Weak* solutions. I: A new perspective on solutions to systems of conservation laws (Q1691595)

The authors introduce a new notion of weak* solutions for systems of conservation laws, which allows to treat the systems as ODEs in an appropriate dual Banach space. This notion is defined with the help of Gelfand calculus, which the authors develop on the base of Gelfand integration of vector functions. The notion of weak* solutions is shown to be useful to handle singular situations like vacuums in Lagrangian gas dynamics or cavities in elasticity. The authors also demonstrate that weak* solutions are stronger than usual weak solution, although for BV solutions the notions of weak and weak* solutions are equivalent.