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Rotational degeneracy of hyperbolic systems of conservation laws. (English) Zbl 0722.35055

The paper discusses hyperbolic conservation laws \(u_ t+f(u)_ x=0\) in one space dimension that are rotationally symmetric with respect to a subset of the variables. Assuming that a related radial system is strictly hyperbolic and that the full problem is only as degenerate as the symmetry forces, the existence of unique stable (in the sense of \(L^ 1\)-closure of linearly stable) solutions to the Riemann problem near any constant state on the center of symmetry is proved. For the construction of these solutions, the well-known procedure of Lax for the nondegenerate case is adapted. An extension of the main result deals with the system \(u_ t=v_ x\), \(v_ t=f(u)_ x\), rotationally symmetric versions of which occur in isentropic magnetohydrodynamics and in isotropic nonlinear elasticity.
Reviewer: H.Engler (Bonn)

MSC:

35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
76W05 Magnetohydrodynamics and electrohydrodynamics
74B20 Nonlinear elasticity
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