AROUND THE ASYMPTOTIC PROPERTIES OF A TWO-DIMENSIONAL PARAMETRIZED EULER FLOW

We study the two-dimensional Euler flow X (<middle dot>, x) for x in the torus T-2 := R-2/2 pi Z(2), solution to the ODE: partial derivative X-t(<middle dot>, x) = b(X(<middle dot>, x)) for t >= 0, with X(0, x) = x, where b is the vector field defined on T-2 by: b(x) = b(x(1), x(2)) := (- A cos x(1) - B sin x(2) , A sin x(1) + B cos x(2)), A, B is an element of R\ {0}. We derive for any x is an element of T-2, the asymptotics of X(t, x) as t tends to infinity, depending on whether |A| = |B| or |A| not equal |B|. In the first case, the orbits of the flow are all bounded. In the second case, it turns out that one of the coordinates of X(t, x) is bounded with an explicit bound, while the other one is equivalent to a(x) t. The function a does not vanish in T-2 and satisfies uniform bounds which depend on parameters A, B. When |A| not equal |B|, we also prove that for any global first integral u of the flow X with a periodic gradient, Vu has at least a cluster point of roots in T-2, which implies the non-existence of any real -analytic first -integral of the flow.