Methods and Applications of Analysis

Volume 24 (2017)

Number 3

Weak* Solutions I: A New Perspective on Solutions to Systems of Conservation Laws

Pages: 351 – 382

DOI: https://dx.doi.org/10.4310/MAA.2017.v24.n3.a1

Authors

Alexey Miroshnikov (Department of Biostatistics and Epidemiology, University of Massachusetts Amherst, Mass., U.S.A.; and Department of Mathematics, University of California at Los Angeles)

Robin Young (Department of Mathematics and Statistics, University of Massachusetts Amherst, Mass., U.S.A.)

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.

Keywords

conservation laws, shock waves, Gelfand integral

2010 Mathematics Subject Classification

28B05, 35L40, 35L65, 35L67, 35L90, 46G12

Full Text (PDF format)

Received 13 February 2016

Accepted 7 March 2017

Published 17 January 2018