Long-time Solution Research Articles

Optical non linear characteristics of particles from natural resources

In these work some nano and micro particles such as granite, zeolite, diopside ferrous and sand glass which considered as waste or byproduct from mining industries. The micro and nano particles from natural resources were prepared by sieving and mixing with acidic solution at certain concentration, the particles are soaked and immerged in the acidic solution for long time to achieve reasonable hemogenity, sterring is done directly before measurements. The non linear characteristics of the prepared solution were measured by z-scan open aperture technique. The results explained in terms of the composition and structure of the micro and nano particles. Granite particles does not introduce any emission due to compact structure, while zeolite, diopside ferrous and sand glass have significant non linear spectrum related to their structure. The research introduces new applications of the ceramics materials micro and nano particles in non linear optical devices. The research ends with conclusions and recommendation for future work in this new field of applications. The new techniques and the new materials selected in the current study have good non linear optical characteristics; these are promising for practical application. It can be used for various optical applications such as optical modulators, sensors and other optical protection applications.

Asymptotic analysis of selection-mutation models in the presence of multiple fitness peaks

We study the long-time behaviour of phenotype-structured models describing the evolutionary dynamics of asexual populations whose phenotypic fitness landscape is characterised by multiple peaks. First we consider the case where phenotypic changes do not occur, and then we include the effect of heritable phenotypic changes. In the former case the model is formulated as an integrodifferential equation for the phenotype distribution of the individuals in the population, whereas in the latter case the evolution of the phenotype distribution is governed by a non-local parabolic equation whereby a linear diffusion operator captures the presence of phenotypic changes. We prove that the long-time limit of the solution to the integrodifferential equation is unique and given by a measure consisting of a weighted sum of Dirac masses centred at the peaks of the phenotypic fitness landscape. We also derive an explicit formula to compute the weights in front of the Dirac masses. Moreover, we demonstrate that the long-time solution of the non-local parabolic equation exhibits a qualitatively similar behaviour in the asymptotic regime where the diffusion coefficient modelling the rate of phenotypic change tends to zero. However, we show that the limit measure of the non-local parabolic equation may consist of less Dirac masses, and we provide a sufficient criterion to identify the positions of their centres. Finally, we carry out a detailed characterisation of the speed of convergence of the integral of the solution (i.e. the population size) to its long-time limit for both models. Taken together, our results support a more in-depth theoretical understanding of the conditions leading to the emergence of stable phenotypic polymorphism in asexual populations.

Local curvature estimates of long-time solutions to the Kähler-Ricci flow

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when a regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

On the Type IIb solutions to mean curvature flow

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph (y,u(|y|)) over Rn, n≥2, satisfying there exist positive constants ϵ, c and N such that u′(r)≥crϵ for r≥N, the longtime solution to mean curvature flow with initial data (y,u(|y|)) must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable rescaling sequence for mean-convex Type IIb mean curvature flow satisfying δ-Andrews' noncollapsing condition is translating soliton.

A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions

The Kijima’s type 1 maintenance model, representing the general renewal process, is one of the most important in the reliability theory. The g-renewal equation is central in Kijima’s theory and it is a Volterra integral equation of the second kind. Although these equations are well-studied, a closed-form solution to the g-renewal equation has not yet been obtained. Despite the fact that several semi-empirical techniques to approximate the g-renewal function have been previously developed, analytical approaches to solve this equation for a wide class of underlying distributions is still of current interest. In this paper, a long-time asymptotic for the g-renewal rate is obtained for distributions with nondecreasing hazard functions and for all values of the restoration factor $$q\in [0,1]$$. The obtained analytical result is compared with the numerical solutions for two types of underlying distributions, showing a good asymptotic match. The obtained approximate g-renewal rate is employed for maintenance optimization, considering the repair cost as a function of the restoration factor. Several numerical examples are performed in order to show the efficiency of our results.

Replica wormhole and information retrieval in the SYK model coupled to Majorana chains

Motivated by recent studies of the information paradox in (1+1)-D anti-de Sitter spacetime with a bath described by a (1+1)-D conformal field theory, we study the dynamics of second Ŕenyi entropy of the Sachdev-Ye-Kitaev (SYK) model (χ) coupled to a Majorana chain bath (ψ). The system is prepared in the thermofield double (TFD) state and then evolved by HL + HR. For small system-bath coupling, we find that the second Rényi entropy {S}_{upchi L,upchi R}^{(2)} of the SYK model undergoes a first order transition during the evolution. In the sense of holographic duality, the long-time solution corresponds to a “replica wormhole”. The transition time corresponds to the Page time of a black hole coupled to a thermal bath. We further study the information scrambling and retrieval by introducing a classical control bit, which controls whether or not we add a perturbation in the SYK system. The mutual information between the bath and the control bit shows a positive jump at the Page time, indicating that the entanglement wedge of the bath includes an island in the holographic bulk.

Geodesic completeness of the $$H^{3/2}$$ metric on $$\mathrm {Diff}(S^{1})$$

Of concern is the study of the long-time existence of solutions to the Euler–Arnold equation of the right-invariant \(H^{\frac{3}{2}}\)-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order s of the metric is greater than \(\frac{3}{2}\), but the behaviour for the critical Sobolev index \(s=\frac{3}{2}\) has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. We show that the behaviour is the same for all Sobolev metrics of order \(\frac{3}{2}\) regardless of lower-order terms.

Graphene under bichromatic driving: commensurability and spatio-temporal symmetries

We study the non-linear current response of a Dirac model that is coupled to two time-periodic electro-magnetic fields with different frequencies. We distinguish between incommensurable and commensurable frequencies, the latter characterized by their ratio p/q with co-prime integers p and q. Coupling the (effective) two-level system to a dissipative bath ensures a well-defined long-time solution for the reduced density operator and, thus, the current. We then analyze the spatio-temporal symmetries that force certain current components to vanish and close with conclusions for directed average currents.

Integrability-preserving regularizations of Laplacian Growth

The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.

Convergence Analysis of Stochastic Structure-Preserving Schemes for Computing Effective Diffusivity in Random Flows

In this paper, we develop efficient stochastic structure-preserving schemes to compute the effective diffusivity for particles moving in random flows. We first introduce the motion of a passive tracer particle in random flows using the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). Then we propose stochastic structure-preserving schemes to solve the SDEs and provide rigorous convergence analysis for the numerical schemes in computing effective diffusivity. The convergence analysis follows a probabilistic approach, which interprets the solution process generated by our numerical schemes as a Markov process. By exploring the ergodicity of the solution process, we obtain a convergence analysis of our method in computing long-time solutions of the SDEs. Most importantly, our analysis result reveals the equivalence of the definition of the effective diffusivity by solving discrete-type and continuous-type (i.e., Eulerian) corrector problems, which is fundamental and interesting. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method and investigate the convection-enhanced diffusion phenomenon in two- and three-dimensional incompressible random flows.

A contrastive investigation on anticorrosive performance of laser-induced super-hydrophobic and oil-infused slippery coatings

Nature inspired super-hydrophobic surface has aroused extensive attention on corrosion prevention. More works have investigated the instant corrosion resistance of fresh super-hydrophobic surface, but less research focused on its stability when immersed into corrosive solution for long time. Recently, another bio-mimic surface with slippery coating also presented great corrosion inhibition due to its excellent insulation between the corrosive medium and base material. However, which coating is suitable for long-term immersion into corrosive environments still remains to investigate. Herein, we firstly fabricated a super-hydrophobic coating on 316 L stainless steel by nanosecond laser ablation combining fluoroalkylsilane modification. Then a slippery coating was obtained by further infusing silicon oil into the as-prepared super-hydrophobic substrate. The PDP and EIS measurements indicate that both fresh super-hydrophobic and slippery surface were observed to improve the corrosion inhibition for the base substrate, and the performance of slippery surface was superior to that of super-hydrophobic surface. Immersion experiments infer that the air cushion of the super-hydrophobic surface was easily destroyed, resulting in a rapid failure of super-hydrophobicity and instability of corrosion resistance. In contrast, after immersed into 3.5 wt.% NaCl solution, the wetting property of oil-impregnated slippery surface almost remained the same. In addition, the electrochemical measurements reveal that the slippery coating presented exceptional anticorrosion behavior after long-term immersion into corrosive liquid. Therefore, the authors believe this study is expected to promote the development of anticorrosive coating for the steel materials. This work is of great importance for employing laser ablation to fabricate slippery surface, which is convenient and effective for metallic corrosion prevention in industrial fields.

Initial-boundary value problems for multi-term time-fractional diffusion equations with <inline-formula><tex-math id="M1">\begin{document}$ x $\end{document}</tex-math></inline-formula>-dependent coefficients

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with $ x $-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

Spectral Characterization of Couplings in a Mixed Optomechanical Model**Supported by the National Natural Science Foundation of China under Grant Nos. 11505055, 11822501, 11774087, and Natural Science Foundation of Hunan Province China under Grant No. 2017JJ1021, and Scientific Research Fund of Hunan Provincial Education Department under Grant No. 18A007

We study the spectrum of single-photon emission and scattering in a mixed optomechanical model which consists of both linear and quadratic optomechanical interactions. The spectra are calculated based on the exact long-time solutions of the single-photon emission and scattering processes in this system. We find that there exist some phonon sideband peaks in the spectra and there are some sub peaks around the phonon sideband peaks under proper parameter conditions. The correspondence between the spectral features and the optomechanical interactions is confirmed, and the optomechanical coupling strengths can be inferred by analyzing the resonance peaks and dips in the spectra.

Coexistence of multiple long-time solutions for two-dimensional laminar flow past a linearly sprung circular cylinder with a rotational nonlinear energy sink

Phase diagram showing the variation of the number of long-time solutions with $R\phantom{\rule{0}{0ex}}e$ and a dimensionless measure of spring stiffness. Each combination of symbol shape and color indicates a different set of such solutions, with the pentagons corresponding to three unsteady solutions and one steady one.

Symmetries and Critical Phenomena in Fluids

AbstractWe study two‐dimensional active scalar systems arising in fluid dynamics in critical spaces in the whole plane. We prove an optimal well‐posedness result that allows for the data and solutions to be scale‐invariant. These scale‐invariant solutions are new and their study seems to have far‐reaching consequences.More specifically, we first show that the class of bounded vorticities satisfying a discrete rotational symmetry is a global existence and uniqueness class for the two‐dimensional Euler squation. That is, in the well‐known L1∩L∞ theory of Yudovich, the L1‐assumption can be dropped upon having an appropriate symmetry condition. We also show via explicit examples the necessity of discrete symmetry for the uniqueness. This already answers problems raised by Lions in 1996 and Bendetto, Marchioro, and Pulvirenti in 1993.Next, we note that merely bounded vorticity allows for one to look at solutions that are invariant under scaling—the class of vorticities that are 0‐homo‐geneous in space. Such vorticity is shown to satisfy a new one‐dimensional evolution equation on 𝕊1. Solutions are also shown to exhibit a number of interesting properties. In particular, using this framework, we construct time quasi‐periodic solutions to the two‐dimensional Euler equation exhibiting pendulum‐like behavior. Finally, using the analysis of the one‐dimensional equation, we exhibit strong solutions to the two‐dimensional Euler equation with compact support for which angular derivatives grow at least (almost) quadratically in time (in particular, superlinear) or exponential in time (the latter being in the presence of a boundary).A similar study can be done for the surface quasi‐geostrophic (SQG) equation. Using the same symmetry condition, we prove local existence and uniqueness of solutions that are merely Lipschitz continuous near the origin—though, without the symmetry, Lipschitz initial data is expected to lose its Lipschitz continuity immediately. Once more, a special class of radially homogeneous solutions is considered, and we extract a one‐dimensional model that bears great resemblance to the so‐called De Gregorio model. We then show that finite‐time singularity formation for the one‐dimensional model implies finite‐time singularity formation in the class of Lipschitz solutions to the SQG equation that are compactly support.While the study of special infinite energy (i.e., nondecaying) solutions to fluid models is classical, this appears to be the first case where these special solutions can be embedded into a natural existence/uniqueness class for the equation. Moreover, these special solutions approximate finite‐energy solutions for long time and have direct bearing on the global regularity problem for finite‐energy solutions. © 2019 Wiley Periodicals, Inc.

Ergodic properties of heterogeneous diffusion processes in a potential well.

Heterogeneous diffusion processes can be well described by an overdamped Langevin equation with space-dependent diffusivity D(x). We investigate the ergodic and nonergodic behavior of these processes in an arbitrary potential well U(x) in terms of the observable-occupation time. Since our main concern is the large-x behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms D(x) = D0|x|α and U(x) = U0|x|β for simplicity. Based on the competition roles played by D(x) and U(x), three different cases, β > α, β = α, and β < α, are discussed. The system is ergodic for the first case β > α, where the time average agrees with the ensemble average, both determined by the steady solution for long times. By contrast, the system is nonergodic for β < α, where the relation between time average and ensemble average is uncovered by infinite-ergodic theory. For the middle case β = α, the ergodic property, depending on the prefactors D0 and U0, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases is also evaluated from Monte Carlo simulations.

The transient force profile of low-speed droplet impact: measurements and model

The transient force exerted by a low-speed liquid droplet impinging onto a flat rigid surface is investigated experimentally. The measurements employ a high-sensitivity piezo-electric sensor, along with a high-speed camera, and cover four decades in droplet Reynolds number and greater than two decades in Weber number. Across these ranges, the peak of individual force profiles span from 3 mN to over 1300 mN. Once normalised, the force–time profiles support the existence of an inertially dominated self-similar regime. Within this regime, previous numerical and theoretical studies predict a$\sqrt{t}$dependence of impact normal force during the initial pre-peak rise. While our measurements confirm this finding, they also indicate that, after the peak force the profiles exhibit an exponential decay. This long-time decay law suggests treatment of the momentum transport from the droplet using a lumped model. An observed linear dependence between the force and force decay rate supports this approach. The reason for the efficacy of treating this system via a lumped model apparently connects to the physics right at the surface that limit the rate of momentum transport from the droplet to the surface. This is explored by estimating the momentum transfer by solely using the deforming droplet shape, but under the condition of negligible momentum gradients within the droplet. The short- and long-time solutions are combined and the resulting model equation is shown to accurately cover the entire force–time profile.

On the detection, measurement, and characterization of slip-velocity in Couette-rheology involving viscoelastic liquids

We present a robust calculation leading to experimentally convenient and accurate ways of detecting, measuring, and characterizing velocity-slippage in viscoelastic-fluids, unlike previous inaccurate Mooney-type techniques. Herein, the unsteady Navier-Stokes equation for viscoelastic-fluid is solved while highlighting the rheological ramification. Our results emphasize phase-lags amongst shear stresses and strains, key output used in differing slip-types. The viscoelastic fluid is non-aging and isothermal, and we obtain an exact solution of the non-trivial flow profile, without assuming a linear Couette profile as customarily used. Moreover, the Navier-type slip boundary condition is considered. This approach is similar to that done in our earlier paper [Azese, “Measurement and characterization of slippage and slip-law using a rigorous analysis in dynamics of oscillating rheometer: Newtonian fluid,” Phys. Fluids 30, 023103 (2018)] where instead it was a Newtonian fluid. Accordingly, the sample fluid is trapped in the Couette-gap, where one of them is stationary and the other is steadily oscillating with an amplitude Re (Reynolds-number) and angular speed Ω = Ro (Roshko number), thus Couette-rheometry. We showcase an alternative way to obtain a steady-periodic solution, matching the long-time solution obtained in our earlier paper. We obtain the unsteady solution for this viscoelastic case and also use the alternative method to obtain the steady-periodic version, later used in obtaining the velocity and stress at the walls. Interestingly, we note the influence of Re, Ro, and Wi (Weissenberg Number) on this analysis. The equations and plots presented evidently show the influence of the slip. We conclude with reverse algorithms, Fourier-transform, Lissajous-figures, and Mooney-like procedures, capable of reproducing the slip-parameters, leading to a systematic measurement-and-characterization of the slip, useful in the calibration of rheological devices.

Self-organization on Riemannian manifolds

We consider an aggregation model that consists of an active transport equation for the macroscopic population density, where the velocity has a nonlocal functional dependence on the density, modelled via an interaction potential. We set up the model on general Riemannian manifolds and provide a framework for constructing interaction potentials which lead to equilibria that are constant on their supports. We consider such potentials for two specific cases (the two-dimensional sphere and the two-dimensional hyperbolic space) and investigate analytically and numerically the long-time behaviour and equilibrium solutions of the aggregation model on these manifolds. Equilibria obtained numerically with other interaction potentials and an application of the model to aggregation on the rotation group $ SO(3) $ are also presented.