Long-time Behavior Research Articles

Regularity and Long-Time Behavior of Global Weak Solutions to a Coupled Cahn–Hilliard System: The Off-Critical Case

We consider a diffuse interface model that describes the macro- and micro-phase separation processes of a polymer mixture. The resulting system consists of a Cahn–Hilliard equation and a Cahn–Hilliard–Oono type equation endowed with the singular Flory–Huggins potential. For the initial boundary value problem in a bounded smooth domain of [Formula: see text] ([Formula: see text]) with homogeneous Neumann boundary conditions for the phase functions as well as chemical potentials, we study the regularity and long-time behavior of global weak solutions in the off-critical case, that is, the mass is not conserved during the micro-phase separation of diblock copolymers. By investigating an auxiliary system with viscous regularizations, we show that every global weak solution regularizes instantaneously for [Formula: see text]. In two dimensions, we obtain the instantaneous strict separation property under a mild growth condition on the first derivative of potential functions near pure phases [Formula: see text], while in three dimensions, we establish the eventual strict separation property for sufficiently large time. Finally, we prove that every global weak solution converges to a single equilibrium as [Formula: see text].

Hyperuniformity in mass transport processes with center-of-mass conservation: some exact results

Abstract We characterize the steady-state static and dynamic properties in a broad class of mass transport processes on a periodic hypercubic lattice of volume Ld , where both the mass and center-of-mass (CoM) remain conserved and the detailed balance is violated in the bulk; we specifically consider the models in the d = 1 and 2 dimensions. Using a microscopic approach, we exactly determine the decay (or growth) exponents for various dynamic and static correlation functions. We show that despite constrained dynamics due to CoM conservation (CoMC), the density relaxation is indeed diffusive. However, the fluctuation properties are strikingly different from those in diffusive systems with a single (mass) conservation law as dynamic and static fluctuations are more suppressed in systems with CoMC, resulting in an extreme (‘class-I’) hyperuniformity in certain cases. In the thermodynamic limit, the steady-state variance ⟨ Q 2 ( T ) ⟩ c of time-integrated bond current Q ( T ) across a bond in the time interval T exhibits the following long-time behavior: ⟨ Q 2 ( T ) ⟩ c ≃ A 1 T + A 2 + A 3 T − d / 2 . The exponents governing the small-frequency behavior of the power spectra S J ( f ) ≃ A 1 + Const . f ψ J for bond current are exactly determined as ψ J = 3 / 2 and 2 in d = 1 and 2 dimensions, respectively; the corresponding unequal-time current-current correlation function decay as t − 5 / 2 and t −3 as a function of time t in d = 1 and 2 dimensions, respectively. Remarkably, depending on the dimensions and microscopic details, the prefactor A 1 can vanish, causing the variance to eventually saturate, implying a class-I ‘dynamic hyperuniformity’. We also compute the static structure factor S(q), which varies as the square of the wave number q in the small-q limit, i.e. S ( q ) ∼ q 2 , implying a class-I spatial hyperuniformity.

On momentum map for the Maxwell–Lorentz equations with spinning particle

We develop the theory of the momentum map for the Maxwell–Lorentz equations for a spinning extended charged particle. The development relies on the Hamilton–Poisson structure of the system. This theory is indispensable for the study of long-time behavior and radiation of the solitons of this system, in particular for the proof of orbital stability of the corresponding solitons moving with constant speed and rotating with constant angular velocity. We apply the theory to the translation and rotation symmetry groups of the system. The general theory of the momentum map results in formal equations for the corresponding invariants. We solve these equations obtaining expressions for the momentum and the angular momentum. We check the conservation of these expressions and their coincidence with known classical invariants. For the first time, we specify a general class of finite energy initial states which provide the conservation of angular momentum.

Joint design and joint performance of Plain Concrete Pavements (JPCP) - Investigations and experiences in Germany -

Unreinforced concrete pavements have to be equipped with longitudinal and transversal kerbs to guarantee controlled cracking (JPCP). The designs of such joints depending on the technical requirements are given in respective specifications. Main features for the design are aiming first for the controlled cracking at the foreseen position during young age of the concrete as well as avoidance of wild cracking during service life, second for sufficient load transfer at the joint to reduce edge load effects and third for a good long time performance, which requires joint sealing. This paper focus on research works performed in the field of joint efficiency (dowelling) of transversal joints, which is especially related to the design parameters and the positioning of dowels. New developments and alternative solutions concerning the today’s usage of coated steel bars must be done within the area of conflicts of following headings: Load transfer efficiency, longitudinal resistance, long-time behaviour (corrosion). FEM models and laboratory tests offer evaluation criteria for modified parameters. Especially the coating-material foreseen to cover the steel bar must be evaluated within the area of conflicts shown above. Additionally the actual dowel position and alignment is of importance. Non-destructive inspection tools are under development. The outputs of these investigations are strengthening the background for the updating work concerning the respective specifications.

Numerical dynamics and optimal control for multi-strain age-structured epidemic model.

In this paper, a novel age-structured epidemiological model that simultaneously considers multiple viral strains is proposed. We develop a numerical framework for the study of the dynamics and optimal control by a linearly implicit Euler method, in which the biological meaning is unconditionally preserved. The first order convergence of numerical solutions in a finite time is derived from a uniform numerical boundedness. Moreover, the numerical dynamics are determined by a numerical basic reproduction number , which reflects the asymptotic stability of the equilibrium points. The abstract framework offers an effective and unified approach to study the long-time behaviour of multi-strain epidemic models that cover a wide variety of well-known models, which also provides a numerical optimal control strategy of the multi-strain age-structured SIR model. Finally, some numerical simulations illustrate the verification and the efficiency of our results.

Ergodicity for the 2D stochastic electrokinetic flow

. We investigate the long-time behavior of 2D stochastic electrokinetic flow modeled by the stochastic Nernst-Planck-Navier-Stokes system with a blocking boundary for ionic species concentrations in a smooth bounded domain 𝒟. The existence of invariant measure is established for the multiplicative noise case corresponding to two situations: two opposite charged ionic species, multiple species with equal diffusivity and same magnitude for all valences. When the noise is additive case, our findings demonstrate that the invariant measure exhibits the ergodicity and exponentially mixing. This conclusion is based on the result of exponentially stability analysis.

Long-time asymptotic behavior and bound state soliton solutions for a generalized derivative nonlinear Schrödinger equation

Long-time asymptotic behavior and bound state soliton solutions for a generalized derivative nonlinear Schrödinger equation

Stochastic models of population growth

<p>We considered three types of stochastic models of a single population growth: with diffusion-type noise; with parameters replaced by stochastic processes; and with random jumps describing a sudden decrease in population size. We presented methods for studying stochastic processes modeling population growth, in particular, the long-time behavior of sample paths and their distributions. We were especially interested in the asymptotic stability of the density of the distributions of these processes. We gave biological interpretations, examples, and numerical simulations of theoretical methods and results.</p>

Long-time behaviour of the correlated random walk system

Long-time behaviour of the correlated random walk system

Electrochemical Nucleation and Growth of Silver on Carbon Screen Printed Electrodes: Interpretation of Current Transients Obtained at Low Overpotentials

Potentiostatic current transients of silver nucleation and growth were obtained on carbon screen printed electrodes (SPE) consisting of single walled carbon nanotubes, carbon nanofibers, and mesoporous carbon. Data for the number of deposited silver particles were obtained by field-emission scanning electron microscopy. All three types of SPE show the presence of large (μm-sized) crystals with surface density of about 105 cm−2 together with much smaller silver particles with sizes typically below 80 nm and surface density in the 108 cm−2 range. The different carbon structures do not affect the shape and size of the large crystals but influence the size and extent of aggregation of the nanometer-sized ones. The current transients are interpreted by the Scharifker-Mostany model for multiple nucleation and diffusion limited growth and data for the nucleation rate per active site (A) and number of active sites for nucleation (N0) were obtained. It was found that the numbers of micron-sized crystals correspond to the calculated data for N0. The physical meaning of the quantities A and N0, obtained by fitting current transients with maxima, was critically assessed. The long-time behavior of the current transients was analyzed and evidence for diffusion at nonzero surface concentration of the silver ions was obtained.

Dynamics of nanoparticle tracers in supercooled nanoparticle matrices.

We investigate the dynamics of tracer nanoparticles in bulk supercooled nanoparticle matrices using confocal microscopy. We mix fluorescent (tracer) and undyed (matrix) charged-stabilized polystyrene nanoparticles with tracer-to-matrix particle size ratios δ = 0.34, 0.36, 0.45, 0.71 at various matrix volume fractions ϕ. Single-particle and collective dynamics were obtained from particle-tracking algorithms and differential dynamic microscopy (DDM), respectively. The long-time behavior of the tracer mean-square displacement (MSD) and the shape of the distributions of particle displacements depend on δ and ϕ. At sufficiently large ϕ, small tracers (δ ≤ 0.36) remain mobile and subdiffusive but large tracers (δ ≥ 0.45) are dynamically arrested. The relaxation times determined from the intermediate scattering function (ISF) increase with δ and ϕ. Anomalous logarithmic decays in the ISF are observed for tracers of size δ ≤ 0.36 over a length scale of four to ten matrix particle diameters. These results provide insight into how penetrant size affects the transport of nanoparticles in porous media with soft interparticle interactions.

Modelling combined diffusion and surface resistances in adsorbent particles: zero length column for spherical and slab geometries

Mass transport in nanoporous materials is a key property that allows to improve the performance of many gas separation processes and design more efficient heterogeneous catalytic reactors. In many instances a combination of surface resistance and internal diffusion are present. The combined model for surface barrier and diffusion in a ZLC system is discussed in detail and the analytical solutions valid for the traditional and the partial loading experiments have been derived for the spherical and slab geometries. The model reduces to the limiting forms of pure diffusion when 100$$\\end{document}]]>, and pure surface barrier when . This study has shown that most literature studies have analysed ZLC responses incorrectly based on an effective combined dimensionless parameter. Two methods are described to obtain the parameters from the long-time asymptotic behaviour of the response curves. Both approaches have been demonstrated on curves generated from the full model solution and experimental data on an etched sample of Y zeolite. Both the analysis of the model and of the experimental results confirm that to characterize combined surface barriers and diffusion one should perform at least experiments at two different flowrates where the system is kinetically controlled, and crucially a partial loading experiment with a time to the switch which should be at least an order of magnitude smaller than the smallest of the diffusion and surface barrier times.

Quantum Langevin dynamics and long-time behaviour of two charged coupled oscillators in a common heat bath

Quantum Langevin dynamics and long-time behaviour of two charged coupled oscillators in a common heat bath

Typical Macroscopic Long-Time Behavior for Random Hamiltonians

Typical Macroscopic Long-Time Behavior for Random Hamiltonians

Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay

Sequence of pseudoequilibria describes the long-time behavior of the nonlinear noisy leaky integrate-and-fire model with large delay

Brownian non-Gaussian polymer diffusion in non-static media.

In nature, essentially, almost all the particles move irregularly in non-static media. With the advance of observation techniques, various kinds of new dynamical phenomena are detected, e.g., Brownian non-Gaussian diffusion. This paper focuses on the dynamical behavior of the center of mass (CM) of a polymer in non-static media and investigates the effect of polymer size fluctuations on the diffusion behavior. First, we establish a diffusing diffusivity model for polymer size fluctuations, linking the polymer size variation to the birth and death process, and introduce co-moving and physical coordinate systems to characterize the position of the CM for a polymer in non-static media. Next, the important statistical quantities for the CM diffusing diffusivity model in non-static media, such as mean square displacement (MSD) and kurtosis, are obtained by adopting the subordinate process approach, and the long-time asymptotic behavior of the MSD in the media of different types is specifically analyzed. Finally, the bivariate Fokker-Planck equation and the Feynman-Kac equation corresponding to the diffusing diffusivity model are detailedly derived and solved through the deep backward stochastic differential equation (BSDE) method to confirm the correctness of the derived equations.

Long-time behavior of damped 2D Boussinesq equations in a half space

We established the global well-posedness and long-time behavior of damped 2D no temperature dissipation Boussinesq equations in half space around the equilibrium state. We first obtained the explicit solution for the linearized system in half space and constructed the relationship between half space and whole space. Then, the resolvent estimates are builded by precious anisotropy analysis for different frequencies. Finally, we got the global solution for the weak dissipation system in half infinite domain and does not required the assistance of decay rate.

Global attractor for the weakly damped forced Kawahara equation on the torus

We study the long-time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in L 2 , to which as time passes all solutions draw closer. In fact, we show that the global attractor turns out to lie in a smoother space H 2 and be bounded therein. Further, we give an upper bound of the size of the attractor in H 2 that depends only on the damping parameter and the norm of the forcing term.

Long-time behavior of solutions for time-periodic reaction-diffusion equations and applications

This article concerns the asymptotic behavior of solutions for time-periodic reaction-diffusion equations with a drift term in one dimensional space. Assuming that drift term is decaying, we analyze the effect of the decaying rate of the drift term on the propagation speed of solutions. Also we show some applications of our results in high-dimensional domains. For more information see https://ejde.math.txstate.edu/Volumes/2024/69/abstr.html

Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups et(T+Ke)t≥0 and et(T+Kc+Ke)t≥0, generated by the operators T+Ke and T+Kc+Ke, respectively, have the same essential type. According to Proposition 1, it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that R2(t) is compact on L2(Ω×V,dxdv), and, using an interpolation argument (see Proposition 2), we establish the compactness of R2(t) on Lp(Ω×V,dxdv)-spaces for 1<p<+∞. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem 1, is new for periodic boundary conditions.