Archive for Rational Mechanics and Analysis最新文献

{"title":"Flat Blow-up Solutions for the Complex Ginzburg Landau Equation","authors":"Giao Ky Duong,&nbsp;Nejla Nouaili,&nbsp;Hatem Zaag","doi":"10.1007/s00205-024-02052-1","DOIUrl":"10.1007/s00205-024-02052-1","url":null,"abstract":"<div><p>In this paper, we consider the complex Ginzburg-Landau equation </p><div><div><span>$$begin{aligned} partial _t u = (1 + i beta ) Delta u + (1 + i delta ) |u|^{p-1}u - alpha u, quad text {where } beta , delta , alpha in {mathbb {R}}. end{aligned}$$</span></div></div><p>The study focuses on investigating the finite-time blow-up phenomenon, which remains an open question for a broad range of parameters, particularly for <span>(beta )</span> and <span>(delta )</span>. Specifically, for a fixed <span>(beta in {mathbb {R}})</span>, the existence of finite-time blow-up solutions for arbitrarily large values of <span>( |delta | )</span> is still unknown. According to a conjecture made by Popp et al. (Physica D Nonlinear Phenom 114:81–107 1998), when <span>(beta = 0)</span> and <span>(delta )</span> is large, blow-up does not occur for <i>generic initial data</i>. In this paper, we show that their conjecture is not valid for all types of initial data, by presenting the existence of blow-up solutions for <span>(beta = 0)</span> and any <span>(delta in {mathbb {R}})</span> with different types of blowup.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672492","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":"Nonlinear Stability of Static Néel Walls in Ferromagnetic Thin Films","authors":"Antonio Capella,&nbsp;Christof Melcher,&nbsp;Lauro Morales,&nbsp;Ramón G. Plaza","doi":"10.1007/s00205-024-02074-9","DOIUrl":"10.1007/s00205-024-02074-9","url":null,"abstract":"<div><p>The paper establishes the nonlinear (orbital) stability of static 180-degree Néel walls in ferromagnetic films under the reduced wave-type dynamics for the in-plane magnetization proposed by Capella et al. (Nonlinearity 20:2519–2537, 2007). The result follows from the spectral analysis of the linearized operator around the Néel wall’s phase, which features a challenging non-local operator. As part of the proof, we show that the non-local linearized operator is a compact perturbation of a suitable non-local linear operator at infinity, a result that is interesting in itself.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679560","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":"Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes","authors":"Max Weissenbacher","doi":"10.1007/s00205-024-02060-1","DOIUrl":"10.1007/s00205-024-02060-1","url":null,"abstract":"<div><p>We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner–Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis’ results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02060-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672494","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":"Observability for Heat Equations with Time-Dependent Analytic Memory","authors":"Gengsheng Wang,&nbsp;Yubiao Zhang,&nbsp;Enrique Zuazua","doi":"10.1007/s00205-024-02058-9","DOIUrl":"10.1007/s00205-024-02058-9","url":null,"abstract":"<div><p>This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672511","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":"Nonuniqueness of Trajectories on a Set of Full Measure for Sobolev Vector Fields","authors":"Anuj Kumar","doi":"10.1007/s00205-024-02063-y","DOIUrl":"10.1007/s00205-024-02063-y","url":null,"abstract":"<div><p>In this paper, we resolve an important long-standing question of Alberti (Rend Lincei 23:477–491, 2012) that asks whether or not if there is a continuous vector field with bounded divergence and of class <span>(W^{1, p})</span> for some <span>(p ge 1)</span> such that the ODE with this vector field has nonunique trajectories on a set of initial conditions with positive Lebesgue measure. This question belongs to the realm of well-known DiPerna–Lions theory for Sobolev vector fields <span>(W^{1, p})</span>. In this work, we design a divergence-free vector field in <span>(W^{1, p})</span> with <span>(p &lt; d)</span> such that the set of initial conditions for which trajectories are not unique is a set of full measure. The construction in this paper is quite explicit; we can write down the expression of the vector field at any point in time and space. Moreover, our vector field construction is novel. We build a vector field <span>(varvec{u})</span> and a corresponding flow map <span>(X^{varvec{u}})</span> such that after finite time <span>(T &gt; 0)</span>, the flow map takes the whole domain <span>(mathbb {T}^d)</span> to a Cantor set <span>(mathcal {C}_Phi )</span>, i.e., <span>(X^{varvec{u}}(T, mathbb {T}^d) = mathcal {C}_Phi )</span> and the Hausdorff dimension of this Cantor set is strictly less than <i>d</i>. The flow map <span>(X^{varvec{u}})</span> constructed as such is not a regular Lagrangian flow. The nonuniqueness of trajectories on a full measure set is then deduced from the existence of the regular Lagrangian flow in the DiPerna–Lions theory.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645656","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":"Coagulation Equations for Non-spherical Clusters","authors":"Iulia Cristian,&nbsp;Juan J. L. Velázquez","doi":"10.1007/s00205-024-02061-0","DOIUrl":"10.1007/s00205-024-02061-0","url":null,"abstract":"<div><p>In this work, we study the long time asymptotics of a coagulation model which describes the evolution of a system of particles characterized by their volume and surface area. The aggregation mechanism takes place in two stages: collision and fusion of particles. During the collision stage, the two particles merge at a contact point. The newly formed particle has volume and area equal to the sum of the respective quantities of the two colliding particles. After collision, the fusion phase begins and during it the geometry of the interacting particles is modified in such a way that the volume of the total system is preserved and the surface area is reduced. During their evolution, the particles must satisfy the isoperimetric inequality. Therefore, the distribution of particles in the volume and area space is supported in the region where <span>({age (36pi )^{frac{1}{3}}v^{frac{2}{3}}})</span>. We assume the coagulation kernel has a weak dependence on the area variable. We prove existence of self-similar profiles for some choices of the functions describing the fusion rate for which the particles have a shape that is close to spherical. On the other hand, for other fusion mechanisms and suitable choices of initial data, we show that the particle distribution describes a system of ramified-like particles.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02061-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645685","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":"The (L^p) Estimate for the Gain Term of the Boltzmann Collision Operator and Its Application","authors":"Ling-Bing He,&nbsp;Jin-Cheng Jiang,&nbsp;Hung-Wen Kuo,&nbsp;Meng-Hao Liang","doi":"10.1007/s00205-024-02067-8","DOIUrl":"10.1007/s00205-024-02067-8","url":null,"abstract":"<div><p>We prove the Hardy–Littlewood–Sobolev type <span>(L^p)</span> estimates for the gain term of the Boltzmann collision operator including Maxwellian molecule, hard potential and hard sphere models. Combined with the results of Alonso et al. (Comm Math Phys 298: 293–322, 2010) for the soft potential and Maxwellian molecule models, we provide a unified form of <span>(L^p)</span> estimate for all cutoff models which are sharp in the sense of scaling. The most striking feature of our new estimates for the hard potential and hard sphere models is that they do not increase the moment, the same as Maxwellian molecule and soft potential models. Based on these novelties, we prove the global existence and scattering of the non-negative unique mild solution for the Cauchy problem of the Boltzmann equation when the positive initial data is small in the weighted <span>(L^3_{x,v})</span> space.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645686","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":"A Top-Down Approach to Algebraic Renormalization in Regularity Structures Based on Multi-indices","authors":"Yvain Bruned,&nbsp;Pablo Linares","doi":"10.1007/s00205-024-02041-4","DOIUrl":"10.1007/s00205-024-02041-4","url":null,"abstract":"<div><p>We provide an algebraic framework to describe renormalization in regularity structures based on multi-indices for a large class of semi-linear stochastic PDEs. This framework is “top-down”, in the sense that we postulate the form of the counterterm and use the renormalized equation to build a canonical smooth model for it. The core of the construction is a generalization of the Hopf algebra of derivations in Linares et al. (Commun Am Math Soc 3:1–64, 2023, https://doi.org/10.1090/cams/16), which is extended beyond the structure group to describe the model equation via an exponential map; this allow us to implement a renormalization procedure which resembles the preparation map approach in our context.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142598940","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":"Homogenisation Problems for Free Discontinuity Functionals with Bounded Cohesive Surface Terms","authors":"Gianni Dal Maso,&nbsp;Rodica Toader","doi":"10.1007/s00205-024-02053-0","DOIUrl":"10.1007/s00205-024-02053-0","url":null,"abstract":"<div><p>We study stochastic homogenisation problems for free discontinuity functionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimisation problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587838","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":"Transverse Magnetic ENZ Resonators: Robustness and Optimal Shape Design","authors":"Robert V. Kohn,&nbsp;Raghavendra Venkatraman","doi":"10.1007/s00205-024-02023-6","DOIUrl":"10.1007/s00205-024-02023-6","url":null,"abstract":"<div><p>We study certain “geometric-invariant resonant cavities” introduced by Liberal, Mahmoud, and Engheta in a 2016 Nature Communications paper. They are cylindrical devices modeled using the transverse magnetic reduction of Maxwell’s equations, so the mathematics is two-dimensional. The cross-section consists of a dielectric inclusion surrounded by an “epsilon-near-zero” (ENZ) shell. When the shell has just the right area, its interaction with the inclusion produces a resonance. Mathematically, the resonance is a nontrivial solution of a 2D divergence-form Helmoltz equation <span>(nabla cdot left( varepsilon ^{-1}(x,omega ) nabla u right) + omega ^2 mu u = 0)</span>, where <span>(varepsilon (x,omega ))</span> is the (complex-valued) dielectric permittivity, <span>(omega )</span> is the frequency, <span>(mu )</span> is the magnetic permeability, and a homogeneous Neumann condition is imposed at the outer boundary of the shell. This is a nonlinear eigenvalue problem, since <span>(varepsilon )</span> depends on <span>(omega )</span>. Use of an ENZ material in the shell means that <span>(varepsilon (x,omega ))</span> is nearly zero there, so the PDE is rather singular. Working with a Lorentz model for the dispersion of the ENZ material, we put the discussion of Liberal et. al. on a sound foundation by proving the existence of the anticipated resonance when the loss parameter of the Lorentz model is sufficiently small. Our analysis is perturbative in character, using the implicit function theorem despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, its quality (that is, the rate at which the resonance decays) depends on the shape of the shell. It is therefore natural to consider an associated optimal design problem: what shape shell gives the slowest-decaying resonance? We prove that if the dielectric inclusion is a ball then the optimal shell is a concentric annulus. For an inclusion of any shape, we study a convex relaxation of the design problem using tools from convex duality. Finally, we discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595484","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}