Stability of the Two-Dimensional Point Vortices in Euler Flows

Abstract

We consider the two-dimensional incompressible Euler equation

\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t\omega +u·\mathrm{\nabla }\omega =0,\hfill \\ \omega (x,0)={\omega }_0(x).\hfill \end{array}\right)\end{array}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}{\left\{ \begin{array}{ll} \partial _t \omega + u\cdot \nabla \omega =0, \\ \omega (x,0)=\omega _0(x). \end{array}\right. }\end{aligned}$$\end{document}

We are interested in the cases when the initial vorticity has the form

{\omega }_0={\omega }_{0,ϵ}+{\omega }_{0p,ϵ}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _0=\omega _{0,\epsilon }+\omega _{0p,\epsilon }$$\end{document}

, where

{\omega }_{0,ϵ}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _{0,\epsilon }$$\end{document}

is concentrated near M disjoint points

p_m^0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_m^0$$\end{document}

and

{\omega }_{0p,ϵ}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _{0p,\epsilon }$$\end{document}

is a small perturbation term. We prove that for such initial vorticities, the solution

\omega (x,t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega (x,t)$$\end{document}

admits a decomposition

\omega (x,t)={\omega }_{ϵ}(x,t)+{\omega }_{p,ϵ}(x,t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega (x,t)=\omega _{\epsilon }(x,t)+\omega _{p,\epsilon }(x,t)$$\end{document}

, where

{\omega }_{ϵ}(x,t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _{\epsilon }(x,t)$$\end{document}

remains concentrated near M points

p_m(t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_m(t)$$\end{document}

and

{\omega }_{p,ϵ}(x,t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _{p,\epsilon }(x,t)$$\end{document}

remains small for

t\in [0,T]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t \in [0,T]$$\end{document}

. As a consequence of such decomposition, we are able to consider the initial vorticity of the form

{\omega }_0(x)={\sum }_{m=1}^M\frac{{\gamma }_m}{{ϵ}^2}\eta (\frac{x-p_m^0}{ϵ})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega _0(x)=\sum _{m=1}^M \frac{\gamma _m}{\epsilon ^2}\eta (\frac{x-p_m^0}{\epsilon })$$\end{document}

, where we do not assume

\eta

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta $$\end{document}

to have compact support. Finally, we prove that if

p_m(t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_m(t)$$\end{document}

remains separated for all

t\in [0,+\infty )

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t\in [0,+\infty )$$\end{document}

, then

\omega (x,t)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega (x,t)$$\end{document}

remains concentrated near M points at least for

t\le c_0|log{A}_{ϵ}|

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$t \le c_0 |\log A_{\epsilon }|$$\end{document}

, where

A_{ϵ}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_{\epsilon }$$\end{document}

is small and converges to 0 as

ϵ\to 0

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\epsilon \rightarrow 0$$\end{document}

.

Keywords

Point vortices / Vortex interaction

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