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A Stable-Compact Method for Qualitative Properties of Semilinear Elliptic Equations 半线性椭圆方程定性性质的稳定-紧化方法
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-26 DOI: 10.1007/s00205-026-02168-6
Henri Berestycki, Cole Graham
{"title":"A Stable-Compact Method for Qualitative Properties of Semilinear Elliptic Equations","authors":"Henri Berestycki,&nbsp;Cole Graham","doi":"10.1007/s00205-026-02168-6","DOIUrl":"10.1007/s00205-026-02168-6","url":null,"abstract":"<div><p>We study the uniqueness of reaction-diffusion steady states in general domains with Dirichlet boundary data. Here we consider “positive” (monostable) reactions. We describe geometric conditions on the domain that ensure uniqueness and we provide complementary examples of nonuniqueness. Along the way, we formulate a number of open problems and conjectures. To derive our results, we develop a general framework, the <i>stable-compact method</i>, to study qualitative properties of nonlinear elliptic equations.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02168-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147560960","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 Continuation Criterion for the Landau Equation with Very Soft and Coulomb Potentials 具有极软和库仑势的朗道方程的延拓判据
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-25 DOI: 10.1007/s00205-026-02182-8
Stanley Snelson, Caleb Solomon
{"title":"A Continuation Criterion for the Landau Equation with Very Soft and Coulomb Potentials","authors":"Stanley Snelson,&nbsp;Caleb Solomon","doi":"10.1007/s00205-026-02182-8","DOIUrl":"10.1007/s00205-026-02182-8","url":null,"abstract":"<div><p>We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, <span>(gamma in [-3,-2])</span>. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in <i>t</i> and <i>x</i>: (1) the mass density, (2) the velocity moment of order <i>s</i> for any small <span>(s&gt;0)</span>, and (3) the <span>(L^p_v)</span> norm for any <span>(p&gt;3/(5+gamma ))</span>. In particular, we do not require a bound on the energy density.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147561835","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
Global Strong Solutions to the Compressible Navier–Stokes–Coriolis System for Large Data 大数据可压缩Navier-Stokes-Coriolis系统的全局强解
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-24 DOI: 10.1007/s00205-026-02172-w
Mikihiro Fujii, Keiichi Watanabe
{"title":"Global Strong Solutions to the Compressible Navier–Stokes–Coriolis System for Large Data","authors":"Mikihiro Fujii,&nbsp;Keiichi Watanabe","doi":"10.1007/s00205-026-02172-w","DOIUrl":"10.1007/s00205-026-02172-w","url":null,"abstract":"<div><p>We consider the compressible Navier–Stokes system with the Coriolis force on the 3D whole space. In this model, the Coriolis force causes the linearized solution to behave like a 4th order dissipative semigroup <span>({ e^{-tDelta ^2} }_{t&gt;0})</span> with slower time decay rates than the heat kernel, which creates difficulties in nonlinear estimates in the low-frequency part and prevents us from constructing the global strong solutions by following the classical method. On account of this circumstance, the existence of unique global strong solutions has been open even in the classical Matsumura–Nishida framework. In this paper, we overcome the aforementioned difficulties and succeed in constructing a unique global strong solution in the framework of the scaling critical Besov space. Furthermore, our result also shows that the global solution is constructed for <i>large</i> initial data provided that the speed of the rotation is high and the Mach number is low enough by focusing on the dispersive effect due to the mixture of the Coriolis force and acoustic wave.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147560918","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 Long Way of a Viscous Vortex Dipole 粘性涡旋偶极子的漫漫长路
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-22 DOI: 10.1007/s00205-026-02169-5
Michele Dolce, Thierry Gallay
{"title":"The Long Way of a Viscous Vortex Dipole","authors":"Michele Dolce,&nbsp;Thierry Gallay","doi":"10.1007/s00205-026-02169-5","DOIUrl":"10.1007/s00205-026-02169-5","url":null,"abstract":"<div><p>We consider the evolution of a viscous vortex dipole in <span>(mathbb {R}^2)</span> originating from a pair of point vortices with opposite circulations. At high Reynolds number <span>(textrm{Re}gg 1)</span>, the dipole can travel a very long way, compared to the distance between the vortex centers, before being slowed down and eventually destroyed by diffusion. In this regime we construct an accurate approximation of the solution in the form of a two-parameter asymptotic expansion involving the aspect ratio of the dipole and the inverse Reynolds number. We then show that the exact solution of the Navier–Stokes equations remains close to the approximation on a time interval of length <span>(mathcal {O}(textrm{Re}^sigma ))</span>, where <span>(sigma &lt; 1)</span> is arbitrary. This improves upon previous results which were essentially restricted to <span>(sigma = 0)</span>. As an application, we provide a rigorous justification of an existing formula which gives the leading order correction to the translation speed of the dipole due to finite size effects.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147560791","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
A Free Boundary Monge–Ampère Equation and Applications to Complete Calabi–Yau Metrics 自由边界monge - ampantere方程及其在完全Calabi-Yau度量中的应用
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-19 DOI: 10.1007/s00205-026-02175-7
Tristan C. Collins, Freid Tong, Shing-Tung Yau
{"title":"A Free Boundary Monge–Ampère Equation and Applications to Complete Calabi–Yau Metrics","authors":"Tristan C. Collins,&nbsp;Freid Tong,&nbsp;Shing-Tung Yau","doi":"10.1007/s00205-026-02175-7","DOIUrl":"10.1007/s00205-026-02175-7","url":null,"abstract":"<div><p>Let <i>P</i> be a convex body containing the origin in its interior. We study a real Monge–Ampère equation with singularities along <span>(partial P)</span> which is Legendre dual to a certain free boundary Monge–Ampère equation. This is motivated by the existence problem for complete Calabi–Yau metrics on log Calabi–Yau pairs (<i>X</i>, <i>D</i>) with <i>D</i> an ample, simple normal crossings divisor. We prove the existence of solutions in <span>(C^{infty }(P)cap C^{1,alpha }(overline{P}))</span>, and establish the strict convexity of the free boundary. When <i>P</i> is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension 1 faces of <span>(partial P)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147560108","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: Free Boundary Problem for a Gas Bubble in a Liquid, and Exponential Stability of the Manifold of Spherically Symmetric Equilibria 修正：液体中气泡的自由边界问题，以及球对称平衡流形的指数稳定性
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-16 DOI: 10.1007/s00205-025-02158-0
Chen-Chih Lai, Michael I. Weinstein
{"title":"Correction to: Free Boundary Problem for a Gas Bubble in a Liquid, and Exponential Stability of the Manifold of Spherically Symmetric Equilibria","authors":"Chen-Chih Lai,&nbsp;Michael I. Weinstein","doi":"10.1007/s00205-025-02158-0","DOIUrl":"10.1007/s00205-025-02158-0","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147559568","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
Global Solution of 2D Hyperbolic Liquid Crystal System for Small Initial Data 二维双曲型液晶系统的小初始数据全局解
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-14 DOI: 10.1007/s00205-026-02178-4
Xuecheng Wang
{"title":"Global Solution of 2D Hyperbolic Liquid Crystal System for Small Initial Data","authors":"Xuecheng Wang","doi":"10.1007/s00205-026-02178-4","DOIUrl":"10.1007/s00205-026-02178-4","url":null,"abstract":"<div><p>We prove the global stability of small perturbation near the constant equilibrium for the two dimensional simplified Ericksen-Leslie hyperbolic system an incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao (J Funct Anal, 288:110858, 2025). As a byproduct, we obtain the sharp (same as the linear solution) <span>(L^infty _x)</span>-decay estimates for both the heat part and the wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity. This paper’s main contribution is the discovery of a novel null structure within the velocity equation’s wave-type quadratic self-interaction. This structure compensates the insufficient decay rate in 2D, which previously hindered the establishment of global regularity for small data.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147441391","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
Extinction Profiles for the Sobolev Critical Fast Diffusion Equation in Bounded Domains. I. One Bubble Dynamics Sobolev临界快速扩散方程的消光曲线。一、一个气泡动力学
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-11 DOI: 10.1007/s00205-026-02177-5
Tianling Jin, Jingang Xiong
{"title":"Extinction Profiles for the Sobolev Critical Fast Diffusion Equation in Bounded Domains. I. One Bubble Dynamics","authors":"Tianling Jin,&nbsp;Jingang Xiong","doi":"10.1007/s00205-026-02177-5","DOIUrl":"10.1007/s00205-026-02177-5","url":null,"abstract":"<div><p>In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold assumption on the initial data, we prove the dichotomy that every solution converges uniformly, in terms of relative error, to either a steady state or a blowing-up bubble.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02177-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147441326","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
Fujita-Kato solutions and optimal time decay for the Vlasov-Navier–Stokes system in the whole space 全空间Vlasov-Navier-Stokes系统的Fujita-Kato解和最优时间衰减
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-03-07 DOI: 10.1007/s00205-026-02171-x
Raphaël Danchin
{"title":"Fujita-Kato solutions and optimal time decay for the Vlasov-Navier–Stokes system in the whole space","authors":"Raphaël Danchin","doi":"10.1007/s00205-026-02171-x","DOIUrl":"10.1007/s00205-026-02171-x","url":null,"abstract":"<div><p>We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier–Stokes system in the whole three-dimensional space. Our primary goal is to establish that small initial velocities with critical Sobolev regularity <span>(H^{1/2})</span> and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier–Stokes equations (NS) that has been proved by Fujita and Kato in [11]. Assuming also that the initial velocity is in <span>(L^1,)</span> we establish that the total energy <span>(E_0)</span> of the system decays to 0 with the same rate <span>(t^{-3/2})</span> as for the weak solutions of (NS), see [22, 24]. Our results partly rely on the use of a higher order energy functional <span>(E_1)</span> that controls the regularity <span>(H^1)</span> of the velocity. This idea seems to originate from the recent paper [18] by Li, Shou and Zhang, devoted to the <i>inhomogeneous</i> Vlasov-Navier–Stokes system. Here we show that <span>(E_1)</span> decays with the rate <span>(t^{-5/2})</span> which, in particular, allows us to prove that the density of the particles has a strong limit when the time goes to infinity.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02171-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147441034","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
Wave Packets Propagation in the Subwavelength Regime Near the Dirac Point 狄拉克点附近亚波长区域的波包传播
IF 2.4 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2026-02-06 DOI: 10.1007/s00205-026-02164-w
Habib Ammari, Xin Fu, Wenjia Jing
{"title":"Wave Packets Propagation in the Subwavelength Regime Near the Dirac Point","authors":"Habib Ammari,&nbsp;Xin Fu,&nbsp;Wenjia Jing","doi":"10.1007/s00205-026-02164-w","DOIUrl":"10.1007/s00205-026-02164-w","url":null,"abstract":"<div><p>In Ammari et al. (SIAM J Math Anal 52:5441–5466, 2020), the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packet, which are spectrally concentrated near such conical points. We prove that the wave packet dynamic is governed by a time-dependent effective Dirac system, which still depends, but in a simple way, on the subwavelength scale.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147337486","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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