Conservation Laws and Global Solutions of Linear First Order PDEs with Distributional Coefficients

We treat linear partial differential equations of first order with distributional coefficients naturally related to physical conservation laws in the spirit of our preceding papers Žwhich concern ordinary differential equations.: the solutions are consistent with the classical ones. Under compatibility conditions we prove uniqueness and existence results. As an example we consider the problem ut t ux 0, uŽ x, 1. hŽ x. Žh C2 Ž . is given.; our theory grants that the unique solution in C2 Ž 2 . D Ž 2 . is uŽ x, t. hŽ x. h Ž0. Ž x, t. and this has a physical mean- l ing Ž D l Ž 2 . is the space of distributions with discrete support and is the Dirac measure at Ž0, 0...