Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics
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- by N. D. Kopachevsky, R. Mennicken, Ju. S. Pashkova and C. Tretter PDF
- Trans. Amer. Math. Soc. 356 (2004), 4737-4766 Request permission
Abstract:
We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space ${\mathcal H}$ of the form \[ \frac {d^2u}{dt^2}+(F+\textrm {i}K)\frac {du}{dt}+Bu=f,\quad u(0)=u^0, \quad u’(0)=u^1. \] 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., \[ {\mathcal D}(F)\subset {\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \] 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.References
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Additional Information
- N. D. Kopachevsky
- Affiliation: Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine
- Email: kopachevsky@tnu.crimea.ua
- R. Mennicken
- Affiliation: NWF I – Mathematik, University of Regensburg, 93040 Regensburg, Germany
- Email: reinhard.mennicken@mathematik.uni-regensburg.de
- Ju. S. Pashkova
- Affiliation: Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine
- Email: kromsh@crimea.com
- C. Tretter
- Affiliation: FB 3 – Mathematik, University of Bremen, Bibliothekstr. 1, 28359 Bremen, Germany
- Email: ctretter@math.uni-bremen.de
- 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. 436 UKR 113/38/0 and No. TR368/4-1, and of the British Engineering and Physical Sciences Research Council, EPSRC, Grant No. GR/R40753.
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4737-4766
- MSC (2000): Primary 35A05; Secondary 35Q30, 47D06, 47F05, 47B44, 47B50
- DOI: https://doi.org/10.1090/S0002-9947-04-03693-1
- MathSciNet review: 2084396