The Stability of a Transonic Pro le Arising from Divergent Detonations

We establish the existence of the viscous proole of an undriven divergent detonation wave for a model problem with small viscosity. It is known that there is a sonic point inside the reaction zone of a divergent detonation wave. As a consequence, the detonation wave proole is a transonic proole and the wave speed cannot be determined before the problem is solved. The wave speed may be interpreted as a nonlinear eigenvalue. The detonations exhibiting this type of behavior are sometimes termed eigenvalue detonations. The shooting method and an asymptotic analysis are performed to prove the existence of the viscous proole for small viscosity. The condition we shoot at is the compatibility condition at the sonic point. In the construction of the viscous proole, an iteration is employed to treat the source term 1 arising from the chemical reaction. It is a consequence of the constructive proof that as viscosity tends to zero the viscous proole tends to the inviscid proole of the undriven divergent detonation wave 12]. Thus the undriven divergent detonation wave found in 12] is structurally stable. Finally, we study the nonlinear stability of the transonic proole. We prove that the solution exists globally and approaches to a shifted traveling wave solution as t! + 1 for 'large' perturbations of the transonic proole.