On the Cauchy Problem of Some Dissipative Flows

The Cauchy problem is studied for a system of nonlinear partial differential equations for some dissipative flows in Lagrangian formulation including heat conduction, damping relaxation, and coupling to electric field. The well-posedness of smooth solutions is investigated. It is proved that, for certain large initial data, the solution will develop singularities and shock waves in finite time, which indicates that the Cauchy problem does not have global smooth solutions even if the initial data are smooth, and one has to seek weak solutions