摘要
The asymptotic expansions are studied for the vorticity $\{ \omega ^ \in (t,x)\} $ to 2D incompressible Euler equations with-initial vorticity $\omega _0^ \in (x) = \omega _0 (x) + \varepsilon \omega _0^1 \left( {x,\frac{{\varphi _0 (x)}}{\varepsilon }} \right)$ , where ?0(x) satisfies |d ?0(x)|≠0 on the support of $\omega _0^1 \left( { \cdot ,\theta } \right),\theta \in {\rm T}$ and $\omega _0 \left( x \right)(resp. \omega _0^1 (x,\theta ))$ is sufficiently smooth and with compact support in ?2 (resp. ?2×T) The limit,v(t,x), of the corresponding velocity fields {v ?(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω0(x). Moreover, $\omega ^ \in (t,x) = \omega (t,x) + \varepsilon \omega ^1 \left( {t,x,\frac{{\varphi (t,x)}}{\varepsilon }} \right) + o(\varepsilon ){\text{ }}in{\text{ }}C([0,\infty ),{\text{ L}}^p $ (?2)) for all 1?p∞, where $\omega (t,x) = \partial _1 \upsilon _2 (t,x) - \partial _2 \upsilon _1 \left( {t,x} \right),\omega ^1 (t,x,\theta )$ and ?(t,x) satisfy some modulation equation and eikonal equation, respectively.
The asymptotic expansions are studied for the vorticity {ω ε(t,x)} to 2D incompressible Euler equations with initial vorticity ω ε 0(x)=ω 0(x)+εω 1 0x,φ 0(x)ε , where φ 0(x) satisfies d φ 0(x)≠0 on the support of ω 1 0(·,θ),θ∈T , and ω 0(x) (resp. ω 1 0(x,θ)) is sufficiently smooth and with compact support in R 2 (resp. R 2× T ). The limit, v(t,x) , of the corresponding velocity fields v ε(t,x) is obtained, which is the unique solution of (E) with initial vorticity ω 0(x) . Moreover, ω ε(t,x)=ω(t,x)+εω 1t,x,φ(t,x)ε+o(ε) in C([0,∞), L p([KX1] R 2)) for all 1≤p<∞ , where ω(t,x)= 1v 2(t,x)- 2v 1(t,x),ω 1(t,x,θ) and φ(t,x) satisfy some modulation equation and eikonal equation, respectively.
