Nečas, J.; Málek, J.; Rokyta, M.; Růžička, M. Weak and measure-valued solutions to evolutionary PDEs. (English) Zbl 0851.35002 Applied Mathematics and Mathematical Computation. 13. London: Chapman & Hall. vii, 317 p. (1996). 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. 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. Reviewer: T.Roubíček (Praha) Cited in 2 ReviewsCited in 286 Documents MSC: 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35Lxx Hyperbolic equations and hyperbolic systems 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics Keywords:nonlinear hyperbolic systems; non-Newtonian fluids; Young measures; uniqueness; regularity PDF BibTeX XML Cite \textit{J. Nečas} et al., Weak and measure-valued solutions to evolutionary PDEs. London: Chapman \& Hall (1996; Zbl 0851.35002)