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.
References
Azizov T.Ya., Iokhvidov I.S. Linear Operators in Spaces with Indefinite Metric. N.Y., Wiley, 1989.
Benilov E.S., O'Brien S.B.G., Sazonov I.A. A New Type of Instability: Explosive Disturbances in a Liquid Fild Inside a Rotating Horizontal Cylinder. J. Fluid Mech., 2003, vol. 497, pp. 201-224.
Benilov E.S., Kopteva N., O'Brien S.B.G. Does Surface Tension Stabilise Liquid Films Inside a Rotating Horizontal Cylinder. Q. J. Mech. Appl. Math., 2005, vol. 58, pp. 158-200.
Boulton L., Levitin M., Marletta M. On a Class of Non-Self-Adjoint Periodic Eigenproblems with Boundary and Interior Singularities. J. of Differential Equations, 2010, vol. 249, no. 12, pp. 3081-3098.
Chugunova M., Karabash I.M., Pyatkov S.G. On the Nature of Ill-Posedness of the Forward-Backward Heat Equation. Integral Equations and Operator Theory, 2009, vol. 65, pp. 319-344.
Chugunova M., Pelinovsky D. Spectrum of a Non-Self-Adjoint Operator Associated with the Periodic Heat Equation. J. Math. Anal. Appl., 2008, vol. 342, pp. 970-988.
Chugunova M., Strauss V. Factorization of the Indefinite Convection-Diffusion Operator. C. R. Math. Rep. Acad. Sci. Canada, 2008, vol. 30, no. 2, pp. 40-47.
Davies E.B. An Indefinite Convection-Diffusion Operator. LMS J. Comp. Math., 2007, vol. 10, pp. 288-306.
Weir J. An Indefinite Convection-Diffusion Operator with Real Spectrum. Appl. Math. Lett., 2008, vol. 22, pp. 280-283.
