The refined inviscid stability condition and cellular instability of viscous shock waves

Combining the work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct, the violation of the refined planar stability condition of Zumbrun–Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we show for a model problem involving flow in a rectangular duct with artificial periodic boundary conditions that transition to multidimensional instability through violation of the refined stability condition of planar viscous shock waves on the whole space generically implies for a duct of sufficiently large cross-section, a cascade of Hopf bifurcations involving more and more complicated cellular instabilities. The refined condition is numerically calculable as described by Benzoni-Gavage–Serre-Zumbrun.