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.
Highlights
The governing equations of an incompressible flow of a Newtonian fluid are described by the Navier–Stokes equations [43]
We study the stochastic version of operator differential algebraic equations (DAEs) considered in the framework of white noise analysis and chaos expansions of generalized stochastic processes [18,21,39]
The same ideas can be applied to our equations once we have performed the regularization to the deterministic system, the convergence of the truncated expansion is, in general, guaranteed by the stability result of Theorem 6
Summary
The governing equations of an incompressible flow of a Newtonian fluid are described by the Navier–Stokes equations [43]. We study the stochastic version of operator DAEs considered in the framework of white noise analysis and chaos expansions of generalized stochastic processes [18,21,39]. We combine the polynomial chaos expansion approach from the white noise theory with the deterministic theory of operator DAEs. in the fluid flow case, we deal with the stochastic equations of the form u(t) − u(t) + ∇ p(t) = F(t) + “noise”, div u(t). For this reason, in the present paper, we develop a general abstract setting based on white noise analysis and chaos expansions. 3. we discuss stochastic noise terms in the differential as well as in the constraint equation and the systems which result from the chaos expansions, Theorems 6, 8 and 9. We discuss extensions of our results to specific types of nonlinear equations
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