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The Unique Limit of the Glimm–Lax Construction for Sobolev Data and Obstructions to 1-d Convex Integration 一维凸积分中Sobolev数据和障碍的Glimm-Lax构造的唯一极限。
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-24 DOI: 10.1007/s00205-026-02232-1
Jeffrey Cheng, Cooper Faile, Sam G. Krupa
{"title":"The Unique Limit of the Glimm–Lax Construction for Sobolev Data and Obstructions to 1-d Convex Integration","authors":"Jeffrey Cheng,&nbsp;Cooper Faile,&nbsp;Sam G. Krupa","doi":"10.1007/s00205-026-02232-1","DOIUrl":"10.1007/s00205-026-02232-1","url":null,"abstract":"<div><p>We consider a genuinely nonlinear 1-d system of hyperbolic conservation laws with two unknowns. A famous construction of Glimm &amp; Lax shows that global-in-time “Glimm–Lax” weak entropy solutions exist in this setting for any initial data with small <span>(L^infty )</span> norm [Mem. Amer. Math. Soc. (1970), no. 101]. Recent work in the <span>(L^1)</span>-stability theory by Bressan, Marconi &amp; Vaidya has given the first partial uniqueness and stability results for these solutions [Arch. Ration. Mech. Anal. (2025), vol. 249]. In this paper, we build on these results by combining them with recent advances in the <span>(L^2)</span>-theory. We show that solutions with initial data in the Sobolev space <span>(H^s)</span> for <span>(s&gt;0)</span> are unique in the full class of Glimm–Lax solutions that decay in total variation at a rate of 1/<i>t</i>. As a secondary result, our techniques are also used to show the recent non-uniqueness result of Chen, Vasseur &amp; Yu for continuous solutions [preprint (2024)] cannot extend to <span>(C^alpha )</span> solutions for <span>(alpha &gt; 1/2)</span>, alongside some appropriate fractional Sobolev spaces <span>(W^{s,p})</span>. An auxiliary result of independent interest is the development of a weighted relative entropy contraction for perturbations of rarefaction waves.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13503483/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148820344","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
Gradient Flow Dynamics for Cell Membranes in the Canham–Helfrich Model Canham-Helfrich模型中细胞膜的梯度流动动力学。
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-24 DOI: 10.1007/s00205-026-02235-y
Fabian Rupp, Christian Scharrer, Manuel Schlierf
{"title":"Gradient Flow Dynamics for Cell Membranes in the Canham–Helfrich Model","authors":"Fabian Rupp,&nbsp;Christian Scharrer,&nbsp;Manuel Schlierf","doi":"10.1007/s00205-026-02235-y","DOIUrl":"10.1007/s00205-026-02235-y","url":null,"abstract":"<div><p>The most energetically efficient way that a deformed red blood cell regains equilibrium is mathematically described by the gradient flow of the Canham–Helfrich functional, including a spontaneous curvature and the conservation of surface area and enclosed volume. Using a recently discovered multiplicity inequality, we prove the global existence and convergence of smooth solutions for spheres and axisymmetric tori, provided that the initial energy lies below explicit thresholds.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13503399/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148820318","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
Multisoliton Solutions and Blow Up for the (L^2)-Critical Hartree Equation (L^2)临界Hartree方程的多孤子解和爆破
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-22 DOI: 10.1007/s00205-026-02229-w
Jaime Gómez, Tobias Schmid, Yutong Wu
{"title":"Multisoliton Solutions and Blow Up for the (L^2)-Critical Hartree Equation","authors":"Jaime Gómez,&nbsp;Tobias Schmid,&nbsp;Yutong Wu","doi":"10.1007/s00205-026-02229-w","DOIUrl":"10.1007/s00205-026-02229-w","url":null,"abstract":"<div><p>We construct multisoliton solutions for the <span>(L^2)</span>-critical Hartree equation with trajectories asymptotically obeying a many-body law for an inverse square potential. More precisely, we consider the <i>m</i>-body hyperbolic and parabolic non-trapped dynamics. The pseudo-conformal symmetry then implies finite-time collision blow-up in the latter case and a solution blowing up at <i>m</i> distinct points in the former case. The approach we take is based on the ideas of Krieger–Martel–Raphaël [10] and the third author’s recent extension [36]. The approximation scheme requires new aspects in order to deal with a certain degeneracy for generalized root space elements.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02229-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783159","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
Structure of the Free Interfaces Near Triple Junction Singularities in Harmonic Maps and Optimal Partition Problems 调和映射中三结奇点附近自由界面的结构及最优配分问题
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-21 DOI: 10.1007/s00205-026-02221-4
Roberto Ognibene, Bozhidar Velichkov
{"title":"Structure of the Free Interfaces Near Triple Junction Singularities in Harmonic Maps and Optimal Partition Problems","authors":"Roberto Ognibene,&nbsp;Bozhidar Velichkov","doi":"10.1007/s00205-026-02221-4","DOIUrl":"10.1007/s00205-026-02221-4","url":null,"abstract":"<div><p>We consider energy-minimizing harmonic maps into trees and we prove the regularity of the singular part of the free interface near triple junction points. Precisely, by proving a new epiperimetric inequality, we show that around any point of frequency <span>(3/2)</span>, the free interface is composed of three <span>(C^{1,alpha })</span>-smooth <span>((d-1))</span>-dimensional manifolds (composed of points of frequency 1) with common <span>(C^{1,alpha })</span>-regular boundary (made of points of frequency <span>(3/2)</span>) that meet along this boundary at 120 degree angles. Our results also apply to spectral optimal partition problems for the Dirichlet eigenvalues.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783399","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
Least Total Curvature Solutions to Steady Euler System and Monotone Solutions to Semilinear Equations in a Strip 稳定欧拉系统的最小总曲率解及条上半线性方程的单调解
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-21 DOI: 10.1007/s00205-026-02237-w
Changfeng Gui, David Ruiz, Chunjing Xie, Huan Xu
{"title":"Least Total Curvature Solutions to Steady Euler System and Monotone Solutions to Semilinear Equations in a Strip","authors":"Changfeng Gui,&nbsp;David Ruiz,&nbsp;Chunjing Xie,&nbsp;Huan Xu","doi":"10.1007/s00205-026-02237-w","DOIUrl":"10.1007/s00205-026-02237-w","url":null,"abstract":"<div><p>This paper focuses on establishing the existence of a class of steady solutions, termed <i>least total curvature solutions</i>, to the incompressible Euler system in a strip. The solutions obtained in this paper <i>complement</i> the least total curvature solutions already known. Our approach employs a minimization procedure to identify a monotone heteroclinic solution for a conveniently chosen semilinear elliptic PDE. This method also enables us to construct positive and monotone (and consequently stable) solutions to semilinear elliptic PDEs with non-convex superlevel sets in a strip domain. This can be regarded as a negative answer to a generalized problem raised in [28].</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783338","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
Boundedness and Decay for the Teukolsky Equation on Kerr in the Full Subextremal Range (|a| 全次极值范围内Kerr上Teukolsky方程的有界性和衰减(|a|<M)：频率空间分析
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-21 DOI: 10.1007/s00205-026-02173-9
Yakov Shlapentokh-Rothman, Rita Teixeira da Costa
{"title":"Boundedness and Decay for the Teukolsky Equation on Kerr in the Full Subextremal Range (|a|<M): Frequency Space Analysis","authors":"Yakov Shlapentokh-Rothman,&nbsp;Rita Teixeira da Costa","doi":"10.1007/s00205-026-02173-9","DOIUrl":"10.1007/s00205-026-02173-9","url":null,"abstract":"<div><p>In this paper, and in the companion ([152]), we study the gauge-invariant components of linear perturbations of Kerr black hole spacetimes in the full subextremal range of parameters <span>(|a|&lt;M)</span>. We show that, for all <span>(|a|&lt;M)</span>, solutions of the Teukolsky equation of spin <span>(pm 1)</span> and spin <span>(pm 2)</span> arising from regular initial data remain bounded and decay in time. Our result makes completely explicit the dependence of these bounds on the Kerr black hole parameters, while being entirely self-contained. For completeness, we also include the case of the scalar wave equation, <span>(s=0)</span>. The present paper, which is an abridged version of our preprint ([151]), considers fixed frequency solutions of the transformed system of equations introduced by Dafermos et al. (Acta Math 222(1):1–214, 2019). We obtain estimates which are uniform in the separation parameters for subextremal <span>(|a|&lt;M)</span>, as well as estimates away from the superradiant frequency threshold up to and including the extremal <span>(|a|=M)</span>. In the subextremal <span>(|a|&lt;M)</span> case, these estimates are summed in our companion paper ([152]) to yield the conclusion of the proof of our main result.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02173-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783400","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 Regularity of Solutions to the Hamilton–Jacobi Equations for the N-Body Problem n体问题Hamilton-Jacobi方程解的正则性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-04 DOI: 10.1007/s00205-026-02227-y
Diego Berti, Davide Polimeni, Susanna Terracini
{"title":"On the Regularity of Solutions to the Hamilton–Jacobi Equations for the N-Body Problem","authors":"Diego Berti,&nbsp;Davide Polimeni,&nbsp;Susanna Terracini","doi":"10.1007/s00205-026-02227-y","DOIUrl":"10.1007/s00205-026-02227-y","url":null,"abstract":"<div><p>We prove that certain renormalized value functions associated with the <i>d</i>-dimensional (<span>(dgeqq 2)</span>) <i>N</i>-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton–Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its <span>(mathcal {H}^{d(N-1)-1})</span>-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02227-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148750740","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
Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation 曲面拟地转方程的整体存在性、哈密顿守恒和消失黏性
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-03 DOI: 10.1007/s00205-026-02231-2
Luigi De Rosa, Mickaël Latocca, Jaemin Park
{"title":"Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation","authors":"Luigi De Rosa,&nbsp;Mickaël Latocca,&nbsp;Jaemin Park","doi":"10.1007/s00205-026-02231-2","DOIUrl":"10.1007/s00205-026-02231-2","url":null,"abstract":"<div><p>For any initial datum <span>(theta _0in L^{frac{4}{3}}_x)</span> it is proven that the existence of a global-in-time weak solution <span>(theta in L^infty _t L^{frac{4}{3}}_x)</span> to the surface quasi-geostrophic equation whose Hamiltonian, i.e. the <span>(dot{H}^{-frac{1}{2}}_x)</span> norm, is constant in time. The solution is obtained as a vanishing viscosity limit. The main idea is to propagate in time the non-concentration of the <span>(L^{frac{4}{3}}_x)</span> norm of the initial data, from which the strong compactness in the Hamiltonian norm is deduced. Minimal Onsager supercritical conditions preventing anomalous dissipation are given.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02231-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148750632","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
Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation 逆幂势对硬球玻尔兹曼方程极限的最优收敛估计
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-08-02 DOI: 10.1007/s00205-026-02218-z
Zheng-Nan Hu, Jin Woo Jang, Zheng-An Yao, Yu-Long Zhou
{"title":"Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation","authors":"Zheng-Nan Hu,&nbsp;Jin Woo Jang,&nbsp;Zheng-An Yao,&nbsp;Yu-Long Zhou","doi":"10.1007/s00205-026-02218-z","DOIUrl":"10.1007/s00205-026-02218-z","url":null,"abstract":"<div><p>The inverse power potential <span>(U(r)=r^{-1/s}, 0&lt;s&lt;1)</span> generates the Boltzmann kernel <span>(B^{s}=|v-v_*|^{1-4s} b_s(theta ))</span> with an angular singularity as <span>(theta rightarrow 0)</span>. Jang et al. [7] proved the limit <span>(B^{s}rightarrow frac{1}{4}|v-v_*|)</span> as <span>(srightarrow 0)</span>, as well as weak convergence of solutions, but without a rate. In this work we establish the sharp estimate </p><div><div><span>$$ |b_s(theta )-tfrac{1}{4}| le C, s,theta ^{-2-2s}. $$</span></div></div><p>In particular, this sharp estimate yields the <i>optimal</i> <i>O</i>(<i>s</i>) convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any <span>(tin [0,T])</span>, we have </p><div><div><span>$$begin{aligned} f^s(t)=f^0(t)+O(s), end{aligned}$$</span></div></div><p>quantified by the <span>(L^1_k)</span> norm for <span>(kge 2.)</span></p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148652324","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
Correction to: Periodic Solutions of Symmetric Hamiltonian Systems 修正：对称哈密顿系统的周期解
IF 2.9 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-07-30 DOI: 10.1007/s00205-026-02226-z
Daniel Strzelecki
{"title":"Correction to: Periodic Solutions of Symmetric Hamiltonian Systems","authors":"Daniel Strzelecki","doi":"10.1007/s00205-026-02226-z","DOIUrl":"10.1007/s00205-026-02226-z","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 4","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614994","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
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