Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1701
Gabriel S. Koch
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.
{"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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1589
Li-Juan Cheng, Anton Thalmaier
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.
{"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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1621
Juan Carlos Cantero, Joan Mateu, Joan Orobitg, Joan Verdera
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.
{"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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1651
Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, Ioannis Parissis, Luz Roncal
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.
Fefferman的球乘法器反例的定量公式自然地与圆锥乘法器和方向乘法器的平方函数估计联系在一起。在本文中，我们基于Carleson序列的方向嵌入定理和多参数时频分析技术，为这些平方函数估计开发了一个新的框架。作为应用，我们证明了Rubio de Francia型圆锥乘法器和沿N方向的矩形乘法器的平方函数的尖锐边界或量化边界。这些估计的一个合适的组合产生了一个新的和目前最著名的傅里叶限制到$N$-gon的对数界，改进了A. Cordoba以前的结果。我们的定向Carleson嵌入扩展到加权设置，对定向极大函数和奇异积分产生以前未知的加权估计。
{"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}
Pub Date : 2023-06-15DOI: 10.2140/apde.2023.16.927
Andrea Merlo
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.
{"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}
Pub Date : 2021-11-10DOI: 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}
Pub Date : 2021-09-07DOI: 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}
Pub Date : 2021-07-06DOI: 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}
Pub Date : 2021-05-18DOI: 10.2140/APDE.2021.14.667
F. Cakoni, P. Monk, V. Selgás
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.
{"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}
Pub Date : 2021-05-18DOI: 10.2140/APDE.2021.14.689
Haodi Chen, Shibing Chen, Qi-Rui Li
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.
{"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}
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