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Limiting Behavior of Minimizing p-Harmonic Maps in 3d as p Goes to 2 with Finite Fundamental Group 有限基本群p→2时三维p调和映射最小化的极限行为
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-24 DOI: 10.1007/s00205-025-02086-z
Bohdan Bulanyi, Jean Van Schaftingen, Benoît Van Vaerenbergh
{"title":"Limiting Behavior of Minimizing p-Harmonic Maps in 3d as p Goes to 2 with Finite Fundamental Group","authors":"Bohdan Bulanyi,&nbsp;Jean Van Schaftingen,&nbsp;Benoît Van Vaerenbergh","doi":"10.1007/s00205-025-02086-z","DOIUrl":"10.1007/s00205-025-02086-z","url":null,"abstract":"<div><p>We study the limiting behavior of minimizing <i>p</i>-harmonic maps from a bounded Lipschitz domain <span>(Omega subset mathbb {R}^{3})</span> to a compact connected Riemannian manifold without boundary and with finite fundamental group as <span>(p nearrow 2)</span>. We prove that there exists a closed set <span>(S_{*})</span> of finite length such that minimizing <i>p</i>-harmonic maps converge to a locally minimizing harmonic map in <span>(Omega setminus S_{*})</span>. We prove that locally inside <span>(Omega )</span> the singular set <span>(S_{*})</span> is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in <span>(overline{Omega })</span> the set <span>(S_{*})</span> is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and <span>(Omega )</span>.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02086-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865414","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
Hölder Regularity of the Pressure for Weak Solutions of the 3D Euler Equations in Bounded Domains Hölder有界区域内三维欧拉方程弱解压力的规律性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-17 DOI: 10.1007/s00205-025-02090-3
Claude Bardos, Daniel W. Boutros, Edriss S. Titi
{"title":"Hölder Regularity of the Pressure for Weak Solutions of the 3D Euler Equations in Bounded Domains","authors":"Claude Bardos,&nbsp;Daniel W. Boutros,&nbsp;Edriss S. Titi","doi":"10.1007/s00205-025-02090-3","DOIUrl":"10.1007/s00205-025-02090-3","url":null,"abstract":"<div><p>We consider the three-dimensional incompressible Euler equations on a bounded domain <span>(Omega )</span> with <span>(C^4)</span> boundary. We prove that if the velocity field <span>(u in C^{0,alpha } (Omega ))</span> with <span>(alpha &gt; 0)</span> (where we are omitting the time dependence), it follows that the corresponding pressure <i>p</i> of a weak solution to the Euler equations belongs to the Hölder space <span>(C^{0, alpha } (Omega ))</span>. We also prove that away from the boundary <i>p</i> has <span>(C^{0,2alpha })</span> regularity. In order to prove these results we use a local parametrisation of the boundary and a very weak formulation of the boundary condition for the pressure of the weak solution, as was introduced in Bardos and Titi (Philos Trans R Soc A 380, 20210073, 2022), which is different than the commonly used boundary condition for classical solutions of the Euler equations. Moreover, we provide an explicit example illustrating the necessity of this new very weak formulation of the boundary condition for the pressure. Furthermore, we also provide a rigorous derivation of this new formulation of the boundary condition for weak solutions of the Euler equations. This result is of importance for the proof of the first half of the Onsager Conjecture, the sufficient conditions for energy conservation of weak solutions to the three-dimensional incompressible Euler equations in bounded domains. In particular, the results in this paper remove the need for separate regularity assumptions on the pressure in the proof of the Onsager conjecture.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143845664","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
The inviscid inflow-outflow problem via analyticity 用解析法求解无粘流入流出问题
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-14 DOI: 10.1007/s00205-025-02095-y
Igor Kukavica, Wojciech Ożański, Marco Sammartino
{"title":"The inviscid inflow-outflow problem via analyticity","authors":"Igor Kukavica,&nbsp;Wojciech Ożański,&nbsp;Marco Sammartino","doi":"10.1007/s00205-025-02095-y","DOIUrl":"10.1007/s00205-025-02095-y","url":null,"abstract":"<div><p>We consider the incompressible Euler equations on an analytic domain <span>(Omega )</span> with a nonhomogeneous boundary condition <span>(ucdot {textsf{n}} = {overline{u}}cdot {textsf{n}})</span> on <span>(partial Omega )</span>, where <span>({overline{u}})</span> is a given divergence-free analytic vector field. We establish the local well-posedness for <i>u</i> in analytic spaces without any compatibility conditions in all space dimensions. We also prove the global well-posedness in the 2D case if <span>({overline{u}})</span> decays in time sufficiently fast.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02095-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143830873","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
Time-Harmonic Maxwell’s Equations in Periodic Waveguides 周期波导中的时谐麦克斯韦方程组
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-10 DOI: 10.1007/s00205-025-02099-8
A. Kirsch, B. Schweizer
{"title":"Time-Harmonic Maxwell’s Equations in Periodic Waveguides","authors":"A. Kirsch,&nbsp;B. Schweizer","doi":"10.1007/s00205-025-02099-8","DOIUrl":"10.1007/s00205-025-02099-8","url":null,"abstract":"<div><p>We study Maxwell’s equations with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and to solve the radiation problem. Furthermore, we characterize the set of all bounded solutions to the homogeneous problem. The case of a compact perturbation of the medium is included, and the scattering problem and the limiting absorption principle are discussed.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02099-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143818183","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
Stability and Large-Time Behavior on 3D Incompressible MHD Equations with Partial Dissipation Near a Background Magnetic Field 背景磁场附近部分耗散的三维不可压缩MHD方程的稳定性和大时间行为
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-10 DOI: 10.1007/s00205-025-02100-4
Hongxia Lin, Jiahong Wu, Yi Zhu
{"title":"Stability and Large-Time Behavior on 3D Incompressible MHD Equations with Partial Dissipation Near a Background Magnetic Field","authors":"Hongxia Lin,&nbsp;Jiahong Wu,&nbsp;Yi Zhu","doi":"10.1007/s00205-025-02100-4","DOIUrl":"10.1007/s00205-025-02100-4","url":null,"abstract":"<div><p>Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in <span>(mathbb R^3)</span>. The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the <span>(x_1)</span>-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting <span>(H^3({mathbb {R}}^3))</span>. In addition, explicit decay rates in <span>(H^2({mathbb {R}}^3))</span> are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in <span>(mathbb R^3)</span> remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02100-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143818214","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 Self-Similar Converging Shock Waves 关于自相似收敛激波
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-04-03 DOI: 10.1007/s00205-025-02096-x
Juhi Jang, Jiaqi Liu, Matthew Schrecker
{"title":"On Self-Similar Converging Shock Waves","authors":"Juhi Jang,&nbsp;Jiaqi Liu,&nbsp;Matthew Schrecker","doi":"10.1007/s00205-025-02096-x","DOIUrl":"10.1007/s00205-025-02096-x","url":null,"abstract":"<div><p>In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for <span>(gamma in (1,3])</span>. These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02096-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769770","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
Hydrodynamic Limit of Multiscale Viscoelastic Models for Rigid Particle Suspensions 刚性颗粒悬浮液多尺度粘弹性模型的水动力极限
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-03-20 DOI: 10.1007/s00205-025-02092-1
Mitia Duerinckx, Lucas Ertzbischoff, Alexandre Girodroux-Lavigne, Richard M. Höfer
{"title":"Hydrodynamic Limit of Multiscale Viscoelastic Models for Rigid Particle Suspensions","authors":"Mitia Duerinckx,&nbsp;Lucas Ertzbischoff,&nbsp;Alexandre Girodroux-Lavigne,&nbsp;Richard M. Höfer","doi":"10.1007/s00205-025-02092-1","DOIUrl":"10.1007/s00205-025-02092-1","url":null,"abstract":"<div><p>We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143668092","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
The Least Action Admissibility Principle 最小行为可采原则
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-03-09 DOI: 10.1007/s00205-025-02094-z
H. Gimperlein, M. Grinfeld, R. J. Knops, M. Slemrod
{"title":"The Least Action Admissibility Principle","authors":"H. Gimperlein,&nbsp;M. Grinfeld,&nbsp;R. J. Knops,&nbsp;M. Slemrod","doi":"10.1007/s00205-025-02094-z","DOIUrl":"10.1007/s00205-025-02094-z","url":null,"abstract":"<div><p>This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided for Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show that the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos’s entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by <span>(p(rho )=rho ^2)</span>, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581275","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
Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications 具有反对称势的p-调和系统的守恒定律及其应用
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-03-06 DOI: 10.1007/s00205-025-02085-0
Francesca Da Lio, Tristan Rivière
{"title":"Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications","authors":"Francesca Da Lio,&nbsp;Tristan Rivière","doi":"10.1007/s00205-025-02085-0","DOIUrl":"10.1007/s00205-025-02085-0","url":null,"abstract":"<div><p>We prove that <i>p</i>-harmonic systems with antisymmetric potentials of the form </p><div><div><span>$$begin{aligned} -,text{ div }left( (1+|nabla u|^2)^{frac{p}{2}-1},nabla uright) =(1+|nabla u|^2)^{frac{p}{2}-1},Omega cdot nabla u, end{aligned}$$</span></div></div><p>(<span>(Omega )</span> is antisymmetric) can be written in divergence form as a conservation law </p><div><div><span>$$begin{aligned} -text{ div }left( (1+|nabla u|^2)^{frac{p}{2}-1},A,nabla uright) =nabla ^perp Bcdot nabla u. end{aligned}$$</span></div></div><p>This extends to the <i>p</i>-harmonic framework the original work of the second author for <span>(p=2)</span> (see Rivière in Invent Math 168(1):1–22, 2007). We give applications of the existence of this divergence structure in the analysis <span>(prightarrow 2)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02085-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554154","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
BCS Critical Temperature on Half-Spaces 半空间上BCS临界温度
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2025-03-02 DOI: 10.1007/s00205-025-02088-x
Barbara Roos, Robert Seiringer
{"title":"BCS Critical Temperature on Half-Spaces","authors":"Barbara Roos,&nbsp;Robert Seiringer","doi":"10.1007/s00205-025-02088-x","DOIUrl":"10.1007/s00205-025-02088-x","url":null,"abstract":"<div><p>We study the BCS critical temperature on half-spaces in dimensions <span>(d=1,2,3)</span> with Dirichlet or Neumann boundary conditions. We prove that the critical temperature on a half-space is strictly higher than on <span>(mathbb {R}^d)</span>, at least at weak coupling in <span>(d=1,2)</span> and weak coupling and small chemical potential in <span>(d=3)</span>. Furthermore, we show that the relative shift in critical temperature vanishes in the weak coupling limit.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-025-02088-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143529998","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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