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The Formation of Shocks in 3-Dimensional Fluids
 
Demetrios Christodoulou Eidgen Technische Hochschule, Zurich, Switzerland
A publication of European Mathematical Society
The Formation of Shocks in 3-Dimensional Fluids
Hardcover ISBN:  978-3-03719-031-9
Product Code:  EMSMONO/2
List Price: $198.00
AMS Member Price: $158.40
Please note AMS points can not be used for this product
The Formation of Shocks in 3-Dimensional Fluids
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The Formation of Shocks in 3-Dimensional Fluids
Demetrios Christodoulou Eidgen Technische Hochschule, Zurich, Switzerland
A publication of European Mathematical Society
Hardcover ISBN:  978-3-03719-031-9
Product Code:  EMSMONO/2
List Price: $198.00
AMS Member Price: $158.40
Please note AMS points can not be used for this product
  • Book Details
     
     
    EMS Monographs in Mathematics
    Volume: 22007; 1000 pp
    MSC: Primary 35; 58; 76;

    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.

  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 22007; 1000 pp
MSC: Primary 35; 58; 76;

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.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.