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

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.