Archive for Rational Mechanics and Analysis最新文献

{"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}

{"title":"Sharp Quantitative Talenti Inequality in Particular Cases","authors":"P. Acampora,&nbsp;J. Lamboley","doi":"10.1007/s00205-026-02179-3","DOIUrl":"10.1007/s00205-026-02179-3","url":null,"abstract":"<div><p>In this paper, we focus on the famous Talenti’s symmetrization inequality, more precisely, its <span>(L^p)</span> corollary which asserts that the <span>(L^p)</span>-norm of the solution to <span>(-Delta v=f^sharp )</span> is higher than the <span>(L^p)</span>-norm of the solution to <span>(-Delta u=f)</span> (we are considering Dirichlet boundary conditions, and <span>(f^sharp )</span> denotes the Schwarz symmetrization of <span>(f:Omega rightarrow mathbb {R}_+)</span>). We focus on the particular case where functions <i>f</i> are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the <span>(L^p)</span>-Talenti inequality with the sharp exponent 2.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147588615","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":"Characterizing the Detailed Balance Property by Means of Measurements","authors":"Eugenia Franco,&nbsp;Bernhard Kepka,&nbsp;Juan J. L. Velázquez","doi":"10.1007/s00205-026-02183-7","DOIUrl":"10.1007/s00205-026-02183-7","url":null,"abstract":"<div><p>In this paper we study how to determine if a linear biochemical network satisfies the detailed balance condition, without knowing the details of all the reactions taking place in the network. To this end, we use the formalism of response functions <span>(R_{ij} (t) )</span> that measure how the system reacts to the injection of the substance <i>j</i> at time <span>(t=0)</span>, by measuring the concentration of the substance <span>(i ne j)</span> for <span>(t &gt;0)</span>. In particular, we obtain a condition involving two reciprocal measurements (that is <span>(R_{ij}(t))</span>, <span>(R_{ji}(t))</span>) that is necessary, but not sufficient, for the detailed balance condition to hold in the network. Moreover, we prove that this necessary condition is also sufficient if a topological condition is satisfied by the graph associated to the network, as well as a stability property that guarantees that the chemical rates are not fine-tuned.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-026-02183-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147588616","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":"Asymptotic Behavior of a Diffused Interface Volume-Preserving Mean Curvature Flow","authors":"Matteo Bonforte,&nbsp;Francesco Maggi,&nbsp;Daniel Restrepo","doi":"10.1007/s00205-026-02185-5","DOIUrl":"10.1007/s00205-026-02185-5","url":null,"abstract":"<div><p>We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single “diffused balls”.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147607339","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":"Existence of Constant Mean Curvature Disks in (mathbb {R}^3) with Capillary Boundary Condition","authors":"Da Rong Cheng","doi":"10.1007/s00205-026-02165-9","DOIUrl":"10.1007/s00205-026-02165-9","url":null,"abstract":"<div><p>We extend Struwe’s result (Acta Math. 160(1–2):19–64, 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in <span>((0, pi ))</span>. Specifically, let <span>(Sigma )</span> be a surface in <span>(mathbb {R}^3)</span> diffeomorphic to the sphere, and let <span>(Sigma ')</span> be a convex surface enclosing <span>(Sigma )</span>. Given <span>(tau in (-1, 1))</span> and a constant <span>(H ge 0)</span> below the infimum of the mean curvature of <span>(Sigma ')</span>, we show that, for almost every <span>(r in (0, 1))</span>, in the region enclosed by <span>(Sigma ')</span>, there exists a branched immersed disk with constant mean curvature <i>rH</i> whose boundary meets <span>(Sigma )</span> at an angle with cosine value <span>(rtau )</span>. Moreover, the constant mean curvature disks we construct have index at most 1.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"250 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147561477","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}