_R(θ)⊂Q_{R0}⊂Ω_T' is not defined; it should presumably be Q_{R0}(θ) for m≥1 and Q_{R0}(ϑ) for 0<m≤1, or the standard cylinder with the scaling made explicit.","section":"Theorem 6.1"},{"comment":"The phrase 'without loss of generosity' should read 'without loss of generality'.","section":"Proof of Lemma 3.9"},{"comment":"Remark 2.2 refers to 'Eq. (3.2)' for the sub/super-solution inequality, but the displayed inequality in Definition 2.1 is not numbered; renumbering would avoid confusion.","section":"Definition 2.1 and Remark 2.2"}],"recommendation":"major_revision","confidential_remarks":"This is a promising paper with a serious gap at the end. The central theorem is likely true and the missing step is fixable in principle by adding a complete proof of the linear oscillation decay for ∂_t W - a(x,t)L'W=0 with bounded measurable a and non-symmetric kernel, or by identifying and verifying a precise theorem from the literature. I therefore recommend major revision rather than rejection. The authors should also clarify the symmetry assumption on K; if K is not intended to be symmetric, the energy estimates in Section 3 need modification. The reliance on [APT], which shares an author with this paper, is not circular as long as the cited results are independently valid, but the overlap should be handled according to the journal's policy and the borrowed estimates stated precisely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is a natural and significant target: interior Hölder regularity for local weak solutions of the nonlocal nonlinear diffusion equation with rough kernels, covering both the porous media (m>1) and fast diffusion (0<m<1) regimes. The paper adapts the De Giorgi-Nash-Moser approach from [APT] and [Lia24] to a solution-dependent geometry near u=0, and the technical work in Sections 3-5 is detailed and internally consistent. The Caccioppoli estimates, the two De Giorgi lemmas, and the switching-radius argument all look credible. The authors are honest about what is new and what is borrowed, and the exposition is clear.\n\nThe soft spot is exactly where the stress-test note points: Section 6.1.2 and its fast-diffusion analogue 6.2.2. After rescaling away from zero, the equation is reduced to ∂_t W - a(x,t)L'W = 0, with a bounded measurable coefficient a that depends on the solution W. The paper states that this is 'well known' and cites [Par23, Par15, CCV11], but it never states or verifies the hypotheses of those theorems for this nonlocal operator with a solution-dependent coefficient. In particular, if a is absorbed into the kernel, the resulting kernel is not symmetric unless a is spatially constant, and standard nonlocal De Giorgi theory typically requires symmetry or a controlled antisymmetric part. This step is load-bearing: it provides the away-from-zero oscillation decay that is glued to the near-zero analysis to prove Theorem 6.1. A referee will need to see either a direct proof or a precise statement of which cited theorem applies, with all hypotheses checked.\n\nThere is also a related minor gap: assumption (1.3) does not assume K(x,y;t)=K(y,x;t), yet several estimates in Section 3 invoke symmetry in an essential way. If the theory really needs symmetry, the assumption should say so; if not, the paper should explain how the antisymmetric part is handled. This is a smaller issue than the linear reduction, but it should be fixed in the same revision.\n\nThe reader's conditional verdict is appropriate. The central De Giorgi machinery is strong and the result is important, but the away-from-zero phase is a genuine gap, not a cosmetic omission. I would send this to a serious referee and require a complete justification of the linear reduction before acceptance. With that gap filled, this will be a valuable reference for the community.","headline":"The De Giorgi machinery in Sections 3-5 is solid and the result is significant, but the away-from-zero phase in Section 6 cites a linear theory it never verifies, so the main theorem is not yet established.","tokens_in":43686,"tokens_out":4732,"would_cite":false,"duration_ms":48997,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R09","35B65","47G20","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that locally bounded local weak solutions to the nonlocal porous medium and fast diffusion equations are locally Hölder continuous for every $m>0$.","keywords":["porous media equations","fast diffusion equations","nonlocal parabolic equations","fractional diffusion equations","fractional porous media equations","fractional fast diffusion equations","De Giorgi-Nash-Moser theory","local Hölder regularity"],"falsifier":"Compute, for a concrete admissible kernel and a solution bounded away from zero on a switching cylinder, the rescaled operator from Section 6.1.2 by setting $X=x/\\rho_l$, $S=t/(\\theta_l\\rho_l^{2s})$, $U=u/M_l$, and $W=\\varphi(U)$; then check whether the time-coefficient and kernel in the resulting linear equation satisfy the ellipticity and measurability assumptions of the cited linear regularity results. If any admissible kernel, for example $K(x,y;t)=\\lambda|x-y|^{-n-2s}$ multiplied by a bounded step function in $t$, produces a transformed coefficient that violates those hypotheses while $u$ remains locally bounded, the oscillation-decay step lacks proof; an explicit locally bounded solution in that class that fails to be Hölder would refute Theorem 1.1.","tokens_in":42645,"feed_emoji":"📐","tokens_out":16078,"duration_ms":126219,"temperature":0.7,"pith_summary":"The paper claims that local boundedness is enough to guarantee local Hölder continuity for solutions of equations modeled on $\\partial_t u + (-\\Delta)^s(|u|^{m-1}u)=0$, for every $s\\in(0,1)$ and $m>0$, when the kernel is any bounded measurable function comparable to $|x-y|^{-n-2s}$. If correct, this completes the interior regularity program for the fractional porous media and fast diffusion families: no boundary values, global boundedness, or coefficient smoothness are needed. The proof controls the set where $u$ is near zero, where the nonlinearity $|u|^{m-1}$ is degenerate ($m>1$) or singular ($0<m<1$), by working in cylinders rescaled by the solution's own amplitude and by iterating De Giorgi-type lemmas with tail control. The paper also records explicit oscillation estimates (Theorem 6.1) and a Liouville corollary for bounded entire solutions.","feed_headline":"Hölder regularity proven for nonlocal porous media and fast diffusion","feed_subtitle":"Interior Hölder bounds now hold for every m>0 with bounded measurable kernels, not only the fractional Laplacian.","key_machinery":"The load-bearing objects are the intrinsic cylinders. For $m\\ge 1$, set $\\theta=M^{1-m}$ and work in $Q_\\rho(\\theta)=K_\\rho\\times(t_0-\\theta\\rho^{2s},t_0]$; for $0<m\\le 1$, set $\\vartheta=M^{(m-1)/(2s)}$ and work in $Q_\\rho(\\vartheta)=K_{\\vartheta\\rho}\\times(t_0-\\rho^{2s},t_0]$, where $M$ measures the essential supremum and a tail of $u$. These geometries match the power of $|u|^{m-1}$ so that the degeneracy or singularity at $u\\approx 0$ becomes uniform. The argument is carried by Caccioppoli estimates built from Steklov averages; a fractional isoperimetric inequality (Lemma 2.5) that supplies the jump term needed in the absence of the classical isoperimetric inequality; De Giorgi lemmas that turn measure density plus smallness of the tail $\\mathrm{Tail}(u;Q)=\\big(r^{2s}\\operatorname*{ess\\,sup}_{t}\\int_{\\mathbb R^n\\setminus K_r}|u(y,t)|^m|y-x_0|^{-n-2s}\\,dy\\big)^{1/m}$ into sup-reduction; and a two-phase iteration that either stays near zero, yielding global sup-reduction, or exits to the linear regime where oscillation decay follows from the cited linear theory.","core_discovery":"On its own terms, the paper's central claim is Theorem 1.1: if $\\varphi(X)=|X|^{m-1}X$, $0<s<1$, and the kernel $K$ is measurable with $\\lambda|x-y|^{-n-2s}\\le K(x,y;t)\\le \\Lambda|x-y|^{-n-2s}$, then every locally bounded local weak solution of $\\partial_t u - \\mathcal{L}\\varphi(u)=0$ lies in $C^{0,\\alpha}_{\\mathrm{loc}}(\\Omega_T)$ for some $\\alpha\\in(0,1)$ depending only on $n,s,m,\\lambda,\\Lambda$. The proof splits into a close-to-zero phase and an away-from-zero phase. Near $\\{u\\approx 0\\}$ the equation is degenerate or singular, and the analysis uses intrinsic cylinders -- time-scaled by $\\theta=M^{1-m}$ when $m\\ge 1$, space-scaled by $\\vartheta=M^{(m-1)/(2s)}$ when $0<m\\le 1$ -- together with a fractional isoperimetric inequality, De Giorgi lemmas, and tail alternatives to force a geometric reduction of the supremum. Once $u$ is bounded away from zero, a rescaling and the change of variable $W=\\varphi(U)$ turn the equation into a linear nonlocal parabolic equation, and known linear regularity supplies oscillation decay.","pith_inferences":["If the linear-reduction step is sound, the same two-phase architecture should work for more general monotone nonlinearities $\\varphi$ that behave like powers near zero and infinity, with only cosmetic changes to the intrinsic cylinder scaling; this is an extension the paper does not state.","The quantitative relation between amplitude and cylinder scaling ($\\theta=M^{1-m}$ versus $\\vartheta=M^{(m-1)/(2s)}$) predicts a specific anisotropic Hölder rate near points where $u$ vanishes, which could be tested numerically against radial Barenblatt solutions for the fractional porous medium equation.","A direct verification that the rescaled operator in Section 6.1.2 satisfies the hypotheses of the cited linear time-coefficient regularity theorems would strengthen the away-from-zero phase; conversely, an admissible kernel for which the rescaled time-coefficient degenerates while $u$ stays away from zero would force a nonlinear substitute.","The tail-condition machinery suggests that the same Hölder estimate should persist under weaker global integrability assumptions on $u$, since tails enter only through the product $(r/R)^{2s/m}\\mathrm{Tail}(u;Q)\\le M$; locating the minimal integrability needed is a natural next question."],"forward_implications":["Interior Hölder continuity holds for sign-changing, locally bounded local weak solutions, with no assumption on initial, boundary, or far-field data beyond the local tail integrability built into the weak-solution definition.","Theorem 6.1 provides a quantitative modulus: the oscillation on $Q_r$ is bounded by $C M (r/R)^\\alpha$, with $M$ the essential supremum plus a tail term, so the Hölder exponent and constant depend only on $n,s,m,\\lambda,\\Lambda$.","For bounded solutions posed on all of $\\mathbb R^n\\times(-\\infty,T)$, the oscillation estimate forces the solution to be constant (Corollary 1.4).","Because the kernel is only assumed bounded and measurable with two-sided power bounds, the result covers rough kernels and is new even for positive solutions of the constant-coefficient fractional Laplacian equation with purely local assumptions.","Taken together, the porous media ($m>1$) and fast diffusion ($0<m<1$) cases give a unified interior regularity theory for all $m>0$."],"supporting_citations":[{"why":"It supplies the linear nonlocal parabolic Hölder theory used in the away-from-zero phase.","marker":"[CCV11]"},{"why":"It provides the Harnack and Hölder result for parabolic equations with coefficients depending on time, used in the linear reduction.","marker":"[Par23]"},{"why":"It supplies the earlier Harnack inequality for mixed-type evolution equations with time-dependent coefficients, also used in the linear reduction.","marker":"[Par15]"},{"why":"It gives the isoperimetric inequality variant and the De Giorgi iteration framework adapted to fractional parabolic problems.","marker":"[APT]"},{"why":"It provides the Steklov-average and Caccioppoli machinery for porous medium systems on which the energy estimates are built.","marker":"[Lia21]"},{"why":"It supplies the test-function computation and regularity framework for parabolic fractional p-Laplacian used in the energy estimates.","marker":"[Lia24]"},{"why":"It contributes the switching-radius and geometric iteration, including the sequence lemma that drives the sup-reduction near the degenerate set.","marker":"[DiB83]"},{"why":"It contains the pointwise Caccioppoli inequality used to control the jump term in the nonlocal energy estimates.","marker":"[Coz17]"}],"fun_headline_variants":["Measurable kernels yield Hölder regularity for nonlocal diffusion","Nonlocal porous media and fast diffusion: Hölder regularity for all m>0","Fractional diffusion: interior Hölder bounds with general measurable kernels","New proof: Hölder continuity for nonlocal diffusion with general kernel","Local Hölder estimates for nonlocal porous media and fast diffusion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the away-from-zero reduction: after rescaling and setting $W=\\varphi(U)$, the paper assumes that the equation is a linear nonlocal parabolic equation with a time-dependent coefficient to which the quoted linear regularity theorems apply directly, but it never verifies that the hypotheses of those theorems are satisfied for the transformed operator.","fun_headline_variants_meta":{"raw":{"variants":["Measurable kernels yield Hölder regularity for nonlocal diffusion","Nonlocal porous media and fast diffusion: Hölder regularity for all m>0","Fractional diffusion: interior Hölder bounds with general measurable kernels","New proof: Hölder continuity for nonlocal diffusion with general kernel","Local Hölder estimates for nonlocal porous media and fast diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00028,"raw_usage":{"total_tokens":1693,"prompt_tokens":1008,"completion_tokens":685,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":595}},"tokens_in":624,"tokens_out":685,"duration_ms":5917,"temperature":1.0,"reasoning_tokens":595,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:19:18.468470+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete admissible kernel and a solution bounded away from zero on a switching cylinder, the rescaled operator from Section 6.1.2 by setting $X=x/\\rho_l$, $S=t/(\\theta_l\\rho_l^{2s})$, $U=u/M_l$, and $W=\\varphi(U)$; then check whether the time-coefficient and kernel in the resulting linear equation satisfy the ellipticity and measurability assumptions of the cited linear regularity results. If any admissible kernel, for example $K(x,y;t)=\\lambda|x-y|^{-n-2s}$ multiplied by a bounded step function in $t$, produces a transformed coefficient that violates those hypotheses while $u$ remains locally bounded, the oscillation-decay step lacks proof; an explicit locally bounded solution in that class that fails to be Hölder would refute Theorem 1.1.","supporting_citations":[],"review_version":1}