Uniqueness Domains for L^ Solutions of 2 2 Hyperbolic Conservation Laws

Abstract For a genuinely nonlinear 2 2 2 × 2 hyperbolic system of conservation laws, assuming that the initial data have a small L^ L ∞ norm but a possibly unbounded total variation, the existence of global solutions was proven in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like t^-1 t - 1. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with a faster decay rate: Tot. Var. \u (t, ) \ C t^ -1 Tot. Var. u (t, ·) ≤ C t α - 1. For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with u ₋^ ₁ ‖ u ¯ ‖ L ∞ ≤ ε 1 small enough, solutions with a fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to 1 by further shrinking the value of ₁>0 ε 1 > 0. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.