Weak and measure-valued solutions to evolutionary PDEs. (English) Zbl 0851.35002

Nečas, J.; Málek, J.; Rokyta, M.; Růžička, M.

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)

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