Communications on Pure and Applied Mathematics最新文献

{"title":"The α$alpha$‐SQG patch problem is illposed in C2,β$C^{2,beta }$ and W2,p$W^{2,p}$","authors":"Alexander Kiselev, Xiaoyutao Luo","doi":"10.1002/cpa.22236","DOIUrl":"https://doi.org/10.1002/cpa.22236","url":null,"abstract":"We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in <jats:italic>every</jats:italic> Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for <jats:italic>every</jats:italic> Sobolev space unless .","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"197 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645950","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":"Mean‐field limit of non‐exchangeable systems","authors":"Pierre‐Emmanuel Jabin, David Poyato, Juan Soler","doi":"10.1002/cpa.22235","DOIUrl":"https://doi.org/10.1002/cpa.22235","url":null,"abstract":"This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"248 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645951","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":"Semiconvexity estimates for nonlinear integro‐differential equations","authors":"Xavier Ros‐Oton, Clara Torres‐Latorre, Marvin Weidner","doi":"10.1002/cpa.22237","DOIUrl":"https://doi.org/10.1002/cpa.22237","url":null,"abstract":"In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"165 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642581","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 to the planar interface for a nonlocal free-boundary evolution","authors":"Felix Otto,&nbsp;Richard Schubert,&nbsp;Maria G. Westdickenberg","doi":"10.1002/cpa.22225","DOIUrl":"10.1002/cpa.22225","url":null,"abstract":"<p>We capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well-prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one-dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"161-208"},"PeriodicalIF":3.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142410","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":"Asymptotics of block Toeplitz determinants with piecewise continuous symbols","authors":"Estelle Basor,&nbsp;Torsten Ehrhardt,&nbsp;Jani A. Virtanen","doi":"10.1002/cpa.22223","DOIUrl":"10.1002/cpa.22223","url":null,"abstract":"<p>We determine the asymptotics of the block Toeplitz determinants <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>det</mo>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ϕ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$det T_n(phi)$</annotation>\u0000 </semantics></math> as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$nrightarrow infty$</annotation>\u0000 </semantics></math> for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>×</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$Ntimes N$</annotation>\u0000 </semantics></math> matrix-valued piecewise continuous functions <span></span><math>\u0000 <semantics>\u0000 <mi>ϕ</mi>\u0000 <annotation>$phi$</annotation>\u0000 </semantics></math> with a finitely many jumps under mild additional conditions. In particular, we prove that\u0000\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"120-160"},"PeriodicalIF":3.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090102","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":"Tight Lipschitz hardness for optimizing mean field spin glasses","authors":"Brice Huang,&nbsp;Mark Sellke","doi":"10.1002/cpa.22222","DOIUrl":"https://doi.org/10.1002/cpa.22222","url":null,"abstract":"&lt;p&gt;We study the problem of algorithmically optimizing the Hamiltonian &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$H_N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of a spherical or Ising mixed &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-spin glass. The maximum asymptotic value &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;OPT&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathsf {OPT}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$H_N/N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing (AMP) algorithms efficiently optimize &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$H_N/N$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; up to a value &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ALG&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathsf {ALG}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a &lt;i&gt;no overlap gap&lt;/i&gt; property (OGP). However, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ALG&lt;/mi&gt;\u0000 &lt;mo&gt;&lt;&lt;/mo&gt;\u0000 &lt;mi&gt;OPT&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;${mathsf {ALG}}&amp;lt; {mathsf {OPT}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; can also occur, and no efficient algorithm producing an objective value exceeding &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ALG&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathsf {ALG}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is known. We prove that for mixed even &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-spin models, no algorithm satisfying an &lt;i&gt;overlap concentration&lt;/i&gt; property can produce an objective larger than &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ALG&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathsf {ALG}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with non-negligible probability.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"60-119"},"PeriodicalIF":3.1,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142665074","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 regularity for critical SQG in bounded domains","authors":"Peter Constantin,&nbsp;Mihaela Ignatova,&nbsp;Quoc-Hung Nguyen","doi":"10.1002/cpa.22221","DOIUrl":"10.1002/cpa.22221","url":null,"abstract":"<p>We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math>. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 1","pages":"3-59"},"PeriodicalIF":3.1,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755298","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}