The semigroup generated by \(2 \times 2\) conservation laws (Q1912922)

The Cauchy problem for a strictly hyperbolic \(2\times 2\) system of conservation laws in one space dimension is considered, assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. A new algorithm is given, based on the wavefront tracking, which yields that the Cauchy sequence of approximate solutions converge to a unique limit, depending continuously on the initial data. The solutions constitute a continuous semigroup, defined on a domain \(D\subset L^1(\mathbb{R}, \mathbb{R}^2)\).