Strong Linear Damping Research Articles

Asymptotic behavior for a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping

<abstract><p>In spaces with any dimension, a class of logarithmic wave equations with Balakrishnan-Taylor damping, nonlinear weak damping and strong linear damping was considered</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {u_{tt}} - M(t)\Delta u + \left( {g * \Delta u} \right)(t) + h({u_t}){u_t} - \Delta {u_t} + f(u) + u = u\ln {\left| u \right|^k} $\end{document} </tex-math></disp-formula></p> <p>with Dirichlet boundary condition. Under a set of specified assumptions, we established the existence of global weak solutions and elucidated the decay rate of the energy function for particular initial data. This contribution extended and surpassed prior investigations, as documented by A. Peyravi (2020), which omitted the consideration of Balakrishnan-Taylor damping and strong linear damping while being confined to three spatial dimensions. Our findings underscored the pivotal role of these overlooked damping factors. Furthermore, our demonstration underscored that Balakrishnan-Taylor damping, weak damping and strong damping collectively induced an exponential decay, although the precise nature of this decay was contingent upon the differentiable function associated with the memory damping term $ {\zeta (t)} $. Consequently, the absence of the damping term in reference by A. Peyravi (2020) was unequivocally shown not to augment the decay rate. This insight enhanced our understanding of the nuanced dynamics involved and contributed to the refinement of existing models.</p></abstract>

Planet–Disk Interaction in Disks with Cooling: Basic Theory

Abstract Gravitational coupling between young planets and their parent disks is often explored using numerical simulations, which typically treat the disk thermodynamics in a highly simplified manner. In particular, many studies adopt the locally isothermal approximation, in which the disk temperature is a fixed function of the stellocentric distance. We explore the dynamics of planet-driven density waves in disks with more general thermodynamics, in which the temperature is relaxed toward an equilibrium profile on a finite cooling timescale t c. We use both linear perturbation theory and direct numerical simulations to examine the global structure of density waves launched by planets in such disks. A key diagnostic used in this study is the behavior of the wave angular momentum flux (AMF), which directly determines the evolution of the underlying disk. The AMF of free waves is constant for slowly cooling (adiabatic) disks but scales with the disk temperature for rapidly cooling (and locally isothermal) disks. However, cooling must be extremely fast, with β = Ωt c ≲ 10−3 for the locally isothermal approximation to provide a good description of density wave dynamics in the linear regime (relaxing to β ≲ 10−2 when nonlinear effects are important). For intermediate cooling timescales, density waves are subject to a strong linear damping. This modifies the appearance of planet-driven spiral arms and the characteristics of axisymmetric structures produced by massive planets: in disks with β ≈ 0.1–1, a near-thermal mass planet opens only a single wide gap around its orbit, in contrast to the several narrow gaps produced when cooling is either faster or slower.

Asymptotic stability of solutions to a semilinear viscoelastic equation with analytic nonlinearity

By a small perturbation of the usual Lyapunov energy and by using the Łojasiewicz–Simon inequality, we show that the dissipation given by the memory term is strong enough to prove the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution to the semilinear viscoelastic equation $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)\mathrm{d}s+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$ in a bounded regular domain of \({\mathbb {R}}^n\) with Dirichlet boundary conditions. Here, the kernel function \(k > 0\) is assumed to decay exponentially at infinity, the nonlinearity f is analytic in the second variable, and the forcing term g is supposed to decay polynomially or exponentially at infinity. The present work extends the previous result where the given equation was studied in the presence of an additional strong linear damping \(-\,\Delta u_t\).

The Exponential Attractor for a Class of Nonlinear Coupled Kirchhoff Equations with Strong Linear Damping

This paper investigates the dynamics for a class of nonlinear higher-order coupled Kirchhoff equations with strong linear damping. By means of the method proposed by Eden et al., the Lipschitz continuity and the discrete squeezing property of its solution semigroup are proved, and thus the existence of the exponential attractor is obtained.

Estimate on the Dimension of Global Attractor for Nonlinear Higher-Order Coupled Kirchhoff Type Equations

In this paper, we investigate the finite dimensions of the global attractor for nonlinear higher-order coupled Kirchhoff type equations with strong linear damping in Hilbert spaces E0 and E1. Under the appropriate assumptions, we acquire a precise estimate of the upper bound for its Hausdorff and Fractal dimensions.

The Inertial Manifold for a Class of Nonlinear Higher-Order Coupled Kirchhoff Equations with Strong Linear Damping

This paper considers the long-time behavior for a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms. Using the Hadamard graph transformation method, we obtain the existence of the inertial manifold while such equations satisfy the spectral interval condition.

The Global and Exponential Attractors for the Higher-order Kirchhoff-type Equation with Strong Linear Damping

In this paper, we study the longtime behavior of solution to the initial boundary value problem for a class of strongly damped Higher-order Kirchhoff type equations: ${u_{tt}} + {( - \Delta )^m}{u_t} + {\left( {\alpha + \beta\left\| {{\nabla ^m}u} \right\|^2} \right)^{q}}{( - \Delta )^m}u + g(u) = f(x)$. At first, we do priori estimation for the equations to obtain two lemmas and prove the existence and uniqueness of the solution by the lemmas and the Galerkin method. Then, we obtain to the existence of the global attractor in $H_0^m(\Omega ) \times {L^2}(\Omega )$ according to some of the attractor theorem. In this case, we consider that the estimation of the upper bounds of Hausdorff for the global attractors are obtained. At last, we also establish the existence of a fractal exponential attractor with the non-supercritical and critical cases.

Strongly Damped Quadratic Matrix Polynomials

We study the eigenvalues and eigenspaces of the quadratic matrix polynomial $M\lambda^2+sD\lambda+K$ as $s\rightarrow\infty$, where $M$ and $K$ are symmetric positive definite and $D$ is symmetric positive semidefinite. This work is motivated by its application to modal analysis of finite element models with strong linear damping. Our results yield a mathematical explanation of why too strong damping may lead to practically undamped modes such that all nodes in the model vibrate essentially in phase.

A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING

Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, <TEX>${\ldots}$</TEX>, order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.

The force transmissibility of MDOF structures with a non-linear viscous damping device

In the present study, the concept of the Output Frequency Response Function (OFRF), recently proposed by the authors, is applied to theoretically investigate the force transmissibility of MDOF structures with a cubic non-linear viscous damping device. The results analytically show that the introduction of cubic non-linear damping can significantly reduce the transmissibility over all resonance regions for a Multiple Degree of Freedom (MDOF) structure and at the same time leave the transmissibility over the isolation region virtually unaffected. The analysis also indicates that a strong linear damping may shift the system resonances and compromise the beneficial effects of cubic non-linear viscous damping on the force transmissibility of MDOF structures. This suggests that a less significant linear damping together with a strong cubic non-linear damping can be used in MDOF structures to achieve a desired vibration isolation performance. This research work has a significant implication for the design of viscously damped MDOF structures for a wide range of practical applications.

An almost two‐dimensional approach to type 2 irregularities in the equatorial electrojet

A recently developed theory of “almost two‐dimensional” turbulence (Albert et al., 1990) is applied to E×B fluctuations in the E region equatorial electrojet. This theory of plasma turbulence in an external magnetic field extends strictly two‐dimensional theory (e.g., Sudan and Keskinen, 1979) to include weak phase variations along the magnetic field. Based on the direct interaction approximation of Kraichnan (1959), it describes the tendency of spectral energy to transfer nonlinearly towards modes with high k∥ despite strong linear damping of these modes. Here, we discuss application of this theory to the aspect angle k∥/k of fluctuations at 3‐m scales, for which radar backscatter measurements have been made (Kudeki and Farley, 1989). Allowing for uncertainty in characterizing the plasma turbulence, we find good agreement except at the lowest altitudes of the electrojet.

Reduction of stimulated brillouin scattering by the generation of ion acoustic harmonics

Ion wave harmonics generated by the primary ion acoustic wave excited in stimulated Brillouin scattering are shown to introduce heavy nonlinear damping of the process and to considerably reduce the anomalous reflectivity in an underdense plasma (n/nc<0.5). The generation of ion wave harmonics acts similarly to strong linear damping and provides an alternative explanation to the heavy damping indicated by experiments.