On double wave discontinuities in fluids and plasmas

Abstract

Simple waves and double waves as exact solutions of quasilinear hyperbolic equations in hydrodynamics may blow up, and, so, generally, there are points at which these solutions are violated at some definite instants of time. Finding the first instant of time for the violation of a simple wave solution is a known method, while this problem for the double waves is not a well defined and an easy task. Here, a two and a three dimensional double wave solutions are reviewed for some hydrodynamical systems to show that, for the 2-D case for some special solutions, one can find a local minimum for the critical time, while for some other solutions not. For the 3-D case, generically, there is no local minimum for the critical time, but at very special situations, the local minimum critical times may exist. Moreover, the dynamics of critical curves and points are briefly discussed.