2D Euler Equations

Vorticity formulation

• $\omega$ is the scalar vorticity field $\omega=\nabla\times u$
• $\psi$ is the stream function
• The spatial domain is $\mathbb{R}^2$
• Global well posedness for $\omega_0\in L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$

Point-vortex Equations

• $x_i$ are the positions of the vortices
• $\Gamma_i$ are the intensities of the vortices
• Well-posed for almost any initial configuration, w.r.t. the Lebesgue measure [Marchioro&Pulvirenti, 1994]
• The initial positions $x_i(0)$ for which the vortices collapse is dense in $\mathbb{R}^{2N}$, for $N>2$

for $i=1,2,\dots,N$

Point-vortex as solutions to the 2D Euler Equations

• Solves the Euler equations in the sense of distributions if the self-interaction terms are neglected [Schochet, 1996]
• Point-vortex equations can be obtained as limit of smooth solutions to the Euler equations
• Point-vortex solutions for a finite dimensional coadjoint orbit in $\Xi_{vol}^*$ [Marsden&Weinstein, 1983]

Empirical measure

Continuation after collapse

Deterministic [Godota&Sakajo, 2016]

for $i=1,2,\dots,N$

where, for any $\varepsilon>0$, $K_\varepsilon = \varrho^\varepsilon\ast K$, $K(x) = \frac{x^\perp}{|x|^2}$,

for $\varrho:\mathbb{R}\rightarrow\mathbb{R}$ convolution kernel $\varrho^\varepsilon(x) = \frac{1}{\varepsilon^2}\varrho(\frac{x}{\varepsilon})$ 

Given $x_1(0), x_2(0), x_3(0)$ collapsing intial configuration, for $\varepsilon\rightarrow 0$

• $x_i^\varepsilon(t)\rightarrow x_i(t)$ for $t\in[0,t_c]$
• $x_i^\varepsilon(t)\rightarrow \overline{x}_i(2t_c-t)$ for $t\in[t_c,2t_c]$

where $\overline{x}_i=(x_{i,1},-x_{i,2})$

Continuation after collapse

Stochastic [Flandoli&Gubinelli&Priola, 2011]

for $i=1,2,\dots,N$

where, for any $\varepsilon>0$, $B^i$ are i.i.d. 2D Brownian motions. Implies pathwise unique strong solution for $t\geq 0$.

Given $x_1(0)^\varepsilon, x_2(0)^\varepsilon, x_3(0)^\varepsilon$ collapsing intial configuration, for $\varepsilon\rightarrow 0$

• $Law(x_1^\varepsilon,x_2^\varepsilon,x_3^\varepsilon)\Rightarrow Law(x_1,x_2,x_3)$, for $\varepsilon\rightarrow 0$ in $t\in[0,2t_c]$, up to subsequences 
• Pathwise convergence before $t_c$ but not after

OPEN QUESTION: What is the limit $Law(x_1,x_2,x_3)$?

First integrals in $\mathbb{R}^2$

The Point-vortex equations in $\mathbb{R}^2$ admit the following first integrals

• Energy: $H=-\frac{1}{2\pi}\sum_{i\neq j} \Gamma_i\Gamma_j \log|x_i-x_j|$
• Barycenter: $M=\sum_{i\neq j} \Gamma_ix_i$
• Moment of intertia: $I =\sum_{i\neq j} \Gamma_i\Gamma_j|x_i-x_j|^2$

For N=3, the point-vortex equations exhibit (self similar) collapse in finite time if and only if

• $\sum_{i\neq j} \Gamma_i\Gamma_j = 0$
• $I=\sum_{i\neq j} \Gamma_i\Gamma_j|x_i-x_j|^2 = 0$

Reduced equations

• For N=3, given a collapsing initial configuration with $M=0$, by invariance under rotation and homotheties of the collapsing condition, we can study $Rx_1$, such that
  $Rx_3=(1,0)$ and $\Gamma_2Rx_2=-\Gamma_3Rx_3 -\Gamma_1Rx_1$, for a suitable matrix $R\in O(2)^+$
   
• Let $Z:=Rx_1$We then have $H=H(Z)$ and $I = I(Z)$
• We study how $H$ and $I$ change after the collapse for $\varepsilon\rightarrow 0$
• The set $\lbrace Z\in\mathbb{R}^2:H(Z)=H_0\cap I(Z)=I_0\rbrace$ determines the possible configurations

Numerical simulations

Trajectories $x^\varepsilon_1,x^\varepsilon_2,x^\varepsilon_3$

Evolution of $Z(t)$ purple, green $I=I_0$, black $H=H_0$

$\varepsilon = 10^{-10}$

Rmk: $H$ and $I$ are local martingales

Numerical scheme

• Adaptive symplectic integrator (midpoint scheme)
• Stochastic midpoint [Burrage&Burrage, 2014], two evaluations of the Wiener process 

where the time-step $h:=h(n)$ is such that $h(n)\sigma(x^n_1,x^n_2,x^n_3)^\alpha=h(0)\sigma(x^0_1,x^0_2,x^0_3)^\alpha$, for $\alpha=1.3$ and $\sigma(x_1,x_2,x_3)=\frac{1}{\frac{1}{\|x_1-x_2\|^2}+\frac{1}{\|x_2-x_3\|^2}+\frac{1}{\|x_1-x_3\|^2}}$

Statistics for the limiting distribution

• Evidences of non-uniform distribution of the reopening angle $\theta$, defined as the angle between $x_2^\varepsilon$ and the $x-$axis at $t=2t_c$, for $\varepsilon\rightarrow 0$:
  - strong fluctuations of CDF for different $\varepsilon$ may indicate non-unique limit distribution;
  - p-values$<<10^{-9}$, for $\varepsilon=10^{-10}$, tested on the uniform distribution hypotesis for $\theta$
   
• Evidences of equal probability in keeping or changing configuration after collapse (4 intersections of $H$ and $I$), p-value=0.295, for $\varepsilon=10^{-10}$

Empirical c.d.f. (solid blue) of the angle of $x_2^\varepsilon(2t_c)$ for $10^4$ samples, $\varepsilon=10^{-10}$ . Dotted blue curves are confidence bounds ($95\%$) for the e.c.d.f. and the solid red line is the c.d.f. of the uniform distribution on $[0,2\pi]$.

Empirical CDF

Empirical c.d.f. (solid blue) of the angle of $x_2^\varepsilon(2t_c)$ for $10^4$ samples, $\varepsilon=10^{-10}$ . Dotted blue curves are confidence bounds ($95\%$) for the e.c.d.f. and the solid red line is the c.d.f. of the uniform distribution on $[0,2\pi]$. Samples being divided (right and left plots) according to the similarity class of the PV position triangle during burst.

Empirical CDF