[Paper Review] Stability of contact discontinuity for the Navier-Stokes-Poisson system with free boundary

Shuangqian Liu, Haiyan Yin+1 more|arXiv (Cornell University)|Aug 6, 2015

Navier-Stokes equation solutions36 references3 citations

TL;DR

This paper establishes the nonlinear stability of a viscous contact discontinuity in the one-dimensional Navier-Stokes-Poisson system with free boundary under quasineutral conditions. By constructing a viscous contact wave via the quasineutral Euler equations and applying an $L^2$ energy method, the authors prove that small perturbations decay over time, ensuring asymptotic stability of the contact discontinuity even when the electric potential takes distinct constant values at the boundary.

ABSTRACT

This paper is concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann relation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact wave through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system. The analysis is based on the techniques developed in \cite{DL} and an elementary $L^2$ energy method.

Motivation & Objective

• To investigate the nonlinear stability of contact discontinuities in the Navier-Stokes-Poisson system with free boundary.
• To analyze the behavior of solutions under quasineutral assumptions where electron density satisfies a Boltzmann-like relation.
• To establish time-asymptotic stability of a non-trivial viscous contact wave profile under small initial perturbations.
• To allow for distinct constant electric potentials at the free boundary, extending stability results beyond homogeneous boundary conditions.

Proposed method

• Construct a viscous contact wave profile using the quasineutral Euler equations derived from the Navier-Stokes-Poisson system.
• Transform the system into Lagrangian coordinates to fix the moving boundary, simplifying analysis.
• Apply an elementary $L^2$ energy method to derive a priori estimates for the perturbation equations.
• Use integration by parts and cancellation techniques to control nonlinear terms arising from the electric potential and density coupling.
• Leverage the quasineutral assumption to reduce the Poisson equation to a balance between ion density and electron density via $ ho_e( heta)$.
• Employ a perturbation framework where the solution is decomposed into a viscous contact wave and a small remainder, with stability proven via energy decay estimates.

Experimental results

Research questions

• RQ1 Can a viscous contact discontinuity in the Navier-Stokes-Poisson system with free boundary remain stable under small perturbations?
• RQ2 How does the presence of a non-zero, constant electric potential at the free boundary affect the stability of the contact wave?
• RQ3 What role does the quasineutral assumption play in constructing and stabilizing the viscous contact wave profile?
• RQ4 Is the stability result robust when the electron density satisfies a generalized Boltzmann-type relation with negative derivative?
• RQ5 Can the $L^2$ energy method be effectively applied to the Navier-Stokes-Poisson system with free boundary and non-trivial electric potential boundary conditions?

Key findings

• A viscous contact wave is constructed as a solution to the quasineutral Euler equations, serving as the asymptotic profile for the Navier-Stokes-Poisson system.
• The viscous contact discontinuity is proven to be time-asymptotically stable under small initial perturbations in the $L^2$ framework.
• The electric potential is allowed to take distinct constant values at the free boundary and at infinity, which is a generalization over previous results.
• The analysis relies on the quasineutral assumption and the condition $ ho_e'( heta) < 0$, which ensures the necessary coercivity in the energy estimates.
• The $L^2$ energy method successfully controls nonlinear terms through integration by parts and careful estimation of derivatives of the contact wave and perturbation.
• Global existence and large-time behavior of solutions are established, with the perturbation decaying to zero as $t \to \infty$.