The strong mean shear in the vicinity of the boundaries in turbulent boundary layer flows preferentially amplifies a particular class of perturbations resulting in the appearance of coherent structures and in characteristic associated spatial and temporal velocity spectra. This enhanced response to certain perturbations can be traced to the nonnormality of the linearized dynamical operator thro...

The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects i...

In this work we construct algebraic equation for elements of spectrum of linearized Fokker Planck differential operator for incompressible fluid. We calculate roots of this equation using simple numeric method. For all these roots real part is positive, that is corresponding solutions are damping. Eigenfunctions of linearized Fokker Planck differential operator for incompressible fluid are expr...

In this note we continue the work in [SL], and give a full description of the essential spectrum for the linearized Euler operator L in dimension two. We prove that the essential spectrum of the operator is one solid vertical strip symmetric with respect to the imaginary axis. The width of the strip is determined by the maximal Lyapunov exponent Λ for the flow induced by the steady state. For c...

This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl’s Theorem and thus does not require Grad’s splitting. The more physical idea of the proof is to use geometr...

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez’s interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upp...

In this article we study the invertibility of the linearized operator coming from the nonlinear elasticity in the special case of a two-dimensional thin beam of thickness 2ε in one direction and of length 2πL and periodic in the other direction. In the context of the displacement approach, we show that the linearized operator is not invertible for some small compressions of orderO(ε/L) in the d...

The linearized stability of stationary solutions for the surface diffusion flow with a triple junction is studied. We derive the second variation of the energy functional under the constraint that the enclosed areas are preserved and show a linearized stability criterion with the help of the H-gradient flow structure of the evolution problem and the analysis of eigenvalues of a corresponding di...