Instabilities in the Mean Field Limit

Abstract

Consider a system of N particles interacting through Newton’s second law with Coulomb interaction potential in one spatial dimension or a $\mathcal {C}^2$ smooth potential in any dimension. We prove that in the mean field limit $N \rightarrow + \infty$, the N particles system displays instabilities in times of order $\log N$, for some configurations approximately distributed according to unstable homogeneous equilibria.

Appendix: Linear Estimates

In this section, we derive estimates on the semigroup $e^{Lt}$, with the linearized operator L defined by

$$\begin{aligned} L f : = - v \cdot \nabla _x f - E \cdot \nabla _v f_\infty , \quad E (x) = - \iint _{\mathbb {T}^d\times \mathbb {R}^d} \nabla \Phi (x-y) f(y,v)\; dy dv,\end{aligned}$$

(4.8)

for a fixed homogenous profile $f_\infty = f_\infty (v)$. The linear problem has been studied for the Coulomb potential; for instance, [6, 13] for one dimension and [12] for higher dimensions. The proof presented in [12] applies to the case of smooth potentials, and we shall reproduce it below for the sake of completeness. The estimates are sharp in terms of the growth in time; however we allow losses of derivatives and weights, but these losses are overcome in the scheme used in Proposition 3.1.

Lemma 4.2

(Sharp semigroup bounds) Let $\Phi$ be a $\mathcal {C}^2$ smooth potential and let $f_\infty (v)$ be a smooth unstable equilibrium of L which decays sufficiently fast as $v \rightarrow \infty$, let $\lambda _0$ be a maximal unstable eigenvalue. Let $n\ge 0, m > \frac{d}{2} +1,$ and h be in the $\langle v \rangle ^{m+2}$-weighted Sobolev spaces $H^{n+2}$. Then, $f = e^{Lt} h$ is well-defined as the solution of the linear problem $(\partial _t - L) f =0$ with the initial data h. Furthermore, there holds

$$\begin{aligned} \Vert \langle v \rangle ^m e^{Lt} h \Vert _{H^{n} } \le C_\beta e^{(\mathfrak {R}\lambda _0+\beta ) t} \big \Vert \langle v \rangle ^{m+2} h\big \Vert _{H^{n+2} }, \quad \forall t\ge 0, \quad \forall \beta >0, \end{aligned}$$

(4.9)

for some constant $C_\beta$ depending on $f_\infty$ and $\beta$.

Proof

Let $n \ge 0$, $m_0 > \frac{d}{2}$, and $m\ge m_0 + 1$. Consider the resolvent equation:

$$\begin{aligned} (\lambda - L)f = h, \quad \, \mathfrak {R}\lambda \ge 0. \end{aligned}$$

Standard $L^2$ energy estimates produce the bound

$$\begin{aligned} \mathfrak {R}\lambda \Vert \langle v\rangle ^m f\Vert _{L^2}\le & {} \Vert \langle v\rangle ^m \nabla _v f_\infty \Vert _{L^2_v } \Vert E\Vert _{L_x^\infty } + \Vert \langle v\rangle ^m h\Vert _{L^2}\\\le & {} C_0 \Vert \langle v\rangle ^m \nabla _v f_\infty \Vert _{L^2_v} \Vert \langle v \rangle ^{m_0} f\Vert _{L^2} + \Vert \langle v\rangle ^m h\Vert _{L^2}, \end{aligned}$$

in which we have used the estimate (3.5) on E in terms of f. Recalling $m\ge m_0+1$, we thus deduce the following weighted $L^2$ resolvent bound

$$\begin{aligned} \Vert \langle v\rangle ^m (\lambda - L)^{-1} h\Vert _{L^2} \le \frac{1}{\mathfrak {R}\lambda - \gamma _{0,m} } \Vert \langle v\rangle ^m h\Vert _{L^2} \end{aligned}$$

for all $\mathfrak {R}\lambda > \gamma _{0,m}:= C_0 \Vert \langle v\rangle ^m \nabla _v f_\infty \Vert _{L^2_v}$. Similarly, estimates for higher order derivatives are obtained, since we observe that we have for $\alpha , \beta \in \mathbb {N}^d$,

$$\begin{aligned} \lambda \partial _v^\beta \partial _x^\alpha f + v \cdot \nabla _x \partial _v^\beta \partial _x^\alpha f - \nabla _v \partial _v^\beta f_\infty \cdot \partial _x^\alpha E + [\partial _v^\beta , v\cdot \nabla _x] \partial _x^\alpha f&= \partial _v^\beta \partial _x^\alpha h \end{aligned}$$

in which $[\partial _v^\beta , v\cdot \nabla _x] = \partial _v^\beta (v\cdot \nabla _x ) - v\cdot \nabla _x \partial _v^\beta$. Combining with the estimate (3.5), we get

$$\begin{aligned} \mathfrak {R}\lambda \Vert \langle v\rangle ^m \partial _x^\alpha f \Vert _{L^2}\le & {} \Vert \langle v\rangle ^m\nabla _v f_\infty \cdot \partial _x^\alpha E \Vert _{L^2} + \Vert \langle v\rangle ^m \partial _x^\alpha h \Vert _{L^2} \\\le & {} C_0 \Vert \langle v\rangle ^m\nabla _v f_\infty \Vert _{L^2_v} \Vert \langle v\rangle ^{m_0} f\Vert _{H^{|\alpha |-1 }_x L^2_v} + \Vert \langle v\rangle ^m \partial _x^\alpha h \Vert _{L^2}. \end{aligned}$$

Likewise, we have

$$\begin{aligned} \mathfrak {R}\lambda \Vert \langle v\rangle ^m \partial _v^\beta \partial _x^\alpha f \Vert _{L^2}\le & {} \Vert \langle v\rangle ^m\nabla _v \partial _v^\beta f_\infty \cdot \partial _x^\alpha E\Vert _{L^2} + \Vert \langle v\rangle ^m[\partial _v^\beta , v\cdot \nabla _x] \partial _x^\alpha f\Vert _{L^2}\\&\quad +\, \Vert \langle v\rangle ^m \partial _v^\beta \partial _x^\alpha h \Vert _{L^2} \\\le & {} C_0 \Vert \langle v\rangle ^m\nabla _v \partial _v^\beta f_\infty \Vert _{L^2_v} \Vert \langle v\rangle ^{m_0} f\Vert _{H^{|\alpha |-1}_x L^2_v} \\&+\,C_0 \Vert \langle v\rangle ^m f \Vert _{ H_x^{|\alpha |+1} H_v^{|\beta |-1}} + \Vert \langle v\rangle ^m \partial _v^\beta \partial _x^\alpha h\Vert _{L^2}. \end{aligned}$$

We therefore obtain by induction

$$\begin{aligned} \mathfrak {R}\lambda \Vert \langle v\rangle ^m \partial _v^\beta \partial _x^\alpha f \Vert _{L^2} \le C'_0 \Vert \langle v\rangle ^m f\Vert _{H^{|\alpha |+|\beta |}_x L^2_v} + C'_0 \Vert \langle v\rangle ^m h\Vert _{H^{n}} , \end{aligned}$$

for all multi-indices $\alpha , \beta$ such that $|\alpha | + |\beta | \le n$. By induction, this proves that there exists $\gamma _{n,m}, C_{n,m}>0$ such that

$$\begin{aligned} \mathfrak {R}\lambda \Vert \langle v \rangle ^m f \Vert _{H^{n}}&\le \gamma _{n,m} \Vert \langle v \rangle ^{m_0} f\Vert _{L^2} + C_{n,m} \Vert \langle v\rangle ^m h\Vert _{H^{n}} , \end{aligned}$$

(4.10)

for all $n\ge 0$ and $\mathfrak {R}\lambda >0$. In particular, this proves that

$$\begin{aligned} \Vert \langle v \rangle ^m (\lambda - L)^{-1} h\Vert _{H^{n}} \le \frac{C_{n,m}}{\mathfrak {R}\lambda - \gamma _{n,m}} \Vert \langle v \rangle ^m h\Vert _{H^{n}}, \end{aligned}$$

for some positive constant $C_{\gamma _{n,m}}$, and for all $\lambda \in \mathbb {C}$ so that $\mathfrak {R}\lambda >\gamma _{n,m}$. The classical Hille–Yosida theorem then asserts that L generates a continuous semigroup $e^{Lt}$ on the Banach space $H^{n}$ with weights; see, for instance, [22] or [26, Appendix A]. We have furthermore the representation formula

$$\begin{aligned} e^{Lt} h= \text {P.V. } \frac{1}{2\pi i} \int _{\gamma - i\infty }^{\gamma + i \infty } e^{\lambda t} (\lambda - L)^{-1} h \; d\lambda \end{aligned}$$

(4.11)

for any $\gamma > \gamma _{n,m}$, where $\text {P.V. }$ denotes the Cauchy principal value.

Next, by assumption, $\lambda _0$ is an unstable eigenvalue with maximal real part, and the resolvent operator $(\lambda - L)^{-1}$ is in fact a well-defined and bounded operator on the weighted space $H^{n}$ for all $\lambda$ so that $\mathfrak {R}\lambda > \mathfrak {R}\lambda _0$. Let $\beta >0$. By Cauchy’s theorem, we can take $\gamma = \mathfrak {R}\lambda _0 + \beta$ in the representation (4.11). Since the resolvent operator is bounded, we obtain at once

$$\begin{aligned} \Big \Vert \frac{\langle v \rangle ^m}{2\pi i} \int _{\gamma - iM}^{\gamma + i M} e^{\lambda t} (\lambda - L)^{-1} h \; d\lambda \Big \Vert _{H^{n}} \le C_{\beta , M} e^{(\mathfrak {R}\lambda _0 + \beta ) t} \Vert \langle v \rangle ^m h\Vert _{H^{n}},\end{aligned}$$

(4.12)

for any large but fixed constant M. For what concerns large values of $\mathfrak {I}\lambda$, we observe directly from the equation $\lambda f = L f + h$, that one has

$$\begin{aligned} |\lambda | \Vert \langle v \rangle ^m f \Vert _{H^{n}} \le \Vert \langle v \rangle ^m (L f + h) \Vert _{H^{n}} \le C ( \Vert \langle v \rangle ^{m+1} \nabla _xf\Vert _{H^{n}} + \Vert E\Vert _{W^{n,\infty }_x})+ \Vert \langle v \rangle ^m h\Vert _{H^{n}}. \end{aligned}$$

Using (4.10), we get, for some $C'_{n+1,m+1}>0$,

$$\begin{aligned} |\lambda | \Vert \langle v \rangle ^m (\lambda - L)^{-1} h\Vert _{H^{n}} \le {C'_{n+1,m+1}} \Vert \langle v \rangle ^{m_0} f \Vert _{L^2} + C'_{n+1,m+1} \Vert \langle v \rangle ^{m+1}h \Vert _{H^{n+1}}. \end{aligned}$$

We take $\mathfrak {R}\lambda = \gamma$, and take

$$\begin{aligned} |\mathfrak {I}\lambda |>\frac{2}{3}C'_{n+1,m+1}. \end{aligned}$$

We end up with

$$\begin{aligned} \Vert \langle v \rangle ^m (\lambda - L)^{-1} h\Vert _{H^{n}} \le \frac{C_\beta }{|\mathfrak {I}\lambda |} \Vert \langle v \rangle ^{m+1} h\Vert _{H^{n+1}}.\end{aligned}$$

(4.13)

Finally, in order to bound the integral for large $|\mathfrak {I}\lambda |$, we write

$$\begin{aligned} (\lambda - L)^{-1} h = \frac{1}{\lambda }(\lambda - L)^{-1} Lh + \frac{h}{\lambda }. \end{aligned}$$

As a consequence, for $\gamma = \mathfrak {R}\lambda _0 + \beta$, we get

$$\begin{aligned}&\text {P.V. } \frac{1}{2\pi i} \int _{\{|\mathfrak {I}\lambda |\ge M\}}e^{\lambda t} (\lambda - L)^{-1} h \; d\lambda \\&\quad = \text {P.V. }\frac{1}{2\pi i} \int _{\{|\mathfrak {I}\lambda |\ge M\}} e^{\lambda t} (\lambda - L)^{-1} \frac{Lh}{\lambda } \; d\lambda + \text {P.V. } \frac{1}{2\pi i} \Big [ \int _{\gamma - i\infty }^{\gamma + i \infty } - \int _{\{|\mathfrak {I}\lambda |< M\}} \Big ] e^{\lambda t} \frac{h}{\lambda } \; d\lambda \end{aligned}$$

in which the second integral on the right-hand side is equal to h, whereas the last integral is bounded by $C_0e^{\gamma t}h$. We consider $M\ge \frac{2}{3}C'_{n+1,m+1}$ so that the bound (4.13) holds. We deduce

$$\begin{aligned} \Big \Vert \langle v \rangle ^m \int _{\{|\mathfrak {I}\lambda |\ge M\}} e^{\lambda t} (\lambda - L)^{-1} \frac{Lh}{\lambda } \; d\lambda \Big \Vert _{H^{n}}\le & {} C_{\beta , M} e^{\gamma t} \Vert \langle v \rangle ^{m+1} L h\Vert _{H^{n+1}} \int _{\{|\mathfrak {I}\lambda |\ge M\}} |\mathfrak {I}\lambda |^{-2} \; d\mathfrak {I}\lambda \\\le & {} C_{\beta , M} e^{\gamma t} \Vert \langle v \rangle ^{m+2} h\Vert _{H^{n+2}}. \end{aligned}$$

Combining this estimate with (4.12) and (4.11), we conclude the proof of the lemma. $\square$