Mathematical Models and Methods in Applied Sciences Vol. 18, No. 8 (2008) 1443–1479 c  World Scientific Publishing Company SHOCK LAYERS FOR TURBULENCE MODELS CHRISTOPHE BERTHON Universit´ e de Nantes, Laboratoire de Math´ ematiques Jean Leray, UMR 6629, 2 rue de la, Houssini` ere, BP 92208, 44322 Nantes Cedex 3, France and INRIA Futur, Projet ScAlApplix, 351 cours de la lib´ eration, 33405 Talence Cedex, France christophe.berthon@math.univ-nantes.fr FR ´ ED ´ ERIC COQUEL Laboratoire Jacques-Louis Lions, UMR 7598, Universit´ e Pierre et Marie Curie, Boˆ ıte courrier, 187, 75252 Paris Cedex 05, France coquel@ann.jussieu.fr Received 27 June 2007 Revised 14 December 2007 Communicated by S. Ukai The present work is devoted to an extension of the Navier–Stokes equations where the fluid is governed by two independent pressure laws. Several turbulence models typically enter this framework. The striking novelty over the usual Navier–Stokes equations stems from the impossibility to recast equivalently the system of interest in full conservation form. Opposing to systems of conservation laws, where the end states of the viscous shock are completely characterized by jump relations, the lack of conservation implies the absence of jump relations. We analyze the traveling wave behaviors according to the ratio of viscosities, and we show that the traveling wave solutions of our system tend to the traveling wave solutions of a fully conservative system. This result is used to exhibit asymptotic expansions of the end states. Such an asymptotic behavior achieves a deep physical interpretation when illustrated in the case of compressible turbulent flows. Keywords : Turbulence models; traveling wave solutions; asymptotic behaviors. AMS Subject Classification: 35L67, 35Q35, 76F50 1. Introduction The present work considers the Navier–Stokes equations for compressible fluid dynamics modeled by two independent pressures. Two independent pressures mean more precisely that each of them is characterized with its own specific entropy. The smooth solutions of the system undergo simultaneously two independent entropy balance equations. 1443