Archive for Rational Mechanics and Analysis最新文献

{"title":"Transport Equations and Flows with One-Sided Lipschitz Velocity Fields","authors":"Pierre-Louis Lions, Benjamin Seeger","doi":"10.1007/s00205-024-02029-0","DOIUrl":"https://doi.org/10.1007/s00205-024-02029-0","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"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}

{"title":"Homogenization of Griffith’s Criterion for Brittle Laminates","authors":"Matteo Negri","doi":"10.1007/s00205-024-02027-2","DOIUrl":"https://doi.org/10.1007/s00205-024-02027-2","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Enhanced Dissipation for Two-Dimensional Hamiltonian Flows","authors":"Elia Bruè, Michele Coti Zelati, Elio Marconi","doi":"10.1007/s00205-024-02034-3","DOIUrl":"https://doi.org/10.1007/s00205-024-02034-3","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Slowly Expanding Stable Dust Spacetimes","authors":"David Fajman, Maximilian Ofner, Zoe Wyatt","doi":"10.1007/s00205-024-02030-7","DOIUrl":"https://doi.org/10.1007/s00205-024-02030-7","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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, L. Miguel Rodrigues, Zhao Yang, Kevin Zumbrun","doi":"10.1007/s00205-024-02033-4","DOIUrl":"https://doi.org/10.1007/s00205-024-02033-4","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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, Nader Masmoudi, Riccardo Montalto","doi":"10.1007/s00205-024-02028-1","DOIUrl":"https://doi.org/10.1007/s00205-024-02028-1","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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, Jan Kristensen","doi":"10.1007/s00205-024-02013-8","DOIUrl":"https://doi.org/10.1007/s00205-024-02013-8","url":null,"abstract":"<p>We establish <span>(textrm{C}^{infty })</span>-partial regularity results for relaxed minimizers of strongly quasiconvex functionals </p><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><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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constraint Maps with Free Boundaries: the Obstacle Case","authors":"Alessio Figalli, Sunghan Kim, Henrik Shahgholian","doi":"10.1007/s00205-024-02032-5","DOIUrl":"https://doi.org/10.1007/s00205-024-02032-5","url":null,"abstract":"<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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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, Jungkyoung Lee, Insuk Seo","doi":"10.1007/s00205-024-02031-6","DOIUrl":"https://doi.org/10.1007/s00205-024-02031-6","url":null,"abstract":"<p>Consider the elliptic operator given by </p><span>$$begin{aligned} {mathscr {L}}_{varepsilon }f,=, {varvec{b}} cdot nabla f ,+, varepsilon , Delta f end{aligned}$$</span>(0.1)<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><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>(0.2)<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><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><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><span>$$begin{aligned} lim _{varepsilon rightarrow 0} u_varepsilon ( varrho _varepsilon , varvec{x} ) ;=; u_0(varvec{m}) end{aligned}$$</span><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>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"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":"A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness","authors":"Cyrill B. Muratov, Matteo Novaga, Philip Zaleski","doi":"10.1007/s00205-024-02012-9","DOIUrl":"https://doi.org/10.1007/s00205-024-02012-9","url":null,"abstract":"<p>This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.</p>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180071","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}