One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics

Abstract

gas dynamics and Freistühler and Trakhinin for compressible magnetohydrodynamics, we study by a combination of asymptotic ODE estimates and numerical Evans function computations the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a γ -law gas with γ ∈ [1, 3]. Other γ -values may be treated similarly, but were not checked numerically. Depending on magnetic field strength, these shocks may be of fast Lax, intermediate (overcompressive), or slow Lax type; however, the shock layer is independent of magnetic field, consisting of a purely gas-dynamical profile. In each case, our results indicate stability. Interesting features of the analysis are the need to renormalize the Evans function in order to pass continuously across parameter values where the shock changes type or toward the large-amplitude limit at frequency λ = 0 and the systematic use of winding number computations on Riemann surfaces.

Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

“…The corresponding spectral stability problem has been studied analytically in [22,28,20], yielding various necessary conditions, and by a numerical Evans function investigation in [22]. An interesting direction for further study would be more systematic numerical investigation along the lines of [10,3,12,11,2] in the viscous shock wave case. A second interesting open problem would be to extend the results for planar waves to the case of solutions with multiple periods, as considered in the reaction-diffusion setting in [26,25,27].…”

“…Substituting (3.5) into (3.1) and computing δ y (ξ, x 1 ) = k e 2π ikx 1 δ y (ξ + 2πke 1 ) = k e 2π ikx 1 e −iξ ·y−2π iky 1 = e −iξ ·y δ y 1 (x 1 ) , (3.8) where the second and third equalities follow from the fact that the Fourier transform (either continuous or discrete) of the delta-function is unity, we obtain (3.6) and the fact that φ is supported on [−π , π]. 2 We now state our main result for this section, which utilizes the spectral representation of G I and [G I ξ ] described in Proposition 3.3 to factor the low-frequency Green kernel into a leading order piece (corresponding to translations) plus a faster decaying residual. Underlying this decomposition is the fundamental relation (3.9) which serves as the crux of the low-frequency analysis in the present context as well as that of [23,13].…”

“…Applying Corollary 3.7 with q = 1, d = 1 to representations (4.18)-(4.19), we obtain for any 2 p < ∞ 10 v(·, t) L p (x) C (1 + t) − 1 2 (1 + t) − 1 2 (1−1/p) . (4.25) Now, by the nonlinear damping estimate given in Proposition 4.6 the size of v in H K (R d ) can be controlled by its size in L 2 together with H K estimates on the shift function ψ .…”

Stability of Hydraulic Shock Profiles

“…Moreover, by [MZ2], we obtain the same linearized and nonlinear stability results for smooth profiles of arbitrary amplitude, provided they are spectrally stable in the sense of a standard Evans function condition, and nondegenerate in the sense that hyperbolic characteristics do not coincide along the profile with the speed of the wave. Hence, the smooth nondegenerate case may be treated by existing analysis, reducing to a standard numerical Evans function study of intermediate-amplitude waves, as carried out for example in [BHRZ,BHZ,BLeZ,BLZ,HLyZ1].…”

Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks

“…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.…”