Abstract
This paper is concerned with the fluid dynamics modeled by the stochastic flow
where the turbulent term is driven by the white noise \(\dot W\). The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler and Navier-Stokes equations for the undetermined components u(t,x) and σ(t,x) of the spatial velocity field are derived from first principles. The resulting equations include as particular cases the deterministic Navier-Stokes and Euler equations as well as these equations with stochastic forcing We also discuss existence of global weak solutions of the stochastic Navier-Stokes equation in Rd for d = 2, 3. Uniqueness of a solution is established in the case d=2.
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Mikulevicius, R., Rozovskii, B. (2001). On Equations of Stochastic Fluid Mechanics. In: Hida, T., Karandikar, R.L., Kunita, H., Rajput, B.S., Watanabe, S., Xiong, J. (eds) Stochastics in Finite and Infinite Dimensions. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0167-0_15
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DOI: https://doi.org/10.1007/978-1-4612-0167-0_15
Publisher Name: Birkhäuser, Boston, MA
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