By Marco Cannone (auth.), Josef Málek, Jindřich Nečas, Mirko Rokyta (eds.)

This ebook comprises six survey contributions which are concerned about numerous open difficulties of theoretical fluid mechanics either for incompressible and compressible fluids. the 1st article "Viscous flows in Besov areas" by way of M quarter Cannone advert attire the matter of world lifestyles of a uniquely outlined method to the 3-dimensional Navier-Stokes equations for incompressible fluids. between others the next issues are intensively handled during this contribution: (i) the systematic description of the areas of preliminary stipulations for which there exists a different neighborhood (in time) answer or a different international resolution for small info, (ii) the lifestyles of ahead self-similar ideas, (iii) the relation of those effects to Leray's susceptible suggestions and backward self-similar ideas, (iv) the extension of the consequences to extra nonlinear evolutionary difficulties. specific cognizance is paid to the severe areas which are invariant lower than the self-similar rework. For small enough Reynolds numbers, the conditional balance within the feel of Lyapunov is usually studied. the thing is endowed through fascinating own and old reviews and an exhaustive bibliography that provides the reader a whole photograph approximately on hand literature. The papers "The dynamical procedure method of the Navier-Stokes equa tions for compressible fluids" through Eduard Feireisl, and "Asymptotic difficulties and compressible-incompressible limits" via Nader Masmoudi are dedicated to the worldwide (in time) houses of strategies to the Navier-Stokes equa and 3 tions for compressible fluids. the worldwide (in time) research of 2 dimensional motions of compressible fluids have been left open for plenty of years.

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E. Schonbek [33], we considered the same question by adapting a well-known work by S. Ukai [142] and giving an explicit formula for the solutions of the 22 Marco Cannone Stokes equations in ~t; in particular we reformulated and generalised part of Voss' work. Finally, N. D. thesis the case of an exterior domain ne , the complementary of a compact set with smooth boundaries [47]. An additional difficulty that arises here is the necessity of introducing the notion of a homogeneous Besov space on ne.

Anal 143 (1) (1998), 29-51. Erratum, ibid. 147 (1999), 363. 140. S. Tourville: An analysis of Numerical Methods for Solving the two-dimensional Navier-Stokes Equations, Ph. D. Thesis, Washington University of St. Louis, MO (1998). 141. S. Tourville: Existence and uniqueness of solutions for a modified NavierStokes equation in R2, Comm. Partial Differential Equations 23 (1-2) (1998), 97-121. 142. S. Ukai: A solution formula for the Stokes equation in R+, Comm. in Pure and Appl. Math. 15 (1987), 611-621.

Kato's paper of 1984. Among the possible generalisations, three seem to be more useful for the applications: an open bounded set Db, the half space 1R~ and finally an exterior domain De, that is the complementary of a compact set. D. thesis in 1996 supervised by R. A. Voss studied the half-space 1R~ problem aiming to prove the existence of self-similar solutions [144]. In collaboration with F. E. Schonbek [33], we considered the same question by adapting a well-known work by S. Ukai [142] and giving an explicit formula for the solutions of the 22 Marco Cannone Stokes equations in ~t; in particular we reformulated and generalised part of Voss' work.