Transverse steady bifurcation of viscous shock solutions of a hyperbolic-parabolic model 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 Franz Achleitner and Peter Szmolyan (Saddle-node bifurcation of viscous profiles) on 1D steady bifurcation of viscous shock profiles; en passant, changing to an appropriate moving coordinate frame, we recover and somewhat sharpen results of Benjamin Texier and Kevin Zumbrun (Galloping instability of viscous shock waves) 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.