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Existence Theory for Stochastic Power Law Fluids

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Abstract

We consider the equations of motion for an incompressible non-Newtonian fluid in a bounded Lipschitz domain \({G {\subset} \mathbb{R}^{d}}\) during the time interval (0, T) together with a stochastic perturbation driven by a Brownian motion W. The balance of momentum reads as

$${\rm {d}\mathbf{v}} = {\rm div}\mathbf{S} {\rm d}t-(\nabla \mathbf{v})\mathbf{v} {\rm d}t + \nabla\pi {\rm d}t + \mathbf{f}{\rm d}t + \Phi(\mathbf{v}) {\rm d}\mathbf{W}_{t},$$

where v is the velocity, \({\pi}\) the pressure and f an external volume force. We assume the common power law model \({\mathbf{S}(\varepsilon(\mathbf{v}))=(1+|\varepsilon(\mathbf{v})|)^{p-2}\varepsilon(\mathbf{v})}\) and show the existence of martingale weak solution provided \({p > \frac{2d+2}{d+2}}\). Our approach is based on the \({L^{\infty}}\)-truncation and a harmonic pressure decomposition which are adapted to the stochastic setting.

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Authors and Affiliations

  1. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Dominic Breit

  2. Mathematical Institute, LMU Munich, Theresienstraβe 39, 80333, Munich, Germany

    Dominic Breit

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  1. Dominic Breit
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Correspondence to Dominic Breit.

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Communicated by H. Beirão da Veiga

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Breit, D. Existence Theory for Stochastic Power Law Fluids. J. Math. Fluid Mech. 17, 295–326 (2015). https://doi.org/10.1007/s00021-015-0203-z

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