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Preprint 31-2015

Operator Differential Algebraic Equations with Noise Arising in Fluid Dynamics

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Author(s) : Robert Altmann , Tijana Levajkovic , Hermann Mena

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 31-2015

MSC 2000

65J10 Equations with linear operators
60H40 White noise theory
60H30 Applications of stochastic analysis
35R60 Partial differential equations with randomness

Abstract :
We study linear semi-explicit stochastic operator differential-algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

Keywords : operators DAE, noise disturbances, chaos expansions, Ito-Skorokhod integral, stochastic convolution, regularization

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