Transverse steady bifurcation of viscous shock solutions of a system of parabolic conservation laws in a strip

Abstract

In this article we derive rigorously a nonlinear, steady, bifurcation through spectral bifurcation (i.e., eigenvalues of the linearized equation crossing the imaginary axis) for a class of hyperbolic-parabolic model in a strip. This is related to "cellular instabilities" occuring in detonation and MHD. Our results extend to multiple dimensions the results of [AS12] on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results of [TZ08a] on transverse Hopf bifurcation, showing that the bifurcating time-periodic solution is in fact a spatially periodic traveling wave. Our technique consists of a Lyapunov-Schmidt type of reduction, which prepares the equations for the application of other bifurcation techniques. For the reduction in transverse modes, a general Fredholm Alternative-type result is derived, allowing us to overcome the unboundedness of the domain and the lack of compact embeddings; this result apply to general closed operators.

Transverse bifurcation of viscous slow MHD shocks

“…2(a)), verifying that there occurs the same stability transition, at the same parameter values, that were seen in the inviscid case. We present, further, numerical results verifying the bifurcation conditions assumed in the abstract results of Texier-Zumbrun [47] and Monteiro [42] (see also the related [43]), consisting of the absence of other neutrally stable eigenvalues (see Fig. 6) and nonzero speed of crossing of the imaginary axis as the magnetic field is varied (see Figure 8).…”

“…In particular, we demonstrate transitions from stability to instability satisfying the bifurcation hypotheses proposed in [42] (parallel magnetic field case) and [47] (nonparallel case), implying bifurcation in a mode transverse to the direction of shock propagation, i.e., lying in the direction parallel to the front, the first examples for which these scenarios have been shown to occur.…”

“…This latter phenomenon has been verified rigorously in the form of a steady transverse bifurcation in a general O(2)-symmetric strictly parabolic system of conservation laws [42] relevant to the parallel MHD case, under appropriate spectral bifurcation conditions, namely, that transition to instability occurs through a pair of real eigenvalues corresponding to nonzero transverse Fourier modes passing through the origin. In the non-O(2) symmetric case, corresponding to nonparallel magnetic field, a similar Hopf bifurcation result was shown in [47], under the assumption that loss of stability occurs through the passage of a complex conjugate pair of eigenvalues associated with nonzero transverse modes.…”

“…In this paper, continuing and extending investigations of [46,47] [24] [6,2], and [42], we study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar viscous slow Lax magnetohydrodynamic (MHD) shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD, a computationally intensive problem representing the current state of the art, and, together with the treatment of gas-dynamical shocks in [29], the first such study for viscous shock profiles of any physical system in multi-D.…”

“…Our main physical conclusions are two: first, we make a mathematical connection between the spectral instability observed for slow inviscid MHD shocks [13,24], and "corrugation instabilities" observed in the astrophysics community [34,45], via a viscous bifurcation analysis as in [46,47,42]. Namely, our results suggest that, rather than a planar shock, a nonplanar "wrinkled" or "corrugated" traveling wave with nearby normal velocity is the typical mode of propagation in the slow Lax mode.…”