A Hopf-Bifurcation Theorem for the Vorticity Formulation of the Navier–Stokes Equations in ℝ³

Abstract

We prove a Hopf-bifurcation theorem for the vorticity formulation of the Navier-Stokes equations in R 3 in case of spatially localized external forcing. The difficulties are due to essential spectrum up to the imaginary axis for all values of the bifurcation parameter which a priori no longer allows to reduce the problem to a finite dimensional one.

Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension

“…Hopf-bifurcation) of incompressible Navier-Stokes equation has attracted much attention, see [3,9,13,22,23], etc. When the linearized operator possesses a continuous spectrum up to the imaginary axis and that a pair of imaginary eigenvalues crosses the imaginary axis, Melcher, A, et al [26] proved Hopf-bifurcation for the vorticity formulation of the incompressible Navier-Stokes equations in R . Their work is mainly motivated by the work of Brand, T, et al [1] who studied the Hopf-bifurcation problem and its exchange of stability for a coupled reaction di usion model in R a .…”

Pacemakers in large arrays of oscillators with nonlocal coupling

“…We will denote the dual space to M k,p γ by M −k,q −γ , where 1/p + 1/q = 1. Such spaces have been used extensively in regularity theory [8], fluids [14,19], and in our prior work on inhomogeneities [6,5]. Fredholm properties for the Laplacian and various generalizations have been established in [15,10,11,12].…”

Inhomogeneities in 3 dimensional oscillatory media

“…Additionally, McOwen and Lockhart used this spaces to study Fredholm properties of elliptic operators and systems of elliptic operators in non-compact manifolds [8,9,10]. Moreover, Kondratiev spaces have also been used in the description of far field asymptotics for fluid problems, in particular when studying the flow past obstacles, since they lend themselves to the study of problems in exterior domains (see [15] for the case of R 3 and [12] for an application towards bifurcation theory). More recently, a variant of these spaces was used in [13] to study Poisson's equation in a one-periodic infinite strip Z = [0, 1] × R.…”