Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	N. D. Kopachevsky R. Mennicken Ju. S. Pashkova C. Tretter

Affiliation:	Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine ; NWF I -- Mathematik, University of Regensburg, 93040 Regensburg, Germany ; Taurida National V. Vernadsky University, Ul. Yaltinskaya, 4, 95007 Simferopol, Crimea, Ukraine ; FB 3 -- Mathematik, University of Bremen, Bibliothekstr. 1, 28359 Bremen, Germany

Abstract:	We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space 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 , , and are unbounded selfadjoint operators. It is assumed that is the dominating operator in the Cauchy problem above, i.e., We also suppose that and 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.

Keywords:	Block operator matrix differential equation in Hilbert space evolution problem Navier--Stokes equations

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