BV Solutions for Systems of Balance Laws

Abstract

The aim here is to discuss the existence and long time behavior of BV solutions to the Cauchy problem for (possibly inhomogeneous) strictly hyperbolic systems of balance laws. Thus, this chapter may be viewed as the counterpart of Sect. 5.5, where the same issues are addressed in the context of classical solutions. For the reasons presented in the preceding chapters, the investigation shall be confined to systems in a single spatial dimension and initial data of small total variation; however, modulo these limitations, the analogy to the results of Sect. 5.5 goes quite far. Thus, the existence of local solutions will be established under moderate restrictions on the flux and on the source, while global existence will hinge on the presence of damping. As in Sect. 5.5, damping shall be induced by a dissipative source incurring nonnegative entropy production.