The Formation of Shocks in 3-Dimensional Fluids
Dealing with the relativistic Euler equations for an ideal fluid with an arbitrary equation of state, this book provides an original, mathematically sound and complete theory of the formation of shocks in a three-dimensional spatial setting. More precisely, starting with initial data that coincide outside of a sphere with the data corresponding to a constant state, and assuming a suitable restriction on the size of an initial perturbation of the constant state, theorems describing maximal classical development are established. It is shown, in particular, that the boundary of the maximal classical development includes a singular part, where the inverse density of the wave fronts vanishes, which indicates a formation of shocks. The central concept used in the book is an acoustic spacetime manifold. Methods from differential geometry play an important role in the book.