Existence of quasilinear relaxation shock profiles in systems with characteristic velocities

Abstract

We revisit the existence problem for shock profiles in quasilinear relaxation systems in the case that the velocity is a characteristic mode, implying that the profile ODE is degenerate. Our result states existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude quasilinear relaxation shocks. Our method of analysis follows the general approach used by Metivier and Zumbrun in the semilinear case, based on Chapman–Enskog expansion and the macro–micro decomposition of Liu and Yu. In the quasilinear case, however, in order to close the analysis, we find it necessary to apply a parameter-dependent Nash-Moser iteration due to Texier and Zumbrun, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.