Abstract
In this work we deal with the Cauchy problem associated to the Brinkman flow, which models fluid flow in certain types of porous media. We study local and global well-posedness in Sobolev spaces Hs(Rn), s>n/2+1, using Kato's theory for quasilinear equations and parabolic regularization.
Highlights
In this article we are interested in the properties of the real valued solutions to the Cauchy problem associated to the Brinkman flow ([4],[28])
K, and μeff denote the fluid viscosity, the porous media permeability and the pure fluid viscosity, respectively, while ρ is the fluid’s density, v its velocity, P (ρ) is the pressure, F is an external mass flow rate, and φ is the porosity of the medium
V = − (1 − ∆)−1 ∇P (ρ), and substitute into the first one to obtain the Cauchy problem for the Brinkman flow equation (BFE)
Summary
In this article we are interested in the properties of the real valued solutions to the Cauchy problem associated to the Brinkman flow ([4],[28]). V = − (1 − ∆)−1 ∇P (ρ) , and substitute into the first one (which describes the variation of mass) to obtain the Cauchy problem for the Brinkman flow equation (BFE). In our case we need to obtain a Comparison Principle for the solutions (see Section 5) to obtain the global estimates in Hs(Rn), n > 1.
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