We construct center-stable and center-unstable manifolds, as well as stable and unstable manifolds, for the nonlinear Klein–Gordon equation with a focusing energy subcritical nonlinearity, associated with a family of solitary waves which is generated from any radial stationary solution by the action of all Lorentz transforms and spatial translations. The construction is based on the graph trans...

The purpose of this article is to survey results concerning the unstable spectrum of the Euler equation linearized about a steady state. The Euler equations of the motion of an inviscid, incompressible fluid are the basic equations of fluid mechanics and they have been the object of much study by mathematicians over the centuries since Euler ”unveiled” them in 1755. However, many significant pr...

In the following work, we consider Boltzmann equation that models a polyatomic gas by representing microscopic internal energy continuous variable I. Under some convenient assumptions on transition function $ \mathcal{B} $, prove linearized operator \mathcal{L} of this model is Fredholm operator. For this, write as perturbation collision frequency multiplication operator, and \mathcal{K} compac...

We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzm...

The existence of a linearized SUSY invariant for N = 8 supergravity whose gravitational components are usually called R was established long ago by on-shell superspace arguments. Superspace and string theory methods have also established analogous higher dimensional DR invariants. However, very little is known about the SUSY completions of these operators which involve other fields of the theor...

For the spatially homogeneous Boltzmann equation with hard potentials and Grad’s cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gaplike estimates valid ...

The linearized Vlasov equation for a plasma system in a uniform magnetic field and the corresponding linear Vlasov operator are studied. The spectrum and the corresponding eigenfunctions of the Vlasov operator are found. The spectrum of this operator consists of two parts: one is continuous and real; the other is discrete and complex. Interestingly, the real eigenvalues are infinitely degenerat...

We consider a simplified version of the kinetic model of simple reacting spheres (SRS) for a quaternary reactive mixture of hard-spheres in the dilute-gas limit. The model mimics a coloring process occurring with probability αR, described by the reversible chemical law A1 +A2 A3 +A4. We provide the linearized collisional operators of our model and investigate some of their mathematical properti...

Global Analysis Of Linearized Inversion For The Acoustic Wave Equation by Cli ord J. Nolan To predict the location of natural resources and reduce the cost of exploration, geophysicists rely on various techniques to map the internal structure of the earth. One common mapping method probes the earth's interior using an acoustic energy source (sound waves). The acoustic waves re ect when they imp...