Petr Girg, Lukas Kotrla
Abstract:
We propose a new mathematical model of groundwater flow in porous medium
layered over inclined impermeable beds.
In its full generality, this is a free-surface problem.
To obtain analytically tractable model, we use generalized
Dupuit-Forchheimer assumption for inclined impermeable bed.
In this way, we arrive at parabolic partial differential equation which
is a generalization of the classical Boussinesq equation.
The novelty of our approach consists in considering nonlinear constitutive
law of the power type. Thus introducing \(p\)-Laplacian-like differential
operator into the Boussinesq equation. Unlike in the classical case of
the Boussinesq equation, the convective term cannot be set aside from the
main part of the diffusive term and remains incorporated within it.
In this article, we analyze qualitative properties of the stationary
solutions of our model. In particular, we study the existence and regularity
of weak solutions for the boundary value problem
$$\displaylines{
-\frac{\rm d}{{\rm d} x}
\Big[(u(x) + H) |\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi)
+ \sin(\varphi) |^{p - 2}
\Big(\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi)
+ \sin(\varphi)\Big)\Big]
= f(x), \quad\ x \in (-1,1)\,,
\cr
u(-1) = u(1) = 0\,,
}$$
where \(p>1\), \(H>0\), \(\varphi\in (0, \pi/2)\), \(f\geq 0\),
\(f\in L^{1}(-1,1)\).
In the case of \(p>2\), we study validity of Weak and Strong Maximum
Principles as well.
We use methods based on the linearization of the
\(p\)-Laplacian-type problems in the vicinity of known solution,
error estimates, and analysis of Green's function of the linearized problem.
Submitted January 3, 2025. Published May 30, 2025.
Math Subject Classifications: 76S05, 35Q35,34B15, 34B27.
Key Words: Porous medium; filtration; nonlinear Darcy's law; \(p\)-Laplacian;
pressure-to-velocity power law.
DOI: 10.58997/ejde.2025.57
Show me the PDF file (717 KB), TEX file for this article.
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Petr Girg Department of Mathematics and NTIS Faculty of Applied Scences University of West Bohemia Univerzitní 8, CZ-301 00 Plzen, Czech Republic email: pgirg@kma.zcu.cz |
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Lukás Kotrla Department of Mathematics and NTIS Faculty of Applied Scences University of West Bohemia Univerzitní 8, CZ-301 00 Plzen, Czech Republic email: kotrla@ntis.zcu.cz |
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