Remarks on the Euler equation

au,/at + 5 U@U$3Xj) = ji + at3pxi ) 1 <i<N, j=l on s;! x (0, T), (1) div u = 0 on Q x (0, T), (2) u-n=0 on aQ x (0, T), (3) u It-0 = 210 on 0, (4) where f(x, t) and U,,(X) are given, while the velocity u(x, t) and the pressure CG(X, t) are to be determined. The Euler equation has been considered by several authors including L. Lichtenstein (1925-30), J. Leray (1932-37), M. Wolibner (1938). T. Kato proved the existence of a global solution for N = 2 [3] and of a local solution for Q = W3 [4]. Recently, D. Ebin and J. Marsden [2] have proved the existence of a local solution in the general case. Their proof relies heavily on techniques of Riemannian geometry on infinite dimensional manifolds. Our purpose is to present