. In this paper, we consider the properties of the solutions for the Oregonator system, which is the mathematical model of the celebrated Belousov–Zhabotinski˘ı reaction. We first investigate the dynamics of the model, and some fundamental analytic properties such as attractive rectangle and stability of the constant solution are estab-lished. Then, we consider the steady states of the model, and the existence and nonexistence of nonconstant steady states under various conditions on the parameters and the size of the reactor.
{"title":"Qualitative analysis of the Oregonator model","authors":"Jun Zhou","doi":"10.4064/ap200321-18-8","DOIUrl":"https://doi.org/10.4064/ap200321-18-8","url":null,"abstract":". In this paper, we consider the properties of the solutions for the Oregonator system, which is the mathematical model of the celebrated Belousov–Zhabotinski˘ı reaction. We first investigate the dynamics of the model, and some fundamental analytic properties such as attractive rectangle and stability of the constant solution are estab-lished. Then, we consider the steady states of the model, and the existence and nonexistence of nonconstant steady states under various conditions on the parameters and the size of the reactor.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70580648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity criteria for a density-dependent incompressible Ginzburg–Landau–Navier–Stokes system in a bounded domain","authors":"Jishan Fan, Lulu Jing, G. Nakamura, T. Tang","doi":"10.4064/ap190616-15-4","DOIUrl":"https://doi.org/10.4064/ap190616-15-4","url":null,"abstract":"","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70579318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the mean field equation α 2 ∆gu+ e u − 1 = 0 on S. We show that under some technical conditions, u has to be constantly zero for 1/3 ≤ α < 1. In particular, this is the case if u(x) = −u(−x) and u is odd symmetric about a plane. In the cases u(x) = −u(−x) with 1/3 ≤ α < 1 and u(x) = u(−x) with 1/4 ≤ α < 1, we analyze the additional symmetries of the nontrivial solution in detail.
{"title":"Trivial solution and symmetries of nontrivial solutions to a mean field equation","authors":"Jiaming Jin, Chuanxi Zhu","doi":"10.4064/ap191126-30-6","DOIUrl":"https://doi.org/10.4064/ap191126-30-6","url":null,"abstract":"We consider the mean field equation α 2 ∆gu+ e u − 1 = 0 on S. We show that under some technical conditions, u has to be constantly zero for 1/3 ≤ α < 1. In particular, this is the case if u(x) = −u(−x) and u is odd symmetric about a plane. In the cases u(x) = −u(−x) with 1/3 ≤ α < 1 and u(x) = u(−x) with 1/4 ≤ α < 1, we analyze the additional symmetries of the nontrivial solution in detail.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70579830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper surveys results on controllable systems with vanishing energy introduced by E. Priola and J. Zabczyk [SIAM J. Control Optim. 42 (2003), 1013–1032]. Applications to space travels and to partial differential equations are discussed.
{"title":"Controllable systems with vanishing energy","authors":"J. Zabczyk","doi":"10.4064/ap200421-29-9","DOIUrl":"https://doi.org/10.4064/ap200421-29-9","url":null,"abstract":"The paper surveys results on controllable systems with vanishing energy introduced by E. Priola and J. Zabczyk [SIAM J. Control Optim. 42 (2003), 1013–1032]. Applications to space travels and to partial differential equations are discussed.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70580697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan’s theorems A and B for quasi-coherent arc-meromorphous sheaves. 0. Introduction. In this note, building on the theory of arc-analytic functions initiated by the second named author [16], we introduce arcmeromorphous functions and arc-meromorphous sheaves on Nash manifolds. Arc-meromorphous functions are analogs for regulous and Nash regulous functions studied in [8] and [13], respectively. The term “regulous” is derived from “regular” and “continuous”, whereas “meromorphous” comes from “meromorphic” and “continuous”. Our theory of arc-meromorphous sheaves is developed in parallel to the theories of regulous sheaves [8] (see also the recent survey [14]) and Nash regulous sheaves [13]. It is established in [8] and [13] that Cartan’s theorems A and B hold for quasi-coherent regulous sheaves and quasi-coherent Nash regulous sheaves. Our main results are Theorem 2.4 (Cartan’s theorem A) and Theorem 2.5 (Cartan’s theorem B) for quasi-coherent arc-meromorphous sheaves. Recall that Cartan’s theorems A and B fail for coherent real algebraic sheaves [6, Example 12.1.5], [7, Theorem 1] and coherent Nash sheaves [11]. We refer to [6] for the general theory of semialgebraic sets, semialgebraic functions, and related concepts. Recall that a Nash manifold is an analytic submanifold X ⊂ Rn, for some n, which is also a semialgebraic set. A realvalued function on X is called a Nash function if it is both analytic and semialgebraic. By [22, Theorem VI.2.1, Remark VI.2.11], each Nash manifold is Nash isomorphic to a nonsingular algebraic set in Rm, for some m. 2020 Mathematics Subject Classification: 14P10, 14P20, 32B10, 58A07.
{"title":"Arc-meromorphous functions","authors":"W. Kucharz, K. Kurdyka","doi":"10.4064/ap200517-7-8","DOIUrl":"https://doi.org/10.4064/ap200517-7-8","url":null,"abstract":"We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan’s theorems A and B for quasi-coherent arc-meromorphous sheaves. 0. Introduction. In this note, building on the theory of arc-analytic functions initiated by the second named author [16], we introduce arcmeromorphous functions and arc-meromorphous sheaves on Nash manifolds. Arc-meromorphous functions are analogs for regulous and Nash regulous functions studied in [8] and [13], respectively. The term “regulous” is derived from “regular” and “continuous”, whereas “meromorphous” comes from “meromorphic” and “continuous”. Our theory of arc-meromorphous sheaves is developed in parallel to the theories of regulous sheaves [8] (see also the recent survey [14]) and Nash regulous sheaves [13]. It is established in [8] and [13] that Cartan’s theorems A and B hold for quasi-coherent regulous sheaves and quasi-coherent Nash regulous sheaves. Our main results are Theorem 2.4 (Cartan’s theorem A) and Theorem 2.5 (Cartan’s theorem B) for quasi-coherent arc-meromorphous sheaves. Recall that Cartan’s theorems A and B fail for coherent real algebraic sheaves [6, Example 12.1.5], [7, Theorem 1] and coherent Nash sheaves [11]. We refer to [6] for the general theory of semialgebraic sets, semialgebraic functions, and related concepts. Recall that a Nash manifold is an analytic submanifold X ⊂ Rn, for some n, which is also a semialgebraic set. A realvalued function on X is called a Nash function if it is both analytic and semialgebraic. By [22, Theorem VI.2.1, Remark VI.2.11], each Nash manifold is Nash isomorphic to a nonsingular algebraic set in Rm, for some m. 2020 Mathematics Subject Classification: 14P10, 14P20, 32B10, 58A07.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70580884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global existence and blowup for a degenerate parabolic equation with a free boundary","authors":"Youpeng Chen, Xingying Liu","doi":"10.4064/ap171230-26-5","DOIUrl":"https://doi.org/10.4064/ap171230-26-5","url":null,"abstract":"","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70574527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generic invariant measures for minimal iterated function systems of homeomorphisms of the circle","authors":"W. Czernous","doi":"10.4064/ap180518-12-4","DOIUrl":"https://doi.org/10.4064/ap180518-12-4","url":null,"abstract":"","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70576542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a regularity criterion for local strong solutions of the StokesMHD equations in terms of the velocity field in Besov space Ḃ ∞,∞.
在Besov空间Ḃ∞，∞上证明了StokesMHD方程的速度场局部强解的正则性判据。
{"title":"A regularity criterion for local strong solutions to the 3D Stokes-MHD equations","authors":"A. Alghamdi, S. Gala, M. Ragusa","doi":"10.4064/ap190307-21-9","DOIUrl":"https://doi.org/10.4064/ap190307-21-9","url":null,"abstract":"We prove a regularity criterion for local strong solutions of the StokesMHD equations in terms of the velocity field in Besov space Ḃ ∞,∞.","PeriodicalId":55513,"journal":{"name":"Annales Polonici Mathematici","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70578821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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