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Obstructions to Topological Relaxation for Generic Magnetic Fields 通用磁场拓扑弛豫的障碍
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-10 DOI: 10.1007/s00205-024-02078-5
Alberto Enciso, Daniel Peralta-Salas
{"title":"Obstructions to Topological Relaxation for Generic Magnetic Fields","authors":"Alberto Enciso,&nbsp;Daniel Peralta-Salas","doi":"10.1007/s00205-024-02078-5","DOIUrl":"10.1007/s00205-024-02078-5","url":null,"abstract":"<div><p>For any (analytic) axisymmetric toroidal domain <span>(Omega subset mathbb {R}^3)</span> we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in <span>(Omega )</span>. Each vector field in this set is Morse–Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02078-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844908","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
On the Converse of Pansu’s Theorem 论潘素定理的逆
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-10 DOI: 10.1007/s00205-024-02059-8
Guido De Philippis, Andrea Marchese, Andrea Merlo, Andrea Pinamonti, Filip Rindler
{"title":"On the Converse of Pansu’s Theorem","authors":"Guido De Philippis,&nbsp;Andrea Marchese,&nbsp;Andrea Merlo,&nbsp;Andrea Pinamonti,&nbsp;Filip Rindler","doi":"10.1007/s00205-024-02059-8","DOIUrl":"10.1007/s00205-024-02059-8","url":null,"abstract":"<div><p>We provide a suitable generalisation of Pansu’s differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures <span>(mu )</span>, then <span>(mu )</span> must be absolutely continuous with respect to the Haar measure of the group.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02059-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844907","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
Regular and Singular Steady States of the 2D Incompressible Euler Equations near the Bahouri–Chemin Patch Bahouri-Chemin Patch附近二维不可压缩Euler方程的正则和奇异稳态
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-09 DOI: 10.1007/s00205-024-02077-6
Tarek M. Elgindi, Yupei Huang
{"title":"Regular and Singular Steady States of the 2D Incompressible Euler Equations near the Bahouri–Chemin Patch","authors":"Tarek M. Elgindi,&nbsp;Yupei Huang","doi":"10.1007/s00205-024-02077-6","DOIUrl":"10.1007/s00205-024-02077-6","url":null,"abstract":"<div><p>We consider steady states of the two-dimensional incompressible Euler equations on <span>({mathbb {T}}^2)</span> and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri–Chemin patch.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844974","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
Asymmetry of MHD Equilibria for Generic Adapted Metrics 一般自适应度量的MHD均衡的不对称性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-08 DOI: 10.1007/s00205-024-02075-8
Robert Cardona, Nathan Duignan, David Perrella
{"title":"Asymmetry of MHD Equilibria for Generic Adapted Metrics","authors":"Robert Cardona,&nbsp;Nathan Duignan,&nbsp;David Perrella","doi":"10.1007/s00205-024-02075-8","DOIUrl":"10.1007/s00205-024-02075-8","url":null,"abstract":"<div><p>Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution <i>X</i> admits a large set of “adapted” metrics on <i>M</i> for which <i>X</i> solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in <span>({mathbb {R}}^3)</span> with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844858","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
Covariance-Modulated Optimal Transport and Gradient Flows 协方差调制优化传输和梯度流动
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-12-03 DOI: 10.1007/s00205-024-02065-w
Martin Burger, Matthias Erbar, Franca Hoffmann, Daniel Matthes, André Schlichting
{"title":"Covariance-Modulated Optimal Transport and Gradient Flows","authors":"Martin Burger,&nbsp;Matthias Erbar,&nbsp;Franca Hoffmann,&nbsp;Daniel Matthes,&nbsp;André Schlichting","doi":"10.1007/s00205-024-02065-w","DOIUrl":"10.1007/s00205-024-02065-w","url":null,"abstract":"<div><p>We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02065-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844752","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
Regularity Structures for Quasilinear Singular SPDEs 准线性奇异 SPDE 的正则结构
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-29 DOI: 10.1007/s00205-024-02069-6
I. Bailleul, M. Hoshino, S. Kusuoka
{"title":"Regularity Structures for Quasilinear Singular SPDEs","authors":"I. Bailleul,&nbsp;M. Hoshino,&nbsp;S. Kusuoka","doi":"10.1007/s00205-024-02069-6","DOIUrl":"10.1007/s00205-024-02069-6","url":null,"abstract":"<div><p>We prove the well-posed character of a regularity structure formulation of the quasilinear generalized (KPZ) equation and give an explicit form for a renormalized equation in the full subcritical regime. Under the assumption that the BPHZ models associated with a non-translation invariant operator converge, we obtain a convergence result for the solutions of the regularized renormalized equations. This conditional result covers the spacetime white noise case.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737043","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
Potential Theory for Nonlocal Drift-Diffusion Equations 非局部漂移-扩散方程的势理论
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-29 DOI: 10.1007/s00205-024-02073-w
Quoc-Hung Nguyen, Simon Nowak, Yannick Sire, Marvin Weidner
{"title":"Potential Theory for Nonlocal Drift-Diffusion Equations","authors":"Quoc-Hung Nguyen,&nbsp;Simon Nowak,&nbsp;Yannick Sire,&nbsp;Marvin Weidner","doi":"10.1007/s00205-024-02073-w","DOIUrl":"10.1007/s00205-024-02073-w","url":null,"abstract":"<div><p>The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow us to derive new estimates for the dissipative surface quasi-geostrophic equation.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737189","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
Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws 超曲线正则化守恒律系统冲击剖面的谱稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-26 DOI: 10.1007/s00205-024-02066-9
Johannes Bärlin
{"title":"Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws","authors":"Johannes Bärlin","doi":"10.1007/s00205-024-02066-9","DOIUrl":"10.1007/s00205-024-02066-9","url":null,"abstract":"<div><p>We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function <span>(mathcal {E}:Lambda rightarrow mathbb {C})</span> with <span>(Lambda subset mathbb {C})</span> an open superset of the closed right half plane <span>(mathbb {H}^+equiv {kappa in mathbb {C}:text {Re},kappa geqq 0})</span> has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02066-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714502","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 New Condition on the Vorticity for Partial Regularity of a Local Suitable Weak Solution to the Navier–Stokes Equations 纳维-斯托克斯方程局部合适弱解的涡度部分正则性新条件
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-25 DOI: 10.1007/s00205-024-02068-7
Dongho Chae, Jörg Wolf
{"title":"A New Condition on the Vorticity for Partial Regularity of a Local Suitable Weak Solution to the Navier–Stokes Equations","authors":"Dongho Chae,&nbsp;Jörg Wolf","doi":"10.1007/s00205-024-02068-7","DOIUrl":"10.1007/s00205-024-02068-7","url":null,"abstract":"<div><p>We provide a new <span>(varepsilon )</span>-condition for the vorticity of a suitable weak solution to the Navier–Stokes equations that leads to partial regularity. This refines the well known limsup condition of the Caffarelli-Kohn-Nirenberg Theorem by a new condition on the vorticity, replacing limsup by a suitable range of the radius <i>r</i> of the parabolic cylinders. As a consequence, the partial regularity is obtained directly from this <span>(varepsilon )</span>-condition of the vorticity without relying on the <span>(varepsilon )</span>-condition of the velocity. Furthermore, by the local nature of the method this result holds for any local suitable weak solution of the Navier–Stokes equations in a general domain.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714239","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
Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition 具有诺伊曼边界条件的黎曼曼体上艾伦-卡恩方程的极限界面的边界行为
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-11-25 DOI: 10.1007/s00205-024-02070-z
Martin Man-chun Li, Davide Parise, Lorenzo Sarnataro
{"title":"Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition","authors":"Martin Man-chun Li,&nbsp;Davide Parise,&nbsp;Lorenzo Sarnataro","doi":"10.1007/s00205-024-02070-z","DOIUrl":"10.1007/s00205-024-02070-z","url":null,"abstract":"<div><p>We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen–Cahn energy functionals on a smooth bounded domain. Given any such sequence with uniform energy bounds, we prove that the limit-interface is a free boundary varifold which is integer rectifiable up to the boundary. This extends earlier work of Hutchinson and Tonegawa on the interior regularity of the limit-interface. A key novelty in our result is that no convexity assumption of the boundary is required and it is valid even when the limit-interface clusters near the boundary. Moreover, our arguments are local and thus work in the Riemannian setting. This work provides the first step towards the regularity theory for the Allen–Cahn min-max theory for free boundary minimal hypersurfaces, which was developed in the Almgren–Pitts setting by the first-named author and Zhou.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02070-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714240","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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