Uniqueness Domains for $$\textbf{L}^\infty $$ Solutions of $$2\times 2$$ Hyperbolic Conservation Laws

For a genuinely nonlinear 2×2 hyperbolic system of conservation laws, assuming that the initial data have a small 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. Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with a faster decay rate: Tot.Var.{u(t,·)}≤Ctα-1. For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with ‖u¯‖L∞≤ε1 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 ε1>0. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.