Optimal Decay Research Articles

Global well-posedness and large time behavior of epitaxy thin film growth model

We consider the global well-posedness and large time behavior of solutions for epitaxy thin film growth model in mathbb{R}^{d} with the dimensional dgeq 3. First, using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution under the condition that the initial data is sufficiently small. Moreover, we also establish the suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Mesofauna community influences litter chemical trajectories during early-stage litter decay

Decomposition of organic material is moderated by both litter chemistry and decomposer communities. Chemical alteration of the litter occurs throughout decomposition, yet our understanding of the nature of chemical change during decomposition is limited, particularly beyond carbon (C) and nitrogen (N). The decomposer food web likely interacts with these chemical changes in their basal resource to alter the trajectories of chemical changes during decay. To investigate how decomposer mesofauna influence patterns of litter chemical change throughout early stages of decay, we tracked mass loss, macro- and micronutrient elements, and fiber chemistry dynamics in Arizona sycamore (Platanus wrightii) leaves decomposed in a mesocosm of optimal decay conditions in a biotically diverse compost pile. By using litterbags of two different mesh sizes to manipulate the mesofauna gaining access to the litter, we identify how the complexity of the soil mesofauna community changes the trajectory of litter chemistry, beyond the typical focus on mass loss and N dynamics. Counter to many examples in the literature, the more abundant and complex arthropod community slowed mass loss and enhanced nutrient immobilization, perhaps due to competition and grazing pressure on the microbial community to immobilize N, P, and Ca sooner from the surrounding compost.

Optimal decay of compressible Navier-Stokes equations with or without potential force

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, we show that the N-th order spatial derivative of global small solution of the CNS equations without potential force tends to zero at the rate (1+t)−(s+N) instead of (1+t)−(s+N−1) stated in [13]. Secondly, we study optimal decay rate for global solution of the CNS equations with potential force. The tricky problem comes from the equilibrium state that will depend on spatial variable rather than constant. Based on energy estimate, spectral analysis, and high-low frequency decomposition, we establish the upper and lower bounds of decay rates for the solution itself and its spatial derivatives. These decay rates are really optimal since the upper bounds of decay rates coincide with the lower ones.

Linear Stability of the 2D Irrotational Circulation Flow around an Elliptical Cylinder

In this article we prove a linear inviscid damping result with optimal decay rates of the 2D irrotational circulation flow around an elliptical cylinder. In our result, all components of the asymptotic velocity field do not vanish and the asymptotic flow lines are not ellipse any more.

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

We investigate time-decay rates of strong solutions to the 3D compressible magnetohydrodynamics equations with large initial data. The main novelty of this paper is two-fold: first, we prove the upper optimal decay rates of the higher-order spatial derivatives of the solution, which are the same as those of the heat equation, and faster than the decay rates in the previous related works. Second, if the initial data satisfy some additional low frequency assumption, we also show the lower optimal decay rates of the solution as well as its all-order spatial derivatives. Therefore, our decay rates are optimal in this sense. Our methods mainly involve the Fourier splitting method, low-frequency and high-frequency decomposition and delicate energy estimates.

Restriction estimates for the flat disks over finite fields

In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions n=4,6. We improve and extend their results to all dimensions n≥6. More precisely, we obtain the sharp L2→Lr estimates, which cannot be proven by applying the usual Stein-Tomas argument over a finite field even with the optimal Fourier decay estimate on the flat disk. One of main ingredients is to discover and analyze an explicit form of the Fourier transform of the surface measure on the flat disk. In addition, based on the recent results on the restriction estimates for the paraboloids, we address improved restriction estimates for the flat disk beyond the L2 restriction estimates.

Optimal decay for a wave-heat system with Coleman–Gurtin thermal law

We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman–Gurtin thermal law. Our approach is based on the asymptotic theory of C0-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman–Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate t−2 as t→∞. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.

Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force

Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force

Decay properties for a density-dependent liquid crystal system

The purpose of this paper is to establish the optimal temporal decay properties for nonhomogeneous incompressible liquid crystal flows in Rn, n = 2, 3. Under a weaker smallness assumption on the initial datum d0, we construct a uniform (in t) energy estimate for the director field d, which plays a crucial role in giving the optimal L2-decay rates for the liquid crystal flows. If the initial data are more regular, the optimal decay rates for the higher spatial derivatives are also obtained.

Optimal decay rate of solutions to the two‐phase flow model

This article is devoted to the study of the global existence and large time behavior of the three‐dimensional two‐phase flow model derived from the Chapman‐Enskog expansion of the Navier‐Stokes‐Vlasov‐Fokker‐Planck equations around the local Maxwellian. When the initial data are a small perturbation of the equilibrium state in , we prove that the strong solution converges to the equilibrium state at an optimal algebraic rate in ‐norm. It is observed that due to the dispersion effect of the drag force term, the difference of velocities decays at a faster rate in ‐norm.

Optimal Decay Result for Wave Equations with Frictional Versus Viscoelastic Damping

Optimal Decay Result for Wave Equations with Frictional Versus Viscoelastic Damping

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.

Intelligent Diagnosis Method of Power Equipment Faults Based on Single‐Stage Infrared Image Target Detection

With the rapid expansion of the scale of the power grid, the efficiency of fault diagnosis has been severely challenged by the large amount of inspection image data generated by intelligent devices such as drones and inspection robots. In order to improve the efficiency of fault diagnosis for power equipment in substations, a new method for intelligently diagnosing different types of faults in power equipment is proposed. For circuit breakers and insulators, YOLOv4 is selected as the target detection model. To improve the detection performance of the YOLOv4 model, this paper improves it: the Cross Stage Partial (CSP) structure is introduced in the Spatial Pyramid Pooling (SPP) module of the neck of the YOLOv4 model. The experimental results show that after using the optimal learning rate decay strategy, the mAP and frames per second (FPS) of the improved YOLOv4 model are better than the original YOLOv4 and PP‐YOLO model. Finally, an intelligent diagnosis terminal system for power equipment faults is developed. Through the target recognition and rapid extraction of equipment temperature, the intelligent diagnosis of thermal faults of equipment is realized. This method is especially suitable for accurate fault diagnosis of more power equipment, and has potential huge applicability in the field of power equipment diagnosis. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Optimal decay rate to 3D reduced gravity two-and-a-half-layer model

Optimal decay rate to 3D reduced gravity two-and-a-half-layer model

Sharp stability of a string with local degenerate Kelvin–Voigt damping

AbstractThis paper is on the asymptotic behavior of the elastic string equation with localized degenerate Kelvin–Voigt damping where on , and on for . It is known that the optimal decay rate of solution is in the limit case and exponential for . When , the damping coefficient is continuous, but its derivative has a singularity at the interface . In this case, the best known decay rate is , which fails to match the optimal one at . In this paper, we obtain a sharper polynomial decay rate . More significantly, it is consistent with the optimal polynomial decay rate at and uniform boundedness of the resolvent operator on the imaginary axis at (consequently, the exponential decay rate at as ). This is a big step toward the goal of obtaining eventually the optimal decay rate.

Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl

We study the long-time behaviour of axisymmetric solutions without swirl for the threedimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate $L^p$-$L^q$ estimates for the linearised evolution equation and its Taylor expansion in self-similar variables.

A GENERAL AND OPTIMAL DECAY RESULT FOR A VISCOELASTIC EQUATION WITH A STRONG TIME DEPENDENT DELAY

In this paper, we establish an optimal and general decay result for the energy of a viscoelastic equation exhibiting a strong time-dependent delay. This is achieved by considering a minimal condition on the relaxation function g. The exponential and polynomial decay rates are obtained as special cases.The theoretical computations are supported with a numerical analysis of the problem under consideration. This work extends and generalizes some recent results in the literature.

Optimal decay rates for higher-order derivatives of solutions to the 3D magneto-micropolar fluid equations

We are concerned with the optimal decay rates for higher-order spatial derivatives of strong solutions to the 3D Cauchy problem of the magneto-micropolar fluid equations. Under some smallness assumptions, it is shown that for any integer N≥3, the N−1-order and N-order spatial derivatives of the solutions converge to zero at the L2-rate (1+t)−(34+N−12) and (1+t)−(34+N2) respectively, which are the same as those of the heat equation, and particularly improve the previous related results.

Joint space-time analyticity of mild solutions to the Navier-Stokes equations

In this paper, we show the optimal decay rate estimates of the space-time derivatives and the joint space-time analyticity of solutions to the Navier-Stokes equations. As it is known from the Hartogs's theorem, for a complex function with two complex variables, the joint analyticity with respect to two variables can be derived from combining of analyticity with respect to each variable. However, as a function of two real variables for space and time, the joint space-time analyticity of solutions to the Navier-Stokes equations cannot be directly obtained from the combination of space analyticity and time analyticity. Our result seems to be the first quantitative result for the joint space-time analyticity of solutions to the Navier-Stokes equations, and the proof only involves real variable methods. Moreover, the decay rate estimates also yield the bounds on the growth (in time) of radius of space analyticity, time analyticity, and joint space-time analyticity of solutions.

Optimal memory-type boundary control of the Bresse system

In this paper, we study the stability of a Bresse system with memory-type boundary conditions. For a wider class of kernel functions, we establish an optimal explicit energy decay result. Our stability result improves many earlier results in the literature. Finally, we also give four numerical tests to illustrate our theoretical results using the conservative Lax–Wendroff method scheme.