Mean Field Limit of Interacting Filaments and Vector Valued Non-linear PDEs

Abstract

Families of N interacting curves are considered, with long range, mean field type, interaction. They generalize models based on classical interacting point particles to models based on curves. In this new set-up, a mean field result is proven, as $N\rightarrow \infty$. The limit PDE is vector valued and, in the limit, each curve interacts with a mean field solution of the PDE. This target is reached by a careful formulation of curves and weak solutions of the PDE which makes use of 1-currents and their topologies. The main results are based on the analysis of a nonlinear Lagrangian-type flow equation. Most of the results are deterministic; as a by-product, when the initial conditions are given by families of independent random curves, we prove a propagation of chaos result. The results are local in time for general interaction kernel, global in time under some additional restriction. Our main motivation is the approximation of 3D-inviscid flow dynamics by the interacting dynamics of a large number of vortex filaments, as observed in certain turbulent fluids; in this respect, the present paper is restricted to smoothed interaction kernels, instead of the true Biot–Savart kernel.

Appendix

In this section, we give some properties of 1-currents that have been defined at the beginning of the paper.

1.1 Completeness of Strong Balls with Respect to the Weak Norm

Lemma 27

If B is a closed ball in $\left( \mathcal {M},\left| \cdot \right| _{\mathcal {M} }\right)$, then $\left( B,d\right)$ is a complete metric space.

Proof

Let $\{\xi _{n}\}_{n\ge 0}$ be a Cauchy sequence in (B, d). This is also a Cauchy sequence in the dual space Lip$_{b}(\mathbb {R}^{d},\mathbb {R} ^{d})^{\prime }$ with the dual operator norm. Hence it converges to some $\xi \in$Lip$_{b}(\mathbb {R}^{d},\mathbb {R}^{d})^{\prime }$. Indeed Lip$_{b}(\mathbb {R}^{d},\mathbb {R}^{d})$ is a Banach space and $\Vert \cdot \Vert$ is the operator norm on his dual, which is complete.

Now we have an operator $\xi$ defined on Lip$_{b}(\mathbb {R}^{d} ,\mathbb {R}^{d})$, we want to extend it to the bigger space $C_{b}\left( \mathbb {R}^{d};\mathbb {R}^{d}\right)$ and to show that this extension is a limit to the sequence $\xi _{n}$ in the norm $\Vert \cdot \Vert$.

Given $\theta \in$Lip$_{b}(\mathbb {R}^{d},\mathbb {R}^{d})$, it holds, for every $n\in \mathbb {N}$,

$$\begin{aligned} |\xi (\theta )|\le |(\xi -\xi _{n})(\theta )|+|\xi _{n}(\theta )|\le \Vert \xi -\xi _{n}\Vert (\Vert \theta \Vert _{\infty }+\text {Lip}(\theta ))+R\Vert \theta \Vert _{\infty } \end{aligned}$$

where R denotes the radius of B. Hence, as $n\rightarrow \infty$, it holds $|\xi (\theta )|\le R\Vert \theta \Vert _{\infty }$. We can thus apply Hahn-Banach theorem to obtain a linear functional $\bar{\xi }$ defined on $C_{b}\left( \mathbb {R}^{d};\mathbb {R}^{d}\right)$ such that $\Vert \bar{\xi }\Vert \le R$ and $\bar{\xi }\equiv \xi$ on Lip$_{b}(\mathbb {R}^{d},\mathbb {R}^{d})^{\prime }$.

It only remains to prove that $\xi _{n}$ converges to $\bar{\xi }$,

$$\begin{aligned}&\Vert \bar{\xi }-\xi _{n}\Vert =\sup \{\bar{\xi }(\theta ) - \xi _{n}(\theta )\;|\;\Vert \theta \Vert _{\infty }+\text {Lip}(\theta )\le 1\}\\&\quad =\sup \{\xi (\theta ) - \xi _{n}(\theta )\;|\;\Vert \theta \Vert _{\infty } +\text {Lip}(\theta )\le 1\}=\Vert \xi -\xi _{n}\Vert \rightarrow 0,\quad \text {as}\;n\rightarrow \infty . \end{aligned}$$

$\square$

1.2 Convolution of 1-Currents and Matrix Valued Operators

If $\xi \in \mathcal {M}$ and $K:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d\times d}$ is a continuous bounded matrix-valued function, then $K*\xi$ is the vector field in $\mathbb {R}^{d}$ with i-component given by

$$\begin{aligned} \left( K*\xi \right) _{i}\left( x\right) =\left( K_{i\cdot }*\xi \right) \left( x\right) :=\xi \left( K_{i\cdot }\left( x-\cdot \right) \right) \end{aligned}$$

(45)

where $K_{i\cdot }\left( z\right)$ is the vector $\left( K_{ij}\left( z\right) \right) _{j=1,...,d}$. We have

$$\begin{aligned} \left| \left( K*\xi \right) \left( x\right) \right| \le \left| \xi \right| _{\mathcal {M}}\Vert K\Vert _{\infty }. \end{aligned}$$

If K, in addition, is also of class $C_{b}^{1}(\mathbb {R}^{d},\mathbb {R} ^{m})$, then

$$\begin{aligned} \left| \left( K*\xi \right) \left( x\right) \right| \le \left\| \xi \right\| \left( \Vert K\Vert _{\infty }+\Vert DK\Vert _{\infty }\right) . \end{aligned}$$

1.3 1-Currents Associated with Curves

One can define the current associated to a curve in the following way. Given a curve $\gamma :\left[ 0,1\right] \rightarrow \mathbb {R}^{d}$ of class $C^{1}$ ($W^{1,1}$ is sufficient), consider the current

$$\begin{aligned} \xi =\int _{0}^{1}\delta \left( \cdot -\gamma \left( \sigma \right) \right) \frac{d\gamma }{d\sigma }\left( \sigma \right) d\sigma \end{aligned}$$

namely the linear functional $\xi :C_{b}\left( \mathbb {R}^{d},\mathbb {R} ^{d}\right) \rightarrow \mathbb {R}$ defined as

$$\begin{aligned} \xi \left( \theta \right) =\int _{0}^{1}\theta \left( \gamma \left( \sigma \right) \right) \cdot \frac{d\gamma }{d\sigma }\left( \sigma \right) d\sigma . \end{aligned}$$

Notice that this is exactly how we defined the empirical measure, which is nothing but the current centered on a family of curves. In this case the push forward can be reformulated in a very specific form. For every $\varphi \in C^{1}\left( \mathbb {R}^{d},\mathbb {R}^{d}\right)$, it is

$$\begin{aligned} \varphi _{\sharp }\xi =\int _{0}^{1}\delta \left( \cdot -\varphi \left( \gamma \left( \sigma \right) \right) \right) D\varphi \left( \gamma \left( \sigma \right) \right) \frac{d\gamma }{d\sigma }\left( \sigma \right) d\sigma . \end{aligned}$$

Remark 28

It is easy to see that the push-forward of $\gamma$ with respect to $\varphi$ is the push-forward of $\varphi \circ \gamma$. This follows from the previous formula and the chain rule, $\frac{d\eta }{d\sigma }\left( \sigma \right) =D\varphi \left( \gamma \left( \sigma \right) \right) \frac{d\gamma }{d\sigma }\left( \sigma \right)$.

1.4 1-Currents Associated with Vector Fields

We have seen that $\varphi _{\sharp }\xi$ has a nice reformulation when $\xi$ is associated to a smooth curve. In Sect. 4 we consider special currents, which are induced by vector fields: we restrict to these particular currents as an intermediate step to prove Lemma .

We briefly describe here how to give a reformulation of $\xi$ and its push-forward when the current is associated to a vector field. Thus, with little abuse of notations, let $\xi : \mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ be an integrable vector field and denote by $\xi$ the associated current defined as

$$\begin{aligned} \xi \left( \theta \right) =\int _{\mathbb {R}^{d}} \theta \left( x\right) \cdot \xi \left( x\right) dx. \end{aligned}$$

Proposition 29

Assume that $\varphi$ is a diffeomorphism of $\mathbb {R}^{d}$ and $\xi$ is a vector field on $\mathbb {R}^{d}$ in $\mathbb {R}^{d}$ of class $L^{1}$. Then $\varphi _{\sharp }\xi$ is the following vector field in $\mathbb {R}^{d}$, of class $L^{1}$:

$$\begin{aligned} \left( \varphi _{\sharp }\xi \right) \left( x\right) =D\varphi \left( \varphi ^{-1}\left( x\right) \right) \xi \left( \varphi ^{-1}\left( x\right) \right) \left| \det D\varphi ^{-1}\left( x\right) \right| . \end{aligned}$$

Proof

By definition we have

$$\begin{aligned} \left( \varphi _{\sharp }\xi \right) \left( \theta \right)&=\xi \left( \varphi _{\sharp }\theta \right) =\int _{\mathbb {R}^{d}} D\varphi \left( x\right) ^{T}\theta \left( \varphi \left( x\right) \right) \cdot \xi \left( x\right) dx\\&=\int _{\mathbb {R}^{d}} \theta \left( \varphi \left( x\right) \right) \cdot D\varphi \left( x\right) \xi \left( x\right) dx\\&= \int _{\mathbb {R}^{d}} \theta \left( y\right) \cdot D\varphi \left( \varphi ^{-1}\left( y\right) \right) \xi \left( \varphi ^{-1}\left( y\right) \right) \left| \det D\varphi ^{-1}\left( y\right) \right| dy. \end{aligned}$$

In the last inequality we used the change of variable $y = \varphi (x)$. $\square$

Remark 30

In the case when $\varphi$ is a diffeomorphismus, if $\xi$ has compact support, then the pursh-forward $\varphi _{\sharp }\xi$ has compact support.