Pub Date : 2021-01-13DOI: 10.1512/iumj.2023.72.9297
T. Haas, B. D. Rijk, G. Schneider
It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.
{"title":"Validity of Whitham's modulation equations for dissipative systems with a conservation law: Phase dynamics in a generalized Ginzburg-Landau system","authors":"T. Haas, B. D. Rijk, G. Schneider","doi":"10.1512/iumj.2023.72.9297","DOIUrl":"https://doi.org/10.1512/iumj.2023.72.9297","url":null,"abstract":"It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46578815","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 : 2021-01-07DOI: 10.1512/iumj.2022.71.9302
R. Lucà, F. Ponce-Vanegas
We consider a fractal refinement of the Carleson problem for the Schr"odinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $alpha$-Hausdorff measure ($alpha$-a.e.). We extend to the fractal setting ($alpha
{"title":"Convergence over fractals for the Schroedinger equation","authors":"R. Lucà, F. Ponce-Vanegas","doi":"10.1512/iumj.2022.71.9302","DOIUrl":"https://doi.org/10.1512/iumj.2022.71.9302","url":null,"abstract":"We consider a fractal refinement of the Carleson problem for the Schr\"odinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $alpha$-Hausdorff measure ($alpha$-a.e.). We extend to the fractal setting ($alpha<n$) a recent counterexample of Bourgain cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($alpha = n$). In doing so we recover the necessary condition from cite{zbMATH07036806} for pointwise convergence~$alpha$-a.e. and we extend it to the range $n/2<alpha leq (3n+1)/4$.","PeriodicalId":50369,"journal":{"name":"Indiana University Mathematics Journal","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42061561","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.2021.70.8121
Xing-Ping Wu
{"title":"Qualitative analysis of solutions for the generalized Zakharov equations with magnetic field in mathbb{R}^d","authors":"Xing-Ping Wu","doi":"10.1512/IUMJ.2021.70.8121","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8121","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":"66764180","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.2021.70.8656
T. Brzeziński, W. Szymański
{"title":"The quantum flag manifold SU_q(3)/mathbb{t}^2 as an example of a noncommutative sphere bundle","authors":"T. Brzeziński, W. Szymański","doi":"10.1512/iumj.2021.70.8656","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.8656","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":"66765051","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.2021.70.8552
Curtis Kent, Russell Ricks
We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. We also demonstrate how maps between asymptotic cones induce maps between Tits boundaries, which we use to show that virtually free Abelian groups are exactly the CAT(0) groups with compact Tits boundary.
{"title":"Asymptotic cones and boundaries of CAT(0) spaces","authors":"Curtis Kent, Russell Ricks","doi":"10.1512/IUMJ.2021.70.8552","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8552","url":null,"abstract":"We explore the relationships between the asymptotic cones of a CAT(0) space and its boundary under both the standard visual (i.e. cone) topology and the Tits metric. We show that the set of asymptotic cones of a proper cocompact CAT(0) space admits canonical connecting maps under which the direct limit is isometric to the Euclidean cone on the Tits boundary. We also demonstrate how maps between asymptotic cones induce maps between Tits boundaries, which we use to show that virtually free Abelian groups are exactly the CAT(0) groups with compact Tits boundary.","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":"66764240","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.2021.70.8574
Leovigildo Alonso Tarrío, A. López, M. Rodríguez
{"title":"On the decomposition of the De Rham complex on formal schemes","authors":"Leovigildo Alonso Tarrío, A. López, M. Rodríguez","doi":"10.1512/IUMJ.2021.70.8574","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8574","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":"66764575","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.2021.70.8749
S. Charpentier, L. Kosinski
Given a domain of holomorphy D in C , N ≥ 2, we show that the set of holomorphic functions in D whose cluster sets along any finite length paths to the boundary of D is maximal, is residual, densely lineable and spaceable in the spaceO(D) of holomorphic functions in D. Besides, if D is a strictly pseudoconvex domain in C , and if a suitable family of smooth curves γ(x, r), x ∈ bD, r ∈ [0, 1), ending at a point of bD is given, then we exhibit a spaceable, densely lineable and residual subset of O(D), every element f of which satisfies the following property: For any measurable function h on bD, there exists a sequence (rn)n ∈ [0, 1) tending to 1, such that f ◦ γ(x, rn)→ h(x), n→∞, for almost every x in bD.
{"title":"Wild boundary behaviour of homomorphic functions in domains of C^N","authors":"S. Charpentier, L. Kosinski","doi":"10.1512/iumj.2021.70.8749","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.8749","url":null,"abstract":"Given a domain of holomorphy D in C , N ≥ 2, we show that the set of holomorphic functions in D whose cluster sets along any finite length paths to the boundary of D is maximal, is residual, densely lineable and spaceable in the spaceO(D) of holomorphic functions in D. Besides, if D is a strictly pseudoconvex domain in C , and if a suitable family of smooth curves γ(x, r), x ∈ bD, r ∈ [0, 1), ending at a point of bD is given, then we exhibit a spaceable, densely lineable and residual subset of O(D), every element f of which satisfies the following property: For any measurable function h on bD, there exists a sequence (rn)n ∈ [0, 1) tending to 1, such that f ◦ γ(x, rn)→ h(x), n→∞, for almost every x in bD.","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":"66765110","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.2021.70.8358
M. Lewicka
{"title":"Noisy tug of war games for the $p$-Laplacian: 1 < p < $infty$","authors":"M. Lewicka","doi":"10.1512/IUMJ.2021.70.8358","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8358","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":"66764441","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.2021.70.8563
J. S. D. Lis, S. S. D. León
{"title":"The limit as $pto 1$ of the higher eigenvalues of the $p$-Laplacian operator $Delta_p$","authors":"J. S. D. Lis, S. S. D. León","doi":"10.1512/IUMJ.2021.70.8563","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8563","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":"66764402","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.2021.70.8719
Xinbao Lu, Ge Xiong
{"title":"The L_p Minkowski problem for the electrostatic p-capacity for $pge n$","authors":"Xinbao Lu, Ge Xiong","doi":"10.1512/iumj.2021.70.8719","DOIUrl":"https://doi.org/10.1512/iumj.2021.70.8719","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":"66765068","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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