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Zbl 0851.35002
Nečas, J.; Málek, J.; Rokyta, M.; Ruužička, M.
Weak and measure-valued solutions to evolutionary PDEs. (English)
[B] Applied Mathematics and Mathematical Computation. 13. London: Chapman \& Hall. vii, 317 p. \sterling 39.00 (1996). ISBN 0-412-57750-X

The monograph addresses evolution partial differential equations of hyperbolic and parabolic types with emphasis on problems arising in nonlinear fluid mechanics. After some auxiliary material summarized in Chapter 1, the theory of multidimensional scalar hyperbolic equations is presented in Chapter 2, using the concept of entropy. Chapter 3 introduces basic notions and results from the theory of Young measures including also the Murat-Tartar relation for nonconvex entropies and illustrating an application on the existence proof of a one-dimensional scalar hyperbolic conservation law. The last two chapters deal with problems where nonlinearities depend on the gradient of the solution, in particular nonlinear scalar hyperbolic second-order equations and a certain class of both compressible and incompressible non-Newtonian fluids. The global-in-time existence of a Young-measure-valued solution is proved. This solution, under suitable data qualification, is shown to be the weak solution and questions about uniqueness and regularity are then addressed, too.\par The book contains a lot of the authors' own results and also points out open problems. As such, it will be found useful both by experts and by advanced students interested in modern mathematical aspects of nonlinear distributed-parameter systems in general and fluid dynamics in particular.
[T.Roubíček (Praha)]

MSC 2000:: *35-02 Research monographs (partial differential equations)
35Lxx Hyperbolic equations and systems
76N10 Compressible fluids, general
76A05 Non-Newtonian fluids
35Q35 Other equations arising in fluid mechanics

Keywords: nonlinear hyperbolic systems; non-Newtonian fluids; Young measures; uniqueness; regularity

Cited in: Zbl 1080.76005 Zbl 0990.35096

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