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Representation of weak limits and definition of nonconservative products. (English) Zbl 0939.35115

Authors’ abstract: The goal of this article is to show that the notion of generalized graphs is able to represent the limit points of the sequence \(\{g(u_n) du_n\}\) in the weak-* topology of measures when \(\{u_n\}\) is a sequence of continuous functions of uniformly bounded variation. The representation theorem induces a natural definition for the nonconservative product \(g(u) du\) in a BV context. Several exiting definitions of nonconservative products are then compared, and the theory is applied to provide a notion of solutions and an existence theory to the Riemann problem for first-order quasilinear (strictly) hyperbolic systems \[ \partial_tu+A(u)\partial_xu=0; \quad x\in \mathbb{R}, \;t>0, \]
\[ u(x,0) = u_- \text{ for } x<0, \text{ respectively } u(x,0) = u_+ \text{ for }x>0, \] where the \(N\times N\) matrix \(A(u)\) is a smooth function of \(u\), and \(u_-, u_+\) are given vectors in \(\mathbb{R}^N\).

MSC:

35L60 First-order nonlinear hyperbolic equations
28A75 Length, area, volume, other geometric measure theory
35L45 Initial value problems for first-order hyperbolic systems
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