One-dimensional systems of conservation laws

In this chapter we introduce the definitions of hyperbolicity and strict hyperbolicity and generalize the definitions given in the previous chapters to the case of general systems of conservation laws in one space dimension. In the last two sections we also show how to construct a weak solution to the Cauchy problem for a strictly hyperbolic system of conservation laws with constant coefficient and to the Riemann problem for a strictly hyperbolic system of conservation laws with genuinely nonlinear or linearly degenerate characteristic fields. We prove also the local existence of a weak solution to the Riemann problem for a general system, when the initial states are sufficiently close. © 2013 Springer International Publishing Switzerland