On the formation of singularities in magnetohydrodynamic waves (Q1343979)

We consider the motion of a compressible perfectly conducting fluid in a magnetic field described by the Lundquist equations. We present two results on the formation of singularities in smooth magnetohydrodynamic waves. The results show that \(C^ 1\)-solutions do not exist in the large under some restrictions on the initial data. The first result is an outcome of ``large'' initial disturbances in three space dimensions. It simply states that the finite-time formation of singularities in three- dimensional magnetohydrodynamic waves is essentially inevitable provided that the initial flow velocity, in some region near the origin, is supersonic relative to the sound speed at infinity. The second result is a ``small'' data result in one space dimension.