Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity, where ϕ0(x) satisfies |d ϕ0(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ2 (resp. ℝ2×T) The limit,v(t,x), of the corresponding velocity fields {vɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω0(x). Moreover,(ℤ2)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.