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Decompositions of \(L^ q\) and \(H^{1,q}_ 0\) with respect to the operator rot. (English) Zbl 0738.46014

This is the second part of the author’s doctoral thesis at the Univ. of Bayreuth, (1989; Zbl 0705.35023) the first part being published in Ann. Univ. Ferrara, Nuova Ser., Sez. VII(1990). The author deals with the so- called decomposition of spaces \((L^ q(G))^ N\) and \((H_ 0^{1,q}(G))^ N\) with respect to the operator rot and extends the results of the first part obtained for \(q=2\) to the case \(1<q<\infty\) in the physically most important case \(N=3\). Instead of the projection theorem the author uses now a priori estimates which for the decomposition of \((L^ q(G))^ 3\) is \[ c\|\hbox{rot }u\|_ q\leq \sup_{\phi\in (H^{1,q'}(G))^ 3}{{\langle\hbox{rot }u,\hbox{rot }\phi\rangle}\over \|\nabla\phi\|_{q'}} \] for all \(u\in (H^{1,q}(G))^ 3\), and employs the idea of a partition of the domain \(G\) into domains \(G_ i\) which are “starlike with respect to an open ball” and admit a “good” Helmholtz decomposition.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q30 Navier-Stokes equations

Citations:

Zbl 0705.35023
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References:

[1] Alt, H.W.: Lineare Funktionalanalysis. Berlin Heidelberg New York: Springer 1985 · Zbl 0577.46001
[2] Amick, Ch.J.: Decomposition theorems for solenoidal vector fields. J. Lond. Math. Soc.15, 288-296 (1977) · Zbl 0363.35034 · doi:10.1112/jlms/s2-15.2.288
[3] Griesinger, R.: Ein Analogon zur Helmholtz-Zerlegung. Diplomarbeit, Universität Bayreuth, 1987
[4] Griesinger, R.: On the boundary value problem rotu=f inL q. Ann. Univ. Ferrara, Nuova Ser., Sez. VII35 (1990) · Zbl 0789.35037
[5] Simader, C.G.: On Dirichlet’s boundary value problem (Lect. Notes Math. vol. 268). Berlin Heidelberg New York: Springer 1972 · Zbl 0242.35027
[6] Simader, C.G., Sohr, H.: The Helmholtz decomposition inL q and related topics. Preprint, 1988
[7] Wahl, W. von: Rudolf Lipschitz-Vorlesungen: Das Außenraumproblem für die instationären Navier-Stokes-Gleichungen. In: Vorlesungsreihe des Sonderforschungsbereichs 256, Institut für Angewandte Mathematik der Universität Bonn 1990
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