10 F. ANCONA AND A. MARSON ut + F(u)x = suxx of (1.1), and by establishing uniform (w.r.t. the viscosity coeffi- cient e) a-priori bounds on its L1 norm. Although this new approach yields definite L1 well-posedness theory for the whole class of strictly hyperbolic systems, our construction remains of interest in itself providing an alternative, true hyperbolic, proof of this result for 2 x 2 systems with NGNL characteristic fields. 2. PRELIMINARIES Let Q C E 2 be an open neighborhood of the origin and F : ft R2 a smooth map. Denote with X\(u) \2(u) the eigenvalues of the Jacobian matrix A(u) = DF(u). Performing a linear change of coordinates in the (£, £)-plane, we can assume that (2.1) Ai 0 A2, 0 A m i n |A |, for some constant Amm. The directional derivative of a function (j) f)(u) in the direction of a vector field r r(u) is written (t(u + hr(u)) d)(u) (2.2) r 4 (u) = Dr(j)(u) = lim -^ \-^ -^-A Since we deal with a 2 x 2 system, it is convenient to work with a set of Riemann coordinates v = (^1,^2) associated with a local diffeomorphism v 1— u(v), defined on a neighborhood of zero V =]ai, bi[x]a2, 62! We recall that {yi, V2) form a coordinate system of Riemann invariants for (1.1) if and only if each Dvi(u), i 1, 2 is a left eigenvector of the Jacobian matrix A(u) = DF(u). We normalize right and left eigenvectors r^(w), h(u), i = 1,2, of A(u) so that (cfr. [D] Section 7.3) Dvi(u)-=U{u) i,j = 1,2, (2 '3) n , ^ (^ J 1 if i = j which, in turn, implies \rl, r3]{u) = rj ri(u) - r-i rj(u) = 0, ( 2 ' 4 ) d u ^ t , ^ w , , -TT-M = n(u(v)) \/v e V, V i z , 2,J = 1,2, i = 1,2. 9u Because of (2.4), the choice of the parameterization in (2.3) offers, in particular, some technical advantages to prove the convergence of the approximate flux func- tions Fl'e{u) that will be introduced in Section 4 (see Proposition 3.1 in [AM2]). In the Riemann coordinates, the rarefaction curves Ri(v), i = 1, 2, through a point v = (^1,^2) £ V are naturally parameterized by { Vi + a if j = i, ^ if J + I , where Rij(v), j = 1,2, denote the components of Ri(v). On the other hand, since shock and rarefaction curves have second order contact, the Hugoniot curves
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