ABSTRACT
This paper aims to present some recent analytic results for a class of moving boundary problems in several space dimensions. Typically, these equations arise in laminar flow of fluids through porous media. More precisely, we consider the following situation: Let Г0 denote a fixed, impermeable layer in a homogeneous and isotropic porous medium. We assume that some portion of the solid matrix above Г0 is occupied by an incompressible Newtonian fluid. In addition, we suppose that there is a sharp interface Гf, separating the wet region Ω f ; enclosed by Г0 and Гf, respectively, from the dry part, i.e. we consider a saturated fluid-air flow. In order to illustrate the above situation, let us introduce the following class of admissible interfaces: https://www.w3.org/1998/Math/MathML"> A 0 ≔ { f ∈ B C 2 ( R n , R ) ; inf x ϵ R n f ( x ) > 0 } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math448.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where https://www.w3.org/1998/Math/MathML"> n ∈ N , n ≥ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math294.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , is fixed. Given https://www.w3.org/1998/Math/MathML"> f ∈ A 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math295.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , let
https://www.w3.org/1998/Math/MathML"> Ω f ≔ { ( x , y ) ∈ R n × ( 0 , ∞ ) ; 0 < y < f ( x ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math296.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/fig9_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>