Stability of Isentropic Navier–Stokes Shocks in the High-Mach Number Limit

Abstract

By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier-Stokes equations with γ-law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ 1 (proved analytically), and (ii) for M ≥ 2, 500, γ ∈ [1, 2.5] (demonstrated numerically). This builds on and completes earlier studies by Matsumura-Nishihara and Barker-Humpherys-Rudd-Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, independent of shock amplitude, of viscous shock waves for γ-law gas dynamics in the range γ ∈ [1, 2.5]. Other γ-values may be treated similarly, but have not been checked numerically. The main idea is to establish convergence of the Evans function in the high-Mach number limit to that of a pressureless, or "infinitely compressible", gas with additional upstream boundary condition determined by a boundarylayer analysis. Recall that low-Mach number behavior is incompressible.1 The result of [21] is obtained by energy estimates combining the techniques of [34] with those of [14,15]; a similar approach has been used in [30] to obtain smallamplitude zero-mass stability of Boltzmann shocks. See [37,12] for an alternative approach based on asymptotic ODE methods. Stability of isentropic viscous shock profiles 3[34], in fact yields stability of small-amplitude shocks of general symmetric hyperbolic-parabolic systems, largely settling the problem of small-amplitude shock stability for continuum mechanical systems.However, there remains the interesting question of large-amplitude stability. The main result in this direction, following a general strategy proposed in [47], is a "refined Lyapunov theorem" 2 established by Mascia and Zumbrun [32,46] for general symmetric hyperbolicparabolic systems, stating that linearized and nonlinear L 1 ∩ H 3 → L 1 ∩ H 3 orbital stability (the standard notions of stability) are equivalent to spectral stability, or nonexistence of nonstable (nonnegative real part) eigenvalues of the linearized operator L about the wave, other than the single zero eigenvalue arising through translational invariance of the underlying equations.This reduces the problem of large-amplitude stability to the study of the associated eigenvalue equation (L−λ)u = 0, a standard analytically and numerically well-posed (boundary value) problem in ODE, which can be attacked by the large body of techniques developed for asymptotic, exact, and numerical study of ODE. In particular, there exist well-developed and efficient numerical algorithms to determine the number of unstable roots for any specific linearized operator L, independent of its origins in the PDE setting; see, e.g., [9,10,11,8,22] and references therein. In this sense, the problem of determining stability of any single wave is satisfactorily resolved, or, for that matter, of any compact family of waves. To determine stability of a family of waves across an unbounded parameter regime, how...

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

“…We note in particular the proof by Mascia-Zumbrun and Humpherys-Zumbrun [62,39] for the first time of small-amplitude (one-dimensional) stability of ordinary gas-dynamical and Laxtype magnetohydrodynamic Navier-Stokes shocks with general equation of state, and the proof by Mascia-Zumbrun and Raoofi-Zumbrun [63,67] of nonlinear (one-dimensional) stability of large-amplitude shock solutions of arbitrary type for a class of systems generalizing the Kawashima class [44,45], including gas dynamics, viscoelasticity, and magnetohydrodynamics (MHD), assuming a numerically verifiable Evans-function condition encoding spectral stability in an appropriate sense; that is, the Evans-function condition accounts for the lack of spectral gap/accumulating essential spectrum at the origin that is an fundamental feature of the shock stability problem. 3 Finally, we note the analytical/numerical studies in [41] and [38], Humpherys-Lafitte-Zumbrun and Humpherys-Lyng-Zumbrun, respectively, verifying spectral and nonlinear stability of arbitrary amplitude polytropic ideal gas shock layers for the isentropic and non-isentropic Navier-Stokes equations with gas constant γ ∈ [1. 2,3].…”

“…2 The study of stability of viscous shock layers, was initiated at the one-dimensional scalar level by Hopf [34] and Il'in-Oleȋnik [42]. For one-dimensional systems, it was begun in the 1980's by Kawashima-Matsumura, Kawashima-Matsumua-Nishihara, Liu, and Goodman [46,47,50,26,27], and essentially concluded in [73,51,24,82,61,62,63,39,41,38,37,67]. We note in particular the proof by Mascia-Zumbrun and Humpherys-Zumbrun [62,39] for the first time of small-amplitude (one-dimensional) stability of ordinary gas-dynamical and Laxtype magnetohydrodynamic Navier-Stokes shocks with general equation of state, and the proof by Mascia-Zumbrun and Raoofi-Zumbrun [63,67] of nonlinear (one-dimensional) stability of large-amplitude shock solutions of arbitrary type for a class of systems generalizing the Kawashima class [44,45], including gas dynamics, viscoelasticity, and magnetohydrodynamics (MHD), assuming a numerically verifiable Evans-function condition encoding spectral stability in an appropriate sense; that is, the Evans-function condition accounts for the lack of spectral gap/accumulating essential spectrum at the origin that is an fundamental feature of the shock stability problem.…”

“…Equivalently, p(ρ, S) = ae S/cv ρ γ , a = constant, where S is thermodynamical entropy, or p(ρ) = aρ γ in the isentropic approximation; see [72,9,41].…”

“…We remark that there are important pre-processing steps built into [5], such as making an analytic choice of initial conditions via Kato's method [43]; see [9,41,38,6] for discussions on these details. Because of the excessive winding in the case where the shock is strong (u + near 1/4 for monatomic and u + near 1/6 for diatomic) andξ is small, we also compute the Evans function using the no-radial option where we throw out the contribution from the scalar ODE (8.4b) 31 that us used to restore analyticity.…”

Pointwise Green Function Bounds and Long-Time Stability of Large-Amplitude Noncharacteristic Boundary Layers

“…The Evans condition is readily checkable numerically, and in some cases analytically; see [Br1,Br2,BrZ,BDG,HuZ,BHRZ,HLZ]. In particular, stability of small-amplitude uniformly noncharacteristic boundary layers has been shown for general hyperbolic-parabolic systems in multi-dimensions in [GMWZ1] using elementary Evans function arguments (convergence to the constant layer).…”