Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

This paper deals with the existence of mild solutions for the following Cauchy problem: dαxt/dtα=Axt+ft,xt,x0=x0+gx,t∈0,τ, where dα./dtα is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 on a Banach space X, f and g are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.

Finite element approximation of an obstacle problem for a class of integro–differential operators

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

Quantum Nonlocality from Higher-Derivative Gravity

A classical origin for the Bohmian quantum potential, as that potential term arises in the quantum mechanical treatment of black holes and Einstein-Rosen (ER) bridges, can be based on 4th-order extensions of Einstein's equations. In Bohm's ontological interpretation, black hole radiation, and the analogous tunneling process of quantum transmission through an ER bridge, are classically allowed if the dynamics are modified to include such a quantum potential. The 4th-order extension of general relativity required to generate the quantum potential is given by adding quadratic curvature terms with coefficients that maintain a fixed ratio, as their magnitudes approach zero. Quantum transmission through the classically non-traversable bridge is replaced by classical transmission through a traversable wormhole. If entangled particles are connected by a Planck-width ER bridge, as conjectured by Maldacena and Susskind, then the classical wormhole transmission effect gives the ontological nonlocal connection between the particles posited in Bohm's interpretation of their entanglement. It is hypothesized that higher-derivative extensions of classical gravity can account for the nonlocal part of the quantum potential generally.

Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

We consider nonlinear Choquard equation [Formula: see text] where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power [Formula: see text] in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if [Formula: see text] then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.

Transient Environmental Sensitivities of Explicitly Simulated Tropical Convection

A three-dimensional cloud-resolving model, maintained in a statistically steady convecting state by tropics-like forcing, is subjected to sudden (10 min) stimuli consisting of horizontally homogeneous temperature and/or moisture sources with various profiles. Ensembles of simulations are used to increase the statistical robustness of the results and to assess the deterministic nature of the model response for domain sizes near contemporary global model resolution. The response to middle- and upper-tropospheric perturbations is predominantly local in the vertical: convection damps the imposed stimulus over a few hours. Low-level perturbations are similarly damped, but also produce a vertically nonlocal response: enhancement or suppression of new deep convective clouds extending above the perturbed level. Experiments show that the “effective inhibition layer” for deep convection is about 4 km deep, far deeper than traditional convective inhibition defined for undilute lifted parcels. Both the local and nonlocal responses are remarkably linear but can be highly stochastic, especially if deep convection is only intermittently present (small domains, weak forcing). Quantitatively, temperature-versus-moisture perturbations in a ratio corresponding to adiabatic vertical displacements produce responses of roughly equal magnitude. However, moisture perturbations seem to provoke the nonlocal (upward spreading) type of response more effectively. This nonlocal part of the response is also more effective when background forcing intensity is weak. Only at very high intensity does the response approach the limits of purely local damping and pure determinism that would be most convenient for theory and parameterization.

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian metricg(0) νμand also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo-Riemannian metricsg(0) νμandg(1) νμare nondegenerate.

Electronic Stopping of Channeled Ions in Silicon

ABSTRACTConcentration profiles of channeling and random implants of boron, phosphorus, and arsenic in silicon are compiled from the literature and are analyzed using Monte Carlo simulations. An empirical 3-parameter model of the electronic stopping power is found which yields excellent results for all channeling directions in the energy range of about 20 keV to 1 MeV. The model contains a local impact parameter dependent part and a nonlocal part, the latter increasing with ion energy. In addition, local electron density dependent stopping power models are investigated, using a realistic electron density distribution obtained by first principles band structure calculations. These models fail to describe the slowing down of ions channeled along the <110> axis.