论文标题
Euler流中二维点涡流的稳定性
Stability of the two-dimensional point vortices in Euler flows
论文作者
论文摘要
我们考虑二维不可压缩的Euler方程\ [\ begin {case} \partial_tΩ+ u \ cdot \ nabla \ nablaω= 0 \\ω（0，x）=ω_0（x）。 \end{cases}\] We are interested in the cases when the initial vorticity has the form $ω_0=ω_{0,ε}+ω_{0p,ε}$, where $ω_{0,ε}$ is concentrated near $M$ disjoint points $p_m^0$ and $ω_{0p,ε}$ is a small perturbation term. First, we prove that for such initial vorticities, the solution $ω(x,t)$ admits a decomposition $ω(x,t)=ω_ε(x,t)+ω_{p,ε}(x,t)$, where $ω_ε(x,t)$ remains concentrated near $M$ points $p_m(t)$ and $ω_{p,ε}(x,t)$ remains small for $ t \ in [0，t] $。其次，当初始涡度具有$ω_0（x）= \ sum_ {m = 1}^m \ frac {γ_M} {ε^2}η（\ frac {x-p_m^0}ε）$时，我们没有假设$η$。最后，我们证明，如果$ p_m（t）$在[0，+\ infty）$中的所有$ t \中保持分开，则$ω（x，t）$保持集中在$ m $点接近$ m $点附近，至少对于$ t \ le c_0 | \ log loga_ε| $，在$a_ε$的情况下，$a_ε$很小，转换为$ 0 $ as $ as $ as $ as $ as $ us $ ^ f. 0 $ f. 0 $ f.s. f. 0. f. 0. f. 0.
We consider the two-dimensional incompressible Euler equation \[\begin{cases} \partial_t ω+ u\cdot \nabla ω=0 \\ ω(0,x)=ω_0(x). \end{cases}\] We are interested in the cases when the initial vorticity has the form $ω_0=ω_{0,ε}+ω_{0p,ε}$, where $ω_{0,ε}$ is concentrated near $M$ disjoint points $p_m^0$ and $ω_{0p,ε}$ is a small perturbation term. First, we prove that for such initial vorticities, the solution $ω(x,t)$ admits a decomposition $ω(x,t)=ω_ε(x,t)+ω_{p,ε}(x,t)$, where $ω_ε(x,t)$ remains concentrated near $M$ points $p_m(t)$ and $ω_{p,ε}(x,t)$ remains small for $t \in [0,T]$. Second, we give a quantitative description when the initial vorticity has the form $ω_0(x)=\sum_{m=1}^M \frac{γ_m}{ε^2}η(\frac{x-p_m^0}ε)$, where we do not assume $η$ to have compact support. Finally, we prove that if $p_m(t)$ remains separated for all $t\in[0,+\infty)$, then $ω(x,t)$ remains concentrated near $M$ points at least for $t \le c_0 |\log A_ε|$, where $A_ε$ is small and converges to $0$ as $ε\to 0$.
