Weak* solutions I: A new perspective on solutions to systems of conservation laws

TL;DR

This paper introduces weak* solutions for systems of conservation laws, enabling the handling of singularities like vacuums and cavities by formulating solutions as Banach space ODEs and utilizing Gelfand calculus.

Contribution

It presents a novel weak* solution framework that extends traditional weak solutions, especially for singular phenomena in conservation law systems.

Findings

Weak* solutions can handle vacuums and cavities.

Weak* solutions are stronger than traditional weak solutions.

For BV solutions, weak* and weak solutions are equivalent.

Abstract

We introduce a new notion of solution, which we call weak* solutions, for systems of conservation laws. These solutions can be used to handle singular situations that standard weak solutions cannot, such as vacuums in Lagrangian gas dynamics or cavities in elasticity. Our framework allows us to treat the systems as ODEs in Banach space. Starting with the observation that solutions act linearly on test functions $\alpha\in X$, we require solutions to take values in the dual space $X^*$ of $X$. Moreover, we weaken the usual requirement of measurability of solutions. In order to do this, we develop the calculus of the Gelfand integral, which is appropriate for weak* measurable functions. We then use the Gelfand calculus to define weak* solutions, and show that they are stronger than the usual notion of weak solution, although for $BV$ solutions the notions are equivalent. It is expected that these solutions will also shed light on vexing issues of ill-posedness for multi-dimensional systems.