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Partial regularity for Navier–Stokes and liquid crystals inequalities without maximum principle 无极大值原理的Navier-Stokes和液晶不等式的部分正则性
1区 数学
Analysis & PDE Pub Date : 2023-09-21 DOI: 10.2140/apde.2023.16.1701
Gabriel S. Koch
{"title":"Partial regularity for Navier–Stokes and liquid crystals inequalities without maximum principle","authors":"Gabriel S. Koch","doi":"10.2140/apde.2023.16.1701","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1701","url":null,"abstract":"In 1985, V. Scheffer discussed partial regularity results for what he called to the inequality. These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the director field $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies a maximum principle for $d$ which they use to establish partial regularity of solutions. For the analogous inequality one loses this maximum principle, but here we establish certain partial regularity results nonetheless. Our results recover in particular the partial regularity results of Caffarelli-Kohn-Nirenberg for suitable weak solutions of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for of the weaker inequality as well.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136101916","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
Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows 共轭热方程的无量纲Harnack不等式及其在几何流动中的应用
1区 数学
Analysis & PDE Pub Date : 2023-09-21 DOI: 10.2140/apde.2023.16.1589
Li-Juan Cheng, Anton Thalmaier
{"title":"Dimension-free Harnack inequalities for conjugate heat equations and their applications to geometric flows","authors":"Li-Juan Cheng, Anton Thalmaier","doi":"10.2140/apde.2023.16.1589","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1589","url":null,"abstract":"Let $M$ be a differentiable manifold endowed with a family of complete Riemannian metrics $g(t)$ evolving under a geometric flow over the time interval $[0,T[$. In this article, we give a probabilistic representation for the derivative of the corresponding conjugate semigroup on $M$ which is generated by a Schr\"{o}dinger type operator. With the help of this derivative formula, we derive fundamental Harnack type inequalities in the setting of evolving Riemannian manifolds. In particular, we establish a dimension-free Harnack inequality and show how it can be used to achieve heat kernel upper bounds in the setting of moving metrics. Moreover, by means of the supercontractivity of the conjugate semigroup, we obtain a family of canonical log-Sobolev inequalities. We discuss and apply these results both in the case of the so-called modified Ricci flow and in the case of general geometric flows.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102133","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: 0
The regularity of the boundary of vortex patches for some nonlinear transport equations 一类非线性输运方程涡旋斑块边界的正则性
1区 数学
Analysis & PDE Pub Date : 2023-09-21 DOI: 10.2140/apde.2023.16.1621
Juan Carlos Cantero, Joan Mateu, Joan Orobitg, Joan Verdera
{"title":"The regularity of the boundary of vortex patches for some nonlinear transport equations","authors":"Juan Carlos Cantero, Joan Mateu, Joan Orobitg, Joan Verdera","doi":"10.2140/apde.2023.16.1621","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1621","url":null,"abstract":"We prove the persistence of boundary smoothness of vortex patches for a non-linear transport equation in $mathbb{R}^n$ with velocity field given by convolution of the density with an odd kernel, homogeneous of degree $-(n-1)$ and of class $C^2(mathbb{R}^nsetminus{0}, mathbb{R}^n).$ This allows the velocity field to have non-trivial divergence. The quasi-geostrophic equation in $mathbb{R}^3$ and the Cauchy transport equation in the plane are examples.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236821","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
Directional square functions 方向平方函数
1区 数学
Analysis & PDE Pub Date : 2023-09-21 DOI: 10.2140/apde.2023.16.1651
Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, Ioannis Parissis, Luz Roncal
{"title":"Directional square functions","authors":"Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, Ioannis Parissis, Luz Roncal","doi":"10.2140/apde.2023.16.1651","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1651","url":null,"abstract":"Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. In this article we develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multi-parameter time-frequency analysis techniques. As applications we prove sharp or quantified bounds for Rubio de Francia type square functions of conical multipliers and of multipliers adapted to rectangles pointing along $N$ directions. A suitable combination of these estimates yields a new and currently best-known logarithmic bound for the Fourier restriction to an $N$-gon, improving on previous results of A. Cordoba. Our directional Carleson embedding extends to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136101917","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
Marstrand–Mattila rectifiability criterion for 1-codimensional measures in Carnot groups Carnot群中一维测度的Marstrand-Mattila可整流判据
1区 数学
Analysis & PDE Pub Date : 2023-06-15 DOI: 10.2140/apde.2023.16.927
Andrea Merlo
{"title":"Marstrand–Mattila rectifiability criterion for 1-codimensional measures in Carnot groups","authors":"Andrea Merlo","doi":"10.2140/apde.2023.16.927","DOIUrl":"https://doi.org/10.2140/apde.2023.16.927","url":null,"abstract":"This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups $mathbb{H}^n$ are $C^1_{mathbb{H}^n}$-rectifiable, providing the first non-Euclidean extension of Preiss's rectifiability theorem and a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135673190","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: 6
On the Ashbaugh–Benguria conjecture aboutlower-order Dirichlet eigenvalues of the Laplacian 关于拉普拉斯算子低阶狄利克雷特征值的Ashbaugh-Benguria猜想
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2021-11-10 DOI: 10.2140/apde.2021.14.2069
Qiaoling Wang, C. Xia
{"title":"On the Ashbaugh–Benguria conjecture about\u0000lower-order Dirichlet eigenvalues of the Laplacian","authors":"Qiaoling Wang, C. Xia","doi":"10.2140/apde.2021.14.2069","DOIUrl":"https://doi.org/10.2140/apde.2021.14.2069","url":null,"abstract":"","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44839610","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
A controlled tangential Julia–Carathéodorytheory via averaged Julia quotients 基于平均茱莉亚商的受控切向茱莉亚-卡拉萨梅多理论
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2021-09-07 DOI: 10.2140/apde.2021.14.1773
J. Pascoe, M. Sargent, R. Tully-Doyle
{"title":"A controlled tangential Julia–Carathéodory\u0000theory via averaged Julia quotients","authors":"J. Pascoe, M. Sargent, R. Tully-Doyle","doi":"10.2140/apde.2021.14.1773","DOIUrl":"https://doi.org/10.2140/apde.2021.14.1773","url":null,"abstract":"","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48472430","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
Multilayer potentials for higher-order systems in rough domains 粗糙域中高阶系统的多层势
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2021-07-06 DOI: 10.2140/APDE.2021.14.1233
G. Hoepfner, Paulo Liboni, D. Mitrea, I. Mitrea, M. Mitrea
{"title":"Multilayer potentials for higher-order systems in rough domains","authors":"G. Hoepfner, Paulo Liboni, D. Mitrea, I. Mitrea, M. Mitrea","doi":"10.2140/APDE.2021.14.1233","DOIUrl":"https://doi.org/10.2140/APDE.2021.14.1233","url":null,"abstract":"","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600850","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: 1
Analysis of the linear sampling method for imaging penetrable obstacles in the time domain 可穿透障碍物时域成像的线性采样方法分析
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2021-05-18 DOI: 10.2140/APDE.2021.14.667
F. Cakoni, P. Monk, V. Selgás
{"title":"Analysis of the linear sampling method for imaging penetrable obstacles in the time domain","authors":"F. Cakoni, P. Monk, V. Selgás","doi":"10.2140/APDE.2021.14.667","DOIUrl":"https://doi.org/10.2140/APDE.2021.14.667","url":null,"abstract":"We consider the problem of locating and reconstructing the geometry of a penetrable obstacle from time domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface enclosing the obstacle, and that the scattered field is measured on the same surface. From these multi-static scattering data we wish to determine the position and shape of the target. To deal with this inverse problem, we propose and analyze the Time Domain Linear Sampling Method (TDLSM) by means of localizing the interior transmission eigenvalues in the FourierLaplace domain. We also prove new time domain estimates for the forward problem and the interior transmission problem, as well as analyze several time domain operators arising in the inversion scheme.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48192962","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: 11
Variations of a class of Monge–Ampère-typefunctionals and their applications 一类monge - amp<e:1> -类型泛函的变体及其应用
IF 2.2 1区 数学
Analysis & PDE Pub Date : 2021-05-18 DOI: 10.2140/APDE.2021.14.689
Haodi Chen, Shibing Chen, Qi-Rui Li
{"title":"Variations of a class of Monge–Ampère-type\u0000functionals and their applications","authors":"Haodi Chen, Shibing Chen, Qi-Rui Li","doi":"10.2140/APDE.2021.14.689","DOIUrl":"https://doi.org/10.2140/APDE.2021.14.689","url":null,"abstract":"We study a class of Monge–Ampere-type functionals arising from the Lp dual Minkowski problem in convex geometry. As an application, we obtain some existence and nonuniqueness results for the problem.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48965910","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: 30
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