Discrete-to-continuum convergence of charged particles in 1D with annihilation (Q2675802)

The authors consider a system of charged particles moving on the real line driven by electrostatic interactions. They write the system as \(\frac{dx_{i} }{dt}=\frac{1}{n}\sum_{j=1,j\neq i}^{n}\frac{b_{i}b_{j}}{x_{i}-x_{j}}\), \( t\in (0,T)\), \(i=1,\ldots ,n\). Here \(n\geq 2\) is the number of particles, \( x=(x_{1},\ldots ,x_{n})\) are the particle positions in \(\mathbb{R}\), and \( b=(b_{1},\ldots ,b_{n})\) are the charges of the particles, initially set as +1 or -1. The authors impose an annihilation rule: Each \(b_{i}\) jumps at most once and if \(b_{i}\) jumps at \(t\in \lbrack 0,T]\), then \(t\in S\), a finite subset in \((0,T]\), \(\left\vert b_{i}(t-)\right\vert |=1\) and \( b_{i}(t)=0\). Moreover, for all \((\tau ,y)\in S\times \mathbb{R}\), \( \sum_{i:x_{i}(\tau )=y}\left[ \left[ b_{i}(\tau )\right] \right] =0\), where the bracket \(\left[ \left[ f\right] \right] (t)\) means the difference between the right and left limits of \(f\) at \(t\). The initial conditions \( (x(0),b(0))=(x^\circ,b^\circ)\) are imposed with \(b^\circ \in \{-1,0,+1\}^{n}\) and \((x^\circ,b^\circ)\in \mathcal{Z}_{n}=\{(x,b)\in \mathbb{R}^{n}\times \{-1,0,+1\}^{n}:\) if \( i>j\) and \(b_{i}b_{j}\neq 0\), then \(x_{i}>x_{j}\}\). The authors define a solution to this problem as a map \((x,b):[0,T]\rightarrow \mathcal{Z}_{n}\) such that there exists a finite subset \(S\subset (0,T]\) and for each \(i\in \{1,n\}\), \(x_{i}\in C([0,T])\cap C^{1}([0,T]\setminus S)\), and \(b_{1},\ldots ,b_{n}:[0,T]\rightarrow \{-1,0,1\}\) are right-continuous, the initial conditions and the annihilation rule are satisfied, and the above ordinary differential system is satisfied on \((0,T)\setminus S\). The authors prove that this system of equations is well posed, that is solutions exist, are unique, and depend continuously on the initial data. In addition they are Lipschitz continuous in time with respect to a metric \(d_{M}\) based on the moments \(M_{k}(x)=.\frac{1}{k}\sum_{i=1}^{n}x_{i}^{k}\), \(k=1,\ldots ,n\), through \(d_{M}(x,y)=\left\Vert M(x)-M(y)\right\Vert _{2}\), with \( Mx):=(M_{1}(x),\ldots ,M_{n}(x))^{T}\). They prove a Lipschitz property of this distance. The purpose of the paper is to establish the ''continuum limit'' to this problem as \(n\rightarrow \infty \). The main result proves that if the initial data satisfy \(u^\circ_{n}\rightarrow u^\circ\) uniformly, and \(u^\circ\) is bounded and uniformly continuous, then \(u_{n}\rightarrow u\) locally uniformly in time-space as \(n\rightarrow \infty \), the function \(u\) is continuous and satisfies the Hamilton-Jacobi equation given formally by \( \partial _{t}u=\mathit{I}[u]\left\vert \partial _{x}u\right\vert \), where \(- \mathit{I}\) is a Lévy operator of order 1 defined as \(\mathit{I} [u](x)=-(-\Delta )^{1/2}u(x)\). For the proof, the authors introduce approximations through a small parameter \(\varepsilon >0\) and \(\rho \)-sub- and \(\rho \)-super-solutions with \(\rho >0\), to the approximate problem and they pass to the limit when \(\varepsilon \rightarrow 0\), then when \(n \rightarrow \infty\).