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Low regularity ill-posedness for elastic waves driven by shock formation 冲击波驱动弹性波的低正则性不适定性
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-03-06 DOI: 10.1353/ajm.2023.a902956
Xinliang An, Hao-Lei Chen, Silu Yin
{"title":"Low regularity ill-posedness for elastic waves driven by shock formation","authors":"Xinliang An, Hao-Lei Chen, Silu Yin","doi":"10.1353/ajm.2023.a902956","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902956","url":null,"abstract":"abstract:In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44630089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 4
Moving plane method for varifolds and applications 变量的移动平面法及其应用
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-03-03 DOI: 10.1353/ajm.2023.a902954
Robert Haslhofer, Or Hershkovits, B. White
{"title":"Moving plane method for varifolds and applications","authors":"Robert Haslhofer, Or Hershkovits, B. White","doi":"10.1353/ajm.2023.a902954","DOIUrl":"https://doi.org/10.1353/ajm.2023.a902954","url":null,"abstract":"abstract:In this paper, we introduce a version of the moving plane method that applies to potentially quite singular hypersurfaces, generalizing the classical moving plane method for smooth hypersurfaces. Loosely speaking, our version for varifolds shows that smoothness and symmetry at infinity (respectively at the boundary) can be promoted to smoothness and symmetry in the interior. The key feature, in contrast with the classical formulation of the moving plane principle, is that smoothness is a conclusion rather than an assumption. We implement our moving plane method in the setting of compactly supported varifolds with smooth boundary and in the setting of varifolds without boundary. A key ingredient is a Hopf lemma for stationary varifolds and varifolds of constant mean curvature. Our Hopf lemma provides a new tool to establish smoothness of varifolds, and works in arbitrary dimensions and without any stability assumptions. As applications of our new moving plane method, we prove varifold uniqueness results for the catenoid, spherical caps, and Delaunay surfaces that are inspired by classical uniqueness results by Schoen, Alexandrov, Meeks and Korevaar-Kusner-Solomon. We also prove a varifold version of Alexandrov's Theorem for compactly supported varifolds of constant mean curvature in hyperbolic space.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42276168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 7
On the order of magnitude of Sudler products 在Sudler产品的数量级上
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-02-16 DOI: 10.1353/ajm.2023.a897495
C. Aistleitner, Niclas Technau, Agamemnon Zafeiropoulos
{"title":"On the order of magnitude of Sudler products","authors":"C. Aistleitner, Niclas Technau, Agamemnon Zafeiropoulos","doi":"10.1353/ajm.2023.a897495","DOIUrl":"https://doi.org/10.1353/ajm.2023.a897495","url":null,"abstract":"abstract:Given an irrational number $alphain(0,1)$, the Sudler product is defined by $P_N(alpha) = prod_{r=1}^{N}2|sinpi ralpha|$. Answering a question of Grepstad, Kaltenb\"ock and Neum\"uller we prove an asymptotic formula for distorted Sudler products when $alpha$ is the golden ratio $(sqrt{5}+1)/2$ and establish that in this case $limsup_{N to infty} P_N(alpha)/N < infty$. We obtain similar results for quadratic irrationals $alpha$ with continued fraction expansion $alpha = [a,a,a,ldots]$ for some integer $a geq 1$, and give a full characterisation of the values of $a$ for which $liminf_{N to infty} P_N(alpha)>0$ and $limsup_{N to infty} P_N(alpha) / N < infty$ hold, respectively. We establish that there is a (sharp) transition point at $a=6$, and resolve as a by-product a problem of the first author, Larcher, Pillichshammer, Saad Eddin, and Tichy.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47208201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 12
On the parity of ideal classes over a fixed prime 关于固定素数上理想类的奇偶性
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-01-23 DOI: 10.1353/ajm.2020.0003
Jeongho Park
{"title":"On the parity of ideal classes over a fixed prime","authors":"Jeongho Park","doi":"10.1353/ajm.2020.0003","DOIUrl":"https://doi.org/10.1353/ajm.2020.0003","url":null,"abstract":"abstract:For real quadratic number fields, we consider the order of ideal classes of split prime ideals, $P$, whose norm is a fixed rational prime. We collect fundamental discriminants satisfying a trivial condition for $P$ to be principal, and show that for a positive density of such discriminants, the cyclic subgroup of the ideal class group generated by $P$ does not have a 2-part.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41842573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 2
Periodicity and decidability of tilings of ℤ2 tilings的周期性和可判定性ℤ2.
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-01-23 DOI: 10.1353/ajm.2020.0006
S. Bhattacharya
{"title":"Periodicity and decidability of tilings of ℤ2","authors":"S. Bhattacharya","doi":"10.1353/ajm.2020.0006","DOIUrl":"https://doi.org/10.1353/ajm.2020.0006","url":null,"abstract":"abstract:We prove that any finite set $Fsubset{Bbb Z}^2$ that tiles ${Bbb Z}^2$ by translations also admits a periodic tiling. As a consequence, the problem whether a given finite set $F$ tiles ${Bbb Z}^2$ is decidable.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2020.0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 24
Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds 散焦能量-超临界NLS和定量全局散射边界的放大标准低于标度
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2020-01-15 DOI: 10.1353/ajm.2023.0013
Aynur Bulut
{"title":"Blow-up criteria below scaling for defocusing energy-supercritical NLS and quantitative global scattering bounds","authors":"Aynur Bulut","doi":"10.1353/ajm.2023.0013","DOIUrl":"https://doi.org/10.1353/ajm.2023.0013","url":null,"abstract":"We establish quantitative blow-up criteria below the scaling threshold for radially symmetric solutions to the defocusing nonlinear Schrodinger equation with nonlinearity $|u|^6u$. This provides to our knowledge the first generic results distinguishing potential blow-up solutions of the defocusing equation from many of the known examples of blow-up in the focusing case. Our main tool is a quantitative version of a result showing that uniform bounds on $L^2$-based critical Sobolev norms imply scattering estimates. \u0000As another application of our techniques, we establish a variant which allows for slow growth in the critical norm. We show that if the critical Sobolev norm on compact time intervals is controlled by a slowly growing quantity depending on the Stricharz norm, then the solution can be extended globally in time, with a corresponding scattering estimate.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46129609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
A short proof of ℓ2 decoupling for the moment curve 对于力矩曲线，一个关于l2解耦的简短证明
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2019-12-20 DOI: 10.1353/ajm.2021.0048
LI Zanekun, Po-Lam Yung, Pavel Zorin-Kranich
{"title":"A short proof of ℓ2 decoupling for the moment curve","authors":"LI Zanekun, Po-Lam Yung, Pavel Zorin-Kranich","doi":"10.1353/ajm.2021.0048","DOIUrl":"https://doi.org/10.1353/ajm.2021.0048","url":null,"abstract":"abstract:We give a short and elementary proof of the $ell^{2}$ decoupling inequality for the moment curve in $hat{Bbb{R}}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2019-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46400960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 14
Families of commuting automorphisms, and a characterization of the affine space 交换自同构族和仿射空间的一个性质
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2019-12-03 DOI: 10.1353/ajm.2023.0009
Serge Cantat, A. Regeta, Junyi Xie
{"title":"Families of commuting automorphisms, and a characterization of the affine space","authors":"Serge Cantat, A. Regeta, Junyi Xie","doi":"10.1353/ajm.2023.0009","DOIUrl":"https://doi.org/10.1353/ajm.2023.0009","url":null,"abstract":"In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2019-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45817402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 14
Rationality of Fano threefolds over non-closed fields 非封闭场上法诺三倍的合理性
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2019-11-20 DOI: 10.1353/ajm.2023.0008
A. Kuznetsov, Yuri Prokhorov
{"title":"Rationality of Fano threefolds over non-closed fields","authors":"A. Kuznetsov, Yuri Prokhorov","doi":"10.1353/ajm.2023.0008","DOIUrl":"https://doi.org/10.1353/ajm.2023.0008","url":null,"abstract":"We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48893246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 20
Joints tightened 接头已拧紧
IF 1.7 1区 数学
American Journal of Mathematics Pub Date : 2019-11-19 DOI: 10.1353/ajm.2023.0014
H. Yu, Yufei Zhao
{"title":"Joints tightened","authors":"H. Yu, Yufei Zhao","doi":"10.1353/ajm.2023.0014","DOIUrl":"https://doi.org/10.1353/ajm.2023.0014","url":null,"abstract":"In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $binom{k}{d-1}$ lines and their $d$-wise intersections to form $binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2019-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46016767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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: 8
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