By Marco Cannone (auth.), Josef Málek, Jindřich Nečas, Mirko Rokyta (eds.)
This booklet contains six survey contributions which are fascinated by a number of open difficulties of theoretical fluid mechanics either for incompressible and compressible fluids. the 1st article "Viscous flows in Besov areas" through M quarter Cannone advert clothes the matter of world life of a uniquely outlined approach to the 3-dimensional Navier-Stokes equations for incompressible fluids. between others the subsequent issues are intensively taken care of during this contribution: (i) the systematic description of the areas of preliminary stipulations for which there exists a different neighborhood (in time) resolution or a distinct worldwide answer for small information, (ii) the lifestyles of ahead self-similar options, (iii) the relation of those effects to Leray's vulnerable recommendations and backward self-similar ideas, (iv) the extension of the consequences to additional nonlinear evolutionary difficulties. specific cognizance is paid to the severe areas which are invariant below the self-similar rework. For small enough Reynolds numbers, the conditional balance within the feel of Lyapunov is additionally studied. the thing is endowed by way of fascinating own and ancient reviews and an exhaustive bibliography that offers the reader a whole photograph approximately to be had literature. The papers "The dynamical approach method of the Navier-Stokes equa tions for compressible fluids" via Eduard Feireisl, and "Asymptotic difficulties and compressible-incompressible limits" via Nader Masmoudi are dedicated to the worldwide (in time) homes of recommendations 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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Additional info for Advances in Mathematical Fluid Mechanics: Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic, Sept. 19–26, 1999
Weissler [35-37] follow the same direction. These works analyse the existence and uniqueness of global solutions for the nonlinear Schrodinger equation iVt + Llv = 1'lvlCl
Terraneo: Sur la non-unicite des solutions faibles de l'equation de la chaleur non lineaire avec non-linearite u 3 , C. R. Acad. Sci. Paris Ser. I Math. 328 (9) (1999), 759-762. 138. G. Tian and Z. Xin: One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal. 11 (1) (1998), 135-145. 139. T. P. Tsai: On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal 143 (1) (1998), 29-51. Erratum, ibid.
Kozono and M. Yamazaki: On a larger class of stable solutions to the Navier-Stokes equations in exterior domains, Math. Z. 228 (1998), 751-785. 85. H. Kozono and Y. Taniuchi: Bilinear estimates in BMO and the Navier-Stokes equations, preprint (1999). 86. H. Kozono and Y. Taniuchi: Limiting case of the Sobolev inequality in BMO, with application to the Euler equation, preprint (1999). 87. L. D. Landau and E. M. Lifchitz: Fluid mechanics, Translated from the Russian by J. B. Sykes and W. H. Reid.