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A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness 介电匹配二元流体中带电液滴的变量模型：电荷不均匀性的影响
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-31 DOI: 10.1007/s00205-024-02012-9
Cyrill B. Muratov, Matteo Novaga, Philip Zaleski
{"title":"A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness","authors":"Cyrill B. Muratov,&nbsp;Matteo Novaga,&nbsp;Philip Zaleski","doi":"10.1007/s00205-024-02012-9","DOIUrl":"10.1007/s00205-024-02012-9","url":null,"abstract":"<div><p>This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02012-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Instability and Spectrum of the Linearized Two-Phase Fluids Interface Problem at Shear Flows 剪切流下线性化两相流体界面问题的不稳定性和频谱
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-28 DOI: 10.1007/s00205-024-02024-5
Xiao Liu
{"title":"Instability and Spectrum of the Linearized Two-Phase Fluids Interface Problem at Shear Flows","authors":"Xiao Liu","doi":"10.1007/s00205-024-02024-5","DOIUrl":"10.1007/s00205-024-02024-5","url":null,"abstract":"<div><p>This paper is concerned with the 2-dim two-phase interface Euler equation linearized at a pair of monotone shear flows in both fluids. We extend the Howard’s Semicircle Theorem and study the eigenvalue distribution of the linearized Euler system. Under certain conditions, there are exactly two eigenvalues for each fixed wave number <span>(kin mathbb {R})</span> in the whole complex plane. We provide sufficient conditions for spectral instability arising from some boundary values of the shear flow velocity. A typical mode is the ocean-air system in which the density ratio of the fluids is sufficiently small. We give a complete picture of eigenvalue distribution for a certain class of shear flows in the ocean-air system.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180060","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
Matrix Displacement Convexity Along Density Flows 沿密度流的矩阵位移凸度
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-27 DOI: 10.1007/s00205-024-02021-8
Yair Shenfeld
{"title":"Matrix Displacement Convexity Along Density Flows","authors":"Yair Shenfeld","doi":"10.1007/s00205-024-02021-8","DOIUrl":"10.1007/s00205-024-02021-8","url":null,"abstract":"<div><p>A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schrödinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180061","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
Local Well-Posedness of the Capillary-Gravity Water Waves with Acute Contact Angles 具有锐接触角的毛细管-重力水波的局部良好假设性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-26 DOI: 10.1007/s00205-024-02019-2
Mei Ming, Chao Wang
{"title":"Local Well-Posedness of the Capillary-Gravity Water Waves with Acute Contact Angles","authors":"Mei Ming,&nbsp;Chao Wang","doi":"10.1007/s00205-024-02019-2","DOIUrl":"10.1007/s00205-024-02019-2","url":null,"abstract":"<div><p>We consider the two-dimensional capillary-gravity water waves problem where the free surface <span>(Gamma _t)</span> intersects the bottom <span>(Gamma _b)</span> at two contact points. In our previous works (Ming and Wang in SIAM J Math Anal 52(5):4861–4899; Commun Pure Appl Math 74(2), 225–285, 2021), the local well-posedness for this problem has been proved with the contact angles less than <span>(pi /16)</span>. In this paper, we study the case where the contact angles belong to <span>((0, pi /2))</span>. It involves much worse singularities generated from corresponding elliptic systems, which have this strong influence on the regularities for the free surface and the velocity field. Combining the theory of singularity decompositions for elliptic problems with the structure of the water waves system, we obtain a priori energy estimates. Based on these estimates, we also prove the local well-posedness of the solutions in a geometric formulation.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180062","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
Parabolic Boundary Harnack Inequalities with Right-Hand Side 带右侧的抛物线边界哈纳克不等式
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-26 DOI: 10.1007/s00205-024-02017-4
Clara Torres-Latorre
{"title":"Parabolic Boundary Harnack Inequalities with Right-Hand Side","authors":"Clara Torres-Latorre","doi":"10.1007/s00205-024-02017-4","DOIUrl":"10.1007/s00205-024-02017-4","url":null,"abstract":"<div><p>We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side <span>(f in L^q)</span> for <span>(q &gt; n+2)</span>. In the case of the heat equation, we also show the optimal <span>(C^{1-varepsilon })</span> regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are <span>(C^{1,alpha })</span> in the parabolic obstacle problem and in the parabolic Signorini problem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11347492/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142116908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantitative Homogenization for the Obstacle Problem and Its Free Boundary 障碍问题及其自由边界的定量均质化
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-18 DOI: 10.1007/s00205-024-02015-6
Gohar Aleksanyan, Tuomo Kuusi
{"title":"Quantitative Homogenization for the Obstacle Problem and Its Free Boundary","authors":"Gohar Aleksanyan,&nbsp;Tuomo Kuusi","doi":"10.1007/s00205-024-02015-6","DOIUrl":"10.1007/s00205-024-02015-6","url":null,"abstract":"<div><p>In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11330955/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142010016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation 最小均方差曲面的存在性和规则性的 PDE 方法
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-08-05 DOI: 10.1007/s00205-024-02016-5
Luis A. Caffarelli, Pablo Raúl Stinga, Hernán Vivas
{"title":"A PDE Approach to the Existence and Regularity of Surfaces of Minimum Mean Curvature Variation","authors":"Luis A. Caffarelli,&nbsp;Pablo Raúl Stinga,&nbsp;Hernán Vivas","doi":"10.1007/s00205-024-02016-5","DOIUrl":"10.1007/s00205-024-02016-5","url":null,"abstract":"<div><p>We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem </p><div><div><span>$$begin{aligned} min _mathcal {M}frac{1}{2}int _mathcal {M}|nabla _{mathcal {M}}H|^2,{text {d}}A, end{aligned}$$</span></div></div><p>where <span>(mathcal {M})</span> ranges over all <i>n</i>-dimensional manifolds in <span>(mathbb {R}^{n+1})</span> with a prescribed boundary, <span>(nabla _{mathcal {M}}H)</span> is the tangential gradient along <span>(mathcal {M})</span> of the mean curvature <i>H</i> of <span>(mathcal {M})</span> and d<i>A</i> is the differential of surface area. The minimizers, called surfaces of minimum mean curvature variation, are central in applications of computer-aided design, computer-aided manufacturing and mechanics. Our main results show the existence of both smooth surfaces and of variational solutions to the minimization problem together with geometric regularity results in the case of graphs. These are the first analytic results available for this problem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945709","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
Quantified Legendreness and the Regularity of Minima 量化传奇和最小值的规律性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-29 DOI: 10.1007/s00205-024-02008-5
Cristiana De Filippis, Lukas Koch, Jan Kristensen
{"title":"Quantified Legendreness and the Regularity of Minima","authors":"Cristiana De Filippis,&nbsp;Lukas Koch,&nbsp;Jan Kristensen","doi":"10.1007/s00205-024-02008-5","DOIUrl":"10.1007/s00205-024-02008-5","url":null,"abstract":"<div><p>We introduce a new quantification of nonuniform ellipticity in variational problems via convex duality, and prove higher differentiability and 2<i>d</i>-smoothness results for vector valued minimizers of possibly degenerate functionals. Our framework covers convex, anisotropic polynomials as prototypical model examples—in particular, we improve in an essentially optimal fashion Marcellini’s original results (Marcellini in Arch Rat Mech Anal 105:267–284, 1989).\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02008-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Gradient Flow for Entropy on Closed Planar Curves 封闭平面曲线上的熵梯度流
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-27 DOI: 10.1007/s00205-024-02014-7
Lachlann O’Donnell, Glen Wheeler, Valentina-Mira Wheeler
{"title":"The Gradient Flow for Entropy on Closed Planar Curves","authors":"Lachlann O’Donnell,&nbsp;Glen Wheeler,&nbsp;Valentina-Mira Wheeler","doi":"10.1007/s00205-024-02014-7","DOIUrl":"10.1007/s00205-024-02014-7","url":null,"abstract":"<div><p>In this paper we consider the steepest descent <span>(L^2)</span>-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class <span>(C^2)</span> or embedded of class <span>(W^{2,2})</span> bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02014-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Interior Regularity for Two-Dimensional Stationary Q-Valued Maps 二维静态 Q 值映射的内部正则性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-22 DOI: 10.1007/s00205-024-02011-w
Jonas Hirsch, Luca Spolaor
{"title":"Interior Regularity for Two-Dimensional Stationary Q-Valued Maps","authors":"Jonas Hirsch,&nbsp;Luca Spolaor","doi":"10.1007/s00205-024-02011-w","DOIUrl":"10.1007/s00205-024-02011-w","url":null,"abstract":"<div><p>We prove that 2-dimensional <i>Q</i>-valued maps that are stationary with respect to outer and inner variations of the Dirichlet energy are Hölder continuous and that the dimension of their singular set is at most one. In the course of the proof we establish a strong concentration-compactness theorem for equicontinuous maps that are stationary with respect to outer variations only, and which holds in every dimensions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02011-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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