Solutions with shocks in several variables (Q1918510)

The author considers overdetermined hyperbolic \(C^1\)-systems (with \(\ell\) unknown functions in \(n\) variables given by \(r\geq\ell\) equations) which include certain magneto-hydrodynamical inviscid fluid flow equations. A qualitative theory is derived for the construction of the image \(u(D)\) (for any solution which exists on a domain \(D\)) and the reconstruction of solutions by giving conical parametrizations of \(u(D)\) by \(n\) independent variables. The methods consist in constructing certain \(k\)-dimensional manifolds \(M_k\) \((k<l, k\leq n)\) with singularities such that after a conical parametrization of \(M_k\) by \(n\) independent variables one can obtain solutions with a prescribed system of shocks.