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Stochastic forcing of the linearized Navier-Stokes equations. (English) Zbl 0809.76078
This paper studies a stochastically driven Navier-Stokes equations, linearized about a mean shear flow: \[ {d\over dt} x = A x + F \xi; \] with \(\xi = {d \over dt} W\) the white noise. Since the operator \(A\) is non- normal, the operators \(B^ t = \int_ 0^ t \exp ((t - s)A^ {\dagger}) \exp ((t - s) A)ds\) and \(C^ t = \int_ 0^ t \exp ((t - s) A) \exp ((t - s)A^ {\dagger})ds\) are different (assuming \(FF^ {\dagger} = 1\)) and neither of the eigenanalysis for \(B^ {\infty}\) or \(C^{\infty}\) is to be identified with the normal modes of the system. The Karhunen-Loéve decomposition of \(C^{\infty}\) determines the response structures of the system, while the Karhunen-Loéve decomposition of \(B^{\infty}\) determines the contributions to the variance of the statistically steady state. There is found a great amplification of perturbation variance for Couette and Poiseuille flows, from relatively small intrinsic and extrinsic forcing disturbances.
Reviewer: W.-Z.Yang (Taipei)

MSC:
76M35 Stochastic analysis applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
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