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Definition and weak stability of nonconservative products. (English) Zbl 0853.35068
Summary: We propose a definition for the nonconservative product $$g(u) du/dx$$, where $$g: \mathbb{R}^p\to \mathbb{R}^p$$ is a locally bounded Borel function and $$u: ]a, b[\to \mathbb{R}^p$$ is a function of bounded variation. This definition generalizes the one previously given by Vol’pert [A. I. Vol’pert, Math. USSR, Sb. 2, 225-267 (1967); translation from Mat. Sb., n. Ser. 73(115), 255-302 (1967; Zbl 0168.07402)] and is based on a Lipschitz continuous completion of the graphs of functions of bounded variation. We study the stability of this product for the weak convergence. As an application, the nonlinear hyperbolic systems in nonconservative form are considered: we give a notion of weak solution for the Riemann problem, and extend Lax’s construction.

##### MSC:
 35L60 First-order nonlinear hyperbolic equations 26A45 Functions of bounded variation, generalizations
##### Keywords:
nonconservative product; Riemann problem