Archive for Rational Mechanics and Analysis最新文献

We establish a new sharp estimate of the order of the vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we prove a localised estimate of the nodal set, at a given time-level, that generalises the celebrated one of Donnelly and Fefferman. We also establish Landis type results for global solutions.

We consider the problem of the asymptotic stability of constant flows in the Aw-Rascle-Zhang traffic model. By using a perturbative approach, we are able to compute where the perturbation is mainly localised in space for a given time, based on the localisation of the perturbation initially. These new ideas can be applied to various other one dimensional models of hyperbolic conservation laws with relaxations.

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. Using the Itô–Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime.

A fundamental result in global analysis and nonlinear elasticity asserts that given a solution (mathfrak {S}) to the Gauss–Codazzi–Ricci equations over a simply-connected closed manifold ((mathcal {M}^n,g)), one may find an isometric immersion (iota ) of ((mathcal {M}^n,g)) into the Euclidean space (mathbb {R}^{n+k}) whose extrinsic geometry coincides with (mathfrak {S}). Here the dimension n and the codimension k are arbitrary. Abundant literature has been devoted to relaxing the regularity assumptions on (mathfrak {S}) and (iota ). The best result up to date is (mathfrak {S} in L^p) and (iota in W^{2,p}) for (p>n ge 3) or (p=n=2). In this paper, we extend the above result to (iota in mathcal {X}) the topology of which is strictly weaker than (W^{2,n}) for (n ge 3). Indeed, (mathcal {X}) can be taken as the Morrey space (L^{p, n-p}_{2}) with arbitrary (p in ]2,n]). This appears to be the first supercritical result in the literature on the existence of isometric immersions with low regularity, given the solubility of the Gauss–Codazzi–Ricci equations. Our proof essentially utilises the theory of Uhlenbeck gauges—in particular, Rivière–Struwe’s work (Rivière and Struwe in Comm Pure Appl Math 61:451–463, 2008) on harmonic maps in arbitrary dimensions and codimensions—and the theory of compensated compactness.

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.