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Systems of conservation laws with incomplete sets of eigenvectors everywhere. (English) Zbl 0929.35089
Chen, Gui-Qiang (ed.) et al., Advances in nonlinear partial differential equations and related areas. Proceedings of a conference dedicated to Professor Xiaqi Ding on the occasion of his 70th birthday, Beijing, China, August 9–10, 1997. Singapore: World Scientific. 399-426 (1998).

Summary: We present a class of systems of conservation laws which have repeated eigenvalues and incomplete sets of eigenvectors in an open set of the state space. These systems contain, as a prototypical example, a one-dimensional $2×2$ system of equations with names such as transport equations in the context of flux-splitting schemes, pressureless equations in isentropic gas dynamics, or adhesion particle dynamics equations in astrophysics. A distinctive feature of the prototypical example and our class of systems is the singular concentration in one of the dependent variables in the form of a weighted Dirac delta function at the same space-time location as a shock wave forms, – the so-called delta-shock wave. We describe these general systems from the perspective of classical systems of conservation laws with strict hyperbolicity and genuine nonlinearity.

In the $2×2$ situation (${u}_{t}+f{\left(u,v\right)}_{x}=0$, ${v}_{t}+g{\left(u,v\right)}_{x}=0$), our general systems are such that the following four conditions hold in an open set of the state space of dependent variables: (i) the eigenvalue is of multiplicity two, (ii) the system is not diagonalizable, (iii) the eigenvalue has vanishing directional derivative along a nonzero right eigenvector, (iv) the eigenvalue has nonvanishing directional derivative along a nonzero generalized right eigenvector. We reduce these general systems to simple forms for continuous solutions, and also for discontinuous solutions when a partial linearity (affinity) condition is further assumed. Definition of distributional solutions for these systems in the space of bounded measures is made possible through the simple forms. Solutions to Riemann problems for the systems are given.

##### MSC:
 35L65 Conservation laws