Particles interacting with a vibrating medium: existence of solutions and convergence to the Vlasov-Poisson system (Q2836024)

The authors wish to analyze several aspects of the Vlasov wave system NEWLINE\[NEWLINE \partial_tf+v\cdot\nabla_xf-\nabla_x(V+\Phi)\cdot\nabla_vf=0,\quad t\geq 0,\quad \mathbb{R}^d\quad v\in\mathbb{R}^d, NEWLINE\]NEWLINE here \(V\) stands for the external potential, while \(\Phi\) is the self-consistent potential describing the interaction with the environment and is defined by the convolution formula NEWLINE\[NEWLINE \Phi(t,x)=\int_{\mathbb{R}^d\times\mathbb{R}^n} \Psi(t,z,y)\sigma_1(x-z)\sigma_2(y)dydz,\quad t\geq 0,\quad \mathbb{R}^d, NEWLINE\]NEWLINE where the vibrating field \(\Psi\) is driven by the following wave equation: NEWLINE\[NEWLINE (\partial^2_{tt}\Psi-c^2\Delta_y\Psi)(t,x,y)=-\sigma_2(y)\int_{\mathbb{R}^d} \sigma_1(x-z)\rho(t,z)dz,\quad t\geq 0,\quad x\in\mathbb{R}^d,\quad y\in\mathbb{R}^n ,NEWLINE\]NEWLINE NEWLINE\[NEWLINE \rho(t,x)=\int_{\mathbb{R}^d}f(t,x,v)dv. NEWLINE\]NEWLINE The system is completed by initial data NEWLINE\[NEWLINE f(0,x,v)=f_0(x,v),\quad \Psi(0,x,v))=\Psi_0(x,y),\quad \partial_t\Psi(0,x,y)=\Psi_1(x,y). NEWLINE\]NEWLINE The authors establish the existence of weak solutions for a wide class of initial data and external forces.