{"id":"03d49ec0-d701-40b3-9c2b-9815f4a0be6f","arxiv_id":"2508.20523","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For aggregation-diffusion equations with nonlinear (s,p) Riesz potentials, radial stationary states coincide (up to scaling) with extremals of a Hardy-Littlewood-Sobolev inequality, and as s tends to 0 they converge to a ball in the diffusion-dominated case.","lead":"This mathematics paper finds the stable equilibrium shapes for a model where particles both spread out and clump together through a nonlocal, nonlinear attraction, and shows what happens when the attraction becomes extremely short-ranged. It extends a well-known family of aggregation models, used in biology for chemotaxis and swarming, to a broader class of interactions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.6, the compactness step for Theorem 2.2, is deferred to [28, Lemma 3.7] without proof; the transfer to the nonlinear potential case p≠2 is not automatic.","rationale":"The paper is a serious, largely self-contained advance: Theorem 2.1 is proved with detailed arguments for existence, Euler–Lagrange equations, regularity, and the correspondence with minimizers, and the scaling algebra checks out after accounting for the free energy convention used throughout Section 5. The reader's weakest-assumption choice correctly identifies Proposition 6.6 as the main unproved transfer. My read of the surrounding estimates confirms that the s→0 compactness in Theorem 6.7 depends on that proposition, and that the one-sentence appeals to [28, Lemma 3.7] and [28, Proposition 2.10] leave unexamined exactly the p≠2 features: the nonlinearity in the Riesz potential makes ∇K_{s,p}(ρ_s) a singular integral of (K_{s/2}*ρ_s)^{p′−1}, whose uniform L^1 control is not a formal consequence of the linear estimates. At the same time, the concern is not an observed contradiction: the radiality and uniform bounds give a plausible alternative route via Helly selection, and the claimed W^{1,1} bound may well be true. Therefore the verdict should move from unconditional acceptance to conditional acceptance: the s→0 part should be accepted once Proposition 6.6 is proved for p≠2 or replaced by an explicit compactness argument. The rest of the paper's claims do not hinge on this gap.","tokens_in":33285,"tokens_out":50898,"duration_ms":471756,"concrete_test":"Independently re-derive Proposition 6.6 from (5.9) and Propositions 6.4–6.5: prove the uniform estimate ∥∇ρ_s∥_{L^1} ≤ C(N,p,m,χ,M) for all small s, isolating the term ∥∇K_{s/2}*((K_{s/2}*ρ_s)^{p′−1})∥_{L^1(supp ρ_s)}. Check whether the estimate closes for every 1<p<∞ or only when p′=2 (e.g., because the linear proof uses the identity ∇(K_s*ρ)=R[K_s*ρ] with a fixed Fourier multiplier). If the nonlinear estimate cannot be closed, replace Prop. 6.6 by a Helly-selection compactness argument for the uniformly bounded, equi-supported radial nonincreasing minimizers; if even that alternative fails, Theorem 2.2 is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The s→0 convergence theorem (Theorem 6.7, hence Theorem 2.2 for m>p′) rests on Proposition 6.6, whose proof is omitted with the sentence “The proof is the same of [28, Lemma 3.7].” This is not a routine substitution: in the linear case p=2 the argument can use that K_s*ρ is a Bessel/Riesz potential of ρ and that ∇(K_s*ρ) is controlled through a Fourier multiplier applied to K_s*ρ. In the nonlinear case one must control ∇K_{s/2}*((K_{s/2}*ρ_s)^{p′−1}), a singular integral applied to a nonlinear power of a bounded function. The uniform L∞ bound (Prop. 6.4) and uniform support radius (Prop. 6.5) do not by themselves yield a uniform W^{1,1} bound: Hölder continuity does not imply absolute continuity, and radial monotone Hölder functions can be singular (Cantor-type examples). Corollary 4.8, which would reduce the desired bound to the elementary inequality ∥∇ρ∥_{L^1} ≤ C∥ρ∥∞ R^{N−1} for radial nonincreasing functions, is itself justified only by “arguing as in [28, Proposition 2.10].” Thus the W^{1,1} input is deferred twice, and if either transfer fails for p≠2 or for small s, the compactness step of Theorem 6.7—and hence the m>p′ part of Theorem 2.2—is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies radial stationary states of the aggregation-diffusion equation (1.1) with the nonlinear (s,p) Riesz potential S=K_{s,p}. It proves that, for m>(p*_s)', the sharp constant in the HLS-type inequality (2.3) is attained; every optimizer is radially nonincreasing, compactly supported, Hölder regular, and satisfies the Euler-Lagrange equation (2.4). In the diffusion-dominated and critical regimes these optimizers correspond exactly to minimizers of the free energy F_{s,p}, with an explicit critical mass M_c in the fair-competition case. The second main result describes the limit s→0: for m>p' minimizers converge strongly in L^q to the characteristic function of a ball, while for m=p' they either vanish uniformly (0<χ<p) or concentrate to a point mass (χ>p), with a separate treatment of χ=p.","tokens_in":33542,"tokens_out":32097,"duration_ms":285247,"significance":"If fully established, this is a substantial contribution: it extends the linear-potential theory of [10,19,28] to nonlinear (s,p) potentials, connects HLS extremals to free-energy minimizers via an explicit scaling correspondence, and provides the first s→0 asymptotic analysis in this nonlinear setting. The strategy is coherent and mostly well executed: existence by the Lieb-Oxford symmetrization method, Euler-Lagrange analysis, a regularity bootstrap, and a Γ-convergence framework. The paper is also honest about open problems and states its novel contributions clearly. The main weakness is that a load-bearing compactness argument in the s→0 section is deferred to previous work without showing that the nonlinear case p≠2 is covered by the same proof.","major_comments":[{"comment":"The W^{1,1} equiboundedness and L^1-compactness step needed for Theorem 6.7, and hence for the m>p' part of Theorem 2.2, is not proved in the manuscript: Proposition 6.6 states that the proof is the same as [28, Lemma 3.7], and Corollary 4.8 is justified only by 'arguing as in [28, Proposition 2.10]'. This is not a routine substitution. In the linear case p=2 one controls derivatives of K_s*ρ through Fourier multipliers; here one must control ∇K_{s/2}*((K_{s/2}*ρ_s)^{p'−1}), a singular integral applied to a nonlinear power of a bounded function. The uniform L∞ bound (Prop. 6.4) and uniform support bound (Prop. 6.5), together with Hölder regularity, do not by themselves imply a uniform W^{1,1} bound, since radial monotone Hölder functions need not be absolutely continuous and may have Cantor-type singular parts. The authors should either reproduce the argument in the p≠2 setting or supply a different proof of the L^1-compactness. As written, Theorem 2.2 (m>p') rests on an unsupported step.","section":"§6 (Prop. 6.6); §4.3 (Cor. 4.8)"},{"comment":"In the proof of compact support, the paper asserts that K_{s,p}(h_s)∈L^{p*_s}(R^N) 'in particular vanishes at infinity'. A function in L^{p*_s} need not tend to zero pointwise. The intended statement may follow from the radial monotonicity of h_s (for instance h_s(x)≤C|x|^{-N}), but this argument is not given. Since compact support is part of Theorem 2.1 and is also used in Corollary 4.8 and Proposition 6.5, this gap should be filled.","section":"§4.3, Lemma 4.6"}],"minor_comments":[{"comment":"The displayed identity F_{s,p}(ρ_s)=−sp/(N−sp)∥ρ_s∥_{p'}^{p'} is missing a factor (p−1); the correct coefficient is −sp(p−1)/(N−sp) for general p. The conclusion of the proof is unaffected, but the formula should be corrected.","section":"§6.3, Prop. 6.10"},{"comment":"In the estimate for A1, the power of (R_s−1) is displayed with the wrong sign in the text; also the reference '(6.2)' should be '(3.1)'.","section":"§6.2, Prop. 6.5"},{"comment":"The displayed statement 'lim ∥ρ_s∥∞ = −lim F_{s,p}(ρ_s)' is not a meaningful equality of real-valued limits; it should be phrased as both expressions tending to the same value in [0,∞].","section":"§6.3, Theorem 6.9"},{"comment":"The final step of the Hölder-regularity bootstrap reduces the argument to [19, Theorem 8] and only sketches the iteration; please expand this reduction or state precisely which parts of [19] transfer verbatim to the nonlinear potential case.","section":"§4.3, Lemma 4.7"},{"comment":"The notation for the sharp constant is inconsistent: H^*_{m,s,p} in (2.5) versus H^*_{m,s} in (4.1); please unify.","section":"§2 and §4"},{"comment":"There are several small typos, e.g. 'Corollay 3.3' in Theorem 6.7, 'wich' in Proposition 6.8, and the exponent in the fair-competition energy computation in Proposition 5.3 appears garbled in the displayed formula.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The work is within the journal's scope and the main ideas are sound. My main concern is the deferred W^{1,1} compactness argument in Proposition 6.6; if the authors can supply the missing proof or convincingly justify the transfer from [28, Lemma 3.7] to the p≠2 case, I would support publication. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the nonlinear (s,p) Riesz potential analogue of the well-studied linear case, and the authors actually carry it through. The results are new for p != 2: existence and regularity of HLS-type optimizers, the Euler-Lagrange equation, the correspondence with free-energy minimizers, the critical mass formula, and the s -> 0 asymptotics. The proof structure is coherent. Existence via Lieb-Oxford, the L-infinity and Holder bootstrap, and the dilation argument connecting HLS extremals to energy minimizers all look sound. The critical mass is computed from the HLS best constant, which is a supremum, not a fit. There are no data, no fitted parameters, and nothing circular. The citations to the authors' earlier work and to [28] point to the right prior results.\n\nThe soft spot is exactly the one flagged in the stress test. Proposition 6.6, which supplies equiboundedness in W^{1,1} and strong L^1 compactness for the minimizer family, is disposed of with one sentence: \"The proof is the same of [28, Lemma 3.7].\" Corollary 4.8 is similarly parked on [28, Proposition 2.10]. The concern is not pedantic: radial nonincreasing Holder functions need not be in W^{1,1} (Cantor-type examples), and the p != 2 structure changes the potential, so the transfer is not automatic. That said, I do not think this is a fatal flaw. The ingredients for a direct proof--boundedness, compact support, the Euler-Lagrange equation, and the potential regularity arguments already in Section 4--are in the paper. I would want the authors to write out Proposition 6.6 for the nonlinear potential rather than cite it, but I would be surprised if it fails. Minor: the constant in Proposition 6.10 has a typo; the asymptotics are unaffected.\n\nWho should read this: anyone working on aggregation-diffusion equations, Hardy-Littlewood-Sobolev inequalities, or the fractional plasma equation. It deserves a serious referee. I would accept it for peer review, with the expectation that the referee checks the W^{1,1} compactness argument or asks the authors to supply it.","headline":"A genuine and mostly rigorous extension of Riesz-potential aggregation-diffusion theory to the nonlinear (s,p) case; the main theorems hold up, with one compactness step that needs a real proof.","tokens_in":34129,"tokens_out":4642,"would_cite":true,"duration_ms":46035,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K44","35R11","49K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that stationary states of an aggregation-diffusion model with a nonlinear Riesz potential are exactly optimizers of a Hardy-Littlewood-Sobolev inequality, and describes their sharp limit as the fractional parameter goes…","keywords":["aggregation-diffusion equations","nonlinear Riesz potential","stationary states","Hardy-Littlewood-Sobolev inequality","critical mass","fair competition regime","porous medium diffusion","s-to-0 limit"],"falsifier":"For a test case with $p\\neq 2$ (for instance $p=3$, $N=2$, $m>p'$), compute the $W^{1,1}$ norm of the minimizers $\\rho_s$ as $s\\to 0$; an unbounded supremum $\\sup_s\\|\\nabla\\rho_s\\|_{L^1}$ would falsify Proposition 6.6 and with it the diffusion-dominated part of Theorem 2.2.","tokens_in":33018,"feed_emoji":"⚖️","tokens_out":14268,"duration_ms":125084,"temperature":0.7,"pith_summary":"An aggregation-diffusion equation in which porous-medium diffusion competes with a nonlocal attraction described by a nonlinear Riesz potential has stationary states that are exactly the extremal profiles of a Hardy-Littlewood-Sobolev inequality. The paper proves that these extremals are radially decreasing, compactly supported, Hölder-regular, and satisfy an explicit Euler-Lagrange equation. In the diffusion-dominated regime each extremal can be rescaled by a unique dilation into a global minimizer of the free energy for any prescribed mass, while in the fair-competition regime minimizers exist only at a critical mass. As the fractional parameter $s$ goes to zero the interaction becomes a backward diffusion, and the minimizers converge to the characteristic function of a ball, or, at the critical exponent, either vanish uniformly or concentrate into a Dirac mass according to the value of the sensitivity. The interest is that equilibrium questions for a nonlocal evolution equation are settled by a sharp classical inequality, with an explicit asymptotic picture.","feed_headline":"Aggregation-diffusion equilibria reduce to HLS optimizers","feed_subtitle":"As the fractional range vanishes, minimizers become a ball, or collapse to a point at critical mass.","key_machinery":"The object that carries the argument is the nonlinear $(s,p)$ Riesz potential $K_{s,p}(\\rho)=K_{s/2}*\\left(K_{s/2}*\\rho\\right)^{p'-1}$, with $(s,p)$ energy $I_{s,p}(\\rho)=\\frac{1}{p'}\\int\\left(K_{s/2}*\\rho\\right)^{p'}$ entering $F_{s,p}(\\rho)=\\frac{1}{m-1}\\int\\rho^m-\\frac{\\chi}{p'}I_{s,p}(\\rho)$. The Hardy-Littlewood-Sobolev inequality (2.3) with sharp constant $H^*_{m,s,p}$ is the tool that controls the aggregation term; existence of extremals comes from the Lieb-Oxford rearrangement-compactness method, the Euler-Lagrange equation from first variations, and regularity from a bootstrap mixing Riesz-potential bounds with $C^{0,\\gamma}$ estimates. Mass-invariant dilations $\\rho_\\lambda(x)=\\lambda^N\\rho(\\lambda x)$ and the critical exponent $m_c=p'-\\frac{sp'}{N}$ fix the balance between diffusion and aggregation, and the optimal dilation factor (5.1) converts each HLS extremal into a minimizer of $F_{s,p}$. In the $s\\to 0$ passage, Kurokawa's approximation theorem, $K_{s/2}*h\\to h$ in $L^p$, is what turns the nonlocal interaction term into a backward diffusion.","core_discovery":"The paper's central claim is Theorem 2.1: for $1<p<\\infty$, $0<sp<N$, and $m>(p^{*}_s)'$ with $p^{*}_s=\\frac{Np}{N-sp}$, the best constant in the Hardy-Littlewood-Sobolev-type inequality $\\|K_{s/2}*h\\|_{p'}^{p'} \\le H\\|h\\|_1^{p'\\vartheta_0}\\|h\\|_m^{p'(1-\\vartheta_0)}$ is attained. Every optimizer is, up to translation, radially nonincreasing, compactly supported, Hölder regular, smooth in the interior of its support, and satisfies $\\rho^{m-1}=a\\left(K_{s,p}(\\rho)-C\\right)_+$ for positive constants $a,C$. In the diffusion-dominated case $m>m_c=p'-\\frac{sp'}{N}$, each optimizer of mass $M$ has a unique mass-invariant dilation that minimizes $F_{s,p}$ over $Y_M$, and minimizers exist for every mass; in the fair-competition case $m=m_c$, minimizers exist exactly at the critical mass $M_c=\\left(\\frac{p^{*}_s}{\\chi H^{*}_{m_c,s,p}}\\right)^{N/(sp')}$ and have zero energy. Theorem 2.2 describes the limit $s\\to 0$: for $m>p'$ the minimizers converge strongly in every $L^q$, $1<q<\\infty$, to the characteristic function of a ball, the unique minimizer of the limiting functional, while for $m=p'$ they converge uniformly to zero if $0<\\chi<p$ and to the point mass $M\\delta_0$ in the sense of measures if $\\chi>p$.","pith_inferences":["Beyond the paper: a quantitative stability estimate for the nonlinear HLS inequality, analogous to the one known for $p=2$, would turn the $s\\to 0$ convergence into a rate.","Beyond the paper: the threshold behaviour at $m=p'$ suggests a phase transition at $\\chi=p$ that could be probed by expanding the energy in powers of $s$; the paper does not perform that expansion.","Beyond the paper: uniqueness of stationary states of fixed mass is left open for $p\\neq 2$; if established, the one-parameter dilation family would give a complete classification of equilibria.","Beyond the paper: the compactness step for $s\\to 0$ is taken from the $p=2$ case; checking that the $W^{1,1}$ bound transfers to general $p$ is a concrete way to confirm Theorem 2.2."],"forward_implications":["For every mass $M>0$ in the diffusion-dominated regime, the free energy has a minimizer, so the evolution equation admits radial, compactly supported, Hölder-regular stationary states.","At fair competition, minimizers exist if and only if $M=M_c$; the minimum energy at the critical mass is zero, extending the Keller-Segel critical-mass phenomenon to the nonlinear potential.","Every minimizer is an HLS extremal, so stationary states inherit radial monotonicity, compact support, interior smoothness, and the pointwise Euler-Lagrange equation.","In the limit $s\\to 0$ with $m>p'$, the minimizers converge strongly in $L^q$ to the characteristic function of the explicit ball $\\rho_0(x)=\\left(\\frac{\\chi}{p}\\right)^{1/(m-p')}\\mathbf{1}_{B_{R_0}}(x)$.","In the limiting fair-competition case $m=p'$, the threshold $\\chi=p$ separates uniform vanishing from concentration to $M\\delta_0$ as $s\\to 0$."],"supporting_citations":[{"why":"Supplies the linear-potential ($p=2$) theory whose $s\\to 0$ machinery and $W^{1,1}$ compactness lemma are invoked without proof for the nonlinear case.","marker":"[28]"},{"why":"Establishes the critical-mass phenomenon for fair-competition Keller-Segel in the Newtonian case, the pattern generalised here.","marker":"[6]"},{"why":"Gives the fair-competition link between HLS optimizers and energy minimizers for Riesz potentials with $p=2$.","marker":"[10]"},{"why":"Supplies the diffusion-dominated ground-state existence and the Hölder-regularity bootstrap adapted to nonlinear $p$.","marker":"[19]"},{"why":"Provides the rearrangement inequalities and Hardy-Littlewood-Sobolev theorems used to obtain radial extremals and compactness.","marker":"[35]"},{"why":"Provides the Lieb-Oxford method for existence of maximisers of the sharp constant.","marker":"[34]"},{"why":"Supplies the first-variation argument used to derive the Euler-Lagrange equation for extremals.","marker":"[16]"},{"why":"Kurokawa's approximation-of-identity theorem for Riesz kernels, used to pass to the $s\\to 0$ limit.","marker":"[32]"},{"why":"Defines the nonlinear Riesz potential that is the paper's central object.","marker":"[38]"}],"fun_headline_variants":["Aggregation-diffusion equilibria are HLS inequality extremals","Vanishing fractional range turns minimizers into a ball or point mass","Nonlinear potential equilibria optimize Hardy-Littlewood-Sobolev","Critical mass yields zero-energy minimizers and point collapse"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $s\\to 0$ compactness step is inherited from the linear-potential case $p=2$ by asserting that the proof of the $W^{1,1}$ bound is the same; if that bound does not transfer to $p\\neq 2$, Theorem 2.2 loses its compactness and does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Aggregation-diffusion equilibria are HLS inequality extremals","Vanishing fractional range turns minimizers into a ball or point mass","Nonlinear potential equilibria optimize Hardy-Littlewood-Sobolev","Critical mass yields zero-energy minimizers and point collapse"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000764,"raw_usage":{"total_tokens":3439,"prompt_tokens":1041,"completion_tokens":2398,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":657,"completion_tokens_details":{"reasoning_tokens":2325}},"tokens_in":657,"tokens_out":2398,"duration_ms":17953,"temperature":1.0,"reasoning_tokens":2325,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:45:01.556430+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a test case with $p\\neq 2$ (for instance $p=3$, $N=2$, $m>p'$), compute the $W^{1,1}$ norm of the minimizers $\\rho_s$ as $s\\to 0$; an unbounded supremum $\\sup_s\\|\\nabla\\rho_s\\|_{L^1}$ would falsify Proposition 6.6 and with it the diffusion-dominated part of Theorem 2.2.","supporting_citations":[{"cited_title":"Huang, E","cited_arxiv_id":null,"evidence_quote":"Supplies the linear-potential ($p=2$) theory whose $s\\to 0$ machinery and $W^{1,1}$ compactness lemma are invoked without proof for the nonlinear case."},{"cited_title":"Blanchet, J.A","cited_arxiv_id":null,"evidence_quote":"Establishes the critical-mass phenomenon for fair-competition Keller-Segel in the Newtonian case, the pattern generalised here."},{"cited_title":"Calvez, J","cited_arxiv_id":null,"evidence_quote":"Gives the fair-competition link between HLS optimizers and energy minimizers for Riesz potentials with $p=2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the diffusion-dominated ground-state existence and the Hölder-regularity bootstrap adapted to nonlinear $p$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the rearrangement inequalities and Hardy-Littlewood-Sobolev theorems used to obtain radial extremals and compactness."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Lieb-Oxford method for existence of maximisers of the sharp constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the first-variation argument used to derive the Euler-Lagrange equation for extremals."},{"cited_title":"Kurokawa, On the Riesz and Bessel kernels as approximations of the identity, Sci","cited_arxiv_id":null,"evidence_quote":"Kurokawa's approximation-of-identity theorem for Riesz kernels, used to pass to the $s\\to 0$ limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the nonlinear Riesz potential that is the paper's central object."}],"review_version":1}