[Paper Review] Slow motion for compressible isentropic Navier--Stokes equations

Corrado Mascia, Marta Strani|arXiv (Cornell University)|Mar 22, 2013

Navier-Stokes equation solutions11 references4 citations

TL;DR

This paper derives a differential equation governing the slow motion of a viscous transition layer in compressible isentropic Navier--Stokes flows on a bounded interval, using a manifold of approximate steady states and spectral projection. The key result shows the layer's speed is exponentially small in viscosity ε, quantifying boundary-induced drift in shock-like solutions.

ABSTRACT

We consider the compressible Navier-Stokes equations for isentropic dynamics with real viscosity on a bounded interval. In the case of boundary data defining an admissible shock wave for the corresponding unviscous hyperbolic system, we determine a scalar differential equation describing the motion of the internal transition layer. In particular, for small viscosity, the velocity of the motion is exponentially small. The approach is based on the construction of a one-parameter manifold of approximate solutions and on an appropriate projection of the evolution of the complete Navier-Stokes system towards such manifold.

Motivation & Objective

• To describe the slow motion of internal transition layers in compressible isentropic Navier--Stokes equations on a bounded interval.
• To quantify the effect of boundary conditions on the dynamics of viscous shock profiles when viscosity ε is small.
• To derive a scalar ordinary differential equation for the position ξ(t) of the transition layer using a 1-parameter manifold of approximate solutions.
• To establish that the layer motion is exponentially slow in ε, consistent with metastable behavior.

Proposed method

• Construct a 1-parameter manifold of approximate steady states W(·,ξ) by matching exact solutions in (−ℓ,ξ) and (ξ,ℓ) with continuity conditions at x=ξ.
• Linearize the Navier--Stokes system around each W(·,ξ) to define the operator 𝒪_ξ, and assume its first eigenvalue λ₁(ξ) is real and simple.
• Use spectral projection onto the first eigenfunction (φ(·,ξ),ψ(·,ξ)) of the adjoint operator 𝒪_ξ* to derive the evolution equation for ξ(t).
• Derive an explicit expression for dξ/dt involving κ₊(ξ)−κ₋(ξ), which is approximated via asymptotic integration of the defining integral equations for small ε.
• Approximate F(u,vₐ) and ∂ᵤG(u,vₐ) near u₊ and u₋ to obtain exponential decay/growth terms in the expression for dξ/dt.
• Obtain the final equation (37) for dξ/dt as a sum of two exponentially small terms, each modulated by ξ-dependent weights and the first eigenvalue λ₁(ξ).

Experimental results

Research questions

• RQ1 How does the boundary data induce a slow drift of the internal transition layer in the viscous compressible Navier--Stokes system?
• RQ2 What is the precise asymptotic form of the layer motion speed as viscosity ε→0?
• RQ3 Can the dynamics of the transition layer be described by a scalar ODE on a manifold of approximate solutions?
• RQ4 What determines the sign and magnitude of the first eigenvalue λ₁(ξ) of the linearized operator, and how does it affect stability?
• RQ5 To what extent can this formal approach be extended to non-isentropic or half-line settings?

Key findings

• The motion of the transition layer is governed by a scalar ODE (37) that describes its slow drift due to boundary effects.
• The layer velocity dξ/dt is exponentially small in ε, with the leading-order behavior dominated by two terms decaying exponentially as e^{−C/ε}.
• The dominant contribution to dξ/dt comes from the exponential decay term near the boundary where the profile is less steep, with the rate determined by ∂ᵤF⁺/∂ᵤG⁺.
• The expression for dξ/dt includes a prefactor involving (u₊−uₐ) and (uₐ−u₋), reflecting the asymmetry in the shock structure.
• The first eigenvalue λ₁(ξ) of the linearized operator 𝒪_ξ controls the amplitude of the motion, and its sign is expected to be negative, implying stability.
• The derived equation is formally valid and flexible, suggesting potential extension to other hyperbolic–parabolic systems, including non-isentropic Navier--Stokes.