THE SEMIGROUP GENERATED BY 2X2 CONSERVATION-LAWS

Consider the Cauchy problem for a strictly hyperbolic 2 x 2 system of conservation laws in one space dimension: u(t) + [F(u)](x) = 0, u(0, x ) = (u) over bar(x), assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. This paper develops a new algorithm, based on wave-front tracking, which yields a Cauchy sequen ce of approximate solutions, converging to a unique limit depending co ntinuously on the initial data. The solutions that we obtain constitut e a semigroup S, defined on a set D of integrable functions with small total variation. For some Lipschitz constant L, we have the estimatep arallel to S-t (u) over bar - S-s (v) over bar parallel to(L1) less th an or equal to L(\t - s\ + parallel to (u) over bar - (v) over bar par allel to(L1)) For All t, s greater than or equal to 0, For All (u) ove r bar, (v) over bar is an element of D.