Compressible flow and Euler's equations / Demetrios Christodoulou, Shuang Miao.
| Author/creator | Christodoulou, Demetrios, 1951- author. |
| Other author | Miao, Shuang (Mathematician), author. |
| Format | Book |
| Publication | Somerville, Mass : International Press ; Beijing, China : Higher Education Press, [2014] |
| Copyright Date | ©2014 |
| Description | x, iv, 580 pages, 1 unnumbered pages ; 26 cm. |
| Subjects |
| Series | Surveys of modern mathematics ; volume 9 Surveys of modern mathematics ; v.9. ^A1257461 |
| Contents | 1. Compressible flow and non-linear wave equations -- 2. The basic geometric construction -- 3. The acoustical structures equations -- 4. The acoustical curvature -- 5. The fundamental energy estimate -- 6. Construction of commutation vector fields -- 7. Outline of the derived estimates of each order -- 8. Regularization of the propagation equation for /dtrx· estimates for the top order angular derivatives of chi -- 9. Regularization of the propagation equation for /delta mu -- 10. Control of the angular derivatives of the first derivatives of the chi [superscript i]. Assumptions and estimates in regard to chi -- 11. Control of the spatial derivatives of the first derivatives of the chi [superscript i]. Assumptions and estimates in regard to mu -- 12. Recovery of the acoustical assumptions, estimates for up to the next the top order angular derivatives of chi and spatial derivatives of mu -- 13. Derivatives of the basic properties of mu -- 14. The error estimates involving the top order spatial derivatives of the acoustical entities -- 15. The top order energy estimates -- 16. The descent scheme -- 17. The isoperimetric inequality. Recovery of assumption J. Recovery of the bootstrap assumption. Proof of the main theorem -- 18. Sufficient conditions on the initial data for the formation of a shock in the evolution -- 19. The structure of the boundary of the domain of the maximal solution. |
| Summary | This monograph considers the classical compressible Euler Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the initial departure from the constant state, the authors establish theorems which give a complete description of the maximal development. In particular, the boundary of the domain of the maximal solution contains a singular part where the density of the wave fronts blows up and shocks form. The authors obtain a detailed description of the geometry of this singular boundary, and a detailed analysis of the behavior of the solution there. The approach is geometric, the central concept being that of the acoustical spacetime manifold. Compared to a previous monograph treating the relativistic fluids by the first author, the present monograph not only gives simpler and self-contained proofs but also sharpens some of the results. In addition, it explains in depth the ideas on which the approach is based. Moreover, certain geometric aspects which pertain only to the non-relativistic theory are discussed. Compressible Flow and Euler's Equations will be of interest to scholars working in partial differential equations in general and in fluid mechanics in particular. |
| Bibliography note | Includes bibliographical references. |
| ISBN | 157146297X |
| ISBN | 9781571462978 |
Availability
| Library | Location | Call Number | Status | Item Actions |
|---|---|---|---|---|
| Joyner | General Stacks | QA861 .C47 2014 | ✔ Available | Place Hold |