The class of 2×2 nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional BV spaces BVs is proved. The exponent s is related to the usual fractional Sobolev derivative. Riemann invariants w and z corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates (w0,z0) are small in BVs×L∞,1/3≤s<1. The restriction on the exponent s is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant z on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant w.
