On the L Well Posedness of Systems of Conservation Laws near Solutions Containing Two Large Shocks

As in the classical paper of Lax [L], we assume here that the system is strictly hyperbolic with each characteristic field either linearly degenerate or genuinely nonlinear. In this setting, the recent progress in the field has shown that within the class of initial data ū ∈ L ∩BV (R,R) having the total variation suitably small, the problem (1.1) (1.2) is well posed in L(R,R). Namely, as proved in [BC1] [BCP] [BLY], the entropy solutions of (1.1) (1.2) constitute a semigroup which is Lipschitz continuous with respect to time and initial data. A major question which remains open is whether the uniqueness of solutions also holds for arbitrarily large initial data. We observe that, because of the finite propagation speed, this is essentially a local problem. Moreover, given any BV function ū : R −→ R, for each point x0 ∈ R one can find left and right neighbourhoods [x0 − δ, x0) and (x0, x0 + δ] on which ū has arbitrarily small variation. By the previous remarks the problem is thus reduced to proving the well posedness of the Cauchy problem (1.1) (1.2) with the initial data ū being a small perturbation of a fixed Riemann problem (u0 , u + 0 ). The solution of the latter consists of m (large) waves of different characteristic families.