Hydrodynamic limit for compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force
Yunfei Su, Lei Yao
TL;DR
The paper proves a hydrodynamic limit for weak solutions of the compressible Navier-Stokes–Vlasov–Poisson system with a local alignment force on $\mathbb{T}^3$ by employing a relative entropy method. It shows that as $\epsilon\to0$, the kinetic distribution concentrates to a Dirac in velocity, $f^{\epsilon} \to \rho_f\otimes\delta_{\xi=u_f}$, and the fluid moments converge to a smooth limit $(\rho,u,\rho_f,u_f)$ solving a coupled two-phase fluid model with a Poisson interaction $-\Delta\Phi = \rho_f-1$ and a convolution with the interaction kernel $K$. The analysis handles the lack of dissipation in the kinetic equation by exploiting the Poisson coupling and carefully controlling cross terms, establishing a relative entropy inequality and passing to the limit. The result extends the hydrodynamic limit framework to the compressible NS-Vlasov-Poisson system with local alignment, providing convergence rates and convergence in distribution for the kinetic density and momentum.
Abstract
We investigate the hydrodynamic limit of weak solutions to compressible Navier-Stokes-Vlasov-Poisson equations with local alignment force in three-dimensional torus domain. Due to the absence of dissipation terms in particle equation, it is difficult to study this problem. Based on the relative entropy method, it is shown that the global weak solutions of the compressible Navier-Stokes-Vlasov-Poisson equations converge to the smooth solutions of the limiting two-phase fluid model.We obtained that the distribution function $f^ε$ converges to a Dirac distribution in velocity, the fluid density $ρ^ε$ and velocity $u^ε$ converge to $ρ$ and $u$, respectively.