2 x 2 hyperbolic systems of conservation laws in classes of functions of bounded p-variation

In this paper, we consider $2\times 2$ hyperbolic systems of conservation laws in one space dimension with characteristic fields satisfying a condition that encompasses genuine nonlinearity and linear degeneracy as well as intermediate cases, namely, with standard notations, $r_i\cdot \mathrm{\nabla }{\lambda }_i\ge 0$. We prove the existence of entropy solutions in the fractional $BV$ spaces $W_p(\mathbb{R})$ of functions of bounded $p$-variation, $p\in [1,\frac{3}{2}]$, for small initial data.