Explicit Decay Research Articles

Stabilizing effect of the magnetic field and decay estimates for the 3D MHD equations with only one direction dissipation

This paper focuses on the stability and decay estimates of the 3D MHD equations with only one direction dissipation near a background magnetic field. The global stability and decay estimates of the 3D Navier-Stokes equations with only one direction dissipation remain an outstanding open problem. However, when coupled with the magnetic field in the MHD system involved here, the velocity field is shown to stabilize and decay in time. This result reveals that the magnetic field stabilizes and damps the electrically conducting fluids. By utilizing time-weighted methods and the method of bootstrapping argument, we establish that any perturbation near a magnetic field is globally stable in H3(R3). Furthermore, explicit decay rates in H2(R3) are obtained.

Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures.In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.

Stability for the magnetic Bénard system with partial dissipation

We prove the stability of the magnetic Bénard system with partial dissipation on perturbations near a background magnetic field in T2. Neglecting the effect of the temperature, the stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.

Stability and optimal decay for the 3D anisotropic magnetohydrodynamic equations

AbstractThis paper investigates the stability problem and large time behavior of solutions to the three‐dimensional magnetohydrodynamic equations with horizontal velocity dissipation and magnetic diffusion only in the direction. By applying the structure of the system, time‐weighted methods, and the method of bootstrapping argument, we prove that any perturbation near the background magnetic field (1, 0, 0) is globally stable in the Sobolev space . Furthermore, explicit decay rates in are obtained. Motivated by the stability of the three‐dimensional Navier–Stokes equations with horizontal dissipation, this paper aims to understand the stability of perturbations near a magnetic background field and reveal the mechanism of how the magnetic field generates enhanced dissipation and helps stabilize the fluid.

General Decay Rates of the Solution Energy in a Viscoelastic Wave Equation With Boundary Feedback and a Nonlinear Source

In a bounded domain, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and certain initial data. We establish an explicit and general decay rate result, using some properties of the convex functions. Our new results substantially improve several earlier related results in the literature. Keywords: General decay, nonlinear source, viscoelastic, wave equation, relaxation function.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

New decay results for Timoshenko system in the light of the second spectrum of frequency with infinite memory and nonlinear damping of variable exponent type

In this study, we consider a one-dimensional Timoshenko system with two damping terms in the context of the second frequency spectrum. One damping is viscoelastic with infinite memory, while the other is a non-linear frictional damping of variable exponent type. These damping terms are simultaneously and complementary acting on the shear force in the domain. We establish, for the first time to the best of our knowledge, explicit and general energy decay rates for this system with infinite memory. We use Sobolev embedding and the multiplier approach to get our decay results. These results generalize and improve some earlier related results in the literature.

On the interaction between the viscoelasticity and the boundary variable-exponent nonlinearity in plate systems

In this paper, we consider a viscoelastic plate system with a boundary frictional damping having a variable exponent m ( x ) . We investigate the competition between the two types of damping and their effects on the energy decay rates. Using the multiplier method, we establish explicit decay results depending on both m and the relaxation function g. Our results address both the optimality and generality and improve earlier related results in the literature.

Well-Posedness and Asymptotic Stability of a Non-Linear Porous System with a Delay Term

Our interest in this work is to treat a one-dimensional Porous system with a non-linear damping and a delay in the non-linear internal feedback. We prove the global existence and uniqueness of its solution in suitable function spaces by means of the Faedo-Galerkin procedure combined with the energy method under a suitable relation between the weight of the delayed feedback and the weight of the non-delayed feedback. Also, we give an explicit and general decay rate estimate by applying the well-known multiplier method integrated with some properties of convex functions and for two opposites cases with respect to the speeds of wave propagation.

Decay for thermoelastic laminated beam with nonlinear delay and nonlinear structural damping

<abstract><p>This paper discussed the decay of a thermoelastic laminated beam subjected to nonlinear delay and nonlinear structural damping. We provided explicit and general energy decay rates of the solution by imposing suitable conditions on both weight delay and wave speeds. To achieve this, we leveraged the properties of convex functions and employed the multiplier technique as a specific approach to demonstrate our stability results.</p></abstract>

Indirect stability of a 2D wave-plate coupling system with memory viscoelastic damping

<abstract><p>We performed a stability analysis of a 2D wave-plate coupling system equipped with memory viscoelastic damping. The study highlights the unique functionality of the damping mechanism, which operates indirectly and exclusively within either the wave or plate subsystem. The opposing subsystem receives dissipative signals indirectly through the coupling component. The primary objective of this study was to determine whether the indirect memory damping is sufficient to ensure the overall stability of the coupled system. To address this question, a frequency domain analysis was employed to establish explicit decay rates of the coupled system. Notably, a polynomial decay rate is observed when the memory damping is applied solely to either the plate or wave subsystem, which provides a conclusive answer to the posed question.</p></abstract>

Existence and General Decay of Solutions for a Weakly Coupled System of Viscoelastic Kirchhoff Plate and Wave Equations

In this paper, a weakly coupled system (by the displacement of symmetric type) consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain is considered. Under the assumptions on a more general type of relaxation functions, an explicit and general decay rate result is established by using the multiplier method and some properties of the convex functions.

Feedback stabilization of non-homogeneous distributed bilinear systems with varying time delay

This paper deals with the feedback stabilization for a class of non-homogeneous distributed bilinear systems, with periodically varying time delay, evolving in Hilbert state space. Sufficient conditions for polynomial and exponential stabilization are given. Explicit energy decay rates are provided in both cases. Illustrative examples are displayed.

Decay of a Thermoelastic Laminated Beam with Microtemperature Effects, Nonlinear Delay, and Nonlinear Structural Damping

This article deals with a non-classical model, namely a thermoelastic laminated beam along with microtemperature effects, nonlinear delay, and nonlinear structural damping, where the last two terms both affect the equation which depicts the dynamics of slip. With the help of convenient conditions in both weight delay and wave speeds, we demonstrate explicit and general energy decay rates of the solution. To attain our interests, we highlight useful properties regarding convex functions and apply a specific approach known as the multiplier technique, which enables us to prove the stability results. Our results here aim to show the impact of different types of damping by taking into account the interaction between them, which extends recent publications in the literature.

Computational Model of the Transition from Novice to Expert Interaction Techniques

Despite the benefits of expert interaction techniques, many users do not learn them and continue to use novice ones. This article aims at better understanding if, when and how users decide to learn and ultimately adopt expert interaction techniques. This dynamic learning process is a complex skill-acquisition and decision-making problem. We first present and compare three generic benchmark models, inspired by the neuroscience literature, to explain and predict the learning process for shortcut adoption. Results show that they do not account for the complexity of users’ behavior. We then introduce a dedicated model, Transition, combining five cognitive mechanisms: implicit and explicit learning, decay, planning and perseveration. Results show that our model outperforms the three benchmark models both in terms of model fitting and model simulation. Finally, a post-analysis shows that each of the five mechanisms contribute to goodness-of-fit, but the role of perseveration is unclear regarding model simulation.

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

Stability for the 2D incompressible MHD equations with only magnetic diffusion

This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion on the periodic domain. The stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.

Theoretical and numerical decay results of a viscoelastic suspension bridge with variable exponents nonlinearity

AbstractStrong vibrations can cause considerable damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exists evidence that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes and winds, features and modifications such as dampers are used to stabilize these bridges. In this work, we study a nonlinear viscoelastic plate equation with variable exponents, which models the deformation of a suspension bridge. First, we establish explicit and general decay results of the system depending on the decay rate of the relaxation function and the nature of the variable‐exponents nonlinearity. Then, we perform several numerical tests to illustrate our theoretical decay results. Our results extend and generalize many earlier works in the literature.

Self-induced flow over a cylinder in a stratified fluid

In this paper we study the self-induced low-Reynolds-number flow generated by a cylinder immersed in a stratified fluid. In the low Péclet limit, where the Péclet number is the ratio of the radius of the cylinder and the Phillips length scale, the flow is captured by a set of linear equations obtained by linearising the governing equations with respect to the prescribed far field conditions. We specifically focus on the low Péclet regime and develop a Green's function approach to solve the linearised equations governing the flow over the cylinder. We cross check our analytical solution against numerical solution of the nonlinear equations to obtain the range of the Péclet numbers for which the linear solution is valid. We then take advantage of the analytical solution to find explicit far-field decay rates of the flow. Our detailed analysis points out that the streamfunction and the velocity field decays algebraically in the far field. Intriguingly, this algebraic decay of the flow is much slower when compared with the exponential decay of the flow generated by a slow moving cylinder in the homogeneous Stokes regime, in the absence of stratification. Consequently, the flow generated by a cylinder in the stratified Stokes regime will have a larger domain of influence when compared with the flow generated by a cylinder in the homogeneous Stokes regime.

Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes

We obtain explicit formulas for the solution of the wave equation in certain Friedmann–Lemaître–Robertson–Walker (FLRW) spacetimes. Our method, pioneered by Klainerman and Sarnak, consists in finding differential operators that map solutions of the wave equation in these FLRW spacetimes to solutions of the conformally invariant wave equation in simpler, ultra-static spacetimes, for which spherical mean formulas are available. In addition to recovering the formulas for the dust-filled flat and hyperbolic FLRW spacetimes originally derived by Klainerman and Sarnak and generalizing them to the spherical case, we obtain new formulas for the radiation-filled FLRW spacetimes and also for the de Sitter, anti-de Sitter, and Milne universes. We use these formulas to study the solutions with respect to the Huygens principle and the decay rates and to formulate conjectures about the general decay rates in flat and hyperbolic FLRW spacetimes. The positive resolution of the conjecture in the flat case is seen to follow from known results in the literature.