Archive for Rational Mechanics and Analysis最新文献

We study a class of radially symmetric Coulomb gas ensembles at inverse temperature (beta =2), for which the droplet consists of a number of concentric annuli, having at least one bounded “gap” G, i.e., a connected component of the complement of the droplet, which disconnects the droplet. Let n be the total number of particles. Among other things, we deduce fine asymptotics as (n rightarrow infty ) for the edge density and the correlation kernel near the gap, as well as for the cumulant generating function of fluctuations of smooth linear statistics. We typically find an oscillatory behaviour in the distribution of particles which fall near the edge of the gap. These oscillations are given explicitly in terms of a discrete Gaussian distribution, weighted Szegő kernels, and the Jacobi theta function, which depend on the parameter n.

We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equation for Non-Equilibrium Reversible Irreversible Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein-Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

We demonstrate the existence of topologically nontrivial energy minimizing maps of a given positive degree from bounded domains in the plane to ({mathbb {S}}^2) in a variational model describing magnetizations in ultrathin ferromagnetic films with Dzyaloshinskii–Moriya interaction. Our strategy is to insert tiny truncated Belavin–Polyakov profiles in carefully chosen locations of lower degree objects such that the total energy increase lies strictly below the expected Dirichlet energy contribution, ruling out loss of degree in the limits of minimizing sequences. The argument requires that the domain be either sufficiently large or sufficiently slender to accommodate a prescribed degree. We also show that these higher degree minimizers concentrate on point-like skyrmionic configurations in a suitable parameter regime.

We consider the (phi ^4_1) measure in an interval of length (ell ), defined by a symmetric double-well potential W and inverse temperature (beta ). Our results concern its asymptotic behavior in the joint limit (beta , ell rightarrow infty ), both in the subcritical regime (ell ll textrm{e}^{beta C_W}) and in the supercritical regime (ell gg textrm{e}^{beta C_W}), where (C_W) denotes the surface tension. In the former case, in which the measure concentrates on the pure phases, we prove the corresponding large deviation principle. The associated rate function is the Modica–Mortola functional modified to take into account the entropy of the locations of the interfaces. Furthermore, we provide the sharp asymptotics of the probability of having a given number of transitions between the two pure phases. In the supercritical regime, the measure no longer concentrates and we show that the interfaces are asymptotically distributed according to a Poisson point process.

A static variational model for shape formation in heteroepitaxial crystal growth is considered. The energy functional takes into account surface energy, elastic misfit-energy and nucleation energy of dislocations. A scaling law for the infimal energy is proven. The results quantify the expectation that in certain parameter regimes, island formation or topological defects are favorable. This generalizes results in the purely elastic setting from [23]. To handle dislocations in the lower bound, a new variant of a ball-construction combined with thorough local estimates is presented.

We show that buoyancy driven flows can be steered in an arbitrary time towards any state by applying as control only an external temperature profile in a subset of small area. More specifically, we prove that the 2D incompressible Boussinesq system on the torus is globally approximately controllable via physically localized heating or cooling. In addition, our controls have an explicitly prescribed structure; even without such structural requirements, large data controllability results for Boussinesq flows driven merely by a physically localized temperature profile were so far unknown. The presented method exploits various connections between the model’s underlying transport-, coupling-, and scaling mechanisms.

In this paper, we study the nonlinear asymptotic stability of Couette flow for the two-dimensional Navier-Stokes equation with small viscosity (nu >0) in (mathbb {T}times mathbb {R}). It is well known that the nonlinear asymptotic stability of the Couette flow depends closely on the size and regularity of the initial perturbation, which yields the stability threshold problem. This work studies the relationship between the regularity and the size of the initial perturbation that makes the nonlinear asymptotic stability hold. More precisely, we prove that if the initial perturbation is in some Gevrey-(frac{1}{s}) class with size (varepsilon nu ^{beta }) where (sin [0,frac{1}{2}]) and (beta ge frac{1-2s}{3-3s}), then the nonlinear asymptotic stability holds. We think this index is sharp.

We develop a Birman–Schwinger principle for the spherically symmetric, asymptotically flat Einstein–Vlasov system. The principle characterizes the stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert–Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman–Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

We resolve a question of Carrapatoso et al. (Arch Ration Mech Anal 243(3):1565–1596, 2022) on Gaussian optimality for the sharp constant in Poincaré-Korn inequalities, under a moment constraint. We also prove stability, showing that measures with a near-optimal constant are quantitatively close to standard Gaussian.