0 On the L 2 - instability and L 2 - controllability of steady flows of an ideal incompressible fluid

On the L 2-instability and L 2-controllability of steady flows of an ideal incompressible fluid 1. In this work we are studying the flows of an ideal incompressible fluid in a bounded 2-d domain M ⊂ R 2 , described by the Euler equations ∂u ∂t + (u, ∇)u + ∇p = 0; (1) ∇ · u = 0. (2) Here u = u(x, t), x ∈ M, t ∈ [0, T ], and u| ∂M is tangent to ∂M. It has been known for a long time, that if the initial velocity field u(x, 0) is smooth, then there exists unique smooth solution u(x, t) of the Euler equations , which is defined for all t ∈ R; see[M-P]. The next natural question is, what may be the behavior of this solution, as t → ∞. This is a problem of indefinite complexity. A restricted problem is the following: suppose that the initial flow field u 0 (x) is in close to a steady solution u 0 (x). What may happen with this flow for big t? Does it stay always close to u 0 , or it can escape far away? Which flows are available, if we start from different initial velocities, close to u 0 ? These are problems of a global, nonlinear perturbation theory of steady solutions of the Euler equations. The first idea is to develop a linear stability theory. The spectrum of a linearized operator is always symmetric w.r.t. both the real and the imaginary axes, for the system is Hamiltonian. Therefore we can never prove an asymptotic stability by the linear method; at best we can prove the absence of a linear instability, which, in its turn, may be a tricky business. The true, nonlinear stability of some classes of steady flows was first proven by V. Arnold (see [A1], [A2], [AK]). He considered a very strong 1