The Essential Spectrum of the Linearized 2d Euler Operator Is a Vertical Band
نویسنده
چکیده
We prove that the essential spectrum of the operator obtained by linearization about a steady state of the Euler equations governing the motion of inviscid ideal fluid in dimension two is a vertical strip whose width is determined by the maximal Lyapunov exponent of the flow induced by the steady state.
منابع مشابه
Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents
The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator L in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov–Oseledets exponent for the flow induced by the steady state contributes a vert...
متن کاملEssential Spectrum of the Linearized
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...
متن کاملIntegrable Structures for 2D Euler Equations of Incompressible Inviscid Fluids
The governing equation of turbulence, that we are interested in, is the incompressible 2D Navier– Stokes equation under periodic boundary conditions. We are particularly interested in investigating the dynamics of 2D Navier–Stokes equation in the infinite Reynolds number limit and of 2D Euler equation. Our approach is different from many other studies on 2D Navier–Stokes equation in which one s...
متن کاملOn Recent Developments in the Spectral Problem for the Linearized Euler Equation
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...
متن کاملSpectrum and essential spectrum of linear combinations of composition operators on the Hardy space H2
Let -----. For an analytic self-map --- of --- , Let --- be the composition operator with composite map --- so that ----. Let --- be a bounded analytic function on --- . The weighted composition operator --- is defined by --- . Suppose that --- is the Hardy space, consisting of all analytic functions defined on --- , whose Maclaurin cofficients are square summable. .....
متن کامل