Archive for Rational Mechanics and Analysis最新文献

We prove global stability for nonlinear wave equations satisfying a generalized null condition. The generalized null condition is made to allow for null forms whose coefficients have bounded (C^k) norms. We prove both the pointwise decay and improved decay of good derivatives using bilinear energy estimates and duality arguments. Combining this strategy with the (r^p) estimates of Dafermos–Rodnianski then allows us to prove the global stability. The proof requires analyzing the geometry of intersecting null hypersurfaces adapted to solutions of wave equations.

We study the Stefan problem with surface tension and radially symmetric initial data. In this context, the notion of a so-called physical solution, which exists globally despite the inherent blow-ups of the melting rate, has been recently introduced in [21]. The paper in hand is devoted to the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. In the course of the proof, we establish a wide variety of regularity estimates for the free boundary and for the temperature function, of interest in their own right.

We construct a new one-parameter family, indexed by (epsilon ), of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant (Lambda ) and bifurcating off a Reissner–Nordström-(dS/AdS) interior ((epsilon =0)). For all small (epsilon ne 0), we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelike Kasner singularity foliated by spheres of zero radius r. Moreover, smaller perturbations (i.e. smaller (|epsilon |)) are more singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form (r^{-O(epsilon ^{-2})}) at the singularity ({r=0}). This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity ({r=0}). This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of (Lambda in mathbb {R}), the case (Lambda <0) is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes.

We prove that if an N-vortex pair nearly minimizes the Yang–Mills–Higgs energy, then it is second order close to a minimizer. First, we use new weighted inequalities in two dimensions and compactness arguments to show stability for sections with some regularity. Second, we define a selection principle using a penalized functional and by the elliptic regularity and smooth perturbation of complex polynomials, we generalize the stability to all nearly minimizing pairs. With the same method, we also prove the analogous second order stability for nearly minimizing pairs on nontrivial line bundles over arbitrary compact smooth surfaces.

We obtain a full resolution result for minimizers in the exterior isoperimetric problem with respect to a compact obstacle in the large volume regime (vrightarrow infty ). 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 isoperimetric residue) in the energy expansion, as (vrightarrow infty ), 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.

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.

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 (hbox {weak}^*) 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.

Let (Hin C^1cap W^{2,p}) be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field (b=nabla ^perp H). We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T(h) of the closed orbit ({H=h}). Specifically, if (0<nu ll 1) is the diffusion coefficient, the enhanced dissipation rate can be at most (O(nu ^{1/3})) in general, the bound improves when H has isolated, non-degenerate elliptic points. Our result provides the better bound (O(nu ^{1/2})) for the standard cellular flow given by (H_{textsf{c}}(x)=sin x_1 sin x_2), 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 b.

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.

Extending the work of Yang–Zumbrun for the hydrodynamically stable case of Froude number (F<2), 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.