The asymptotic behavior of the smooth solutions of the Euler equations in R2R2

In this paper, we consider the smooth solutions u(x,t), with suitable decay at infinity, of the Euler equations on R2×(0,∞)R2×(0,∞). We assume that the initial vorticity ω0=∇×u0 has a compact support which leads by Marchioro (1994, 1996) to a consequence that the support of the vorticity ω(t)ω(t) for any finite time is a bounded set. Then, we will show that the decay(growth) rate, in Lp(R2)Lp(R2) spaces for 2