Stability of the Two-Dimensional Point Vortices in Euler Flows

Dengjun Guo

Communications in Mathematics and Statistics ›› : 1 -42.

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Communications in Mathematics and Statistics ›› : 1 -42. DOI: 10.1007/s40304-024-00436-z
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Stability of the Two-Dimensional Point Vortices in Euler Flows

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Abstract

We consider the two-dimensional incompressible Euler equation

t ω + u · ω = 0 , ω ( x , 0 ) = ω 0 ( x ) .
We are interested in the cases when the initial vorticity has the form
ω 0 = ω 0 , ϵ + ω 0 p , ϵ
, where
ω 0 , ϵ
is concentrated near M disjoint points
p m 0
and
ω 0 p , ϵ
is a small perturbation term. 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 [ 0 , T ]
. As a consequence of such decomposition, we are able to consider the initial vorticity of the form
ω 0 ( x ) = m = 1 M γ m ϵ 2 η ( 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 [ 0 , + )
, then
ω ( x , t )
remains concentrated near M points at least for
t c 0 | log A ϵ |
, where
A ϵ
is small and converges to 0 as
ϵ 0
.

Keywords

Point vortices / Vortex interaction

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Dengjun Guo. Stability of the Two-Dimensional Point Vortices in Euler Flows. Communications in Mathematics and Statistics 1-42 DOI:10.1007/s40304-024-00436-z

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Funding

NSFC Grant of China(12271497)

National Key Research and Development Program of China(2020YFA0713100)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature

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