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Communications on Pure and Applied Mathematics最新文献

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On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates 关于二维重力波的波湍流理论，I：确定性能量估计
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-09-11 DOI: 10.1002/cpa.22224
Yu Deng, Alexandru D. Ionescu, Fabio Pusateri
{"title":"On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates","authors":"Yu Deng, Alexandru D. Ionescu, Fabio Pusateri","doi":"10.1002/cpa.22224","DOIUrl":"https://doi.org/10.1002/cpa.22224","url":null,"abstract":"Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus of size . Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain ‐type norm is small. This energy inequality is of “quintic” type: if the norm is , then the increment of the high‐order energies is controlled for times of the order , consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order , in a suitable scaling regime relating and . For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable ‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size are regular for times of the order . In particular, on the real line, solutions exist for times of order . This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142170872","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}
引用次数: 0
Convergence to the planar interface for a nonlocal free‐boundary evolution 收敛到非局部自由边界演化的平面界面
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-09-05 DOI: 10.1002/cpa.22225
Felix Otto, Richard Schubert, Maria G. Westdickenberg
{"title":"Convergence to the planar interface for a nonlocal free‐boundary evolution","authors":"Felix Otto, Richard Schubert, Maria G. Westdickenberg","doi":"10.1002/cpa.22225","DOIUrl":"https://doi.org/10.1002/cpa.22225","url":null,"abstract":"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.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotics of block Toeplitz determinants with piecewise continuous symbols 具有片断连续符号的块托普利兹行列式的渐近论
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-08-28 DOI: 10.1002/cpa.22223
Estelle Basor, Torsten Ehrhardt, Jani A. Virtanen
{"title":"Asymptotics of block Toeplitz determinants with piecewise continuous symbols","authors":"Estelle Basor, Torsten Ehrhardt, Jani A. Virtanen","doi":"10.1002/cpa.22223","DOIUrl":"https://doi.org/10.1002/cpa.22223","url":null,"abstract":"We determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that <jats:disp-formula/>where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global regularity for critical SQG in bounded domains 有界域中临界 SQG 的全局正则性
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-07-24 DOI: 10.1002/cpa.22221
Peter Constantin, Mihaela Ignatova, Quoc‐Hung Nguyen
{"title":"Global regularity for critical SQG in bounded domains","authors":"Peter Constantin, Mihaela Ignatova, Quoc‐Hung Nguyen","doi":"10.1002/cpa.22221","DOIUrl":"https://doi.org/10.1002/cpa.22221","url":null,"abstract":"We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . 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.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"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}
引用次数: 0
Almost sharp lower bound for the nodal volume of harmonic functions 谐函数节点体积的近似尖锐下界
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-29 DOI: 10.1002/cpa.22207
Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori
{"title":"Almost sharp lower bound for the nodal volume of harmonic functions","authors":"Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori","doi":"10.1002/cpa.22207","DOIUrl":"https://doi.org/10.1002/cpa.22207","url":null,"abstract":"This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove <jats:disp-formula/>where the doubling index is a notion of growth defined by <jats:disp-formula/>This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are the notion of <jats:italic>stable growth</jats:italic>, and a multiscale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant imuprovement over the previous best‐known bound , which implied Nadirashvili's conjecture.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141177297","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}
引用次数: 0
Allen–Cahn solutions with triple junction structure at infinity 无穷远处具有三重结点结构的艾伦-卡恩解
IF 3.1 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-17 DOI: 10.1002/cpa.22204
Étienne Sandier, Peter Sternberg
{"title":"Allen–Cahn solutions with triple junction structure at infinity","authors":"Étienne Sandier,&nbsp;Peter Sternberg","doi":"10.1002/cpa.22204","DOIUrl":"10.1002/cpa.22204","url":null,"abstract":"<p>We construct an entire solution <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>U</mi>\u0000 <mo>:</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$U:mathbb {R}^2rightarrow mathbb {R}^2$</annotation>\u0000 </semantics></math> to the elliptic system\u0000\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140953873","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}
引用次数: 0
Multiplicative chaos measures from thick points of log-correlated fields 来自对数相关场厚点的乘法混沌度量
IF 3.1 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-17 DOI: 10.1002/cpa.22205
Janne Junnila, Gaultier Lambert, Christian Webb
{"title":"Multiplicative chaos measures from thick points of log-correlated fields","authors":"Janne Junnila,&nbsp;Gaultier Lambert,&nbsp;Christian Webb","doi":"10.1002/cpa.22205","DOIUrl":"10.1002/cpa.22205","url":null,"abstract":"<p>We prove that multiplicative chaos measures can be constructed from extreme level sets or <i>thick points</i> of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix (CUE), using known asymptotics for Toeplitz determinant with (merging) Fisher–Hartwig singularities. Hence, this proves a conjecture of Fyodorov and Keating concerning the fluctuations of the volume of thick points of the CUE characteristic polynomial.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22205","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140954009","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}
引用次数: 0
Twisted Kähler–Einstein metrics in big classes 大类中的扭曲凯勒-爱因斯坦度量
IF 3 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-17 DOI: 10.1002/cpa.22206
Tamás Darvas, Kewei Zhang
{"title":"Twisted Kähler–Einstein metrics in big classes","authors":"Tamás Darvas, Kewei Zhang","doi":"10.1002/cpa.22206","DOIUrl":"https://doi.org/10.1002/cpa.22206","url":null,"abstract":"We prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K‐energy in our arguments, and our techniques provide a simple roadmap to prove YTD existence theorems for KE type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li–Tian–Wang's existence theorem in the log Fano setting.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140953951","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}
引用次数: 0
Infinite-width limit of deep linear neural networks 深度线性神经网络的无穷宽极限
IF 3.1 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-06 DOI: 10.1002/cpa.22200
Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, Alessio Figalli
{"title":"Infinite-width limit of deep linear neural networks","authors":"Lénaïc Chizat,&nbsp;Maria Colombo,&nbsp;Xavier Fernández-Real,&nbsp;Alessio Figalli","doi":"10.1002/cpa.22200","DOIUrl":"10.1002/cpa.22200","url":null,"abstract":"<p>This paper studies the infinite-width limit of deep linear neural networks (NNs) initialized with random parameters. We obtain that, when the number of parameters diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear NN. Moreover, even if the weights remain random, we get their precise law along the training dynamics, and prove a quantitative convergence result of the linear predictor in terms of the number of parameters. We finally study the continuous-time limit obtained for infinitely wide linear NNs and show that the linear predictors of the NN converge at an exponential rate to the minimal <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <annotation>$ell _2$</annotation>\u0000 </semantics></math>-norm minimizer of the risk.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22200","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845799","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}
引用次数: 0
The Calogero–Moser derivative nonlinear Schrödinger equation 卡洛吉罗-莫泽导数非线性薛定谔方程
IF 3.1 1区 数学
Communications on Pure and Applied Mathematics Pub Date : 2024-05-06 DOI: 10.1002/cpa.22203
Patrick Gérard, Enno Lenzmann
{"title":"The Calogero–Moser derivative nonlinear Schrödinger equation","authors":"Patrick Gérard,&nbsp;Enno Lenzmann","doi":"10.1002/cpa.22203","DOIUrl":"10.1002/cpa.22203","url":null,"abstract":"<p>We study the Calogero–Moser derivative nonlinear Schrödinger NLS equation\u0000\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845697","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}
引用次数: 0
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