Fundamental Directions in Mathematical Fluid Mechanics by Giovanni P. Galdi (auth.), Giovanni P. Galdi, John G.

By Giovanni P. Galdi (auth.), Giovanni P. Galdi, John G. Heywood, Rolf Rannacher (eds.)

This quantity contains six articles, each one treating a huge subject within the concept ofthe Navier-Stokes equations, on the study point. a few of the articles are in most cases expository, placing jointly, in a unified atmosphere, the result of fresh examine papers and convention lectures. numerous different articles are dedicated regularly to new effects, yet current them inside a much broader context and with a fuller exposition than is common for journals. The plan to post those articles as a ebook all started with the lecture notes for the quick classes of G.P. Galdi and R. Rannacher, given at the start of the foreign Workshop on Theoretical and Numerical Fluid Dynamics, held in Vancouver, Canada, July 27 to August 2, 1996. A renewed power for this undertaking got here with the founding of the magazine of Mathematical Fluid Mechanics, via G.P. Galdi, J. Heywood, and R. Rannacher, in 1998. at the moment it was once made up our minds that this quantity might be released in organization with the magazine, and improved to incorporate articles through J. Heywood and W. Nagata, J. Heywood and M. Padula, and P. Gervasio, A. Quarteroni and F. Saleri. the unique lecture notes have been additionally revised and updated.

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For instance, regularity would trivially follow if we could take (J' = 2 and T a decreasing function of Ilvol12. However, the existence theory known so far, with data in Lq, requires (J' 2': n. 30, 31 In order to avoid technical difficulties, in what follows we shall assume that 0= rn;n, referring to Giga (1986, Theorem 4) for the more general case when 0 has a (non-empty) compact boundary. 32 The results we shall prove will be then an immediate consequence of suitable estimates for solutions to the heat equation and of the classical successive approximation method applied to the linearized Stokes 28Throughout this section, for the sake of simplicity, we shall assume f == O.

1, we show that u satisfies the following relation lot {(u, ~~) - v(Vu, V

20 Proof. To avoid unessential technical difficulties, we limit ourselves to give the proof in the case D bounded and n = 3, referring the reader to Galdi and Maremonti (1988) for the proof in the general case. 3). Thus, we shall look for approximating solutions Uk of the form k Uk(X, t) = L Ckr(t)ar(x), kEN, r=l 19n is said uniformly of class em, m ~ 0, if n lies on one part of its boundary an and, for each there exists a ball B centered at Xo and of radius independent of xo, such that an n B admits a Cartesian representation of the form Xn = ,),(X1, ...

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