ON FACTORIZATION OF A DIFFERENTIAL OPERATOR ARISING IN FLUID DYNAMICS

Abstract

Spectral properties of linear operators are very important in stability analysis of
dynamical systems. The paper studies the non-selfadjoint second order differential operator
that originated from a steady state stability problem in dynamic of viscous Newtonian
fluid on the inner surface of horizontally rotating cylinder in the presence of gravitational
field. The linearization of the thin liquid film flow in the lubrication limit about the
uniform coating steady state results into the operator which domain couples two subspaces
spanned by positive and negative Fourier exponents which are not invariant subspaces of the
operator. We prove that the operator admits factorization and use this new representation
of the operator to prove compactness of its resolvent and to find its domain.