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Case study of a nonlinear, nonconservative, nonstrictly hyperbolic system. (English) Zbl 0790.35065

The Cauchy problems for the system \(u_ t+ uu_ x=0\), \(v_ t+uv_ x=0\) and for its parabolic approximations are discussed. These systems may be viewed as a rough model of one-dimensional elastic and visco- elastic material of high density in a nearly plastic state. Using the concept of the functions with bounded variation, the author proves the existence and uniqueness of parabolic approximations and discusses their convergence to the first system. Then these problems are treated in the space of generalized functions, which gives very interesting results.

MSC:

35L45 Initial value problems for first-order hyperbolic systems
35L67 Shocks and singularities for hyperbolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
74B20 Nonlinear elasticity
74Hxx Dynamical problems in solid mechanics
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