Archive for Rational Mechanics and Analysis最新文献

{"title":"Isoperimetric Residues and a Mesoscale Flatness Criterion for Hypersurfaces with Bounded Mean Curvature","authors":"Francesco Maggi,&nbsp;Michael Novack","doi":"10.1007/s00205-024-02039-y","DOIUrl":"10.1007/s00205-024-02039-y","url":null,"abstract":"<div><p>We obtain a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime <span>(vrightarrow infty )</span>. This is achieved by the study of a Plateau-type problem with a free boundary (both on the compact obstacle and at infinity), which is used to identify the first obstacle-dependent term (called <i>isoperimetric residue</i>) in the energy expansion, as <span>(vrightarrow infty )</span>, of the exterior isoperimetric problem. A crucial tool in the analysis of isoperimetric residues is a new “mesoscale flatness criterion” for hypersurfaces with bounded mean curvature, which we obtain as a development of ideas originating in the theory of minimal surfaces with isolated singularities.</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":"142412492","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":"Homogenization of Griffith’s Criterion for Brittle Laminates","authors":"Matteo Negri","doi":"10.1007/s00205-024-02027-2","DOIUrl":"10.1007/s00205-024-02027-2","url":null,"abstract":"<div><p>We consider a periodic, linear elastic laminate with a brittle crack, evolving along a prescribed path according to Griffith’s criterion. We study the homogenized limit of this evolution, as the size of the layers vanishes. The limit evolution is governed again by Griffith’s criterion, in terms of the energy release (of the homogenized elastic energy) and an effective toughness, which, in general, differs from the <span>(hbox {weak}^*)</span> limit of the periodic toughness. We provide a variational characterization of the effective toughness and, by the energy identity, we link the toughening effect (in the limit) to the micro-instabilities of the evolution (in the periodic laminate). Finally, we provide a couple of explicit calculations of the effective toughness in the anti-plane setting, showing, in particular, an example of toughening by elastic contrast.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02027-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267450","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":"Enhanced Dissipation for Two-Dimensional Hamiltonian Flows","authors":"Elia Bruè,&nbsp;Michele Coti Zelati,&nbsp;Elio Marconi","doi":"10.1007/s00205-024-02034-3","DOIUrl":"10.1007/s00205-024-02034-3","url":null,"abstract":"<div><p>Let <span>(Hin C^1cap W^{2,p})</span> be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field <span>(b=nabla ^perp H)</span>. We give sharp upper bounds on the enhanced dissipation rate of <i>b</i> in terms of the properties of the period <i>T</i>(<i>h</i>) of the closed orbit <span>({H=h})</span>. Specifically, if <span>(0&lt;nu ll 1)</span> is the diffusion coefficient, the enhanced dissipation rate can be at most <span>(O(nu ^{1/3}))</span> in general, the bound improves when <i>H</i> has isolated, non-degenerate elliptic points. Our result provides the better bound <span>(O(nu ^{1/2}))</span> for the standard cellular flow given by <span>(H_{textsf{c}}(x)=sin x_1 sin x_2)</span>, for which we can also prove a new upper bound on its mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by <i>b</i>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02034-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267451","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":"Slowly Expanding Stable Dust Spacetimes","authors":"David Fajman,&nbsp;Maximilian Ofner,&nbsp;Zoe Wyatt","doi":"10.1007/s00205-024-02030-7","DOIUrl":"10.1007/s00205-024-02030-7","url":null,"abstract":"<div><p>We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which, in particular implies that the expansion rate of the respective models is linear, i.e. has zero acceleration. The resulting spacetimes are future globally regular. These solutions constitute the first generic class of future regular Einstein-Dust spacetimes not undergoing accelerated expansion and are thereby the slowest expanding generic family of future complete Einstein-Dust spacetimes currently known.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02030-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267452","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":"Existence and Stability of Nonmonotone Hydraulic Shocks for the Saint Venant Equations of Inclined Thin-Film Flow","authors":"Grégory Faye,&nbsp;L. Miguel Rodrigues,&nbsp;Zhao Yang,&nbsp;Kevin Zumbrun","doi":"10.1007/s00205-024-02033-4","DOIUrl":"10.1007/s00205-024-02033-4","url":null,"abstract":"<div><p>Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number <span>(F&lt;2)</span>, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams, are all rigorously obtained by mathematical analysis and explicit calculation.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02033-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180066","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":"Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow","authors":"Luca Franzoi,&nbsp;Nader Masmoudi,&nbsp;Riccardo Montalto","doi":"10.1007/s00205-024-02028-1","DOIUrl":"10.1007/s00205-024-02028-1","url":null,"abstract":"<div><p>We prove the existence of steady <i>space quasi-periodic</i> stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel <span>({{mathbb {R}}}times [-1,1])</span>. These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash–Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure. These solutions exist for most values of the parameters characterizing the shear equilibrium. As a by-product, the streamlines of the nonlinear flow exhibit Kelvin’s cat eye-like trajectories arising from the finitely many stagnation lines of the shear equilibrium.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02028-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180068","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":"Quasiconvex Functionals of (p, q)-Growth and the Partial Regularity of Relaxed Minimizers","authors":"Franz Gmeineder,&nbsp;Jan Kristensen","doi":"10.1007/s00205-024-02013-8","DOIUrl":"10.1007/s00205-024-02013-8","url":null,"abstract":"<div><p>We establish <span>(textrm{C}^{infty })</span>-partial regularity results for relaxed minimizers of strongly quasiconvex functionals </p><div><div><span>$$begin{aligned} mathscr {F}[u;Omega ]:=int _{Omega }F(nabla u)textrm{d}x,qquad u:Omega rightarrow mathbb {R}^{N}, end{aligned}$$</span></div></div><p>subject to a <i>q</i>-growth condition <span>(|F(z)|leqq c(1+|z|^{q}))</span>, <span>(zin mathbb {R}^{Ntimes n})</span>, and natural <i>p</i>-mean coercivity conditions on <span>(Fin textrm{C}^{infty }(mathbb {R}^{Ntimes n}))</span> for the basically optimal exponent range <span>(1leqq pleqq q&lt;min {frac{np}{n-1},p+1})</span>. With the <i>p</i>-mean coercivity condition being stated in terms of a strong quasiconvexity condition on <i>F</i>, our results include pointwise (<i>p</i>, <i>q</i>)-growth conditions as special cases. Moreover, we directly allow for signed integrands which is natural in view of coercivity considerations and hence the direct method, but is novel in the study of relaxed problems. In the particular case of classical pointwise (<i>p</i>, <i>q</i>)-growth conditions, our results extend the previously known exponent range from <span>Schmidt</span>’s foundational work (Schmidt in Arch Ration Mech Anal 193:311–337, 2009) for non-negative integrands to the maximal range for which relaxations are meaningful, moreover allowing for <span>(p=1)</span>. We also emphasize that our results apply to the canonical class of signed integrands and do not rely in any way on measure representations à la <span>Fonseca</span> and <span>Malý</span> (Ann Inst Henri Poincaré Anal Non Linéaire 14:309–338, 1997).</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02013-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180067","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":"Constraint Maps with Free Boundaries: the Obstacle Case","authors":"Alessio Figalli,&nbsp;Sunghan Kim,&nbsp;Henrik Shahgholian","doi":"10.1007/s00205-024-02032-5","DOIUrl":"10.1007/s00205-024-02032-5","url":null,"abstract":"<div><p>This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, concerns the properties of the projected image and its higher regularity, which we show to be at most of class <span>(C^{2,1})</span>. In arbitrary dimensions, we prove that the image map is globally of class <span>(W^{3,BMO})</span>, and locally of class <span>(C^{2,1})</span> around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02032-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180069","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":"Metastability and Time Scales for Parabolic Equations with Drift 1: The First Time Scale","authors":"Claudio Landim,&nbsp;Jungkyoung Lee,&nbsp;Insuk Seo","doi":"10.1007/s00205-024-02031-6","DOIUrl":"10.1007/s00205-024-02031-6","url":null,"abstract":"<div><p>Consider the elliptic operator given by </p><div><div><span>$$begin{aligned} {mathscr {L}}_{varepsilon }f,=, {varvec{b}} cdot nabla f ,+, varepsilon , Delta f end{aligned}$$</span></div><div>\u0000 (0.1)\u0000 </div></div><p>for some smooth vector field <span>(varvec{b}:{mathbb R}^drightarrow {mathbb R}^d)</span> and a small parameter <span>(varepsilon &gt;0)</span>. Consider the initial-valued problem </p><div><div><span>$$begin{aligned} left{ begin{aligned}&amp;partial _ t u_varepsilon ,=, {mathscr {L}}_varepsilon u_varepsilon , &amp;u_varepsilon (0, cdot ) = u_0(cdot ) , end{aligned} right. end{aligned}$$</span></div><div>\u0000 (0.2)\u0000 </div></div><p>for some bounded continuous function <span>(u_0)</span>. Denote by <span>(mathcal {M}_0)</span> the set of critical points of <span>(varvec{b})</span> which are stable stationary points for the ODE <span>(dot{varvec{x}} (t) = varvec{b} (varvec{x}(t)))</span>. Under the hypothesis that <span>(mathcal {M}_0)</span> is finite and <span>(varvec{b} = -(nabla U + varvec{ell }))</span>, where <span>(varvec{ell })</span> is a divergence-free field orthogonal to <span>(nabla U)</span>, the main result of this article states that there exist a time-scale <span>(theta ^{(1)}_varepsilon )</span>, <span>(theta ^{(1)}_varepsilon rightarrow infty )</span> as <span>(varepsilon rightarrow 0)</span>, and a Markov semigroup <span>({p_t: tge 0})</span> defined on <span>(mathcal {M}_0)</span> such that </p><div><div><span>$$begin{aligned} lim _{varepsilon rightarrow 0} u_varepsilon ( t , theta ^{(1)}_varepsilon , varvec{x} ) ;=; sum _{varvec{m}'in mathcal {M}_0} p_t(varvec{m}, varvec{m}'), u_0(varvec{m}'); end{aligned}$$</span></div></div><p>for all <span>(t&gt;0)</span> and <span>(varvec{x})</span> in the domain of attraction of <span>(varvec{m})</span> [for the ODE <span>(dot{varvec{x}}(t) = varvec{b}(varvec{x}(t)))</span>]. The time scale <span>(theta ^{(1)})</span> is critical in the sense that, for all time scales <span>(varrho _varepsilon )</span> such that <span>(varrho _varepsilon rightarrow infty )</span>, <span>(varrho _varepsilon /theta ^{(1)}_varepsilon rightarrow 0)</span>, </p><div><div><span>$$begin{aligned} lim _{varepsilon rightarrow 0} u_varepsilon ( varrho _varepsilon , varvec{x} ) ;=; u_0(varvec{m}) end{aligned}$$</span></div></div><p>for all <span>(varvec{x} in mathcal {D}(varvec{m}))</span>. Namely, <span>(theta _varepsilon ^{(1)})</span> is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [20] we extend this result finding all critical time-scales at which the solution of the initial-valued problem (0.2) evolves smoothly in time and we show that the solution <span>(u_varepsilon )</span> is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of <span>(varvec{b})</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180070","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":"Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System","authors":"Helmut Abels,&nbsp;Julian Fischer,&nbsp;Maximilian Moser","doi":"10.1007/s00205-024-02020-9","DOIUrl":"10.1007/s00205-024-02020-9","url":null,"abstract":"<div><p>We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility <span>(m_varepsilon &gt;0)</span> in the Allen–Cahn equation tends to zero in a subcritical way, i.e., <span>(m_varepsilon = m_0 varepsilon ^beta )</span> for some <span>(beta in (0,2))</span> and <span>(m_0&gt;0)</span>. The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term <span>(m_varepsilon H_{Gamma _t})</span> in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11371890/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142141788","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}