DISCRETE AND CONTINUOUS Website http://aimSciences.org DYNAMICAL SYSTEMS Volume 23, Number 1&2, January & February 2009 pp. 1–XX ON THE CONVERGENCE OF VISCOUS APPROXIMATIONS AFTER SHOCK INTERACTIONS Alberto Bressan Department of Mathematics, Penn State University University Park, PA 16802, USA Carlotta Donadello S.I.S.S.A.–I.S.A.S., Via Beirut 4, Trieste 34014, Italy ABSTRACT. We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient. 1. Introduction. Consider a strictly hyperbolic system of conservation laws u t + f (u) x =0, (1.1) together with its viscous approximations u ε t + f (u ε ) x = εu ε xx . For a fixed initial data with small total variation u(0, ·)=¯ u(·), (1.2) the convergence u ε → u, as ε → 0+, was proved in [2]. An estimate on the convergence rate   u ε (t) − u(t)   L 1 (IR) = O(1) · (t + 1) √ ε ln ε, was later provided in [4]. In the scalar case, more detailed results can be found in [7], [10], and [11]. Also for computational purposes, it is interesting to examine whether viscous approximations admit a power series expansion in the viscosity coefficient ε. In the 2000 Mathematics Subject Classification. Primary: 35L65, 35B25. Key words and phrases. scalar conservation laws with viscosity, viscous shock profiles, singular perturbation expansion. This research was partially supported by NSF, Grant No. 0505430. 1