Communications on Pure and Applied Mathematics最新文献

{"title":"Issue Information ‐ TOC","authors":"","doi":"10.1002/cpa.70052","DOIUrl":"https://doi.org/10.1002/cpa.70052","url":null,"abstract":"","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"8 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147739341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of Viscous Three‐Dimensional Stratified Couette Flow via Dispersion and Mixing","authors":"Michele Coti Zelati, Augusto Del Zotto, Klaus Widmayer","doi":"10.1002/cpa.70048","DOIUrl":"https://doi.org/10.1002/cpa.70048","url":null,"abstract":"This article explores the stability of stratified Couette flow in the viscous Boussinesq equations. In this system, mixing effects arise from the shearing background, and gravity acts as a restoring force leading to dispersive internal gravity waves. These mechanisms are of fundamentally different nature and relevant in complementary dynamical regimes. Our study combines them to establish a bound for the non‐linear transition threshold, which is quantitatively larger than the inverse Reynolds number , and increases with stronger stratification resp. gravity.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"33 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147739342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global Well‐Posedness and Relaxation for Solutions of the Fokker–Planck–Alignment Equations","authors":"Roman Shvydkoy","doi":"10.1002/cpa.70049","DOIUrl":"https://doi.org/10.1002/cpa.70049","url":null,"abstract":"In this paper, we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker–Planck–Alignment models, which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no‐vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker–Smale model, we demonstrate that any bounded data with finite higher moment, , , gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness‐based renormalization procedure, which circumvents the problem of degenerate diffusion in nonperturbative regime.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147739343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effective Velocities in the Toda Lattice","authors":"Amol Aggarwal","doi":"10.1002/cpa.70046","DOIUrl":"https://doi.org/10.1002/cpa.70046","url":null,"abstract":"In this paper, we consider the Toda lattice at thermal equilibrium, meaning that its variables and are independent Gaussian and Gamma random variables, respectively. This model can be thought of a dense collection of many “quasiparticles” that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"50 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147719935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Convergence Rates in Multiscale Elliptic Homogenization","authors":"Weisheng Niu, Yao Xu, Jinping Zhuge","doi":"10.1002/cpa.70047","DOIUrl":"https://doi.org/10.1002/cpa.70047","url":null,"abstract":"This paper is devoted to the quantitative homogenization of multiscale elliptic operator , where , , and . We assume that is 1‐periodic in each and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios . In the present paper, under the assumption of real analytic coefficients, we introduce the so‐called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to . This convergence rate is optimal in the sense that cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double‐log scale‐separation condition.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"7 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147681955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Eventual regularization of fractional mean curvature flow","authors":"Stephen Cameron","doi":"10.1002/cpa.70028","DOIUrl":"10.1002/cpa.70028","url":null,"abstract":"<p>We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 6","pages":"1399-1448"},"PeriodicalIF":2.7,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145955149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias","authors":"Yifan Chen,&nbsp;Xiaoou Cheng,&nbsp;Jonathan Niles-Weed,&nbsp;Jonathan Weare","doi":"10.1002/cpa.70032","DOIUrl":"10.1002/cpa.70032","url":null,"abstract":"&lt;div&gt;\u0000 \u0000 &lt;p&gt;The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of the problem increases, the number of iterations required to ensure convergence within a desired error in the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$W_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; metric scales in proportion to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; or &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msqrt&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msqrt&gt;\u0000 &lt;annotation&gt;$sqrt {d}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we argue that, despite this poor scaling of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$W_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; error for the full set of variables, the behavior for a &lt;i&gt;small number&lt;/i&gt; of variables can be significantly better: A number of iterations proportional to &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, up to logarithmic terms in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;annotation&gt;$d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, often suffices for the algorithm to converge to within a desired &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$W_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; error for all &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;annotation&gt;$K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-marginals. We refer to this effect as &lt;i&gt;delocalization of bias&lt;/i&gt;. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;W&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;mi&gt;∞&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 6","pages":"1467-1491"},"PeriodicalIF":2.7,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146089380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fourier Mass Lower Bounds for Batchelor-Regime Passive Scalars","authors":"William Cooperman,&nbsp;Keefer Rowan","doi":"10.1002/cpa.70030","DOIUrl":"10.1002/cpa.70030","url":null,"abstract":"<p>Batchelor predicted that a passive scalar <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>ψ</mi>\u0000 <mi>ν</mi>\u0000 </msup>\u0000 <annotation>$psi ^nu$</annotation>\u0000 </semantics></math> with diffusivity <span></span><math>\u0000 <semantics>\u0000 <mi>ν</mi>\u0000 <annotation>$nu$</annotation>\u0000 </semantics></math>, advected by a smooth fluid velocity, should typically have Fourier mass distributed as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <msup>\u0000 <mover>\u0000 <mi>ψ</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <mi>ν</mi>\u0000 </msup>\u0000 <msup>\u0000 <mo>|</mo>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≈</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>k</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$|widehat{psi }^nu |^2(k) approx |k|^{-d}$</annotation>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>k</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mo>≪</mo>\u0000 <msup>\u0000 <mi>ν</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$|k| ll nu ^{-1/2}$</annotation>\u0000 </semantics></math>. For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"79 6","pages":"1449-1466"},"PeriodicalIF":2.7,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.70030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145993108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Benjamin–Ono Equation in the Zero‐Dispersion Limit for Rational Initial Data: Generation of Dispersive Shock Waves","authors":"Elliot Blackstone, Louise Gassot, Patrick Gérard, Peter D. Miller","doi":"10.1002/cpa.70044","DOIUrl":"https://doi.org/10.1002/cpa.70044","url":null,"abstract":"The leading‐order asymptotic behavior of the solution of the Cauchy initial‐value problem for the Benjamin–Ono equation in is obtained explicitly for generic rational initial data . An explicit asymptotic wave profile is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data , such that the solution of the Benjamin–Ono equation with dispersion parameter and initial data satisfies in the locally uniform sense as , provided a discriminant inequality holds implying that certain caustic curves in the ‐plane are avoided. In some cases, this convergence implies strong convergence. The asymptotic profile is consistent with the modulated multiphase wave solutions described by Dobrokhotov and Krichever.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"61 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147586397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant Measure and Universality of the 2D Yang–Mills Langevin Dynamic","authors":"Ilya Chevyrev, Hao Shen","doi":"10.1002/cpa.70043","DOIUrl":"https://doi.org/10.1002/cpa.70043","url":null,"abstract":"We prove that the Yang–Mills (YM) measure for the trivial principal bundle over the two‐dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge‐fixing and Bourgain's method for invariant measures. Several corollaries are presented including a gauge‐fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge‐covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"25 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147518739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}