2022/01/26 by Guo, Dengjun
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2201.11158
We consider the two-dimensional incompressible Euler equation \begincases ∂t ω+ u⋅ ∇ ω=0
ω(0,x)=ω0(x). \endcases We are interested in the cases when the initial vorticity has the form ω0=ω0,ε+ω0p,ε, where ω0,ε is concentrated near M disjoint points pm0 and ω0p,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)=ωε(x,t)+ωp,ε(x,t), where ωε(x,t) remains concentrated near M points pm(t) and ωp,ε(x,t) remains small for t ∈ [0,T]. Second, we give a quantitative description when the initial vorticity has the form ω0(x)=∑m=1M (γm)/(ε2)η(\fracx-pm0ε), where we do not assume η to have compact support. Finally, we prove that if pm(t) remains separated for all t∈[0,+∞), then ω(x,t) remains concentrated near M points at least for t ≤ c0 |log Aε|, where Aε is small and converges to 0 as ε→ 0.