The global existence for semilinear wave equations with space-dependent critical damping ∂ t u−∆u+ V0 |x| ∂tu = f(u) in an exterior domain is dealt with, where f(u) = |u|p−1u and f(u) = |u| are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition. Mathematics Subject Classification (2010): Primary:35L71, 35A01, Secondary:35L20, 35B40,
{"title":"On global existence for semilinear wave equations with space-dependent critical damping","authors":"M. Sobajima","doi":"10.2969/jmsj/87388738","DOIUrl":"https://doi.org/10.2969/jmsj/87388738","url":null,"abstract":"The global existence for semilinear wave equations with space-dependent critical damping ∂ t u−∆u+ V0 |x| ∂tu = f(u) in an exterior domain is dealt with, where f(u) = |u|p−1u and f(u) = |u| are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition. Mathematics Subject Classification (2010): Primary:35L71, 35A01, Secondary:35L20, 35B40,","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41972379","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}
Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an integer invariant for virtual knots. The sequence of n-writhes {Jn}n̸=0 of a virtual knot K satisfies ∑ n̸=0 nJn(K) = 0. They showed that for any sequence of integers {cn}n̸=0 with ∑ n̸=0 ncn = 0, there exists a virtual knot K with Jn(K) = cn for any n ̸= 0. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number and is denoted by uv . In this paper, we show that if {cn}n̸=0 is a sequence of integers with ∑ n̸=0 ncn = 0, then there exists a virtual knot K such that uv(K) = 1 and Jn(K) = cn for any n ̸= 0.
{"title":"A virtual knot whose virtual unknotting number equals one and a sequence of $n$-writhes","authors":"Y. Ohyama, Migiwa Sakurai","doi":"10.2969/JMSJ/84478447","DOIUrl":"https://doi.org/10.2969/JMSJ/84478447","url":null,"abstract":"Satoh and Taniguchi introduced the n-writhe Jn for each non-zero integer n, which is an integer invariant for virtual knots. The sequence of n-writhes {Jn}n̸=0 of a virtual knot K satisfies ∑ n̸=0 nJn(K) = 0. They showed that for any sequence of integers {cn}n̸=0 with ∑ n̸=0 ncn = 0, there exists a virtual knot K with Jn(K) = cn for any n ̸= 0. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number and is denoted by uv . In this paper, we show that if {cn}n̸=0 is a sequence of integers with ∑ n̸=0 ncn = 0, then there exists a virtual knot K such that uv(K) = 1 and Jn(K) = cn for any n ̸= 0.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44270488","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 sharp estimates for the renewal measure of a strongly nonlattice probability measure on the real line. In particular we consider the case where the measure has finite moments between 1 and 2. The proof uses Fourier analysis of tempered distributions.
{"title":"Estimates of the renewal measure","authors":"H. Carlsson","doi":"10.2969/JMSJ/83298329","DOIUrl":"https://doi.org/10.2969/JMSJ/83298329","url":null,"abstract":"We prove sharp estimates for the renewal measure of a strongly nonlattice probability measure on the real line. In particular we consider the case where the measure has finite moments between 1 and 2. The proof uses Fourier analysis of tempered distributions.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49098759","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 that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macr`i and Stellari. When the degree of $X$ is at least $3$, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.
{"title":"Bridgeland stability of minimal instanton bundles on Fano threefolds","authors":"Xuqiang Qin","doi":"10.2969/jmsj/89238923","DOIUrl":"https://doi.org/10.2969/jmsj/89238923","url":null,"abstract":"We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macr`i and Stellari. When the degree of $X$ is at least $3$, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43170506","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 compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces Ḃ n p p,1 × Ḃ n p −1 p,1 for all 1 ≤ p < 2n. However, if the data is in a larger scaling invariant class such as p > 2n, then the system is not well-posed. In this paper, we demonstrate that for the critical case p = 2n the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin [10] and Haspot [18] are indeed sharp in the framework of the homogeneous Besov spaces.
考虑临界Besov空间中的可压缩Navier-Stokes系统。已知系统在齐次Besov空间的缩放半不变空间中是(半)适定的Ḃ n p p,1 × Ḃ n p−1p,1对于所有1≤p < 2n。然而，如果数据是一个更大的尺度不变类，如p bbb20n，那么系统不是适定的。在本文中，我们通过构造一个初始数据序列来表示解映射在临界空间中的不连续，证明了对于临界情况p = 2n，系统是不适定的。我们的结果表明，在齐次Besov空间的框架下，由Danchin[10]和Haspot[18]引起的适定性结果确实是尖锐的。
{"title":"Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces","authors":"T. Iwabuchi, T. Ogawa","doi":"10.2969/JMSJ/81598159","DOIUrl":"https://doi.org/10.2969/JMSJ/81598159","url":null,"abstract":"We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces Ḃ n p p,1 × Ḃ n p −1 p,1 for all 1 ≤ p < 2n. However, if the data is in a larger scaling invariant class such as p > 2n, then the system is not well-posed. In this paper, we demonstrate that for the critical case p = 2n the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin [10] and Haspot [18] are indeed sharp in the framework of the homogeneous Besov spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574279","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}
Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators L = ∆ + V where ∆ is the Laplace-Beltrami operator and V is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of L together with its time derivatives and then apply them to obtain the weak type (1, 1) estimate of the functional calculus of Laplace transform type of √ L which is defined by M( √ L)f(x) := ́∞ 0 [√ Le−t √ Lf(x) ] m(t)dt where m(t) is a bounded function on [0,∞). In the setting of our study, both doubling condition of the measure on M and the smoothness of the operators’ kernels are missing. The purely imaginary power Lis, s ∈ R, is a special case of our result and an example of weak type (1, 1) estimates of a singular integral with non-smooth kernels on non-doubling spaces.
{"title":"Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends","authors":"Hong Chuong Doan","doi":"10.2969/JMSJ/83348334","DOIUrl":"https://doi.org/10.2969/JMSJ/83348334","url":null,"abstract":"Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators L = ∆ + V where ∆ is the Laplace-Beltrami operator and V is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of L together with its time derivatives and then apply them to obtain the weak type (1, 1) estimate of the functional calculus of Laplace transform type of √ L which is defined by M( √ L)f(x) := ́∞ 0 [√ Le−t √ Lf(x) ] m(t)dt where m(t) is a bounded function on [0,∞). In the setting of our study, both doubling condition of the measure on M and the smoothness of the operators’ kernels are missing. The purely imaginary power Lis, s ∈ R, is a special case of our result and an example of weak type (1, 1) estimates of a singular integral with non-smooth kernels on non-doubling spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574353","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 notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled KählerEinstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.
{"title":"Deformation for coupled Kähler–Einstein metrics","authors":"Satoshi X. Nakamura","doi":"10.2969/JMSJ/84408440","DOIUrl":"https://doi.org/10.2969/JMSJ/84408440","url":null,"abstract":"The notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled KählerEinstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47300822","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 show finiteness results on torsion points of commutative algebraic groups over a p-adic field K with values in various algebraic extensions L/K of infinite degree. We mainly study the following cases: (1) L is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) L is a certain iterate extension of K associated with Lubin-Tate formal groups, which is familiar with Kummer theory.
{"title":"Torsion of algebraic groups and iterate extensions associated with Lubin–Tate formal groups","authors":"Yoshiyasu Ozeki","doi":"10.2969/jmsj/87238723","DOIUrl":"https://doi.org/10.2969/jmsj/87238723","url":null,"abstract":"We show finiteness results on torsion points of commutative algebraic groups over a p-adic field K with values in various algebraic extensions L/K of infinite degree. We mainly study the following cases: (1) L is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) L is a certain iterate extension of K associated with Lubin-Tate formal groups, which is familiar with Kummer theory.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43446645","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 Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.
{"title":"Ohno relation for regularized multiple zeta values","authors":"M. Hirose, H. Murahara, Shingo Saito","doi":"10.2969/jmsj/89088908","DOIUrl":"https://doi.org/10.2969/jmsj/89088908","url":null,"abstract":"The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47935591","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 a homogenization problem for symmetric jumpdiffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.
{"title":"Homogenization of symmetric Dirichlet forms","authors":"M. Tomisaki, T. Uemura","doi":"10.2969/JMSJ/85268526","DOIUrl":"https://doi.org/10.2969/JMSJ/85268526","url":null,"abstract":"We consider a homogenization problem for symmetric jumpdiffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41317674","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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