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Transactions of the American Mathematical Society

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Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics

Authors: N. D. Kopachevsky, R. Mennicken, Ju. S. Pashkova and C. Tretter
Journal: Trans. Amer. Math. Soc. 356 (2004), 4737-4766
MSC (2000): Primary 35A05; Secondary 35Q30, 47D06, 47F05, 47B44, 47B50
Published electronically: June 29, 2004
MathSciNet review: 2084396
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Abstract: We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space ${\mathcal H}$ of the form

\begin{displaymath}\frac{d^2u}{dt^2}+(F+{\rm i}K)\frac{du}{dt}+Bu=f,\quad u(0)=u^0, \quad u'(0)=u^1. \end{displaymath}

Problems of this kind arise, e.g., in hydrodynamics where the coefficients $F$, $K$, and $B$ are unbounded selfadjoint operators. It is assumed that $F$ is the dominating operator in the Cauchy problem above, i.e.,

\begin{displaymath}{\mathcal D}(F)\subset{\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \end{displaymath}

We also suppose that $F$ and $B$ are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.

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Additional Information

N. D. Kopachevsky
Affiliation: Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine

R. Mennicken
Affiliation: NWF I – Mathematik, University of Regensburg, 93040 Regensburg, Germany

Ju. S. Pashkova
Affiliation: Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine

C. Tretter
Affiliation: FB 3 – Mathematik, University of Bremen, Bibliothekstr. 1, 28359 Bremen, Germany

Keywords: Block operator matrix, differential equation in Hilbert space, evolution problem, Navier--Stokes equations
Received by editor(s): January 29, 2002
Published electronically: June 29, 2004
Additional Notes: N. D. Kopachevsky, R. Mennicken, and C. Tretter gratefully acknowledge the support of the German Research Foundation, DFG, Grants No. 436UKR113/38/0 and No. TR368/4-1, and of the British Engineering and Physical Sciences Research Council, EPSRC, Grant No. GR/R40753.
Article copyright: © Copyright 2004 American Mathematical Society

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