A dissipative top in a weakly compact lie algebra and stability of basic flows in a plane channel

Abstract

Based on the equation of an abstract dissipative top, it is shown that the linear and parabolic velocity profiles are stable with respect to arbitrary smooth per turbations of the initial data in a flat periodic channel filled with an ideal or viscous incompressible medium, under the impermeability OR no slip boundary condi tions, respectively.The stability of the Couette flow with a linear velocity profile and the instability of the Poiseuille flow with a parabolic profile for a viscous incompressible fluid in a plane channel had been proved at restricted three dimensional (the less the molecular viscosity, the less they are) [1] and indefinitely small two dimensional perturbations of the velocity field [2], respectively.Here, we consider arbitrary two dimensional per turbations of the initial data in a periodic cell V = {0 ≤ x ≤ l, 0 ≤ y ≤ h} of the ideal or viscous incompressible medium with impermeability or no slip conditions at the cell walls y = 0, h, respectively. Allowing for equiv alency of weak and strong solutions of the Euler or Navier equations for plane parallel flows in a bounded region and the impossibility of increasing the smooth ness of the flow due to the boundary and initial condi tions [3-5], we below show the nonlinear stability of the mentioned basic flows with respect to two dimen sional smooth perturbations at any norm of the initial vorticity of the velocity field in the space of quadrati cally integrable functions L 2 (V). The functional algebraic constructions used can be considered as an alternative continuation of the known group theory analogy between differomor phisms of the flow region conserving the element of the volume and rotations of the top (an absolutely rigid body with one fastened point) [6]. They were already useful in obtaining the conditions of nonlinear stabil ity of basic flows on smooth two dimensional varieties with compact closing and the Kolmogorov flow in a channel at the no slip conditions [7-9].In contrast with the latter, the flows under study are free of the rotor of the field of external body forces and correspond to normal (in the direct and indirect sense) rotation of a typical top: the "kinetic moment" (a vor tex, or vorticity of the velocity field) of the basic flow of a continuous top generated by hydrodynamic equa tions is normal, i.e., orthogonal (in the sense of the scalar product of space L 2 (V)) to all its gyroscopic moments as "vector products" (the Poisson brackets) of "angular velocities" (functions of the flow) and their "kinetic moments" (vorticities).For example, rotation of the whirligig is normal. This simplest gyroscope has two equal moments of inertia μ 2 = μ 3 = μ (the top is symmetric), the residual moment μ 1 ≠ μ unequal to them (the top is nontrivial), and a constant vector of angular velocity ψ = (ψ 1 , 0, 0), ψ 1 ≠ 0, the kinetic moment of which Aψ = μ 1 ψ is directed along the axis of the isolated basic moment (the top is normal).The gyroscopic moment [χ, Aχ], which is deter mined by the angular velocity χ = (χ 1 , …), by the k...

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“…In order to analyze the stability of the vortex reverse flow (21)- (23), which corresponds to the main rotation of the top (20), with account for (24), we rewrite Eq. (20) as a variational equation (25) Following [21,42,47] and taking the scalar product of the variational equation with and , with account for the scalar square derivative, the symmetry of inertia , the anti-symmetry of commutator , and the quasi-compactness of in the following equalities we obtain the energy identities .…”

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