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Zbl 1117.35001
Christodoulou, Demetrios
The formation of shocks in 3-dimensional fluids. (English)
[B] EMS Monographs in Mathematics. Zürich: European Mathematical Society Publishing House. viii, 992~p. EUR~148.00 (2007). ISBN 978-3-03719-031-9/hbk

This monograph deals with the relativistic Euler equations in three-space dimensions for a perfect fluid with an arbitrary equation of state, and relies more heavily on differential geometric concepts and methods. The general treatment is based on the concept of variation; to each variation, there are energy estimates associated with it. It is through energy currents and their associated integral identities that the estimates essential to the approach are derived. Indeed, the monograph is a welcome addition to the literature of shock waves. The monograph comprises fifteen chapters. The first chapter deals with the mechanics of a perfect fluid in the framework of the Minkowski space time of special relativity. Chapters 2 and 3 deal with basic geometric construction, and formulate the basic equations which govern the geometry of the 2-parameter foliation of the space time manifold. Chapter 4 deals with the regular form of acoustical structure equations, and analyses the acoustical curvature. After the first four chapters, which set up the general framework, attention is confined to the irrotational isentropic problem up to chapter thirteen. Chapter 5 considers wave equations and establishes the fundamental energy estimates. The commutator fields are defined in Chapter 6. In chapter seven, a recursion formula is obtained for the source function associated to the higher-order variations; the error integrals arising form the contributions to the source functions are estimated. Chapters 8 and 9 deal with estimates for the top order derivatives of the acoustical entities. The major parts of Chapters 10 and 11 consist of the derivations of estimates for the spatial derivatives of the first derivative of the spatial rectangular coordinates in terms of the acoustical entities. Chapter 12 establishes the acoustical bootstrap assumptions using the method of continuity which consists only of pointwise estimates for the variations up to certain order. Chapter 13 begins by establishing the basic assumption on the behaviour of an acoustical function related to the geometry of the foliation of space time on which the energy estimates rely, and then formulates the final bootstrap assumption and derives an explicit formula for the source functions. The monograph considers regular initial data on a spacelike hyperplane in Minkowski spacetime, which outside a sphere coincide with the data corresponding to a constant state. Under a suitable restriction on the size of the departure of the initial data from those of the constant state, certain results are established which describe the maximal solution. In Chapter 14, certain sharp sufficient conditions on the initial data for the formation of a shock in the evolution are established. Chapter 15 deals with the geometry of the boundary of the domain of the maximal solution. Finally, the monograph concludes with a derivation of a formula for the jump in vorticity across a hypersurface, which shows that while the flow is irrotational ahead of the shock it acquires vorticity immediately behind, which is tangential to the shock front and is associated to the gradient along the shock front of the entropy jump. This is a well written monograph which contains valuable information on shock waves; it should be of interest to anyone interested on shock formation in a nonlinear medium.
[V. D. Sharma (Mumbai)]

MSC 2000:: *35-02 Research monographs (partial differential equations)
76-02 Research monographs (fluid mechanics)
35L67 Shocks, etc.
35L65 Conservation laws
35L70 Second order nonlinear hyperbolic equations
58J45 Hyperbolic equations
76L05 Shock waves (fluid mechanics)
76N15 Gas dynamics, general
76Y05 Nonclassical hydrodynamics

Keywords: shock waves; relativistic fluids; nonlinear waves; Euler equations in 3-dimensional spaces; singular boundary; spacetime manifold; relativistic Euler equations; 3-space dimensions; differential geometric concepts; irrotational isentropic problem

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