Archive for Rational Mechanics and Analysis最新文献

{"title":"Concentration of Cones in the Alt-Phillips Problem","authors":"Ovidiu Savin,&nbsp;Hui Yu","doi":"10.1007/s00205-026-02204-5","DOIUrl":"10.1007/s00205-026-02204-5","url":null,"abstract":"<div><p>We study minimizing cones in the Alt-Phillips problem for when the exponent <span>(gamma )</span> is for close to 1. For when <span>(gamma )</span> converges to 1, we show that the cones concentrate around <i>symmetric</i> solutions to the classical obstacle problem. To be precise, the limiting profiles are radial in a subspace and invariant in directions perpendicular to that subspace.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147830166","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 Existence of Source-Solutions to the Multidimensional Burgers Equation","authors":"João Fernando Nariyoshi","doi":"10.1007/s00205-026-02199-z","DOIUrl":"10.1007/s00205-026-02199-z","url":null,"abstract":"<div><p>We establish that, in general, there are no entropy solutions to the Cauchy problem for the multidimensional Burgers equation when the initial data is a measure. This answers in the negative a conjecture of <span>D. Serre–L. Silvestre</span> (Arch Rat Mech Anal 234:1391–1411, 2019). This conjecture was motivated by the description of the asymptotic behavior of entropy solutions with integrable initial data. Despite this negative result, we still derive some information on the asymptotic behavior of such solutions. Our method relies on new Laplace transform estimates for the propagation of information.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02199-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147829340","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":"Separation of Time Scales in Weakly Interacting Diffusions","authors":"Zachary P. Adams,&nbsp;Maximilian Engel,&nbsp;Rishabh S. Gvalani","doi":"10.1007/s00205-026-02180-w","DOIUrl":"10.1007/s00205-026-02180-w","url":null,"abstract":"<div><p>We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a “droplet state” which is <i>metastable</i>, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to 0. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is <i>O</i>(1) as the inverse temperature <span>(beta rightarrow infty )</span>. Meanwhile the rate of leakage away from its centre of mass is <span>(O(e^{-beta }))</span>. Furthermore, the quasi-stationary distribution is localised on a length scale of order <span>(O(beta ^{-frac{1}{2}}))</span>. Our proofs rely on understanding the large <span>(beta )</span>-asymptotics of the first two eigenvalues of the generator, which we study using techniques from semiclassical analysis. We thus provide a partial answer to a question posed by Carrillo et al. (see Aggregation–diffusion equations: dynamics, asymptotics, and singular limits. Active particles. Advances in theory, models, and applications, modeling and simulation in science, engineering and technology, vol 2, pp 65–108, Birkhäuser/Springer, Cham, 2019, Section 3.2.2) in the microscopic setting.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02180-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147796957","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":"Long-Time Dynamics for the Kelvin-Helmholtz Equations Close to Circular Vortex Sheets","authors":"Federico Murgante,&nbsp;Emeric Roulley,&nbsp;Stefano Scrobogna","doi":"10.1007/s00205-026-02174-8","DOIUrl":"10.1007/s00205-026-02174-8","url":null,"abstract":"<div><p>We consider the Kelvin-Helmholtz system describing the evolution of a vortex-sheet near the circular stationary solution. Answering previous numerical conjectures in the 1990s physics literature, we prove an almost global existence result for small-amplitude solutions. We first establish the existence of a linear stability threshold for the Weber number, which represents the ratio between the square of the background velocity jump and the surface tension. Then, we prove that, for almost all values of the Weber number below this threshold, any small solution lives for almost all times, remaining close to the equilibrium. Our analysis reveals a remarkable stabilization phenomenon: the presence of both non-zero background velocity jump and capillarity effects enables us to prevent nonlinear instability phenomena, despite the inherently unstable nature of the classical Kelvin-Helmholtz problem. This long-time existence would not be achievable in a setting where capillarity alone provides linear stabilization, without the richer modulation induced by the velocity jump. Our proof exploits the Hamiltonian nature of the equations. More specifically, we employ Hamiltonian Birkhoff normal form techniques for quasi-linear systems together with a general approach for paralinearization of non-linear singular integral operators. This approach allows us to control resonances and quasi-resonances at arbitrary order, ensuring the desired long-time stability result.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13121218/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147789960","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":"Stability of Stochastically Driven Couette Flow in 2D with Navier Boundary Conditions at High Reynolds Number via Averaging Principle","authors":"Ryan Arbon,&nbsp;Jacob Bedrossian","doi":"10.1007/s00205-026-02187-3","DOIUrl":"10.1007/s00205-026-02187-3","url":null,"abstract":"<div><p>We characterize the behavior of stochastic Navier–Stokes on <span>(mathbb {T}times [-1,1])</span> with Navier boundary conditions at high Reynolds number when initialized near Couette flow subject to small additive stochastic forcing. We take additive noise of strength <span>(nu ^{5/6} Phi textrm{d}V_t + nu ^{2/3+alpha } Psi textrm{d}W_t)</span>, where <span>(Phi textrm{d}V_t)</span> has spatial correlation in <span>(H_0^3)</span> and acts only on <i>x</i>-independent modes of the vorticity, while <span>(Psi textrm{d}W_t)</span> has spatial correlation in a lower order, anisotropic, Sobolev space <span>(mathcal {H})</span> and acts on <i>x</i>-dependent-modes. We take the initial <i>x</i>-independent modes in the perturbation to be small in <span>(H_0^3)</span> in a <span>(nu )</span>-independent sense, while the non-zero <i>x</i>-modes are taken to be <span>(O(nu ^{1/2 + alpha }))</span> in <span>(mathcal {H})</span>. The parameter <span>(alpha )</span> is taken to be <span>(alpha &gt; 1/12)</span>. Letting <span>(omega )</span> solve the resulting perturbation equation, we split <span>(omega )</span> into the zero <i>x</i>-modes <span>(omega _0)</span> and the non-zero <i>x</i>-modes <span>(omega _{ne })</span>. We demonstrate that an averaging principle holds wherein <span>(omega _{ne })</span> is the fast variable and <span>(omega _0)</span> is the slow variable, deriving a closed nonlinear evolution equation on <span>(omega _0)</span> that holds over long time-scales (while the fast <span>(omega _{ne })</span> modes solve a ‘pseudo-linearized’ equation to leading order with dynamics dominated by inviscid damping and enhanced dissipation). This work can also be considered the stochastic analogue of the stability threshold problem for shear flows. Furthermore, we explain the connections to the Stochastic Structural Stability Theory (S3T) in the physics literature.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02187-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147737647","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":"Parabolic PDEs with Dynamic Data under a Bounded Slope Condition","authors":"Verena Bögelein,&nbsp;Frank Duzaar,&nbsp;Giulia Treu","doi":"10.1007/s00205-026-02184-6","DOIUrl":"10.1007/s00205-026-02184-6","url":null,"abstract":"<div><p>We establish the existence of Lipschitz-continuous solutions to the Cauchy–Dirichlet problem for a class of evolutionary partial differential equations of the form </p><div><div><span>$$begin{aligned} partial _tu-{{,textrm{div},}}_x nabla _xi f(nabla u)=0 end{aligned}$$</span></div></div><p>in a space-time cylinder <span>(Omega _T=Omega times (0,T))</span>, subject to time-dependent boundary data <span>(g:partial _{mathcal {P}}Omega _Trightarrow mathbb {R})</span> prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data <i>g</i> along the lateral boundary <span>(partial Omega times (0,T))</span>. More precisely, we require that, for each fixed <span>(tin [0,T))</span>, the graph of <span>(g(cdot ,t))</span> over <span>(partial Omega )</span> admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13070980/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147693734","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":"Homogenization of a Vertical Oscillating Neumann Condition","authors":"William M Feldman,&nbsp;Zhonggan Huang","doi":"10.1007/s00205-026-02188-2","DOIUrl":"10.1007/s00205-026-02188-2","url":null,"abstract":"<div><p>We homogenize the Laplace and heat equations with the Neumann data oscillating in the “vertical\" <i>u</i>-variable. These are simplified models for interface motion in heterogeneous media, particularly capillary contact lines. The homogenization limit reveals a pinning effect at zero tangential slope, leading to a novel singularly anisotropic pinned Neumann condition. The singular pinning creates an unconstrained contact set generalizing the contact set in the classical thin obstacle problem. We establish a comparison principle for the heat equation with this new type of boundary condition. The comparison principle enables a proof of homogenization via the method of half-relaxed limits from viscosity solution theory. Our work also demonstrates – for the first time in a PDE problem in multiple dimensions – the emergence of rate-independent pinning from gradient flows with wiggly energies. Prior limit theorems of this type, in rate independent contexts, were limited to ODEs and PDEs in one dimension.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642945","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":"(L^2)-Based Stability of Blowup with Log Correction for Semilinear Heat Equation","authors":"Thomas Y. Hou,&nbsp;Van Tien Nguyen,&nbsp;Yixuan Wang","doi":"10.1007/s00205-026-02191-7","DOIUrl":"10.1007/s00205-026-02191-7","url":null,"abstract":"<div><p>We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted <span>(H^k)</span> stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the <span>(L^2)</span>-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642946","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":"Rigorous Derivation of Damped–Driven Wave Turbulence Theory","authors":"Ricardo Grande,&nbsp;Zaher Hani","doi":"10.1007/s00205-026-02181-9","DOIUrl":"10.1007/s00205-026-02181-9","url":null,"abstract":"<div><p>We provide a rigorous justification of various kinetic regimes exhibited by the nonlinear Schrödinger equation with an additive stochastic forcing and a viscous dissipation. The importance of such damped-driven models stems from their wide empirical use in studying turbulence for nonlinear wave systems. The force injects energy into the system at large scales, which is then transferred across scales, thanks to the nonlinear wave interactions, until it is eventually dissipated at smaller scales. The presence of such scale-separated forcing and dissipation allows for the constant flux of energy in the intermediate scales, known as the inertial range, which is the focus of the vast amount of numerical and physical literature on wave turbulence. Roughly speaking, our results provide a rigorous kinetic framework for this turbulent behavior by proving that the stochastic dynamics can be effectively described by a deterministic damped-driven kinetic equation, which carries the full picture of the turbulent energy dynamic across scales (like cascade spectra or other flux solutions). The analysis extends previous works in the unperturbed setting (Deng and Hani in Forum Math PI 9:e6, 2021; Deng and Hani in Invent math 543–724, 2023; Deng and Hani in Propagation of chaos and the higher order statistics in the wave kinetic theory, 2021. arXiv:2110.04565) to the above empirically motivated damped driven setting. Here, in addition to the size <i>L</i> of the system and the strength <span>(lambda )</span> of the nonlinearity, an extra thermodynamic parameter has to be included in the kinetic limit (<span>(Lrightarrow infty , lambda rightarrow 0)</span>), namely the strength <span>(nu )</span> of the forcing and dissipation. Various regimes emerge depending on the relative sizes of <i>L</i>, <span>(lambda )</span> and <span>(nu )</span>, which give rise to different kinetic equations. Two major novelties of this work is the extension of the Feynman diagram analysis to additive stochastic objects, and the sharp asymptotic development of the leading terms in that expansion.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02181-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147642639","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":"Long-Time Stability of a Stably Stratified Rest State in the Inviscid 2D Boussinesq Equation","authors":"Catalina Jurja,&nbsp;Klaus Widmayer","doi":"10.1007/s00205-026-02166-8","DOIUrl":"10.1007/s00205-026-02166-8","url":null,"abstract":"<div><p>We establish the nonlinear stability on a timescale <span>(O(varepsilon ^{-2}))</span> of a linearly, stably stratified rest state in the inviscid Boussinesq system on <span>(mathbb {R}^2)</span>. Here, <span>(varepsilon &gt;0)</span> denotes the size of an initially sufficiently small, Sobolev regular and localized perturbation. A similar statement also holds for the related dispersive SQG equation.</p><p>At the core of this result is a dispersive effect due to anisotropic internal gravity waves. At the linearized level, this gives rise to amplitude decay at a rate of <span>(t^{-1/2})</span>, as observed in Elgindi and Widmayer (SIAM J. Math. Anal. 47(6):4672–4684, 2015). We establish a refined version of this, and propagate nonlinear control via a detailed analysis of nonlinear interactions using the method of partial symmetries developed in Guo et al. (Invent. Math. 231(1):169–262, 2023).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13050770/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147635251","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}