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Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics 作为黎曼度量漫域上协变导数的客观率
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-15 DOI: 10.1007/s00205-024-02010-x
B. Kolev, R. Desmorat
{"title":"Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics","authors":"B. Kolev,&nbsp;R. Desmorat","doi":"10.1007/s00205-024-02010-x","DOIUrl":"10.1007/s00205-024-02010-x","url":null,"abstract":"<div><p>The subject of so-called objective derivatives in Continuum Mechanics has a long history and has generated varying views concerning their true mathematical interpretation. Several attempts have been made to provide a mathematical definition that would at least partially unify the existing notions. In this paper, we demonstrate that, under natural assumptions, all objective derivatives correspond to covariant derivatives on the infinite-dimensional manifold <span>(textrm{Met}(mathcal {B}))</span> of Riemannian metrics on the body. Furthermore, a natural Leibniz rule enables canonical extensions from covariant to contravariant tensor fields and vice versa. This makes the sometimes-used distinction between objective derivatives of “Lie type” and “co-rotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on <span>(textrm{Met}(mathcal {B}))</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721976","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
Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index 稳定或指数受限的 7 维最小超曲面的退化
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-03 DOI: 10.1007/s00205-024-02003-w
Nick Edelen
{"title":"Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index","authors":"Nick Edelen","doi":"10.1007/s00205-024-02003-w","DOIUrl":"10.1007/s00205-024-02003-w","url":null,"abstract":"<div><p>A 7-dimensional area-minimizing embedded hypersurface <span>(M^7)</span> will in general have a discrete singular set, and the same is true if <i>M</i> is locally stable provided <span>({mathcal {H}}^6(textrm{sing}M) = 0)</span>. We show that if <span>(M_i^7)</span> is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then <span>(M_i rightarrow M)</span> can limit to a singular <span>(M^7)</span> with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence <span>(i')</span> with controlled bi-Lipschitz maps <span>(phi _{i'})</span> taking <span>(phi _{i'}(M_{1'}) = M_{i'})</span>. As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces <i>M</i> in a closed Riemannian 8-manifold <span>((N^8, g))</span> with a priori bounds <span>({mathcal {H}}^7(M) leqq Lambda )</span> and <span>(textrm{index}(M) leqq I)</span> divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric <i>g</i> to vary, or <i>M</i> to be singular.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02003-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551825","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
Stable Recovery of Coefficients in an Inverse Fault Friction Problem 反断层摩擦问题中系数的稳定恢复
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-07-01 DOI: 10.1007/s00205-024-02009-4
Maarten V. de Hoop, Matti Lassas, Jinpeng Lu, Lauri Oksanen
{"title":"Stable Recovery of Coefficients in an Inverse Fault Friction Problem","authors":"Maarten V. de Hoop,&nbsp;Matti Lassas,&nbsp;Jinpeng Lu,&nbsp;Lauri Oksanen","doi":"10.1007/s00205-024-02009-4","DOIUrl":"10.1007/s00205-024-02009-4","url":null,"abstract":"<div><p>We consider the inverse fault friction problem of determining the friction coefficient in the Tresca friction model, which can be formulated as an inverse problem for differential inequalities. We show that the measurements of elastic waves during a rupture uniquely determine the friction coefficient at the rupture surface with explicit stability estimates.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02009-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508634","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
Existence of Minimizers for the Dirac–Fock Model of Crystals 晶体的狄拉克-福克模型存在最小值
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-25 DOI: 10.1007/s00205-024-01988-8
Isabelle Catto, Long Meng, Éric Paturel, Éric Séré
{"title":"Existence of Minimizers for the Dirac–Fock Model of Crystals","authors":"Isabelle Catto,&nbsp;Long Meng,&nbsp;Éric Paturel,&nbsp;Éric Séré","doi":"10.1007/s00205-024-01988-8","DOIUrl":"10.1007/s00205-024-01988-8","url":null,"abstract":"<div><p>Whereas many different models exist in mathematics and physics for the ground states of non-relativistic crystals, the relativistic case has been much less studied, and we are not aware of any mathematical result on a fully relativistic treatment of crystals. In this paper, we introduce a mean-field relativistic energy for crystals in terms of periodic density matrices. This model is inspired both from a recent definition of the Dirac–Fock ground state for atoms and molecules, due to one of us, and from the non-relativistic Hartree–Fock model for crystals. We prove the existence of a ground state when the number of electrons per cell is not too large.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508635","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
Recovery of Coefficients in Semilinear Transport Equations 恢复半线性传输方程中的系数
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-19 DOI: 10.1007/s00205-024-02007-6
Ru-Yu Lai, Gunther Uhlmann, Hanming Zhou
{"title":"Recovery of Coefficients in Semilinear Transport Equations","authors":"Ru-Yu Lai,&nbsp;Gunther Uhlmann,&nbsp;Hanming Zhou","doi":"10.1007/s00205-024-02007-6","DOIUrl":"10.1007/s00205-024-02007-6","url":null,"abstract":"<div><p>We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements, by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508636","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
Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System 凯勒-西格尔系统中无限时间炸裂的存在性和稳定性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-18 DOI: 10.1007/s00205-024-02006-7
Juan Dávila, Manuel del Pino, Jean Dolbeault, Monica Musso, Juncheng Wei
{"title":"Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System","authors":"Juan Dávila,&nbsp;Manuel del Pino,&nbsp;Jean Dolbeault,&nbsp;Monica Musso,&nbsp;Juncheng Wei","doi":"10.1007/s00205-024-02006-7","DOIUrl":"10.1007/s00205-024-02006-7","url":null,"abstract":"<div><p>Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system </p><div><figure><div><div><picture><img></picture></div></div></figure></div><p> We consider the critical mass case <span>(int _{{mathbb {R}}^2} u_0(x), textrm{d}x = 8pi )</span>, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function <span>(u_0^*)</span> with mass <span>(8pi )</span> such that for any initial condition <span>(u_0)</span> sufficiently close to <span>(u_0^*)</span> and mass <span>(8pi )</span>, the solution <i>u</i>(<i>x</i>, <i>t</i>) of (<span>(*)</span>) is globally defined and blows-up in infinite time. As <span>(trightarrow +infty )</span> it has the approximate profile </p><div><div><span>$$begin{aligned} u(x,t) approx frac{1}{lambda ^2(t)} Uleft( frac{x-xi (t)}{lambda (t)} right) , quad U(y)= frac{8}{(1+|y|^2)^2}, end{aligned}$$</span></div></div><p>where <span>(lambda (t) approx frac{c}{sqrt{log t}})</span>, <span>(xi (t)rightarrow q)</span> for some <span>(c&gt;0)</span> and <span>(qin {mathbb {R}}^2)</span>. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02006-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508637","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
Regularity for Nonuniformly Elliptic Equations with (p,!q)-Growth and Explicit (x,!u)-Dependence 具有$$p,!q$$增长和明确$$x,!u$$依赖性的非均匀椭圆方程的正则性
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-17 DOI: 10.1007/s00205-024-01982-0
Giovanni Cupini, Paolo Marcellini, Elvira Mascolo
{"title":"Regularity for Nonuniformly Elliptic Equations with (p,!q)-Growth and Explicit (x,!u)-Dependence","authors":"Giovanni Cupini,&nbsp;Paolo Marcellini,&nbsp;Elvira Mascolo","doi":"10.1007/s00205-024-01982-0","DOIUrl":"10.1007/s00205-024-01982-0","url":null,"abstract":"<div><p>We are interested in the regularity of weak solutions <i>u</i> to the elliptic equation in divergence form as in (1.1), and more precisely in their local boundedness and their local Lipschitz continuity under <i> general growth conditions</i>, the so called <span>(p,!q)</span>-<i>growth conditions</i>, as in (1.2) and (1.3) below. We found a unique set of assumptions to get all of these regularity properties at the same time; in the meantime we also found the way to treat a more general context, with explicit dependence on <span>(left( x,uright) )</span>, in addition to the gradient variable <span>(xi =Du)</span>. These aspects require particular attention, due to the <span>(p,!q)</span>-context, with some differences and new difficulties compared to the standard case <span>(p=q)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529782","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
Conditional (L^{infty }) Estimates for the Non-cutoff Boltzmann Equation in a Bounded Domain 有界域中非截止波尔兹曼方程的条件 $$L^{infty }$$ 估计值
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-06 DOI: 10.1007/s00205-024-02002-x
Zhimeng Ouyang, Luis Silvestre
{"title":"Conditional (L^{infty }) Estimates for the Non-cutoff Boltzmann Equation in a Bounded Domain","authors":"Zhimeng Ouyang,&nbsp;Luis Silvestre","doi":"10.1007/s00205-024-02002-x","DOIUrl":"10.1007/s00205-024-02002-x","url":null,"abstract":"<div><p>We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equation in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from a vacuum, we obtain an estimate of the <span>(L^infty )</span> norm of the solution depending on the macroscopic bounds on these hydrodynamic quantities only. This is a regularization effect in the sense that the initial data is not required to be bounded. We present a proof based on variational ideas, which is fundamentally different to the proof that was previously known for the equation in periodic spatial domains.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141378729","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
Correction to: Stable Singularity Formation for the Keller-Segel System in Three Dimensions 更正：三维凯勒-西格尔系统的稳定奇点形成
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-05 DOI: 10.1007/s00205-024-02004-9
Irfan Glogić, Birgit Schörkhuber
{"title":"Correction to: Stable Singularity Formation for the Keller-Segel System in Three Dimensions","authors":"Irfan Glogić,&nbsp;Birgit Schörkhuber","doi":"10.1007/s00205-024-02004-9","DOIUrl":"10.1007/s00205-024-02004-9","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02004-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141383741","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
Correction to: Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation 更正：修正科特维格-德-弗里斯方程自相似解在膨胀时的扰动
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-06-05 DOI: 10.1007/s00205-024-02005-8
Simão Correia, Raphaël Côte
{"title":"Correction to: Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation","authors":"Simão Correia,&nbsp;Raphaël Côte","doi":"10.1007/s00205-024-02005-8","DOIUrl":"10.1007/s00205-024-02005-8","url":null,"abstract":"","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141383514","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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