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.
{"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}
Felix Otto, Richard Schubert, Maria G. Westdickenberg
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.
{"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}
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 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.
{"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}
Peter Constantin, Mihaela Ignatova, Quoc‐Hung Nguyen
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.
{"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}
Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori
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 where the doubling index is a notion of growth defined by 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 stable growth, 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.
{"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}
We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points 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.
{"title":"Multiplicative chaos measures from thick points of log-correlated fields","authors":"Janne Junnila, Gaultier Lambert, 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}
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.
我们利用除法稳定性条件证明了大同调类中扭曲凯勒-爱因斯坦度量的存在性。特别是，当大同调时，我们得到了凯勒-爱因斯坦（KE）度量的统一游天-唐纳森（YTD）存在定理。为此，我们利用多势理论，从零开始建立了超越大背景下的藤田-大高（Fujita-Odaka）型三角不变式理论。我们在论证中不使用 K 能，我们的技术为证明 KE 类型度量的 YTD 存在性定理提供了一个简单的路线图，它只需要适当丁能的凸性。作为应用，我们给出了对数法诺环境中李天王存在性定理的简化证明。
{"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}
Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, Alessio Figalli
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 -norm minimizer of the risk.
本文研究了以随机参数初始化的深度线性神经网络（NN）的无限宽极限。我们发现，当参数数量发散时，训练动态（在精确意义上）会收敛到无限宽确定性线性神经网络的梯度下降动态。此外，即使权重仍然是随机的，我们也能沿着训练动态得到它们的精确规律，并证明了线性预测器在参数数量上的定量收敛结果。最后，我们研究了无限宽线性 NN 的连续时间极限，并证明 NN 的线性预测器以指数速度收敛到风险的最小正态最小化。
{"title":"Infinite-width limit of deep linear neural networks","authors":"Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, 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}
{"title":"The Calogero–Moser derivative nonlinear Schrödinger equation","authors":"Patrick Gérard, 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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：