Archive for Rational Mechanics and Analysis最新文献

{"title":"Distance Between Minimal Surfaces and Flows","authors":"Tobias Holck Colding,&nbsp;William P. Minicozzi II","doi":"10.1007/s00205-026-02251-y","DOIUrl":"10.1007/s00205-026-02251-y","url":null,"abstract":"<div><p>We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation.\u0000 This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading, in particular, to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149034567","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}

{"title":"Algebraic Localization Implies Exponential Localization in Non-periodic Insulators","authors":"Jianfeng Lu,&nbsp;Kevin D. Stubbs","doi":"10.1007/s00205-026-02242-z","DOIUrl":"10.1007/s00205-026-02242-z","url":null,"abstract":"<div><p>Exponentially-localized Wannier functions are a basis of the Fermi projection of a Hamiltonian consisting of functions which decay exponentially fast in space. In two and three spatial dimensions, it is well understood for periodic insulators that exponentially-localized Wannier functions exist if and only if there exists an orthonormal basis for the Fermi projection with finite second moment (that is all basis elements satisfy <span>(int vert varvec{x}vert ^2 vert w(varvec{x})vert ^2 ,text {d}{varvec{x}} &lt; infty )</span>). In this work, we establish a similar result for non-periodic insulators in two spatial dimensions. In particular, we prove that if there exists an orthonormal basis for the Fermi projection which satisfies <span>(int vert varvec{x}vert ^{5 + varepsilon } vert w(varvec{x})vert ^2 ,text {d}{varvec{x}} &lt; infty )</span> for some <span>(varepsilon &gt; 0)</span>, then there also exists an orthonormal basis for the Fermi projection which decays exponentially fast in space. This result lends support to the Localization Dichotomy Conjecture for non-periodic systems recently proposed by Marcelli, Monaco, Moscolari, and Panati in [1, 2].</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02242-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149034671","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}

{"title":"Contact in Fluid-Plate Interaction: Formation and Detachment","authors":"SrƉan Trifunović","doi":"10.1007/s00205-026-02239-8","DOIUrl":"10.1007/s00205-026-02239-8","url":null,"abstract":"<div><p>In this paper, we study an interaction problem between an elastic plate and a compressible viscous fluid located between the rigid bottom <span>(z=0)</span> and the plate. First, by utilizing the vertical fluid dissipation, we show that the plate displacement <span>(eta )</span> satisfies <span>(ln eta (t) in L^1)</span> for any <span>(t&gt;0)</span>, provided that <span>(ln eta _0in L^1)</span>, ensuring that additional plate contact can form only on a set of a measure zero. Then, by utilizing the expanding capability of compressible fluid pressure, we show that all contact has to detach in finite time provided that the source force acting onto the plate is not pushing down excessively. Finally, we show that contact at any point can be detached in any given time with a strong enough source force localized around that point which is pulling the plate up. The results are based on a novel variational inequality which preserves key information about the total force applied onto the plate, even in presence of contact. This is the first result where detachment of contact is proven in fluid-structure interaction.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986558","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}

{"title":"On the Relaxation of Polyconvex Functionals with Linear Growth Under Strict Convergence in BV","authors":"Riccardo Scala","doi":"10.1007/s00205-026-02243-y","DOIUrl":"10.1007/s00205-026-02243-y","url":null,"abstract":"<div><p>We consider the relaxation of polyconvex functionals with linear growth with respect to the strict convergence in the space of functions of bounded variation. The functionals under relaxation are of the form <span>(F(u,Omega ):=int _Omega f(nabla u)dx)</span>, where <span>(u:Omega rightarrow mathbb {R}^m)</span>, and <i>f</i> is polyconvex. In constrast with the case of relaxation with respect to the standard <span>(L^1)</span>-convergence, in the case that <span>(Omega )</span> is 2-dimensional, we prove that the sets map <span>(Amapsto F(u,A))</span> for <i>A</i> open, is, for every <span>(uin BV(Omega ;mathbb {R}^m))</span>, <span>(mge 1)</span>, the restriction of a Borel measure. This is not true in the case <span>(Omega subset mathbb {R}^n)</span>, with <span>(nge 3)</span>. Using the integral representation formula for a special class of functions, we also give a short proof of the existence of Cartesian maps whose relaxed area functional with respect to the <span>(L^1)</span>-convergence is strictly larger than the area of its graph.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986392","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}

{"title":"Correction to: Navigating the Complex Landscape of Shock Filter Cahn–Hilliard Equation: From Regularized to Entropy Solutions","authors":"Darko Mitrović,&nbsp;Andrej Novak","doi":"10.1007/s00205-026-02233-0","DOIUrl":"10.1007/s00205-026-02233-0","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986226","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}

{"title":"Sharp Macroscopic Blow-up Behavior for the Parabolic-Elliptic Keller–Segel System in Dimensions (nge 3)","authors":"Loth Damagui CHABI,&nbsp;Philippe SOUPLET","doi":"10.1007/s00205-026-02234-z","DOIUrl":"10.1007/s00205-026-02234-z","url":null,"abstract":"<div><p>We study the space-time concentration or blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller–Segel system in the whole space or in a ball. We show that, for any blow-up solution in dimensions <span>(3le nle 9)</span> (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution <i>U</i> such that </p><div><div><span>$$begin{aligned} u(x,t)=(1+o(1))U(x,t),quad hbox {as }(x,t)rightarrow (0,T), end{aligned}$$</span></div></div><p>where <i>T</i> is the blow-up time. This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables (<i>x</i>, <i>t</i>). It strongly improves on existing results, since such behavior was previously known ([26]) to hold only in the microscopic scale <span>(|x|le O(sqrt{T-t}))</span> as <span>(trightarrow T)</span> (and in the whole space case only). As a consequence, we obtain the two-sided global estimate </p><div><div><span>$$begin{aligned} C_1le (T-t+|x|^2)u(x,t)le C_2quad hbox {in }B_Rtimes (T/2,T), end{aligned}$$</span></div></div><p>whose upper part only was known before ([56]), as well as the sharp final profile </p><div><div><span>$$begin{aligned} lim _{xrightarrow 0} |x|^2u(x,T)=Lin (0,infty ). end{aligned}$$</span></div></div><p>The latter improves, with a different proof, the recent result of [1] by excluding the possibility <span>(L=0)</span>. We also give extensions of these results, in higher dimensions, to type I and to time monotone solutions. Moreover, we extend the known results on type I estimates and on convergence in similarity variables, and significantly simplify their proofs.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927717","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}

{"title":"Structural Stability of Prandtl Boundary Layers in the Entire Subsonic Regime","authors":"Shengxin Li,&nbsp;Tong Yang,&nbsp;Zhu Zhang","doi":"10.1007/s00205-026-02241-0","DOIUrl":"10.1007/s00205-026-02241-0","url":null,"abstract":"<div><p>Despite the physical significance of the compressible Navier–Stokes equations in the presence of strong boundary layers, rigorous mathematical theories in this area remain at an early stage of development. In this paper, we establish the structural stability of shear flow boundary layer profiles for the two-dimensional steady compressible Navier–Stokes equations. The stability estimates are uniform across the entire subsonic regime. As a byproduct, we provide the first rigorous justification of the low Mach number limit in the presence of Prandtl-type boundary layers. The proof relies on the quasi-compressible-Stokes iteration introduced in [41], combining a frequency-dependent analysis of the density-velocity coupling, and the identification of structural cancellations in higher-order estimates.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02241-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148926889","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}

{"title":"Hypoelliptic Regularization in the Obstacle Problem for the Kolmogorov Operator","authors":"David Bowman","doi":"10.1007/s00205-026-02244-x","DOIUrl":"10.1007/s00205-026-02244-x","url":null,"abstract":"<div><p>We study the obstacle problem associated with the Kolmogorov operator <span>(mathcal {L}{:}{=}Delta _v )</span> <span>(- partial _t - vcdot nabla _x)</span>, which arises from the theory of optimal control in Asian-American options pricing models. This problem presents a significant departure from the elliptic and parabolic obstacle problems due to the highly degenerate hypoelliptic nature and the non-commutative Galilean group structure underlying the operator <span>(mathcal {L})</span>. First, we improve the known regularity of solutions. The existing interior estimate in the literature is from [14], where it is proven that solutions are <span>(C^{1,1})</span> with respect to the kinetic distance and therefore <span>(C^{0,2/3}_x)</span> Hölder regular in the spatial variable <i>x</i>. This result, which is claimed to be optimal, is the typical regularity obtained in obstacle problems, and is consistent with the typically expected gain of 2/3 derivatives in <i>x</i> from the theory of hypoellipticity for the operator <span>(mathcal {L})</span>. Our first result is rather unexpected then, for we manage to prove an even stronger hypoelliptic regularization effect: solutions enjoy a Lipschitz <span>(C^{0,1}_x)</span> interior estimate in the spatial variable <i>x</i>. We prove <span>(C^{0,1}_{t,x} cap C^{1,1}_v)</span> interior estimates following an approach which is based on Bernstein’s technique and draws on ideas from Evans-Kyrlov theory, and our method seems to be quite robust. We then use this improvement in regularity of the solution to prove the first known free boundary regularity result. We show that under a standard thickness condition, the free boundary is differentiable with respect to the kinetic distance and is a <span>(C^{0,1/2}_{t,x} cap C^{0,1}_v)</span> regular surface. This result constitutes the first step in the program of free boundary regularity. Critically, our arguments rely on a new monotonicity formula and a commutator estimate that are only made possible by the solution’s enhanced regularity in <i>x</i>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02244-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148926848","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}

{"title":"Does Yakhot’s Growth Law for Turbulent Burning Velocity Hold?","authors":"Wenjia Jing,&nbsp;Jack Xin,&nbsp;Yifeng Yu","doi":"10.1007/s00205-026-02238-9","DOIUrl":"10.1007/s00205-026-02238-9","url":null,"abstract":"<div><p>Using formal renormalization theory, Yakhot derived in ([32], 1988) an <span>(Oleft( frac{A}{sqrt{log A}}right) )</span> growth law of the turbulent flame speed with respect to large flow intensity <i>A</i> based on the inviscid G-equation. Although this growth law is widely cited in the combustion literature, there has been no rigorous mathematical discussion to date about its validity. As a first step towards unveiling this mystery, we prove that there is no intermediate growth law between <span>(Oleft( frac{A}{log A}right) )</span> and <i>O</i>(<i>A</i>) for two dimensional incompressible Lipschitz continuous periodic flows with bounded swirl sizes. In particular, we do not assume the non-degeneracy of critical points. In addition, other examples of flows with lower regularity, Lagrangian chaos, and related phenomena are also discussed.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 5","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148926435","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}

{"title":"The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates","authors":"Lennart Machill,&nbsp;Angkana Rüland","doi":"10.1007/s00205-026-02228-x","DOIUrl":"10.1007/s00205-026-02228-x","url":null,"abstract":"<div><p>We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather general lower scaling bounds, both in the setting of prescribed Dirichlet data and for periodic configurations with a mean value constraint. In contrast to the setting without gauge invariances, these lower scaling bounds depend on <i>two</i> parameters – the order of lamination of the boundary data as well as the number of involved (non-)degenerate symmetrized rank-one directions. By discussing upper bounds in specific geometries and for a specific constellation of wells, we give evidence of the sharpness of these lower bound estimates. Hence, it is necessary to keep track of the outlined <i>two</i> parameters in deducing scaling laws within the geometrically linearized theory of elasticity.</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://link.springer.com/content/pdf/10.1007/s00205-026-02228-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148837512","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}