Pub Date : 2021-01-01DOI: 10.1512/iumj.2021.70.9401
L. De Masi
Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature H of V is in L for some p ∈ [1, k], then the set of points where the k-density of V does not exist or is infinite has Hausdorff dimension at most k − p. We use this result to prove, under suitable assumptions, that the part of the first variation of V with positive and finite (k− 1)-density is (k− 1)-rectifiable.
{"title":"Rectifiability of the free boundary for varifolds","authors":"L. De Masi","doi":"10.1512/iumj.2021.70.9401","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.9401","url":null,"abstract":"Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature H of V is in L for some p ∈ [1, k], then the set of points where the k-density of V does not exist or is infinite has Hausdorff dimension at most k − p. We use this result to prove, under suitable assumptions, that the part of the first variation of V with positive and finite (k− 1)-density is (k− 1)-rectifiable.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66765177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1512/IUMJ.2021.70.8075
Dehua Wang, Zhian Wang, Kun Zhao
In this paper, we study the qualitative behavior of solutions to the Cauchy problem of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, in multiple space dimensions. Assuming H2 initial data, it is shown that under the assumption that only some fractions of the total energy associated with the initial perturbation around a prescribed constant ground state are small, the Cauchy problem admits a unique global-in-time solution, and the solution converges to the prescribed ground state as time goes to infinity. In addition, it is shown that solutions of the fully dissipative model converge to that of the corresponding partially dissipative model with certain convergence rates as a specific system parameter tends to zero.
{"title":"Cauchy problem of a system of parabolic conservation laws arising from the singular Keller-Segel model in multi-dimensions","authors":"Dehua Wang, Zhian Wang, Kun Zhao","doi":"10.1512/IUMJ.2021.70.8075","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8075","url":null,"abstract":"In this paper, we study the qualitative behavior of solutions to the Cauchy problem of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, in multiple space dimensions. Assuming H2 initial data, it is shown that under the assumption that only some fractions of the total energy associated with the initial perturbation around a prescribed constant ground state are small, the Cauchy problem admits a unique global-in-time solution, and the solution converges to the prescribed ground state as time goes to infinity. In addition, it is shown that solutions of the fully dissipative model converge to that of the corresponding partially dissipative model with certain convergence rates as a specific system parameter tends to zero.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66763974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1512/IUMJ.2021.70.8573
X. Duong, M. Lacey, Ji Li, B. Wick, Qingyan Wu
In this paper we study the commutator of Cauchy type integrals C on a bounded strongly pseudoconvex domain D in C with boundary bD satisfying the minimum regularity condition C as in the recent result of Lanzani–Stein. We point out that in this setting the Cauchy type integrals C is the sum of the essential part C which is a Calderón–Zygmund operator and a remainder R which is no longer a Calderón–Zygmund operator. We show that the commutator [b,C] is bounded on L(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b, C] is compact on L(bD) (1 < p < ∞) if and only if b is in the VMO space on bD. Our method can also be applied to the commutator of Cauchy–Leray integral in a bounded, strongly C-linearly convex domain D in C with the boundary bD satisfying the minimum regularity C. Such a Cauchy–Leray integral is a Calderón–Zygmund operator as proved in the recent result of Lanzani–Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy–Szegő operator on a bounded strongly pseudoconvex domain D in C with smooth boundary (first established by Krantz–Li).
本文研究了C中的有界强伪凸域D上，边界bD满足最小正则性条件C的柯西型积分C的对易子，这是Lanzani-Stein最近的结果。我们指出，在这种情况下，柯西型积分C是本质部分C的和，它是一个Calderón-Zygmund算子，余数R不再是Calderón-Zygmund算子。我们证明了换向子[b,C]在L(bD) (1 < p <∞)上是有界的当且仅当b在bD上的VMO空间上，并且换向子[b,C]在L(bD) (1 < p <∞)上是紧致的当且仅当b在bD上的VMO空间上。我们的方法也可以应用于有界的Cauchy-Leray积分的换向子。C中的强C-线性凸域D，边界bD满足最小正则性C。Lanzani-Stein最近的结果证明，这样的Cauchy-Leray积分是一个Calderón-Zygmund算子。我们还指出，我们的方法再次证明了C中具有光滑边界的有界强伪凸域D上cauchy - szeger算子的对易子的有界性和紧性(最早由Krantz-Li建立)。
{"title":"Commutators of Cauchy-Szego type integrals for domains in C^n with minimal smoothness","authors":"X. Duong, M. Lacey, Ji Li, B. Wick, Qingyan Wu","doi":"10.1512/IUMJ.2021.70.8573","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8573","url":null,"abstract":"In this paper we study the commutator of Cauchy type integrals C on a bounded strongly pseudoconvex domain D in C with boundary bD satisfying the minimum regularity condition C as in the recent result of Lanzani–Stein. We point out that in this setting the Cauchy type integrals C is the sum of the essential part C which is a Calderón–Zygmund operator and a remainder R which is no longer a Calderón–Zygmund operator. We show that the commutator [b,C] is bounded on L(bD) (1 < p < ∞) if and only if b is in the BMO space on bD. Moreover, the commutator [b, C] is compact on L(bD) (1 < p < ∞) if and only if b is in the VMO space on bD. Our method can also be applied to the commutator of Cauchy–Leray integral in a bounded, strongly C-linearly convex domain D in C with the boundary bD satisfying the minimum regularity C. Such a Cauchy–Leray integral is a Calderón–Zygmund operator as proved in the recent result of Lanzani–Stein. We also point out that our method provides another proof of the boundedness and compactness of commutator of Cauchy–Szegő operator on a bounded strongly pseudoconvex domain D in C with smooth boundary (first established by Krantz–Li).","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66764518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1512/IUMJ.2021.70.8620
M. Cristo, L. Rondi
{"title":"The distance from the boundary in a Riemannian manifold: regularity up to a conformal change of the metric","authors":"M. Cristo, L. Rondi","doi":"10.1512/IUMJ.2021.70.8620","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8620","url":null,"abstract":"","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66764943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-31DOI: 10.1512/iumj.2023.72.9315
K. Grosse-Erdmann, Dimitris Papathanasiou
We study the dynamical behaviour of weighted shifts defined on sequence spaces of a directed tree. In particular, we characterize their boundedness as well as when they are hypercyclic, weakly mixing and mixing.
{"title":"Dynamics of weighted shifts on directed trees","authors":"K. Grosse-Erdmann, Dimitris Papathanasiou","doi":"10.1512/iumj.2023.72.9315","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9315","url":null,"abstract":"We study the dynamical behaviour of weighted shifts defined on sequence spaces of a directed tree. In particular, we characterize their boundedness as well as when they are hypercyclic, weakly mixing and mixing.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42508330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-23DOI: 10.1512/iumj.2023.72.9272
Ryan N. Goh, C. E. Wayne, R. Welter
We study the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of the heat equation in terms of Hermite functions in which $n^{th}$ order approximations can be computed for solutions with $n^{th}$ order moments. We then obtain existence of solutions to the mcNS system and show that the approximation in terms of Hermite functions gives the leading order terms in the long-time asymptotics, and under certain assumptions can be evaluated explicitly.
{"title":"Asymptotic approximation of a modified compressible Navier-Stokes system","authors":"Ryan N. Goh, C. E. Wayne, R. Welter","doi":"10.1512/iumj.2023.72.9272","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9272","url":null,"abstract":"We study the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of the heat equation in terms of Hermite functions in which $n^{th}$ order approximations can be computed for solutions with $n^{th}$ order moments. We then obtain existence of solutions to the mcNS system and show that the approximation in terms of Hermite functions gives the leading order terms in the long-time asymptotics, and under certain assumptions can be evaluated explicitly.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49639886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-20DOI: 10.1512/iumj.2022.71.9239
Annika Bach, M. Cicalese, Leonard Kreutz, G. Orlando
We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice in the vortex regime. Within this regime, the spin system cannot overcome the energetic barrier of chirality transitions, hence one of the two chirality phases is prevalent. We find the order parameter that describes the vortex structure of the spin field in the majority chirality phase and we compute explicitly the $Gamma$-limit of the scaled energy, showing that it concentrates on finitely many vortex-like singularities of the spin field.
{"title":"The antiferromagnetic xy model on the triangular lattice: topological singularities","authors":"Annika Bach, M. Cicalese, Leonard Kreutz, G. Orlando","doi":"10.1512/iumj.2022.71.9239","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9239","url":null,"abstract":"We study the discrete-to-continuum variational limit of the antiferromagnetic XY model on the two-dimensional triangular lattice in the vortex regime. Within this regime, the spin system cannot overcome the energetic barrier of chirality transitions, hence one of the two chirality phases is prevalent. We find the order parameter that describes the vortex structure of the spin field in the majority chirality phase and we compute explicitly the $Gamma$-limit of the scaled energy, showing that it concentrates on finitely many vortex-like singularities of the spin field.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43111046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-04DOI: 10.1512/iumj.2023.72.9320
F. Gancedo, Rafael Granero-Belinch'on, S. Scrobogna
In this paper we study the Peskin problem. This is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. To obtain this result we use a new contour dynamic formulation which reduces the system to a scalar equation. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space $L^2 left( [0,T];H^{3/2} right) $ to satisfy the contour equation pointwise.
{"title":"Global existence in the Lipschitz class for the N-Peskin problem","authors":"F. Gancedo, Rafael Granero-Belinch'on, S. Scrobogna","doi":"10.1512/iumj.2023.72.9320","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9320","url":null,"abstract":"In this paper we study the Peskin problem. This is a fluid-structure interaction problem that describes the motion of an elastic rod immersed in an incompressible Stokes fluid. We prove global in time existence of solution for initial data in the critical Lipschitz space. To obtain this result we use a new contour dynamic formulation which reduces the system to a scalar equation. Using a new decomposition together with cancellation properties, pointwise methods allow us to obtain the desired estimates in the Lipschitz class. Moreover, we perform energy estimates in order to obtain that the solution lies in the space $L^2 left( [0,T];H^{3/2} right) $ to satisfy the contour equation pointwise.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46227721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-23DOI: 10.1512/iumj.2023.72.9243
Zihao Qi, Yongjun Xu, James J. Zhang, Xiangui Zhao
The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $rin {0}cup {1}cup [2,infty)$ or $r=infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads.
{"title":"Growth of nonsymmetric operads","authors":"Zihao Qi, Yongjun Xu, James J. Zhang, Xiangui Zhao","doi":"10.1512/iumj.2023.72.9243","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9243","url":null,"abstract":"The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $rin {0}cup {1}cup [2,infty)$ or $r=infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41787030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-15DOI: 10.1512/iumj.2023.72.9206
Ludovic Godard-Cadillac, Philippe Gravejat, D. Smets
We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.
{"title":"Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation","authors":"Ludovic Godard-Cadillac, Philippe Gravejat, D. Smets","doi":"10.1512/iumj.2023.72.9206","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9206","url":null,"abstract":"We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41341893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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