We investigate the resonant states for the parabolic potential barrier known also as inverted or reversed oscillator. They correspond to the poles of meromorphic continuation of the resolvent operator to the complex energy plane. As a byproduct we establish an interesting relation between parabolic cylinder functions (representing energy eigenfunctions of our system) and a class of Gel'fand distributions used in our recent paper.
Decay rates for the energy of solutions of the damped wave equation on the torus are studied. In particular, damping invariant in one direction and equal to a sum of squares of nonnegative functions with a particular number of derivatives of regularity is considered. For such damping energy decays at rate $1/t^{2/3}$. If additional regularity is assumed the decay rate improves. When such a damping is smooth the energy decays at $1/t^{4/5-δ}$. The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.
We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness m…
We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.
We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $α/x$, $α>0$. We establish the exponential stability of the semigroup for all positive $α$, and determine conditions for the spectrum to consist of a finite number of eigenvalues. As a consequence, we fully characterize the set of initial conditions for which there is extinction of solutions in finite time. Finally, we propose two open problems related to extremal decay rates of solutions.
We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^β$ near the boundary of the support and show decay at rate $1/t^{\frac{β+2}{β+3}}$. In the case where $W$ vanishes exactly like $x^β$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.
We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and t…
It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence -- in the absence of damping the limiting behaviour depends on the exact con…
We study the decay of global energy for the wave equation with Hölder continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.
We determine the class of damped modes \tilde{y} which are related to the common free damping modes y by supersymmetry. They are obtained by employing the factorization of Newton's differential equation of motion for the free damped oscillator by means of the general solution of the corresponding Riccati equation together with Witten's method of constructing the supersymmetric partner operator. This procedure leads to one-parameter families of (transient) modes for each of the three types of free damping, corresponding to a particular type of %time-dependent angular frequency. %time-dependent,…
In this paper the linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is 0 less than or equal to nu which is less than or equal to 1 . At the lower end, nu = 0, the equation represents an un-damped oscillator and at the upper end, nu = 1, the ordinary linearly damped oscillator equation is recovered. A solution is found analytically and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for t…
In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the d…
Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.
The work considers the damped Pinney equation, defined as the model arising when a linear in velocity damping term is included in the Pinney equation. In the general case the resulting equation does not admit Lie point symmetries or is reducible to a simpler form by any obvious coordinate transformation. In this context the method of Kuzmak-Luke is applied to derive a perturbation solution, for weak damping and slow time-dependence of the frequency function. The perturbative and numerical solutions are shown to be in good agreement. The results are applied to examine the time-evolution of Gaus…
Regularization of damped motion under central forces in two and three-dimensions are investigated and equivalent, undamped systems are obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and subjected to a damping force is shown to be regularized a la Levi-Civita. We then generalize this regularization mapping to the case of damped motion in the potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler motion in 3-dimensions is mapped to an oscillator with inverted sextic potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel regularization …
We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov. The main application is to obtain conditions (in terms of the geodesic flow on $X$ and the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. We review various estimates for the high frequency spectrum in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also present a new condi…
Using Koopman's approach to classical dynamical systems we show that the classical damping may be interpreted as appearance of resonant states of the corresponding Koopman's operator. It turns out that simple classical damped systems give rise to discrete complex spectra. Therefore, the corresponding generalized eigenvectors may be interpreted as classical resonant states.
