Kinetic formulation for systems of two conservation laws and elastodynamics
The authors introduce the so-called kinetic formulation for the system of elastodynamics \[ u_t - v_x = 0,\qquad v_t - \sigma(u)_x = 0 \] in one space variable, where \(\sigma\) is an increasing nonlinear function with exactly one inflection point located at zero. The kinetic formulation allows to characterize the entropy solutions of the problem in terms of the kinetic functions \(\Theta(u,v,\xi)\), \(\overline \Theta(u,v,\xi)\) satisfying the system \[ \partial_t \Theta + \partial_x [ (-a(u) \kappa + a(u) (1- \kappa)) \Theta ] = \partial_{\xi} m , \] \[ \partial_t \overline \Theta + \partial_x [ (-a(u) \overline \kappa + a(u) (1- \overline \kappa)) \overline \Theta ] = \partial_{\xi} \overline m, \] where \(a(u) = (\sigma'(u))^{1\over 2}\), \(\kappa\), \(\overline \kappa\) are certain functions of \(u,v\) and of the kinetic variable \(\xi\), and \(m\) and \(\overline m\) are measures. This approach gives a full characterization of convex entropies related to the system in question and simplifies considerably the existing theory based on the concept of the Young measures. The paper provides also a very nice and self-contained exposition of both the kinetic approach to systems of conservation laws and the nonlinear wave equation theory.
scientific article; zbMATH DE number 1556302