Decay Of Solutions Research Articles

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner‐type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.

Mittag–Leffler stability of numerical solutions to time fractional ODEs

The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3(2), 216–240, 1990), we are able to prove that both $$\mathcal {L}$$ 1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on $$\alpha$$ -difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation. The new results and analyses provide not only the rigorous justifications and explanations of the Mittag-Leffler stability of numerical solutions with exact decay rate, but also establish some close connection between the continuous and discrete F-ODEs. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data

In this work we study the inhomogeneous Muskat problem, i.e. the evolution of an internal wave between two different fluids in a porous medium with discontinuous permeability. In particular, under precise conditions on the initial datum and the physical quantities of the problem, our results ensure the decay of the solutions towards the equilibrium state in the Lipschitz norm. In addition, we establish the global existence and decay of Lipschitz solutions.

Optimal economic scheduling of microgrids considering renewable energy sources based on energy hub model using demand response and improved water wave optimization algorithm

Unpredictable nature of renewable energy sources puts power system operators in different conditions in terms of maintaining the system reliability. Microgrids (MGs) can provide an effective solution for the problems of grid-connected renewable energy sources due to their tuning capability and flexibility. In this study, a stochastic optimization strategy was proposed for participating in energy market operations considering demand response (DR). The results indicated the operating cost decreased when the MG implemented DR programs. Also, DR program could shift energy consumption from on- to off-peak hours and flatten the load curve. Therefore, a scheduling model was presented for the operation of energy carriers and reserves considering security constraints of power and natural gas grids in interconnected hubs as well as responsive load participation using the developed water wave optimization (WWO) algorithm. The objective function of this model aimed to minimize the operating cost of sources to supply electrical and thermal loads applied to the proposed MG. WWO is a meta-heuristic algorithm inspired by the behavior of water waves. The waves formed on the sea surface have complex, but interesting, relationships that can be used to solve optimization problems. In this algorithm, each problem solution is encoded as a water wave and undergoes changes in the problem search space based on three behaviors of propagation, refraction, and decay of water waves or problem solutions. The simplicity of the structure of this method, elitism and the ability to escape from local points due to the existence of different search operators and the mutation of generations have led to the use of this method. The results indicated a decline in operating costs through electrical and thermal responsive load participation and thermal energy storage system. Results of the proposed model revealed a correlation between the electricity price and natural gas consumption, indicating multi-carrier energy grids should be examined and optimized simultaneously.

Mathematical modeling and dissipative structure for systems of magnetohydrodynamics with Hall effect

This paper is concerned with the mathematical modeling of electro-magneto-hydrodynamics and magnetohydrodynamics by taking account of the Hall effect. We discuss conservation laws, strict convexity of the negative entropy as a function of conserved quantities, and the associated energy form. Moreover, we investigate the dissipative structure and decay properties of the linearized systems as applications of the general theory developed in Umeda et al. [On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435–457] and Shizuta and Kawashima [Systems of equations of hyperbolic–parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249–275].

General decay for weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions

In this paper, we consider the general decay of solutions for the weak viscoelastic equation of Kirchhoff type containing Balakrishnan–Taylor damping with nonlinear delay and acoustic boundary conditions. By using suitable energy and Lyapunov functionals, we prove the general decay for the energy, which depends on the behavior of both σ and k.

Decay of solutions to the initial boundary value problem of the stokes system in the halfspace

Decay of solutions to the initial boundary value problem of the stokes system in the halfspace

On the Decay of Solutions of the Linearized Equations of Conservation Equations with Viscosity and Relaxation

On the Decay of Solutions of the Linearized Equations of Conservation Equations with Viscosity and Relaxation

Polynomial stability of transmission viscoelastic wave and plate equations on Riemannian manifolds

In this paper, we consider a class of transmission system which coupled viscoelastic wave and plate equations on common boundary. We establish the polynomial decay of solution to the transmission system. Our proof relies on the frequency domain method and geometrical multiplier method.

Propagation and Diffusion of Fluorescent Substances with Footprints in Indoor Environments.

Some studies have shown that contaminants can be transferred between floors and the soles, and there are few studies on pollutant propagation caused by human walking in real-life situations. This study explored the propagation and diffusion law of ground pollutants from rubber soles to poly vinyl chloride (PVC) floor during indoor walking through employing a fluorescent solution as a simulant. The footprint decay (D) and transfer efficiency (τ) of the fluorescent solution transferred from the sole to the indoor floor during walking were analyzed based on the fluorescent footprint imaging. The effects of namely body weight (50–75 kg), walking frequency (80–120 steps/min), and solution viscosity (oil and water) were also investigated. It was found that the total fluorescence gray value on the ground decreased exponentially as the number of walking steps (i) increased. The relationship between the normalized gray value of the fluorescent solution (D) on each floor panel i was , and τ was distributed in the range of 0.51–0.72. All influencing factors had a significant effect on a, and a greater body weight resulted in a smaller a value, while only the body weight had a significant effect on b and τ, and a greater body weight led to larger b and lower τ values.

Global existence and decay of solutions for p-biharmonic parabolic equation with logarithmic nonlinearity

In this paper, we study the initial boundary value problem for a p-biharmonic parabolic equation with logarithmic nonlinearity. By using the potential wells method and logarithmic Sobolev inequality, we obtain the existence of the unique global weak solution. In addition, we also obtain decay polynomially of solutions.

Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities

Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities

Dynamics of phase defects trapped in optically imprinted orbits in dissipative binary polariton condensates

We study the dynamics of phase defects trapped in a finite optically imprinted ring lattice in binary polariton condensates, under the influence of the cross interaction (CI) between the condensate fractions in different spin components and the spin-orbit interaction (SOI). In this configuration, we find that a vortex circulates unidirectionally in optically induced orbits because of the Magnus force acting in the polariton fluid and the vortex' angular velocity is influenced by the SOI and CI. Interestingly, in our system, these two interactions can also lead to elongated and frozen phase defects, forming a frozen dark solution with similarity to a dark soliton but with finite size in both spin components. When the dark solution is stretched further to occupy the entire orbit of a condensate ring, the phase defect triggers a snake instability and induces the decay of the dark ring solution. The circulation direction of a single vortex is determined by the Magnus force. This situation is more complex for the group motion of multiple vortices because of significant vortex-antivortex interaction. The collective motion of such vortex constellations, however, is determined by the SOI.

Threshold Solutions for Nonlocal Reaction Diffusion Equations

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp and describe the behavior of the corresponding threshold solution: it is bounded, symmetric and has a global maximum at zero. Numerical examples are shown to verify our results.

Creation of turbulence in polyatomic gas flow via an intermolecular potential

Turbulence in gases and liquids manifests in the form of spontaneously developing chaotic dynamics in a laminar flow without any measurable external disturbances. For gas flows, we propose that turbulence is created by the average effect of an intermolecular potential, leading to a system of equations with novel coupling between density and momentum. Numerical simulations show that turbulent dynamics in this system emerges spontaneously in the absence of external disturbances, in agreement with observations. In addition, the time-averaged kinetic energy spectra of our numerical solutions decay at the Kolmogorov rate, which is also an intriguing feature of observed turbulent flows.

Dissipative structure of one-dimensional isothermal compressible fluids of Korteweg type

This paper studies the dissipative structure of the system of equations that describes the motion of a compressible, isothermal, viscous-capillar fluid of Korteweg type in one space dimension. It is shown that the system satisfies the genuine coupling condition of Humpherys [23], which is, in turn, an extension to higher order systems of the classical condition by Kawashima and Shizuta [30,42] for second order systems. It is proved that genuine coupling implies the decay of solutions to the linearized system around a constant equilibrium state. For that purpose, the symmetrizability of the Fourier symbol is used in order to construct an appropriate compensating matrix. These linear decay estimates imply the global decay of perturbations to constant equilibrium states as solutions to the full nonlinear system, via a standard continuation argument.

Optimal $$L^2$$-decay of solutions to a nonlinear Schrödinger equation with sub-critical dissipative nonlinearity

Optimal $$L^2$$-decay of solutions to a nonlinear Schrödinger equation with sub-critical dissipative nonlinearity

Qualitative properties of traveling wave solutions in delayed reaction-diffusion equations with degenerate monostable nonlinearity

This paper deals with the asymptotic behavior, uniqueness up to translation and Lyapunov stability of traveling wave solutions with non-critical speeds in a class of delayed reaction-diffusion equations with degenerate monostable nonlinearity. With the help of Harnack inequalities and the detailed asymptotic higher order expansion, the precise asymptotic behavior of traveling wave solutions is established. Our results show that these traveling wave solutions decay algebraically, and the corresponding decaying rates of traveling wave solutions are independent of the delay. To illustrate our results on the asymptotic behavior, several degenerate equations are investigated. For a special equation, we present that the delay could affect the asymptotic behavior of traveling wave solutions. Furthermore, based on the special asymptotic behavior, we obtain the uniqueness up to translation and Lyapunov stability of traveling wave solutions.

Nonadiabatic decay of metastable states on coupled linear potentials

Avoided crossings of level pairs with opposite slopes can form potential-energy minima for the external degree of freedom of quantum particles, giving rise to metastable states on the avoided crossings (MSACs). Nonadiabatic decay of MSACs is studied by solving the two-component Schrödinger equation in diabatic and adiabatic representations. Non-perturbative lifetime values are found by evaluating wave function flux and scattering phases of time-independent solutions, as well as wave-function decay of time-dependent solutions. The values from these methods generally agree well, validating the utilized approaches. As the adiabaticity parameter, V, of the system is increased by about a factor of ten across the mixed diabatic/adiabatic regime, the MSAC character transitions from marginally to highly stable, with the lifetimes increasing by about ten orders of magnitude. The dependence of MSAC lifetime on the vibrational quantum number, ν, is discussed for several regimes of V. Time-dependent perturbation theory yields lifetimes that deviate by ≲30% from non-perturbative results, over the range of V and ν studied, while a semi-classical model based on Landau–Zener tunneling is up to a factor of twenty off. The results are relevant to numerous atomic and molecular systems with metastable states on intersecting, coupled potential energy curves.

Absence of singularities in solutions for the compressible Euler equations with source terms in

The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$ , $d\geq 1$ , which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$ -norm of $\nabla u$ is exponential decay.