Christodoulou D. The Formation of Shocks in 3-Dimensional Fluids 2007

European Mathematical Society, 2007. – 992 p. – ISBN: 3037190310.The equations describing the motion of a perfect fluid were first formulated by Euler in 1752, partially based on earlier work of D. Bernoulli. These equations were among the first partial differential equations to be written down but, after the lapse of two and a half centuries, we are still far from having achieved an adequate understanding of the observed phenomena which are supposed to lie within the domain of validity of the Euler equations. The first work on the formation of shocks, the subject of the present monograph, was done by Riemann in 1858; with Einstein's discovery of the special theory of relativity in 1905, and its final formulation by Minkowski in 1908 the concept of space time with its geometry was introduced and the Euler equations were extended to become compatible with special relativity. This lead to new insight on the shock development, but few general results on the formation of shocks in three-dimensional fluids have been obtained up to this day. This monograph considers the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. We consider initial data for these equations which outside a sphere coincide with the data corresponding to a constant state. Under suitable restriction on the size of the initial departure from the constant state, we establish theorems that give a complete description of the maximal classical development. In particular, it is shown that the boundary of the domain of the maximal classical development has a singular part where the inverse density of the wave fronts vanishes, signalling shock formation. The theorems give a detailed description of the geometry of this singular boundary and a detailed analysis of the behavior of the solution there. A complete picture of shock formation in three-dimensional fluids is thereby obtained. The approach is geometric, the central concept being that of the acoustical space time manifold. The monograph will be of interest to people working in partial differential equations in general and in fluid mechanics in particular.Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variation
The Basic Geometric Construction
The Acoustical Structure Equations
The Acoustical Curvature
The Fundamental Energy Estimate
Construction of the Commutation Vector fields
Outline of the Derived Estimates of Each Order
Regularization of the Propagation Equation for d/tr. Estimates for the Top Order Angular Derivatives of Regularization of the Propagation Equation. Estimates for the Top Order Spatial Derivatives of Control of the Angular Derivatives of the First Derivatives of the xi Assumptions and Estimates in Regard to Control of the Spatial Derivatives of the First Derivatives of the xi . Assumptions and Estimates in Regard to 
Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of  and Spatial Derivatives of 
The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities. The Energy Estimates. Recovery of the Bootstrap Assumptions.
Sufficient Conditions on the Initial Data for the Formation of a Shockinthe Evolution
The Nature of the Singular Hyper surface. The Invariant Curves. The Trichotomy Theorem. The Structure of the Boundary of the Domainof the Maximal Solution