Stability of solutions to generalized Forchheimer equations of any degree
type: Item
instance of: scholarly article
title: Stability of solutions to generalized Forchheimer equations of any degree
zbMATH Open document ID: 1341.35117
author: Thinh T. Kieu
author: Luan Hoang
author: Akif Ibragimov
author: Zeev Sobol
published in: Journal of Mathematical Sciences (New York)
publication date: 2016-01-19T00:00:00Z
full work available at URL: http://hdl.handle.net/11299/181200
review text: The generalized Forchheimer equation \(g(|u|)u = -\nabla p\) is considered, where \(u\) is the velocity of the fluid in a porous medium, \(\nabla p\) is the pressure gradient, and \(g(s)\) is a polynomial of any degree. This equation can be inverted to \(u = -K(|\nabla p|) \nabla p\). The authors study the initial value problem for the parabolic differential equation in the form \[ \frac{\partial p}{\partial t} = \nabla \cdot (K(|\nabla p|) \nabla p). \] They show that the solution of this equation continuously depends on the boundary data and the coefficients of the Forchheimer polynomial in the spaces \(L^{\alpha}\), \(\alpha \geq 1\), and \(W^{1,2-a}\), \(a = \frac{\mathrm{deg}(g)}{(1 + \mathrm{deg}(g))}\). The results are obtained for the DC case, when the technical degree condition \(\mathrm{deg}(g) \leq \frac{n}{n - 2}\) holds, and for the opposite NDC case.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 76N99
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 6532741
zbMATH Keywords: generalized Forchheimer equation
zbMATH Keywords: slightly compressible fluid
zbMATH Keywords: stability
zbMATH Keywords: Poincaré-Sobolev inequalities
reviewed by: Natalia Bondarenko
MaRDI profile type: Publication
OpenAlex ID: W1781385145
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DOI: 10.1007/S10958-015-2576-1
description: scientific article; zbMATH DE number 6532741