For a genuinely nonlinear \(2\times 2\) hyperbolic system of conservation laws, assuming that the initial data have a small \(\textbf{L}^\infty \) 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: \(\hbox {Tot.Var.}\bigl \{u(t,\cdot )\bigr \}\le C t^{\alpha -1}\) . For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with \(\Vert {\bar{u}}\Vert _{\textbf{L}^\infty } \le \varepsilon _1\) 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 \(\varepsilon _1>0\) . An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.
