## On Factorization of a Perturbation of a J-selfadjoint Operator Arising in Fluid Dynamics. (2008)

### BibTeX

@MISC{Chugunova08onfactorization,

author = {Marina Chugunova and Vladimir Strauss},

title = {On Factorization of a Perturbation of a J-selfadjoint Operator Arising in Fluid Dynamics.},

year = {2008}

}

### OpenURL

### Abstract

Abstract: We prove that some perturbation of a J-selfadjoint second order differential operator admits factorization and use this new representation of the operator to prove compactness of its resolvent and to find its domain. AMS classification codes: primary 47B10, 34L40; secondary 35M10. Keywords: factorization, Krein space, J-self-adjoint, fluid mechanics, forward-backward heat equation

### Citations

10 | Foundations of the theory of linear operators in spaces with an indefinite metric - Azizov - 1986 |

10 | An indefinite convection-diffusion operator
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- 2007
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Citation Context ...ions. The spectrum of the linear operator L that is defined by the operation l[.] and periodic boundary conditions y(−π) = y(π) for the special case when the parameter a = 0 was studied rigorously in =-=[8, 6, 9]-=-. Using different approaches they justified that if the parameter b restricted to the interval [0, 2] then the operator L is well defined in the sense that it admits closure in L 2 (−π, π) with non-em... |

10 | An indefinite convection-diffusion operator with real spectrum
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9 | and D Pelinovsky: Spectrum of an non-self-adjoint operator associated with the cylindric heat equation
- Chugunova
- 2007
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Citation Context ...ions. The spectrum of the linear operator L that is defined by the operation l[.] and periodic boundary conditions y(−π) = y(π) for the special case when the parameter a = 0 was studied rigorously in =-=[8, 6, 9]-=-. Using different approaches they justified that if the parameter b restricted to the interval [0, 2] then the operator L is well defined in the sense that it admits closure in L 2 (−π, π) with non-em... |

8 | A PTsymmetric periodic problem with boundary and interior singularities. arXiv:0801.0172
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Citation Context ...lues only. As a result all eigenfunctions have the following symmetry yλ(−x) = yλ(x). The more general operator with the function sin(x) replaced by the arbitrary 2π-periodic functions was studied in =-=[4]-=- and it was proved that this operator multiplied by i belongs to a wide class of PT-symmetric operators which are not similar to self-adjoint but nevertheless possesses purely real spectrum due to som... |

4 | Factorization of the Indefinite Convection-Diffusion Operator
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Citation Context ...al of this paper is to find the factorization of the operator L (under some restrictions on parameters a and b) that would be in some sense similar to one we constructed for the special case a = 0 in =-=[7]-=- (in this case the operator L is J-self-adjoint with the operator J defined as a shift J(f(x)) = f(π − x)) and to examine some properties of the operator L using this factorization. The main difficult... |

3 |
Does surface tension stabilise liquid films inside a rotating horizontal cylinder
- Benilov, Kopteva, et al.
- 2005
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Citation Context ... evolution model of a thin film of liquid on the inner surface of a cylinder rotating in a gravitational field was based on the lubrication approximation and examined by Benilov, O’Brien, and Sazonov =-=[2, 3]-=-. The related Cauchy problem has the following form: yt + l[y] = 0, y(0, x) = y0, y(−π, t) = y(π, t), x ∈ [−π, π], t > 0 (1.1) where l[y] = d ( (1 − a cosx)y(x) + b sin x · d x ) d y(x) , a, b > 0 (1.... |

3 |
On the nature of ill-posedness of the forwardbackward heat equation”, preprint, arXiv:0803.2552v2 [math.AP
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Citation Context ...he ceiling of the cylinder where the effect of the gravity is the strongest) was studied analytically and explained in terms of the absence of the Riesz basis property of the set of eigenfunctions in =-=[5]-=-. The question of a conditional basis property of the set of eigenfunction is still open. For the case when a ̸= 0, as it was discussed in [3], the spectral properties of the operator L are not expect... |