# The Formation of Shocks in 3-Dimensional Fluids

Share this page
*Demetrios Christodoulou*

A publication of the European Mathematical Society

The equations describing the motion of a perfect fluid were first
formulated by Euler in 1752. These equations were among the first
partial differential equations to be written down, but, after a lapse
of two and a half centuries, we are still far from adequately
understanding the observed phenomena which are supposed to lie within
their domain of validity.

These phenomena include the formation and evolution of shocks in
compressible fluids, the subject of the present monograph. The first
work on shock formation was done by Riemann in 1858. However, his
analysis was limited
to the simplified case of one space dimension. Since then, several
deep physical insights have been attained and new methods of
mathematical analysis invented. Nevertheless, the theory of the
formation and evolution of shocks in real three-dimensional fluids has
remained up to this day fundamentally incomplete.

This monograph considers the relativistic Euler equations in three
space dimensions for a perfect fluid with an arbitrary equation of
state. The author considers 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, he establishes 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 spacetime manifold.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in applications and differential equations.