0000000000368418

AUTHOR

Konstantin A. Makarov

showing 3 related works from this author

The Tan 2Θ Theorem in fluid dynamics

2017

We show that the generalized Reynolds number (in fluid dynamics) introduced by Ladyzhenskaya is closely related to the rotation of the positive spectral subspace of the Stokes block-operator in the underlying Hilbert space. We also explicitly evaluate the bottom of the negative spectrum of the Stokes operator and prove a sharp inequality relating the distance from the bottom of its spectrum to the origin and the length of the first positive gap.

Spectral subspacePhysics35Q35 47A67 (Primary) 35Q30 47A12 (Secondary)Spectrum (functional analysis)Mathematical analysisHilbert spaceReynolds numberStatistical and Nonlinear PhysicsMathematics - Spectral TheoryMathematics - Functional AnalysisPhysics::Fluid Dynamicssymbols.namesakeFluid dynamicssymbolsGeometry and TopologyStokes operatorNavier–Stokes equation ; Stokes operator ; Reynolds number ; rotation of subspaces ; quadratic forms ; quadratic numerical rangeRotation (mathematics)Mathematical Physics
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Representation Theorems for Indefinite Quadratic Forms Revisited

2010

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensuring the second representation theorem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

Pure mathematicsGeneral MathematicsFOS: Physical sciencesMathematical proofDirac operator01 natural sciencesMathematics - Spectral Theorysymbols.namesakeOperator (computer programming)Simple (abstract algebra)0103 physical sciencesFOS: Mathematics0101 mathematicsSpectral Theory (math.SP)Mathematical PhysicsMathematicsRepresentation theorem010102 general mathematicsRepresentation (systemics)Mathematical Physics (math-ph)16. Peace & justice47A07 47A55 15A63 46C20Functional Analysis (math.FA)Mathematics - Functional AnalysisTheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGESsymbolsIndefinite quadratic forms ; representation theorems ; perturbation theory ; Krein spaces ; Dirac operator010307 mathematical physicsPerturbation theory (quantum mechanics)
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Diagonalization of indefinite saddle point forms

2020

We obtain sufficient conditions that ensure block diagonalization (by a direct rotation) of sign-indefinite symmetric sesquilinear forms as well as the associated operators that are semi-bounded neither from below nor from above. In the semi-bounded case, we refine the obtained results and, as an example, revisit the block Stokes operator from fluid dynamics.

Saddle pointMathematical analysisFluid dynamicsBlock (permutation group theory)Perturbation theory (quantum mechanics)Stokes operatorRotation (mathematics)Mathematics
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