Stability of undercompressive shock profiles

Original · EN

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under--overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations |u₀(x)|≤ E₀ (1+|x|)⁻³/² in C⁰⁺α, E₀ sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in ZH, extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in SzX, L, ZH, Z.2, Ra, MZ.1--MZ.5, and for special undercompressive profiles in LZ.1--LZ.2, HZ. In particular, together with spectral results of Z.6, our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model Sl.5 for van der Waal gas dynamics or elasticity with viscosity--capillarity.