By Marco Cannone (auth.), Josef Málek, Jindřich Nečas, Mirko Rokyta (eds.)
This publication involves six survey contributions which are fascinated about a number of open difficulties of theoretical fluid mechanics either for incompressible and compressible fluids. the 1st article "Viscous flows in Besov areas" via M sector Cannone advert clothes the matter of worldwide lifestyles of a uniquely outlined option to the third-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 distinct neighborhood (in time) answer or a special international answer for small info, (ii) the life of ahead self-similar strategies, (iii) the relation of those effects to Leray's vulnerable options and backward self-similar ideas, (iv) the extension of the consequences to additional nonlinear evolutionary difficulties. specific recognition is paid to the severe areas which are invariant below the self-similar remodel. For small enough Reynolds numbers, the conditional balance within the feel of Lyapunov is additionally studied. the thing is endowed by means of attention-grabbing own and historic reviews and an exhaustive bibliography that provides the reader an entire photograph approximately to be had literature. The papers "The dynamical method 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) homes 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 lots 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
T ) The existence of forward regular self-similar solutions of the form ~ V ( with V E LP(JR3) and 3 < p < 00 was given in [22,26]. But no result is known if 1 ~ p ~ 3 or p = 00. P. "j x ) with a > 0, 2a(T-t) V E LP(JR3) and 3 ~ p What happens if 1 ~ p Jan-Sznitman space? 4 2a(T-t) < 00 and that, if p = 00, then V is a constant. < 3? g. LP(JR3), 3 < p ~ 00, we know the existence and uniqueness of a local mild solution of the Navier-Stokes equations, but it is not clear whether such a solution can be extended to a global one.
Mat. Iberoamericana (to appear) (2000). 60. Y. Giga: Solutions for semilinear parabolic equations in LP and regularity of weak solutions of the Navier-Stokes system, J. Diff. Eq. 62 (1986), 186-212. 61. Y. Giga: Review of the paper by H. Brezis, "Remarks on the preceding paper by M. html]. 62. Y. -H. Giga: Nonlinear Partial Differential Equations-Asymptotic Behaviour of Solutions and Self-Similar Solutions (in Japanese), Kyoritsu Shuppan, Tokyo, 1999. 63. Y. Giga and T. Miyakawa: Solutions in £T of the Navier-Stokes initial value problem, Arch.
Systems 3(3) (1997), 305-316. 127. F. Ribaud: Cauchy problem for semi linear parabolic equations with initial data in H;(IRn) spaces, Rev. Mat. Iberoamericana 14 (1) (1998), 1-46. 128. F. Ribaud and A. Youssfi: Regular and self-similar solutions of nonlinear Schrodinger equations, J. Math. Pures Appl. 77 (10) (1998),1065-1079. 129. F. Ribaud et A. Youssfi: Solutions globales et solutions auto-similaires de l'equation des ondes non lineaire, C. R. Acad. Sci. Paris 329 (1) (1999), 33-36. 130. F. Ribaud and A.
Advances in Mathematical Fluid Mechanics: Lecture Notes of the Sixth International School Mathematical Theory in Fluid Mechanics, Paseky, Czech Republic, Sept. 19–26, 1999 by Marco Cannone (auth.), Josef Málek, Jindřich Nečas, Mirko Rokyta (eds.)