Core Concepts
Establishing functional integral representations for solutions of motion equations in wall-bounded viscous flows using perturbation techniques.
Abstract
The article introduces a method for exact random vortex dynamics in wall-bounded viscous flows. It proposes numerical schemes and demonstrates convergence with additional force terms. The study focuses on 2D flows past flat plates, extending to 3D with modifications. Key highlights include the random vortex method's application, convergence proof for numerical schemes, and new tools for studying viscous flows near boundaries.
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Introduction
- Establishing Feynman-Kac formulas for incompressible fluid flow past solid walls.
- Devoted to 2D flows past flat plates with applicability to 3D flows.
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Random Vortex Method
- Introduces a stochastic integral representation for unconstrained flows.
- Derives numerical schemes based on interacting SDEs.
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Mean-Field Equation Analysis
- Proves well-posedness of mean-field random vortex dynamics.
- Demonstrates robust approximation of velocity by dynamics with regularized kernels.
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Assumptions and Notations
- Defines conventions and assumptions regarding drift functions and kernel properties.
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Lipschitz Estimate
- Provides a key Lipschitz estimate for the operator K based on drift function bounds.
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Cameron-Martin Theorem
- Summarizes the Radon-Nikodym derivative theorem for diffusion measures.
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Theorem Proof
- Proves the existence of fixed points using Banach fixed point argument.
Stats
"Numerical experiments are carried out"
"Several numerical schemes are proposed"
"Exact random vortex model is put forward"