Analysis & PDEPub Date : 2023-09-21DOI: 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}
Analysis & PDEPub Date : 2023-09-21DOI: 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}
Analysis & PDEPub Date : 2023-09-21DOI: 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}
Analysis & PDEPub Date : 2023-09-21DOI: 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}
Analysis & PDEPub Date : 2023-06-15DOI: 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}
Analysis & PDEPub 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}