1. INTRODUCTION Consider the Cauchy problem for a strictly hyperbolic 2 x 2 system of conser- vation laws in one space dimension (1.1) ut + F{u)x = 0J (1.2) u(0,x) = u(x). Let the flux function F : ft y- R2 be a smooth vector field defined on a neighborhood of the origin Q C R2. Denote by Ai(ix) \2{u) the eigenvalues of the Jacobian matrix A(u) = DF(u), and let {ri(u), ^(w)} be a basis of right eigenvectors. We assume that system (1.1) may admit non genuinely nonlinear (NGNL) characteristic fields, i.e. characteristic fields that are neither genuinely nonlinear (GNL) nor linearly degenerate (LD) in the sense of Lax [La, Sm]. Instead, letting n Q x A , x . r t{u + hrk{u)) -t{u) (1.3) rk • j{u) = h m — ^ -—l h-*0 a denote the directional derivative of a function 0 = j(u) in the direction of the characteristic field r^(ti), we make the following assumption. (A) If the k-th characteristic family is NGNL, then the set (1.4) Tk = {uen rk.Xk(u) = 0} is a smooth curve in the ix-space that is transversal to the vector field rk (see figure 1.1), and there holds (1.5) rk • {rk .\k){u)^0 V u e Tk . FIGURE 1.1. The set Tk i

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