The paper is focused on the loss of smoothness hypothesis which claims that vorticity (or vorticity gradients in the 2D case) grows unboundedly for the substantial part of the inviscid incompressible flows. At least, every steady flow is supposed to belong to the closure of this set (relative to a reasonably strong topology). We approach the problem involving both direct Lyapunov method and some sort of the linearization. We present new (and rather wide) classes of 2D flows in a generic domains which admit the loss of smoothness and related phenomena.

The flow of an ideal fluid in a domain with a permeable boundary may be asymptotically stable. Here the permeability means that the fluid can flow into and out of the domain through some parts of the boundary. This permeability is a principal reason for the asymptotic stability. Indeed, the well-known conservation laws make the asymptotic stability of an inviscid flow impossible, if the usual no flux condition on a rigid wall (or on a free boundary) is employed. We study the stability problem using the direct Lyapunov method in the Arnold's form. We prove the linear and nonlinear Lyapunov stability of a two-dimensional flow through a domain with a permeable boundary under Arnold's conditions. Under certain additional conditions, we amplify the linear result and prove the exponential decay of small disturbances. Here we employ the plan of the proof of the Barbashin-Krasovskiy theorem, established originally only for systems with a finite number of degrees of freedom. (C) 2002 American Institute of Physics.

The stability of a two-dimensional inviscid flow in an annulus between two permeable cylinders is examined. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It is shown that, independent of the direction of the radial flow, the basic flow is unstable to small two-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. The instability is oscillatory and persists if the viscosity of the fluid is taken into consideration.

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.