Mathematical Modelling

On factorization of a differential operator arising in fluid dynamics

M. Chugunova^a, V. Strauss<sup>b

a</sup> Institute of Mathematical Sciences, Claremont Graduate University, Claremont, USA
^b Department of Pure & Applied Mathematics, Simón Bolívar University, Caracas, Venezuela

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.

Keywords: factorization, lubrication approximation, fluid mechanics, forward-backward heat equation.

UDC: 517.984

MSC: Primary 47B10, 34L40; Secondary 35M10

Received: 28.05.2013

Language: English