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Linear stability analysis of the Couette flow for 2D compressible Navier-Stokes-Poisson system

Yurui Lu, Xueke Pu

TL;DR

The article conducts a rigorous linear stability analysis of Couette flow in a 2D compressible Navier-Stokes-Poisson system, separating zero and nonzero Fourier modes to derive sharp decay estimates at high Reynolds numbers. It develops two complementary energy frameworks: one with loss of derivatives (th1) and another without (th2), employing time-dependent Fourier multipliers to capture enhanced dissipation while managing the nonlocal electrostatic coupling. For $k=0$ modes, it proves coercive decay bounds for both ion and electron subsystems, while for $k\neq 0$ modes it establishes dissipative estimates through a carefully crafted energy functional in Fourier space. The results demonstrate transient growth of the irrotational component followed by exponential decay and reveal how viscosity, Mach number, and electric-field nonlocality shape stability, aligning with known enhanced-dissipation phenomena in shear flows and extending them to compressible NSP dynamics. This work provides foundational linear theory that underpins nonlinear stability analyses and plasma-fluid applications in domains with periodic and unbounded directions.

Abstract

In this paper, we study the linear stability of Couette flow for 2D compressible Navier-Stokes-Poisson system at high Reynolds number in the domain $\mathbb{T}\times\mathbb{R}$ with initial perturbation in Sobolev spaces. We establish the upper bounds for the solutions of linearized system near Couette flow. In particular, we show that the irrotational component of the perturbation may have a transient growth, after which it decays exponentially.

Linear stability analysis of the Couette flow for 2D compressible Navier-Stokes-Poisson system

TL;DR

The article conducts a rigorous linear stability analysis of Couette flow in a 2D compressible Navier-Stokes-Poisson system, separating zero and nonzero Fourier modes to derive sharp decay estimates at high Reynolds numbers. It develops two complementary energy frameworks: one with loss of derivatives (th1) and another without (th2), employing time-dependent Fourier multipliers to capture enhanced dissipation while managing the nonlocal electrostatic coupling. For modes, it proves coercive decay bounds for both ion and electron subsystems, while for modes it establishes dissipative estimates through a carefully crafted energy functional in Fourier space. The results demonstrate transient growth of the irrotational component followed by exponential decay and reveal how viscosity, Mach number, and electric-field nonlocality shape stability, aligning with known enhanced-dissipation phenomena in shear flows and extending them to compressible NSP dynamics. This work provides foundational linear theory that underpins nonlinear stability analyses and plasma-fluid applications in domains with periodic and unbounded directions.

Abstract

In this paper, we study the linear stability of Couette flow for 2D compressible Navier-Stokes-Poisson system at high Reynolds number in the domain with initial perturbation in Sobolev spaces. We establish the upper bounds for the solutions of linearized system near Couette flow. In particular, we show that the irrotational component of the perturbation may have a transient growth, after which it decays exponentially.

Paper Structure

This paper contains 10 sections, 9 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Ion case
  3. Electron case
  4. Main results
  5. Analysis of $k=0$ mode
  6. $k=0$ mode of Ion system
  7. $k=0$ mode of Electron system
  8. Analysis of nonzero modes
  9. Dissipation with loss of derivatives
  10. Dissipation without loss of derivatives

Key Result

Theorem 1.1

Suppose $\nu, \lambda\geq 0$ and $M>0$ satisfy $\nu+\lambda\leq1/2$, $M\leq \mathrm{min}\{(\nu+\lambda)^{-1},\lambda^{-\frac{1}{2}},\nu^{-\frac{1}{3}}\}$ and $M^2\leq \mathrm{min} \{ \frac{1}{\nu+\lambda}, \frac{\nu+\lambda}{16\nu }, \left(\frac{\nu+\lambda}{32 \pi \nu } \right)^\frac{1}{2} \}$. Ass

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • ...and 16 more