Archive for Rational Mechanics and Analysis最新文献

{"title":"Remarkable Localized Integral Identities for 3D Compressible Euler Flow and the Double-Null Framework","authors":"Leonardo Abbrescia,&nbsp;Jared Speck","doi":"10.1007/s00205-024-01997-7","DOIUrl":"10.1007/s00205-024-01997-7","url":null,"abstract":"<div><p>We derive new, localized geometric integral identities for solutions to the 3<i>D</i> compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The integral identities are coercive in the derivatives of the specific vorticity (defined to be vorticity divided by density) and the derivatives of the entropy gradient vectorfield, and the error terms exhibit remarkable regularity and null structures. Our framework plays a fundamental role in our companion works (Abbrescia L, Speck J. The emergence of the singular boundary from the crease in 3<i>D</i> compressible Euler flow, 2022; Abbrescia and Speck, The emergence of the Cauchy horizon from the crease in 3<i>D</i> compressible Euler flow (in preparation)) on the structure of the maximal classical development for shock-forming solutions. It allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions, and our approach reveals fundamental new coordinate-invariant structural features of the flow. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution’s higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: (i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric, where the top and bottom boundaries are acoustically spacelike—but not necessarily equal to portions of constant Cartesian-time hypersurfaces; and (ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. The main new contribution of the present article is our analysis of the spacelike, co-dimension one and two boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation and using careful geometric decompositions, we observe several crucial cancellations, which in total show that after a further integration with respect to an acoustical time function, the boundary integrals have a good sign, up to error terms that can be controlled due to their good null structure and regularity properties.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-01997-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142880508","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":"Oscillations in Wave Map Systems and Homogenization of the Einstein Equations in Symmetry","authors":"André Guerra,&nbsp;Rita Teixeira da Costa","doi":"10.1007/s00205-024-02042-3","DOIUrl":"10.1007/s00205-024-02042-3","url":null,"abstract":"<div><p>In 1989, Burnett conjectured that, under appropriate assumptions, the limit of highly oscillatory solutions to the Einstein vacuum equations is a solution of the Einstein–massless Vlasov system. In a recent breakthrough, Huneau–Luk (Ann Sci l’ENS, 2024) gave a proof of the conjecture in <i>U</i>(1)-symmetry and elliptic gauge. They also require control on up to fourth order derivatives of the metric components. In this paper, we give a streamlined proof of a stronger result and, in the spirit of Burnett’s original conjecture, we remove the need for control on higher derivatives. Our methods also apply to general wave map equations.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02042-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859603","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":"On an Angle-Averaged Neumann-to-Dirichlet Map for Thin Filaments","authors":"Laurel Ohm","doi":"10.1007/s00205-024-02079-4","DOIUrl":"10.1007/s00205-024-02079-4","url":null,"abstract":"<div><p>We consider the Laplace equation in the exterior of a thin filament in <span>(mathbb {R}^3)</span> and perform a detailed decomposition of a notion of slender body Neumann-to-Dirichlet (NtD) and Dirichlet-to-Neumann (DtN) maps along the filament surface. The decomposition is motivated by a filament evolution equation in Stokes flow for which the Laplace setting serves as an important toy problem. Given a general curved, closed filament with constant radius <span>(varepsilon &gt;0)</span>, we show that both the slender body DtN and NtD maps may be decomposed into the corresponding operator about a straight, periodic filament plus lower order remainders. For the straight filament, both the slender body NtD and DtN maps are given by explicit Fourier multipliers and it is straightforward to compute their mapping properties. The remainder terms are lower order in the sense that they are small with respect to <span>(varepsilon )</span> or smoother. While the strategy here is meant to serve as a blueprint for the Stokes setting, the Laplace problem may be of independent interest.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142859781","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":"Liquid Filled Elastomers: From Linearization to Elastic Enhancement","authors":"Juan Casado-Díaz,&nbsp;Gilles A. Francfort,&nbsp;Oscar Lopez-Pamies,&nbsp;Maria Giovanna Mora","doi":"10.1007/s00205-024-02064-x","DOIUrl":"10.1007/s00205-024-02064-x","url":null,"abstract":"<div><p>Surface tension at cavity walls can play havoc with the mechanical properties of perforated soft solids when the cavities are filled with a fluid. This study is an investigation of the macroscopic elastic properties of elastomers embedding spherical cavities filled with a pressurized liquid in the presence of surface tension, starting with the linearization of the fully nonlinear model and ending with the enhancement properties of the linearized model when many such liquid filled cavities are present.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844928","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":"Variational Models with Eulerian–Lagrangian Formulation Allowing for Material Failure","authors":"Marco Bresciani,&nbsp;Manuel Friedrich,&nbsp;Carlos Mora-Corral","doi":"10.1007/s00205-024-02076-7","DOIUrl":"10.1007/s00205-024-02076-7","url":null,"abstract":"<div><p>We investigate the existence of minimizers of variational models featuring Eulerian–Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do this, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02076-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142845023","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":"Obstructions to Topological Relaxation for Generic Magnetic Fields","authors":"Alberto Enciso,&nbsp;Daniel Peralta-Salas","doi":"10.1007/s00205-024-02078-5","DOIUrl":"10.1007/s00205-024-02078-5","url":null,"abstract":"<div><p>For any (analytic) axisymmetric toroidal domain <span>(Omega subset mathbb {R}^3)</span> we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in <span>(Omega )</span>. Each vector field in this set is Morse–Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02078-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844908","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":"On the Converse of Pansu’s Theorem","authors":"Guido De Philippis,&nbsp;Andrea Marchese,&nbsp;Andrea Merlo,&nbsp;Andrea Pinamonti,&nbsp;Filip Rindler","doi":"10.1007/s00205-024-02059-8","DOIUrl":"10.1007/s00205-024-02059-8","url":null,"abstract":"<div><p>We provide a suitable generalisation of Pansu’s differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures <span>(mu )</span>, then <span>(mu )</span> must be absolutely continuous with respect to the Haar measure of the group.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02059-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844907","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":"Regular and Singular Steady States of the 2D Incompressible Euler Equations near the Bahouri–Chemin Patch","authors":"Tarek M. Elgindi,&nbsp;Yupei Huang","doi":"10.1007/s00205-024-02077-6","DOIUrl":"10.1007/s00205-024-02077-6","url":null,"abstract":"<div><p>We consider steady states of the two-dimensional incompressible Euler equations on <span>({mathbb {T}}^2)</span> and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri–Chemin patch.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844974","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":"Asymmetry of MHD Equilibria for Generic Adapted Metrics","authors":"Robert Cardona,&nbsp;Nathan Duignan,&nbsp;David Perrella","doi":"10.1007/s00205-024-02075-8","DOIUrl":"10.1007/s00205-024-02075-8","url":null,"abstract":"<div><p>Ideal magnetohydrodynamic (MHD) equilibria on a Riemannian 3-manifold satisfy the stationary Euler equations for ideal fluids. A stationary solution <i>X</i> admits a large set of “adapted” metrics on <i>M</i> for which <i>X</i> solves the corresponding MHD equilibrium equations with the same pressure function. We prove different versions of the following statement: an MHD equilibrium with non-constant pressure on a compact three-manifold with or without boundary admits no continuous Killing symmetries for an open and dense set of adapted metrics. This contrasts with the classical conjecture of Grad which loosely states that an MHD equilibrium on a toroidal Euclidean domain in <span>({mathbb {R}}^3)</span> with pressure function foliating the domain with nested toroidal surfaces must admit Euclidean symmetries.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844858","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":"Covariance-Modulated Optimal Transport and Gradient Flows","authors":"Martin Burger,&nbsp;Matthias Erbar,&nbsp;Franca Hoffmann,&nbsp;Daniel Matthes,&nbsp;André Schlichting","doi":"10.1007/s00205-024-02065-w","DOIUrl":"10.1007/s00205-024-02065-w","url":null,"abstract":"<div><p>We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"249 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02065-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142844752","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}