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On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations 论静力学欧拉方程局部瑞利条件的崩溃和半拉格朗日方程的有限时间膨胀
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-10-04 DOI: 10.1007/s00205-024-02040-5
Victor Cañulef-Aguilar
{"title":"On the collapse of the local Rayleigh condition for the hydrostatic Euler equations and the finite time blow-up of the semi-Lagrangian equations","authors":"Victor Cañulef-Aguilar","doi":"10.1007/s00205-024-02040-5","DOIUrl":"10.1007/s00205-024-02040-5","url":null,"abstract":"<div><p>Local existence and uniqueness for the two-dimensional hydrostatic Euler equations in Sobolev spaces has been established by Masmoudi and Wong (Arch Rational Mech Anal 204:231–271, 2012) under the local Rayleigh condition. Under certain assumptions, we show that such solution will either develop singularities or produce the collapse of the local Rayleigh condition. In addition, we find necessary conditions for global solvability in Sobolev spaces. Finally, for certain class of initial data, we establish the finite time blow-up of solutions of the semi-Lagrangian equations introduced by Brenier (Nonlinearity 12:495–512, 1999). Our proof relies on new monotonicity identities for the solution of the hydrostatic Euler equations under the local Rayleigh condition.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409915","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
Semi-Dilute Rheology of Particle Suspensions: Derivation of Doi-Type Models 颗粒悬浮液的半稀释流变学：推导 Doi-Type 模型
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-10-03 DOI: 10.1007/s00205-024-02047-y
Mitia Duerinckx
{"title":"Semi-Dilute Rheology of Particle Suspensions: Derivation of Doi-Type Models","authors":"Mitia Duerinckx","doi":"10.1007/s00205-024-02047-y","DOIUrl":"10.1007/s00205-024-02047-y","url":null,"abstract":"<div><p>This work is devoted to the large-scale rheology of suspensions of non-Brownian inertialess rigid particles, possibly self-propelling, suspended in a Stokes flow. Starting from a hydrodynamic model, we derive a semi-dilute mean-field description in form of a Doi-type model, which is given by a ‘macroscopic’ effective Stokes equation coupled with a ‘microscopic’ Vlasov equation for the statistical distribution of particle positions and orientations. This accounts for some non-Newtonian effects since the viscosity in the effective Stokes equation depends on the local distribution of particle orientations via Einstein’s formula. The main difficulty is the detailed analysis of multibody hydrodynamic interactions between the particles, which we perform by means of a cluster expansion combined with a multipole expansion in a suitable dilute regime.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409661","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
Optimal Regularity for Lagrangian Mean Curvature Type Equations 拉格朗日平均曲率型方程的最优正则性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-10-03 DOI: 10.1007/s00205-024-02050-3
Arunima Bhattacharya, Ravi Shankar
{"title":"Optimal Regularity for Lagrangian Mean Curvature Type Equations","authors":"Arunima Bhattacharya,&nbsp;Ravi Shankar","doi":"10.1007/s00205-024-02050-3","DOIUrl":"10.1007/s00205-024-02050-3","url":null,"abstract":"<div><p>We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren (J Differ Geom 84(2):267-287, 2010), Huang (J Funct Anal 269(4):1095-1114, 2015), and Wang-Huang-Bao (Calc Var Partial Differ Equ 62(3):74 2023). We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is <span>(C^2)</span> and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are <span>(C^{1,beta })</span> for sufficiently large <span>(beta )</span>. Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409603","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
On Anomalous Diffusion in the Kraichnan Model and Correlated-in-Time Variants 论克莱希南模型中的反常扩散和时间相关变体
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-28 DOI: 10.1007/s00205-024-02045-0
Keefer Rowan
{"title":"On Anomalous Diffusion in the Kraichnan Model and Correlated-in-Time Variants","authors":"Keefer Rowan","doi":"10.1007/s00205-024-02045-0","DOIUrl":"10.1007/s00205-024-02045-0","url":null,"abstract":"<div><p>We provide a concise PDE-based proof of anomalous diffusion in the Kraichan model—a stochastic, white-in-time model of passive scalar turbulence; that is, we show an exponential rate of <span>(L^2)</span> decay in expectation of a passive scalar advected by a certain white-in-time, correlated-in-space, divergence-free Gaussian field, uniform in the initial data and the diffusivity of the passive scalar. Additionally, we provide examples of correlated-in-time versions of the Kraichnan model which fail to exhibit anomalous diffusion despite their (formal) white-in-time limits exhibiting anomalous diffusion. As part of this analysis, we prove that anomalous diffusion of a scalar advected by some flow implies non-uniqueness of the ODE trajectories of that flow.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142414817","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 New Gauge for Gravitational Perturbations of Kerr Spacetimes II: The Linear Stability of Schwarzschild Revisited 克尔时空引力扰动的新量纲 II：重新审视施瓦兹柴尔德的线性稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-27 DOI: 10.1007/s00205-024-02036-1
Gabriele Benomio
{"title":"A New Gauge for Gravitational Perturbations of Kerr Spacetimes II: The Linear Stability of Schwarzschild Revisited","authors":"Gabriele Benomio","doi":"10.1007/s00205-024-02036-1","DOIUrl":"10.1007/s00205-024-02036-1","url":null,"abstract":"<div><p>We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of Benomio (A new gauge for gravitational perturbations of Kerr spacetimes I: the linearised theory, 2022, https://arxiv.org/abs/2211.00602), specialised to the <span>(|a|=0)</span> case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin <span>(pm 2)</span> Teukolsky equations, we make enhanced use of the <i>red-shifted</i> transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an <i>initial-data</i> gauge normalisation suffices to establish both orbital and <i>asymptotic</i> stability for <i>all</i> the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of Dafermos et al. (Acta Math 222:1–214, 2019), which requires a <i>future</i> normalised (double-null) gauge to establish asymptotic stability for the full system.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02036-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142414590","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 Stability for Nonlinear Wave Equations Satisfying a Generalized Null Condition 满足广义零条件的非线性波方程的全局稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-25 DOI: 10.1007/s00205-024-02025-4
John Anderson, Samuel Zbarsky
{"title":"Global Stability for Nonlinear Wave Equations Satisfying a Generalized Null Condition","authors":"John Anderson,&nbsp;Samuel Zbarsky","doi":"10.1007/s00205-024-02025-4","DOIUrl":"10.1007/s00205-024-02025-4","url":null,"abstract":"<div><p>We prove global stability for nonlinear wave equations satisfying a generalized null condition. The generalized null condition is made to allow for null forms whose coefficients have bounded <span>(C^k)</span> norms. We prove both the pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the <span>(r^p)</span> estimates of Dafermos–Rodnianski then allows us to prove the global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413783","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
Stefan Problem with Surface Tension: Uniqueness of Physical Solutions under Radial Symmetry 表面张力的斯特凡问题：径向对称下物理解的唯一性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-23 DOI: 10.1007/s00205-024-02026-3
Yucheng Guo, Sergey Nadtochiy, Mykhaylo Shkolnikov
{"title":"Stefan Problem with Surface Tension: Uniqueness of Physical Solutions under Radial Symmetry","authors":"Yucheng Guo,&nbsp;Sergey Nadtochiy,&nbsp;Mykhaylo Shkolnikov","doi":"10.1007/s00205-024-02026-3","DOIUrl":"10.1007/s00205-024-02026-3","url":null,"abstract":"<div><p>We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [21]. The paper in hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own right.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02026-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413364","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
Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes 带电毛状黑洞内部的暴力非线性坍缩
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-21 DOI: 10.1007/s00205-024-02038-z
Maxime Van de Moortel
{"title":"Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes","authors":"Maxime Van de Moortel","doi":"10.1007/s00205-024-02038-z","DOIUrl":"10.1007/s00205-024-02038-z","url":null,"abstract":"<div><p>We construct a new one-parameter family, indexed by <span>(epsilon )</span>, of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant <span>(Lambda )</span> and bifurcating off a Reissner–Nordström-(dS/AdS) interior (<span>(epsilon =0)</span>). For all small <span>(epsilon ne 0)</span>, we prove that, although the black hole is charged, its terminal boundary is an everywhere-<i>spacelike</i> Kasner singularity foliated by spheres of zero radius <i>r</i>. Moreover, smaller perturbations (i.e. smaller <span>(|epsilon |)</span>) are <i>more singular than larger ones</i>, in the sense that the Hawking mass and the curvature blow up following a power law of the form <span>(r^{-O(epsilon ^{-2})})</span> at the singularity <span>({r=0})</span>. This unusual property originates from a dynamical phenomenon—<i>violent nonlinear collapse</i>—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity <span>({r=0})</span>. This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of <span>(Lambda in mathbb {R})</span>, the case <span>(Lambda &lt;0)</span> is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02038-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412984","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
Quantitative stability of Yang–Mills–Higgs instantons in two dimensions 二维杨-米尔斯-希格斯瞬子的定量稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-20 DOI: 10.1007/s00205-024-02035-2
Aria Halavati
{"title":"Quantitative stability of Yang–Mills–Higgs instantons in two dimensions","authors":"Aria Halavati","doi":"10.1007/s00205-024-02035-2","DOIUrl":"10.1007/s00205-024-02035-2","url":null,"abstract":"<div><p>We prove that if an N-vortex pair nearly minimizes the Yang–Mills–Higgs energy, then it is second order close to a minimizer. First, we use new weighted inequalities in two dimensions and compactness arguments to show stability for sections with some regularity. Second, we define a selection principle using a penalized functional and by the elliptic regularity and smooth perturbation of complex polynomials, we generalize the stability to all nearly minimizing pairs. With the same method, we also prove the analogous second order stability for nearly minimizing pairs on nontrivial line bundles over arbitrary compact smooth surfaces.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412726","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
Transport Equations and Flows with One-Sided Lipschitz Velocity Fields 传输方程与单边 Lipschitz 速度场的流动
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-09-19 DOI: 10.1007/s00205-024-02029-0
Pierre-Louis Lions, Benjamin Seeger
{"title":"Transport Equations and Flows with One-Sided Lipschitz Velocity Fields","authors":"Pierre-Louis Lions,&nbsp;Benjamin Seeger","doi":"10.1007/s00205-024-02029-0","DOIUrl":"10.1007/s00205-024-02029-0","url":null,"abstract":"<div><p>We study first- and second-order linear transport equations, as well as flows for ordinary and stochastic differential equations, with irregular velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In the compressive regime, we characterize the stable continuous distributional solutions of both the first and second-order nonconservative transport equations as the unique viscosity solution, and we also provide new observations and characterizations for the dual, conservative equations. Our results in the expansive regime complement the theory of Bouchut et al. (Ann Sc Norm Super Pisa Cl Sci (5) 4:1–25, 2005), and we develop a complete theory for both the conservative and nonconservative equations in Lebesgue spaces, as well as proving the existence, uniqueness, and stability of the regular Lagrangian flow for the associated ordinary differential equation. We also provide analogous results in this context for second order equations with degenerate noise coefficients that are constant in the spatial variable, as well as for the related stochastic differential equation flows.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267449","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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