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The pressure equation for fluid flow in a stochastic medium

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Abstract

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R d of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation

$$\left\{ {\begin{array}{*{20}c} {div(k(x,{\mathbf{ }}\omega )\diamondsuit \nabla p(x,\omega )){\mathbf{ }} = {\mathbf{ }}--f(x);{\mathbf{ }}x \in D} \\ {\begin{array}{*{20}c} {p(x,{\mathbf{ }}\omega ){\mathbf{ }} = {\mathbf{ }}0;} & {x \in \partial D} \\ \end{array} } \\ \end{array} } \right.$$

wheref is the given source rate of the fluid, ◊ denotes Wick product.

We representk as the positive noise given by the Wick exponential of white noise, and we find an explicit formula for the (unique) solutionp(x, w), which is proved to belong to the space (S)−1 of generalized white noise distributions.

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Authors and Affiliations

  1. Dept. of Mathematical Sciences, Norwegian Institute of Technology, University of Trondheim, N-7034, Trondheim, Norway

    H. Holden

  2. Dept. of Mathematics, University of Oslo, Blindern, Box 1053, N-0316, Oslo, Norway

    T. Lindstrøm

  3. VISTA/Dept. of Mathematics, University of Oslo, Blindern, Box 1053, N-0316, Oslo, Norway

    B. Øksendal

  4. Dept. of Mathematics, Stord/Haugesund College, Skåregt. 103, N-5500, Haugesund, Norway

    J. Ubøe & T. -S. Zhang

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  1. H. Holden
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  4. J. Ubøe
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  5. T. -S. Zhang
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Holden, H., Lindstrøm, T., Øksendal, B. et al. The pressure equation for fluid flow in a stochastic medium. Potential Anal 4, 655–674 (1995). https://doi.org/10.1007/BF02345830

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