{"title":"广义黎曼ζ热流","authors":"Víctor Castillo , Claudio Muñoz , Felipe Poblete , Vicente Salinas","doi":"10.1016/j.jfa.2025.110879","DOIUrl":null,"url":null,"abstract":"<div><div>We consider the PDE flow associated to Riemann zeta and general Dirichlet <em>L</em>-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet <em>L</em>-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet <em>L</em>-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet <em>L</em>-functions and data initially on the right of a possible pole at <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>. Additional global well-posedness and convergence results are proved in the case of the defocussing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the “focusing” model, and prove blow-up of real-valued solutions near the pole <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 10","pages":"Article 110879"},"PeriodicalIF":1.7000,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The generalized Riemann zeta heat flow\",\"authors\":\"Víctor Castillo , Claudio Muñoz , Felipe Poblete , Vicente Salinas\",\"doi\":\"10.1016/j.jfa.2025.110879\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We consider the PDE flow associated to Riemann zeta and general Dirichlet <em>L</em>-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet <em>L</em>-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet <em>L</em>-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet <em>L</em>-functions and data initially on the right of a possible pole at <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>. Additional global well-posedness and convergence results are proved in the case of the defocussing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the “focusing” model, and prove blow-up of real-valued solutions near the pole <span><math><mi>s</mi><mo>=</mo><mn>1</mn></math></span>.</div></div>\",\"PeriodicalId\":15750,\"journal\":{\"name\":\"Journal of Functional Analysis\",\"volume\":\"288 10\",\"pages\":\"Article 110879\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-02-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Functional Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022123625000618\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123625000618","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们考虑与黎曼ζ和一般狄利克雷l函数相关的偏微分方程流。这些模型以出现在经典数论问题中的非线性为特征,并推广了Broughan和Barnett研究的经典全纯黎曼流。狄利克雷l函数的每一个零都是模型的精确解。本文首先证明了任何初始条件远离极点的Dirichlet l -函数流(只要存在)在Duhamel意义上的有界连续解的局部存在性。在第二个结果中,我们证明了Dirichlet l -函数形式的非线性的整体存在性,并且数据最初在s=1的可能极点的右边。在离焦Riemann zeta非线性和初始数据位于实线上且接近zeta的平凡零点的情况下,证明了该方法的全局适定性和收敛性。并证明了任意稳定零点的渐近稳定性。最后,在Riemann zeta情况下,我们考虑了“聚焦”模型,并证明了在极点s=1附近实值解的爆破。
We consider the PDE flow associated to Riemann zeta and general Dirichlet L-functions. These are models characterized by nonlinearities appearing in classical number theory problems, and generalizing the classical holomorphic Riemann flow studied by Broughan and Barnett. Each zero of a Dirichlet L-function is an exact solution of the model. In this paper, we first show local existence of bounded continuous solutions in the Duhamel sense to any Dirichlet L-function flow with initial condition far from the pole (as long as this exists). In a second result, we prove global existence in the case of nonlinearities of the form Dirichlet L-functions and data initially on the right of a possible pole at . Additional global well-posedness and convergence results are proved in the case of the defocussing Riemann zeta nonlinearity and initial data located on the real line and close to the trivial zeros of the zeta. The asymptotic stability of any stable zero is also proved. Finally, in the Riemann zeta case, we consider the “focusing” model, and prove blow-up of real-valued solutions near the pole .
期刊介绍:
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis