ABSTRACT
This chapter considers the existence of global weak solutions for the nonlinear hyperbolic systems with bounded measurable initial data. It presents theorems for the Euler equations of compressible fluid flow. From the system of the quadratic flux, Le Roux system and two special systems of Euler equations, we can see that because of the explicit constructions of flux functions in these systems, we can make some suitable transformations of variables to construct explicit entropy-entropy flux pairs of Lax type via the solutions of Fuchsian equations, and hence obtain necessary estimates about these function pairs. Entropy-entropy flux pairs to more general strictly hyperbolic systems or systems in the strictly hyperbolic domains were well analyzed by Lax. However, to apply the compensated compactness method to some nonstrictly hyperbolic systems just as given in the form of system, some new techniques to construct entropy-entropy flux pairs must be investigated.
