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Rigidity and Functional Properties of (textrm{BD}_{textrm{dev}}(Omega )) 的刚性和功能特性 (textrm{BD}_{textrm{dev}}(Omega ))
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-21 DOI: 10.1007/s00205-026-02210-7
Marco Caroccia, Nicolas Van Goethem
{"title":"Rigidity and Functional Properties of (textrm{BD}_{textrm{dev}}(Omega ))","authors":"Marco Caroccia,&nbsp;Nicolas Van Goethem","doi":"10.1007/s00205-026-02210-7","DOIUrl":"10.1007/s00205-026-02210-7","url":null,"abstract":"<div><p>We provide a structural analysis of the space of functions of bounded deviatoric deformation, <span>(textrm{BD}_{textrm{dev}})</span>, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for <span>(textrm{BD}_{textrm{dev}})</span>-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation, homogenization, and integral representation problems, allowing for integrands with explicit dependence on <i>u</i> as well as <span>({{mathcal {E}}}_d u)</span>. Our approach overcomes several difficulties as compared to the <span>(textrm{BD})</span> case, in particular due to the lack of invariance of <span>({{mathcal {E}}}_d)</span> under orthogonalization of the polar directions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02210-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148323172","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
Scattering from analytic and piecewise analytic inhomogeneities 解析和分段解析非齐次的散射
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-19 DOI: 10.1007/s00205-026-02215-2
Narek Hovsepyan, Michael S. Vogelius
{"title":"Scattering from analytic and piecewise analytic inhomogeneities","authors":"Narek Hovsepyan,&nbsp;Michael S. Vogelius","doi":"10.1007/s00205-026-02215-2","DOIUrl":"10.1007/s00205-026-02215-2","url":null,"abstract":"<div><p>We study scattering for the linear Helmholtz operator in two dimensions and develop a technique which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02215-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281673","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
Singular Flows with Time-Varying Weights 时变权的奇异流
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-17 DOI: 10.1007/s00205-026-02216-1
Immanuel Ben-Porat, José A. Carrillo, Pierre-Emmanuel Jabin
{"title":"Singular Flows with Time-Varying Weights","authors":"Immanuel Ben-Porat,&nbsp;José A. Carrillo,&nbsp;Pierre-Emmanuel Jabin","doi":"10.1007/s00205-026-02216-1","DOIUrl":"10.1007/s00205-026-02216-1","url":null,"abstract":"<div><p>We study the mean field limit for singular dynamics with time evolving weights. Our results are an extension of the work of Serfaty [14] and Bresch-Jabin-Wang [8], which consider singular Coulomb flows with weights which are constant time. The inclusion of time dependent weights necessitates the commutator estimates of [8, 14], as well as a new functional inequality. The well-posedness of the mean field PDE and the associated system of trajectories is also proven.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02216-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281203","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
Condensation and Non-condensation Times for 4-Wave Kinetic Equations 四波动力学方程的凝聚和非凝聚时间
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-16 DOI: 10.1007/s00205-026-02217-0
Gigliola Staffilani, Minh-Binh Tran
{"title":"Condensation and Non-condensation Times for 4-Wave Kinetic Equations","authors":"Gigliola Staffilani,&nbsp;Minh-Binh Tran","doi":"10.1007/s00205-026-02217-0","DOIUrl":"10.1007/s00205-026-02217-0","url":null,"abstract":"<div><p>Inspired by the pioneering work of Escobedo and Velazquez [30, 31], we develop a new framework to prove that solutions of 4-wave kinetic equations, under very general forms of dispersion relations, develop a condensation at the origin in finite time, under weaker conditions on the initial data than the ones considered in [30, 31]. We also provide some estimates on the non-condensation times of the solutions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02217-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281425","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
Ill-Posedness of the Boltzmann–BGK Model in the Exponential Class 指数类中Boltzmann-BGK模型的病态性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-16 DOI: 10.1007/s00205-026-02176-6
Donghyun Lee, Sungbin Park, Seok-Bae Yun
{"title":"Ill-Posedness of the Boltzmann–BGK Model in the Exponential Class","authors":"Donghyun Lee,&nbsp;Sungbin Park,&nbsp;Seok-Bae Yun","doi":"10.1007/s00205-026-02176-6","DOIUrl":"10.1007/s00205-026-02176-6","url":null,"abstract":"<div><p>The BGK (Bhatnagar–Gross–Krook) model is a relaxation-type model of the Boltzmann equation, which is popularly used in place of the Boltzmann equation in physics and engineering. In this paper, we address the ill-posedness problem for the BGK model, in which the solution instantly escapes the initial solution space. For this, we propose two ill-posedness scenarios, namely, the homogeneous and the inhomogeneous ill-posedness mechanisms. In the former case, we find a class of spatially homogeneous solutions to the BGK model, where removing the small velocity part of the initial data triggers ill-posedness by increasing temperature. For the latter, we construct a spatially inhomogeneous solution to the BGK model such that the local temperature constructed from the solution has a polynomial growth in spatial variable. These ill-posedness properties for the BGK model pose a stark contrast with the Boltzmann equation for which the solution map is, at least for a finite time, stable in the corresponding solution spaces.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02176-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281394","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
Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring 切线锚定单元球上谐波映射的最小化
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-14 DOI: 10.1007/s00205-026-02213-4
Lia Bronsard, Andrew Colinet, Dominik Stantejsky
{"title":"Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring","authors":"Lia Bronsard,&nbsp;Andrew Colinet,&nbsp;Dominik Stantejsky","doi":"10.1007/s00205-026-02213-4","DOIUrl":"10.1007/s00205-026-02213-4","url":null,"abstract":"<div><p>Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of <span>(mathbb {S}^2-)</span>valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in <span>(mathbb {R}^{3})</span>. In particular, we obtain a monotonicity formula respecting tangentiality on a curved boundary in order to show optimal regularity up to the boundary. We introduce novel sufficient conditions under which the minimizer must exhibit symmetries. Under a symmetry assumption, we present a delineation of the singularities of minimizers, namely that a mimimizer has exactly two point singularities, located on the boundary at opposite points.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148281085","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
Free Boundary Regularity for Semilinear Variational Problems with a Topological Constraint 具有拓扑约束的半线性变分问题的自由边界正则性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-09 DOI: 10.1007/s00205-026-02211-6
Michael Novack, Daniel Restrepo, Anna Skorobogatova
{"title":"Free Boundary Regularity for Semilinear Variational Problems with a Topological Constraint","authors":"Michael Novack,&nbsp;Daniel Restrepo,&nbsp;Anna Skorobogatova","doi":"10.1007/s00205-026-02211-6","DOIUrl":"10.1007/s00205-026-02211-6","url":null,"abstract":"<div><p>We study a class of semilinear free boundary problems in which admissible functions <i>u</i> have a topological constraint, or <i>spanning condition</i>, on their 1-level set. This constraint forces <span>({u=1})</span>, which is the free boundary, to behave like a surface with some special types of singularities attached to a fixed boundary frame, in the spirit of the Plateau problem [28]. Two such free boundary problems are the minimization of capacity among surfaces sharing a common boundary and an Allen–Cahn formulation of the Plateau problem. We establish the existence of minimizers and study their regularity properties, obtaining the optimal Lipschitz regularity of minimizers and analytic regularity for the free boundaries away from a codimension two singular set. The singularity models for these problems are given by conical critical points of the minimal capacity problem, which are closely related to spectral optimal partition and segregation problems.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148236473","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
Long–Time and Chaotic Dynamics in Microscopic Systems with Singular Interactions 具有奇异相互作用的微观系统的长时间混沌动力学
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-06 DOI: 10.1007/s00205-026-02189-1
Alexis Béjar-López, Alain Blaustein, Pierre-Emmanuel Jabin, Juan Soler
{"title":"Long–Time and Chaotic Dynamics in Microscopic Systems with Singular Interactions","authors":"Alexis Béjar-López,&nbsp;Alain Blaustein,&nbsp;Pierre-Emmanuel Jabin,&nbsp;Juan Soler","doi":"10.1007/s00205-026-02189-1","DOIUrl":"10.1007/s00205-026-02189-1","url":null,"abstract":"<div><p>This paper investigates the long–time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of <i>N</i> interacting point particles, where the time evolution of the joint distribution <span>(f_N(t))</span> is governed by the Liouville equation. Our primary objective is to analyze the system’s behavior over extended time intervals, focusing on the stability, the potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime <span>(N gg 1)</span>, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases we prove that it is possible to derive uniform bounds in <span>( L^2 )</span>, both over time and with respect to the number of particles, on the marginals <span>( left( f_{k,N}right) _{1leqq k leqq N})</span>, irrespective of the initial state’s chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels <span>( K in W^{frac{-2}{d+2}, d+2} )</span> in dimension <span>(dgeqq 2)</span>, surpassing the traditional <span>( L^d )</span> regularity barriers. Finally, we address the highly singular case of <span>( K in H^{-1} )</span> under a threshold temperature regime, offering new insights into the behavior of such systems.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02189-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172750","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
Maximal Dissipation and Well-Posedness of the Euler System of Gas Dynamics 气体动力学欧拉系统的最大耗散和适定性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-03 DOI: 10.1007/s00205-026-02205-4
Eduard Feireisl, Ansgar Jüngel, Mária Lukáčová-Medvid’ová
{"title":"Maximal Dissipation and Well-Posedness of the Euler System of Gas Dynamics","authors":"Eduard Feireisl,&nbsp;Ansgar Jüngel,&nbsp;Mária Lukáčová-Medvid’ová","doi":"10.1007/s00205-026-02205-4","DOIUrl":"10.1007/s00205-026-02205-4","url":null,"abstract":"<div><p>We show that any dissipative (measure–valued) solution of the compressible Euler system that complies with Dafermos’ criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two-step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos’ criterion yielding a unique solution of the problem for any finite energy initial data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02205-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148141679","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
Effective Dynamics and Blow Up in a Model of Magnetic Relaxation 磁松弛模型中的有效动力学和爆炸
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-06-01 DOI: 10.1007/s00205-026-02202-7
Dimitri Cobb, Daniel Sánchez-Simón del Pino, Juan J. L. Velázquez
{"title":"Effective Dynamics and Blow Up in a Model of Magnetic Relaxation","authors":"Dimitri Cobb,&nbsp;Daniel Sánchez-Simón del Pino,&nbsp;Juan J. L. Velázquez","doi":"10.1007/s00205-026-02202-7","DOIUrl":"10.1007/s00205-026-02202-7","url":null,"abstract":"<div><p>In this article we study a one dimensional model for Magnetic Relaxation. This model was introduced by Moffatt [20] and describes a low resistivity viscous plasma in which the pressure and the inertia are much smaller than the magnetic pressure. In the limit of resistivity <span>(varepsilon rightarrow 0)</span>, we prove the existence of two time scales for the evolution of the magnetic field: a fast one for times of order <span>(log (varepsilon ^{-1}))</span> in which the resistivity plays no role and the energy is dissipated only via viscosity; and a slow one for times of order <span>(varepsilon ^{-1})</span> characterized by the influence of the resistivity. We show that in this second time scale, as <span>(varepsilon rightarrow 0)</span>, the modulus of magnetic field approaches a function that depends only on time. We also prove that, in this regime, the magnetic field <span>(b_varepsilon (t,x))</span> can be approximated as <span>(varepsilon rightarrow 0)</span> by the solution of a PDE whose solutions exhibit blow up for some choices of initial data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02202-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172122","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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