Operator differential-algebraic equations with noise arising in fluid dynamics

Area of Science:

• Numerical Analysis
• Stochastic Differential Equations
• Fluid Dynamics

Background:

• Linear semi-explicit stochastic operator differential algebraic equations (DAEs) present challenges, particularly when constraint equations are explicit.
• The Stokes equations in fluid dynamics are a key example requiring robust solution methods for stochastic perturbations.

Purpose of the Study:

• To develop a method for analyzing linear semi-explicit stochastic operator DAEs with explicit constraints.
• To incorporate Gaussian noise and stochastic convolution terms into both differential and constraint equations.
• To establish the existence and uniqueness of solutions for these complex systems.

Main Methods:

• White noise polynomial chaos expansion to model stochastic perturbations.
• Deterministic regularization techniques applied to address sensitivity in constraint equations.
• Reduction of stochastic operator DAEs to infinite systems of deterministic operator DAEs.

Main Results:

• Successfully incorporated Gaussian noise and stochastic convolution terms.
• Demonstrated the reduction to an infinite system of deterministic DAEs.
• Proved the existence and uniqueness of solutions in a weighted space of stochastic processes for a regularized system.

Conclusions:

• The combined approach of polynomial chaos expansion and regularization effectively handles stochastic perturbations in DAEs.
• The method provides a rigorous framework for analyzing and solving stochastic differential algebraic equations, with implications for fluid dynamics.
• Existence and uniqueness of solutions are confirmed under specific conditions, advancing the understanding of these mathematical models.