Energy Decay Rate Research Articles

Energy-stable backward differentiation formula type fourier collocation spectral schemes for the Cahn-Hilliard equation

We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with a Fourier collocation spectral approximation in space. A three-point stencil is applied in the temporal discretization, and the concave term diffusion term is treated explicitly. An addition-al Douglas-Dupont regularization term is introduced, which ensures the energy stability with a mild requirement. Various numerical simulations including the verification of accuracy, coarsening process and energy decay rate are presented to demonstrate the efficiency and the robustness of proposed schemes.

The lack of exponential stability for a weakly coupled wave equations through a variable density term

<p style='text-indent:20px;'>In this paper, we consider a system of two wave equations coupled through zero order terms. One of these equations has an internal damping, and the other has a boundary damping. We investigate stability properties of the system according to the variable strings densities. Indeed, our main result is to show that the corresponding model is not exponentially stable using a spectral theory which forms the center of this work. Otherwise, we establish a polynomial energy decay rate of type <inline-formula><tex-math id="M1">\begin{document}$ \frac{1}{\sqrt{t}}. $\end{document}</tex-math></inline-formula></p>

Polynomial stability in viscoelastic network of strings

<p style='text-indent:20px;'>In this paper we consider star-shaped viscoelastic networks, and study the large-time behaviour of these networks by proving polynomial decay rates. The energy decay rate depends on the irrationality properties of the lengths of the rods.</p>

Sharper and finer energy decay rate for an elastic string with localized Kelvin-Voigt damping

<p style='text-indent:20px;'>This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}(x, t)-[u_{x}(x, t)+b(x)u_{x, t}(x, t)]_{x} = 0, \; x\in(-1, 1), \; t>0, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ b(x) = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M2">\begin{document}$ x\in (-1, 0] $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M3">\begin{document}$ b(x) = a(x)>0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ x\in (0, 1) $\end{document}</tex-math></inline-formula>. It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that <inline-formula><tex-math id="M5">\begin{document}$ a'(x) $\end{document}</tex-math></inline-formula> has a singularity at <inline-formula><tex-math id="M6">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula>, we investigate the decay rate of the solution which depends on the order of the singularity.</p><p style='text-indent:20px;'>When <inline-formula><tex-math id="M7">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> behaves like <inline-formula><tex-math id="M8">\begin{document}$ x^{\alpha}(-\log x)^{-\beta} $\end{document}</tex-math></inline-formula> near <inline-formula><tex-math id="M9">\begin{document}$ x = 0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M10">\begin{document}$ 0\le{\alpha}<1, \;0\le\beta $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M11">\begin{document}$ 0<{\alpha}<1, \;\beta<0 $\end{document}</tex-math></inline-formula>, we show that the system can achieve a mixed polynomial-logarithmic decay rate.</p><p style='text-indent:20px;'>As a byproduct, when <inline-formula><tex-math id="M12">\begin{document}$ \beta = 0 $\end{document}</tex-math></inline-formula>, we obtain the decay rate <inline-formula><tex-math id="M13">\begin{document}$ t^{-\frac{ 3-\alpha-\varepsilon}{2(1-{\alpha})}} $\end{document}</tex-math></inline-formula> of solution for arbitrarily small <inline-formula><tex-math id="M14">\begin{document}$ \varepsilon>0 $\end{document}</tex-math></inline-formula>, which improves the rate <inline-formula><tex-math id="M15">\begin{document}$ t^{-\frac{1}{1-{\alpha}}} $\end{document}</tex-math></inline-formula> obtained in [<xref ref-type="bibr" rid="b14">14</xref>]. The new rate is again consistent with the exponential decay rate in the limit case <inline-formula><tex-math id="M16">\begin{document}$ \alpha\to 1^- $\end{document}</tex-math></inline-formula>. This is a step toward the goal of obtaining the optimal decay rate eventually.</p>

Rational Energy Decay Rate of a Wave Equation: the Case of Dimension ≥ 2

Rational Energy Decay Rate of a Wave Equation: the Case of Dimension ≥ 2

Local indirect stabilization of same coupled evolution systems through resolvent estimates

<p style='text-indent:20px;'>In this paper, we consider same systems of two coupled equations (wave-wave, Schrödinger-Schrödinger) in a bounded domain. Only one of the two equations is directly damped by a localized damping term (indirect stabilization). Under geometric control conditions on both coupling and damping regions (internal or boundary), we establish the energy decay rate by means of a suitable resolvent estimate. The numerical contribution is interpreted to confirm the theoretical result of a wave-wave system.</p>

General decay results for viscoelastic systems with memory and time‐varying delay

In this paper, we consider a wave equation with memory term and time‐varying delayed velocity term. By some subtle analysis techniques, assuming memory kernel function with a minimal and general condition, we establish a general uniform energy decay result for the system. The condition here substantially weakens the conditions in earlier literature. From the result, we can give the explicit energy decay rates directly if the memory kernel is expressed explicitly. The result is new even for the case that memory kernel decays polynomially.

The stable trapping phenomenon for black strings and black rings and its obstructions on the decay of linear waves

The geometry of solutions to the higher dimensional Einstein vacuum equations presents aspects that are absent in four dimensions, one of the most remarkable being the existence of stably trapped null geodesics in the exterior of asymptotically flat black holes. This paper investigates the stable trapping phenomenon for two families of higher dimensional black holes, namely black strings and black rings, and how this trapping structure is responsible for the slow decay of linear waves on their exterior. More precisely, we study decay properties for the energy of solutions to the scalar, linear wave equation $\Box_{g_{\textup{ring}}} \Psi=0$, where $g_{\textup{ring}}$ is the metric of a fixed black ring solution to the five-dimensional Einstein vacuum equations. For a class $\mathfrak{g}$ of black ring metrics, we prove a logarithmic lower bound for the uniform energy decay rate on the black ring exterior $(\mathcal{D},g_{\textup{ring}})$, with $g_{\textup{ring}}\in\mathfrak{g}$. The proof generalizes the perturbation argument and quasimode construction of Holzegel--Smulevici \cite{SharpLogHolz} to the case of a non-separable wave equation and crucially relies on the presence of stably trapped null geodesics on $\mathcal{D}$. As a by-product, the same logarithmic lower bound can be established for any five-dimensional black string. Our result is the first mathematically rigorous statement supporting the expectation that black rings are dynamically unstable to generic perturbations. In particular, we conjecture a new \textit{nonlinear} instability for five-dimensional black strings and thin black rings which is already present at the level of scalar perturbations and clearly differs from the mechanism driven by the well-known Gregory--Laflamme instability.

Energy Envelope and Attenuation Characteristics of High-Frequency (HF) and Very-High-Frequency (VF) Martian Events

ABSTRACTSince its deployment at the surface of Mars, the Seismic Experiment for Interior Structure (SEIS) instrument of the InSight mission has detected hundreds of small-magnitude seismic events. In this work, we highlight some features of two specific families: high-frequency (HF) and very-high-frequency (VF) events. We characterize the shape of the energy envelopes of HF and VF events with two parameters: (1) the delay time td between the onset and the peak of the dominant arrival; and (2) the quality factor Qc, which quantifies the energy decay rate in the coda. We observe that the envelope of HF and VF events is frequency independent. As a consequence, a single delay time suffices to characterize envelope broadening in the 2.5–7.5 Hz band. The typical coda decay time is also frequency independent, as attested by the close to linear increase of Qc with frequency. Finally, we use elastic radiative transfer theory to perform a series of inversion of seismogram envelopes for the attenuation properties of the Martian lithosphere. The good fit between synthetic and observed envelopes confirms that multiple scattering of elastic waves released by internal sources is a plausible explanation of the events characteristics. We quantify scattering and attenuation properties of Mars and highlight the differences and similarities with the Earth and the Moon. The albedo, that is, the contribution of scattering to the total attenuation, derived from VF events is very high, which we interpret as a signature of a mostly dry medium. Our results also suggest a stratification of the scattering and attenuation properties.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t−(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·,t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·,t) will always decay at least as fast as t−(n+2)/4, no matter how slow the decay of ∥u(·,t)∥L2(Rn) might be.

Existence and asymptotic stability of solutions for a hyperbolic equation with logarithmic source

In this paper, we consider a hyperbolic equation with a logarithmic source term. This work is devoted to prove the global existence and the uniform energy decay rates of a weak solution applying the logarithmic Sobolev inequality and modified multiplier method. This result gives us to classify the energy decay rate dependent on the growth near zero on the damping term.

Decay rates of energy of the 1D damped original nonlinear wave equation

We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity.

On a nonlinear problem with Dirichlet and Acoustic boundary conditions

The aims of this paper are to establish theoretical analysis and numerical simulation for a nonlinear wave equation with mixed boundary conditions of Dirichlet and Acoustic type. The theoretical results are about: existence and uniqueness of global solutions, regularity and uniform stability of these global solutions and an exponential decay rate for energy. In the numerical context, simulations are presented using the finite element method in space (with linear and quadratic Lagrange basis), the Crank-Nicolson method in time and, for each discrete time, the Newton’s method is used to solve the nonlinear algebraic system. Furthermore, the energy exponential decay and convergence order (sub-optimal and optimal) are presented numerically.

Curvature perturbations and anomaly explain dark energy

Abstract We investigate the history of dark energy to explain the present magnitude. We assume the dark energy is the residual cosmological constant. The most important channel in the reheating process is gluon pair production by the quantumchromodynamic trace anomaly. We argue that dark energy decays rapidly by gluon pair emissions during the reheating and after the big bang. The reheating temperature is determined by the decay width of dark energy, $\Gamma$, and the Planck mass, $M_\mathrm{p}$, as $\sqrt{M_\mathrm{P}\Gamma} \sim 10^6 \, GeV$. This is a consequence of Friedmann’s equation and the equilibrium condition $\Gamma\sim H$. As the Universe cools below the hadronic scale, the dark energy density is almost frozen. Nevertheless, the dark energy further decreases by emitting two photons. We have estimated the current decay rate of dark energy from the quantum electrodynamic trace anomaly. The consistent solution of the Friedmann equation is in excellent agreement with the observations. The suppression factor of the dark energy scale is the product of the fine structure constant, $\alpha$, and the curvature perturbation, $P$: $10^{-30}=(\alpha^2P/4\pi)^2$. We argue that the conformal symmetry breaking in both ultraviolet and infrared are necessary unless dark energy is subtracted. We also investigate leptogenesis by adding massive right-handed neutrinos: realistic leptogenesis takes place during the reheating process.

English

We are concerned with a transmission problem for the Kirchhoff plate equation where one small part of the domain is made of a viscoelastic material with the Kelvin-Voigt constitutive relation. We obtain the logarithmic stabilization result (explicit energy decay rate), as well as the wellposedness, for the transmission system. The method is based on a new Carleman estimate to obtain information on the resolvent for high frequency. The main ingredient of the proof is some careful analysis for the Kirchhoff transmission plate equation.

Energy decay for a weakly nonlinear damped porous system with a nonlinear delay

This paper considers a one-dimensional porous system damped with weakly nonlinear feedback in the presence of a nonlinear delay. We prove the global existence and uniqueness results using the Faedo–Galerkin procedure. Furthermore, under appropriate assumptions on the weight of the delay and without imposing any restrictive growth assumption on the damping term at the origin, we establish an energy decay rate, using a perturbed energy method and some properties of convex functions in case of the same speed of propagation in the two equations of the system.

Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity

This paper studies the asymptotic behavior of a solution of dispersive shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation. The energy decay rates of the solution to the model are examined through the Fourier transform method. Moreover, we develop a pseudo-compact finite difference scheme for solving the model, study solution behavior, and confirm our theoretical results. The fundamental energy-decreasing property, which is obtained from the model of coupling the Rosenau-RLW equation with the Rosenau-Burgers equation, is derived and preserved by the present numerical scheme. The existence, uniqueness, convergence, and stability of the numerical solution are theoretically analyzed. Some numerical experiments are also conducted to demonstrate the accuracy and robustness of the present method. Finally, to verify the optimal decay rates, the numerical results are carried out at variant time scales by applying the moving boundary technique. The simulations are successfully constructed to support the theoretical results.

Optimal energy decay result for nonlinear abstract viscoelastic dissipative systems

In this paper, we consider the nonlinear abstract equation $$\begin{aligned} u_{tt}+Au-\int _{0}^{t}g(t-s)Au(s)\mathrm{d}s+h(u_{t})=j(u) \end{aligned}$$ subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of g at infinity and the behavior of h near 0, we establish explicit and optimal energy decay result. To the best of our knowledge, this is the first time we have such combination of generality and optimality in one explicit formula for the energy decay rates of this system.

Annihilation of slowly-decaying terms of Navier–Stokes flows by external forcing

The goal of this paper is to provide an algorithm that, for any sufficiently localised, divergence-free small initial data, explicitly constructs a localised external force leading to a rapidly dissipative solutions of the Navier–Stokes equations : namely, the energy decay rate of the flow will be forced to satisfy as t → ∞, which is beyond the usual optimal rate. An important feature of our construction is that this force can always be taken compactly supported in space–time, and its profile arbitrarily prescribed up to a spatial rescaling. Since the forcing term vanishes after a finite time interval, our result suggests that nontrivial interactions between the linear and nonlinear parts occur, annihilating all the slowly decaying terms contained in Miyakawa and Schonbek’s asymptotic profiles.

Numerical Modeling of Freestream Turbulence Decay Using Different Commercial Computational Fluid Dynamics Codes

Abstract This work models the spatial decay of freestream turbulence using three different commercial computational fluid dynamics (CFD) codes: Fluent, star-ccm+, and cfx. The two-equation shear stress transport k–ω (SST-k–ω) steady Reynolds-averaged-Navier–Stokes (RANS) model was used, within each of these three different commercial codes, and the modeling variations were analyzed. Comparison of the results from the SST-k–ω model with experiments and large eddy simulation (LES) (carried out using star-ccm+) were also made, which reveal that all the commercial CFD codes demonstrate either a higher or slower rate of spatial turbulent kinetic energy (TKE) decay. Attempts were then made to unify the resultant modeling approach between these three CFD tools, by careful manipulation of the inlet boundary conditions and subsequent fine-tuning of the SST-k–ω model constant (β∞∗). The results obtained not only displayed uniformity among the three CFD codes but also demonstrated a much better agreement to the experiments and the LES results. Thereafter, the optimized model coefficient (β∞∗) was integrated with the three-equation k–kl–ω transition model to examine its applicability in modeling a turbulent boundary layer flow over a flat plate with low incoming turbulence. The results showed good agreement with the theoretical boundary layer correlations, with correct prediction of the transition location. The findings from this study can be used as a suitable modeling method to accurately model the effects of freestream turbulence on bluff-body and boundary layer flows.