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The Formation of Shocks in 3-Dimensional Fluids
Demetrios Christodoulou, Eidgen Technische Hochschule, Zurich, Switzerland
A publication of the European Mathematical Society.
cover
EMS Monographs in Mathematics
2007; 1000 pp; hardcover
Volume: 2
ISBN-10: 3-03719-031-0
ISBN-13: 978-3-03719-031-9
List Price: US$198
Member Price: US$158.40
Order Code: EMSMONO/2
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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.

Table of Contents

  • Prologue and summary
  • Relativistic fluids and nonlinear wave equations. The equations of variations
  • The basic geometric construction
  • The acoustical structure equations
  • The acoustical curvature
  • The fundamental energy estimate
  • Construction of the commutation vectorfields
  • Outline of the derived estimates of each order
  • Regularization of the propagation equation for \(\not\!\!{d}\mathrm{tr}\chi\). Estimates for the top order spatial derivatives of \(\chi\)
  • Regularization of the propagation equation for \(\not\!\!{\Delta}\mu\). Estimates for the top order spatial derivatives of \(\mu\)
  • Control of the angular derivatives of the first derivatives of the \(x^i\). Assumptions and estimates in regard to \(\chi\)
  • Control of the spatial derivatives of the first derivatives of the \(x^i\). Assumptions and estimates in regard to \(\mu\)
  • Recovery of the acoustical assumptions. Estimates for up to the next to the top order angular derivatives of \(\chi\) and spatial derivatives of \(\mu\)
  • The error estimates involving the top order spatial derivatives of the acoustical entities. The energy estimates. Recovery of the bootstrap assumptions. Statement and proof of the main Theorem: Existence up to shock formation
  • Sufficient conditions on the initial data for the formation of a shock in the evolution
  • The nature of the singular hypersurface. The invariant curves. The trichotomy theorem. The structure of the boundary of the domain of the maximal solution
  • Epilogue
  • Bibliography
  • Index
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