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

AbstractWe treat linear partial differential equations of first order with distributional coefficients naturally related to physical conservation laws in the spirit of our preceding papers (which 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+δtux=0,u(x,−1)=h(x)(h∈C2(R) is given); our theory grants that the unique solution in C2(R2)⊕D′ℓ(R2) is u(x,t)=h(x)−h′(0)δ(x,t) and this has a physical meaning (D′ℓ(R2) is the space of distributions with discrete support and δ is the Dirac measure at (0,0))