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Discrete heat equation with irregular thermal conductivity and tempered distributional data 具有不规则导热系数和回火分布数据的离散热方程
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-04-20 DOI: 10.1017/prm.2023.84
M. Chatzakou, Aparajita Dasgupta, Michael Ruzhansky, Abhilash Tushir
{"title":"Discrete heat equation with irregular thermal conductivity and tempered distributional data","authors":"M. Chatzakou, Aparajita Dasgupta, Michael Ruzhansky, Abhilash Tushir","doi":"10.1017/prm.2023.84","DOIUrl":"https://doi.org/10.1017/prm.2023.84","url":null,"abstract":"In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice \u0000 \u0000 $hbar mathbb {Z}^n$\u0000 \u0000 \u0000 . We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces \u0000 \u0000 $ell ^{2}_{s}(hbar mathbb {Z}^n)$\u0000 \u0000 \u0000 , \u0000 \u0000 $s in mathbb {R}$\u0000 \u0000 \u0000 , which is enough to prove the well-posedness in the space of tempered distributions \u0000 \u0000 $mathcal {S}'(hbar mathbb {Z}^n)$\u0000 \u0000 \u0000 . Notably, when \u0000 \u0000 $s=0$\u0000 \u0000 \u0000 , we show that for \u0000 \u0000 $hbar rightarrow 0$\u0000 \u0000 \u0000 , the classical (resp. very weak) solution of the heat equation in the Euclidean setting \u0000 \u0000 $mathbb {R}^n$\u0000 \u0000 \u0000 is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting \u0000 \u0000 $hbar mathbb {Z}^n$\u0000 \u0000 \u0000 .","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"95 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76589794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 3
Remarks on countable subadditivity 可数子可加性的注释
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-04-16 DOI: 10.1017/prm.2023.77
L. Grafakos, M. Visan
{"title":"Remarks on countable subadditivity","authors":"L. Grafakos, M. Visan","doi":"10.1017/prm.2023.77","DOIUrl":"https://doi.org/10.1017/prm.2023.77","url":null,"abstract":"We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80330319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 for the fractional p-Laplacian and its application to the extended Nirenberg problem 分数阶p-拉普拉斯算子的收敛性及其在扩展Nirenberg问题中的应用
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-04-05 DOI: 10.1017/prm.2023.32
Zhiwen Zhao
{"title":"Convergence for the fractional p-Laplacian and its application to the extended Nirenberg problem","authors":"Zhiwen Zhao","doi":"10.1017/prm.2023.32","DOIUrl":"https://doi.org/10.1017/prm.2023.32","url":null,"abstract":"The main objective of this paper is to establish the convergence for the fractional \u0000 \u0000 $p$\u0000 \u0000 \u0000 -Laplacian of sequences of nonnegative functions with \u0000 \u0000 $p>2$\u0000 \u0000 \u0000 . Furthermore, we show the blow-up phenomena for solutions to the extended Nirenberg problem modelled by fractional \u0000 \u0000 $p$\u0000 \u0000 \u0000 -Laplacian with the prescribed negative functions.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"37 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75432046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
PRM volume 153 issue 2 Cover and Back matter PRM卷153第2期封面和封底
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-29 DOI: 10.1017/prm.2023.30
{"title":"PRM volume 153 issue 2 Cover and Back matter","authors":"","doi":"10.1017/prm.2023.30","DOIUrl":"https://doi.org/10.1017/prm.2023.30","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"153 1","pages":"b1 - b2"},"PeriodicalIF":1.3,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87419417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
PRM volume 153 issue 2 Cover and Front matter PRM第153卷第2期封面和封面问题
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-29 DOI: 10.1017/prm.2023.29
{"title":"PRM volume 153 issue 2 Cover and Front matter","authors":"","doi":"10.1017/prm.2023.29","DOIUrl":"https://doi.org/10.1017/prm.2023.29","url":null,"abstract":"","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"14 1","pages":"f1 - f2"},"PeriodicalIF":1.3,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81467180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system 三维不可压缩Phan-Thien-Tanner系统的消失极限
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-27 DOI: 10.1017/prm.2023.27
Yuhui Chen, Minling Li, Qinghe Yao, Z. Yao
{"title":"Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system","authors":"Yuhui Chen, Minling Li, Qinghe Yao, Z. Yao","doi":"10.1017/prm.2023.27","DOIUrl":"https://doi.org/10.1017/prm.2023.27","url":null,"abstract":"This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time \u0000 \u0000 $t$\u0000 \u0000 \u0000 and coefficients \u0000 \u0000 $a$\u0000 \u0000 \u0000 and \u0000 \u0000 $b$\u0000 \u0000 \u0000 , then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution \u0000 \u0000 $(u,,tau )$\u0000 \u0000 \u0000 of PTT system with \u0000 \u0000 $0leq bleq Ca$\u0000 \u0000 \u0000 converges globally in time to some limit \u0000 \u0000 $(widetilde {u},,widetilde {tau })$\u0000 \u0000 \u0000 in a suitable Sobolev space when \u0000 \u0000 $a$\u0000 \u0000 \u0000 and \u0000 \u0000 $b$\u0000 \u0000 \u0000 go to zero simultaneously (or, only \u0000 \u0000 $b$\u0000 \u0000 \u0000 goes to zero). We may check that \u0000 \u0000 $(widetilde {u},,widetilde {tau })$\u0000 \u0000 \u0000 is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80527328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
A note on the differentiability of discrete Palmer's linearization 关于离散帕尔默线性化的可微性的注记
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-22 DOI: 10.1017/prm.2023.25
Álvaro Castañeda, N'estor Jara
{"title":"A note on the differentiability of discrete Palmer's linearization","authors":"Álvaro Castañeda, N'estor Jara","doi":"10.1017/prm.2023.25","DOIUrl":"https://doi.org/10.1017/prm.2023.25","url":null,"abstract":"In the context of discrete nonautonomous dynamics, we prove that the homeomorphisms in the linearization theorem are \u0000 \u0000 $C^2$\u0000 \u0000 \u0000 diffeomorphisms. In contrast to other related works, our result does not involve non-resonance conditions or spectral gaps. Our approach is based on the interlacing of the properties of nonautonomous hyperbolicity of the linear part, and boundedness and Lipschitzness of the nonlinearities. Moreover, we propose a functional approach to find conditions for regularity of arbitrary degree.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"100 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74823387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Avoidance loci and tropicalizations of real bitangents to plane quartics 真实生物对平面四分体的回避轨迹和热带化
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-14 DOI: 10.1017/prm.2023.67
H. Markwig, S. Payne, Kristin M. Shaw
{"title":"Avoidance loci and tropicalizations of real bitangents to plane quartics","authors":"H. Markwig, S. Payne, Kristin M. Shaw","doi":"10.1017/prm.2023.67","DOIUrl":"https://doi.org/10.1017/prm.2023.67","url":null,"abstract":"We compare two partitions of real bitangents to smooth plane quartics into sets of 4: one coming from the closures of connected components of the avoidance locus and another coming from tropical geometry. When both are defined, we use the Tarski principle for real closed fields in combination with the topology of real plane quartics and the tropical geometry of bitangents and theta characteristics to show that they coincide.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89543522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions 具有实际边界条件的趋化- navier - stokes系统的全局经典解
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-03-02 DOI: 10.1017/prm.2023.19
Chunhua Jin
{"title":"Global classical solution to the chemotaxis-Navier-Stokes system with some realistic boundary conditions","authors":"Chunhua Jin","doi":"10.1017/prm.2023.19","DOIUrl":"https://doi.org/10.1017/prm.2023.19","url":null,"abstract":"In this paper, we consider the chemotaxis-Navier-Stokes model with realistic boundary conditions matching the experiments of Hillesdon, Kessler et al. in a two-dimensional periodic strip domain. For the lower boundary, we impose the usual homogeneous Neumann-Neumann-Dirichlet boundary condition. While, for the upper boundary, since it is open to the atmosphere, we consider three kinds of different mixed non-homogeneous boundary conditions, that is, (i) Neumann-Dirichlet-Navier slip boundary condition; (ii) Zero flux-Dirichlet-Navier slip boundary condition; (iii) Zero flux-Robin-Navier slip boundary condition. For boundary conditions (i) and (iii), the existence and uniqueness of global classical solutions for any initial data and any large chemotactic sensitivity coefficient is established, and for boundary condition (ii), the existence and uniqueness of global classical solutions for any initial data and small chemotactic sensitivity coefficient is proved.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"196 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79846640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 1
On a class of difference operator and its applications to a family of analytic functions 一类差分算子及其在解析函数族中的应用
IF 1.3 3区 数学
Proceedings of the Royal Society of Edinburgh Section A-Mathematics Pub Date : 2023-02-27 DOI: 10.1017/prm.2023.8
R. K. Raina, J. Sokół, K. Tra̧bka-Wiȩcław
{"title":"On a class of difference operator and its applications to a family of analytic functions","authors":"R. K. Raina, J. Sokół, K. Tra̧bka-Wiȩcław","doi":"10.1017/prm.2023.8","DOIUrl":"https://doi.org/10.1017/prm.2023.8","url":null,"abstract":"This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85468952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
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