\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 16, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/16\hfil Origin of the $p$-Laplacian]
{Origin of the $p$-Laplacian and A. Missbach}

\author[J. Benedikt, P. Girg, L. Kotrla, P. Tak\'a\v{c}\hfil EJDE-2018/16\hfilneg]
{Ji\v{r}\'i Benedikt, Petr Girg,  Luk\'{a}\v{s} Kotrla, Peter Tak\'a\v{c}}

\address{Ji\v{r}\'i Benedikt \newline
Department of Mathematics and NTIS,
Faculty of Applied Scences, University of West Bohemia,
Univerzitn\'{\i} 8, CZ-306\,14~Plze\v{n}, Czech Republic}
\email{benedikt@kma.zcu.cz}

\address{Petr Girg \newline
Department of Mathematics and NTIS,
Faculty of Applied Scences, University of West Bohemia,
Univerzitn\'{\i} 8, CZ-306\,14~Plze\v{n}, Czech Republic}
\email{pgirg@kma.zcu.cz}

\address{Luk\'a\v{s} Kotrla \newline
Department of Mathematics and NTIS,
Faculty of Applied Scences, University of West Bohemia,
Univerzitn\'{\i} 8, CZ-306\,14~Plze\v{n}, Czech Republic}
\email{kotrla@ntis.zcu.cz}

\address{Peter Tak\'a\v{c} \newline
Institut f\"ur Mathematik, Universit\"at Rostock, Germany}
\email{peter.takac@uni-rostock.de}


\dedicatory{Communicated by Ratnasingham Shivaji}

\thanks{Submitted October 22, 2017. Published January 13, 2018.}
\subjclass[2010]{76S05, 35Q35}
\keywords{Porous medium; filtration; nonlinear Darcy law;
\hfill\break\indent pressure-to-velocity power law; $p$-Laplacian}

\begin{abstract}
 We describe the historical process of derivation of
 the $p$-Laplace operator from a non\-linear Darcy law and
 the continuity equation.
 The story begins with non\-linear flows in channels and ditches.
 As the non\-linear Darcy law we use the power law discovered by
 Smreker and verified in experiments by Missbach
 for flows through porous media in one space dimension.
 These results were generalized by  Christianovitch and Leibenson
 to porous media in higher space dimensions.
 We provide a brief description of Missbach's experiments.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction} \label{s:Intro}

The authors of this article have often been confronted with
the question on the origin of the $p$-Laplace operator.
The main goal of the present work is to answer this question
at satisfactory technical and historical levels.
We do not attempt to provide or claim complete answers to many questions
that arise in our investigation of the available resources.
In particular, we leave the question of {\em competitiveness\/}
of mathematical models with the $p$-Laplacian to
alternative mathematical models still
widely open in practical applications
\cite{QuinnCherryParker2011, Stark-Volker}.


\section{The Filtration Problem and the Equation}
\label{s:Filer_Probl}

An important task of hydrodynamics engineering throughout
the $18^\mathrm{th}$ century was to build reliable water supplies
for fast growing urban centers.
The need for water sparked a number new directions
in theoretical research on hydrodynamics and hydrology.
Numerous interesting mathematical problems in this area
are derived and formulated in the monograph by
 Jacob Bear \cite{Bear_1972}.
Among them we are interested in
{\it filtration of fluids through porous media\/} and
{\it unsaturated flow\/}; see
\cite[Sect.\ 5.2, 5.10, 5.11]{Bear_1972} and
\cite[Sect.\ 9.4]{Bear_1972}, respectively.
A mathematical model for such phenomena is presented in
J.~I.\ D\'{\i}az and  F.\ de~Th\'elin
\cite{Diaz-Thelin}.
It is described by the following {\em nonlinear\/}
initial\--boundary value problem of parabolic type
for the unknown function $u = u(x,t)$ of the space and time variables,
$x$ and $t$, respectively:
%
\begin{equation}
\label{e:Diaz}
\begin{gathered}
    \frac{\partial}{\partial t}\, b(u)
  - \operatorname{div}
    \phi\bigl( \nabla u - K(b(u))\mathbf{e}\bigr)
  + g(x,u)  = f(x,t)
  \quad\text{in } \Omega\times (0,\infty) \,,\\
  u(x,t)  = 0
  \quad\text{on } \partial\Omega\times (0,\infty) \,,\\
  b(u(x,0))  = b(u_0(x))
  \quad\text{in } \Omega \,.
\end{gathered}
\end{equation}
Here, $\Omega\subset \mathbb{R}^N$ is a bounded open subset of
the $N$-dimensional Euclidean space $\mathbb{R}^N$
with sufficiently smooth boundary $\partial\Omega$,
$b:\mathbb{R}\to \mathbb{R}$,
$K:\mathbb{R}\to \mathbb{R}$, and
$g(x,\,\cdot\,): \mathbb{R}\to \mathbb{R}$
are continuous functions satisfying some additional hypotheses
(\cite[Sect.~1]{Diaz-Thelin}), such as
$b$ being monotone increasing with $b(0) = 0$,
$\mathbf{e}$ denotes a given unit vector in $\mathbb{R}^N$,
and for some $1 < p < \infty$,
\begin{equation}\label{e:p-Laplace}
    \phi(\boldsymbol{\zeta})
  = |\boldsymbol{\zeta}|^{p-2} \boldsymbol{\zeta}
  \quad\text{for every } \boldsymbol{\zeta}\in \mathbb{R}^N \,.
\end{equation}
As usual, $t\in \mathbb{R}_+:= [0,\infty)$.
Finally,
$f: \Omega\times (0,\infty)\to \mathbb{R}$
is (typically) a Lebesgue\--integrable function standing for
sources (if $f(x,t) > 0$) and sinks (if $f(x,t) < 0$), whereas
$u_0: \Omega\to \mathbb{R}$
stands for the prescribed initial data, usually assumed to be
Lebesgue\--measurable and (essentially) bounded.

For filtration of fluids through porous media in laminar regime
one begins with
\begin{gather} \label{e:cont_eq}
\text{the continuity equation}\qquad
 \frac{\partial\theta}{\partial t}
  + \operatorname{div} \mathbf{v} = 0\\
\label{e:p-Darcy}
\text{and the Darcy law}\qquad
\mathbf{v} = - K(\theta)\, \nabla\Phi(\theta) \,,
\end{gather}
where $\theta = \theta(x,t)$ is the volumetric moisture content,
$K = K(\theta)$ is the hydraulic conductivity, and the potential
$\Phi$ is given by
$\Phi(\theta) = \psi(\theta) + z$
with $\psi(\theta)$ being the hydrostatic potential and
$z$ the gravitational potential.
For instance, if $N=3$ then we fix the unit vector
$\mathbf{e} = (0,0,-1)\in \mathbb{R}^N$
in the direction opposite (but parallel) to the gravitational force,
perpendicular to the horizontal plane $(x_1,x_2,0)$,
so that the gravitational potential
$z = z(x) =  g x_3 + \mathrm{const}$ at the point
$x = (x_1,x_2,x_3)\in \mathbb{R}^3$ yields the gravitational force
\begin{equation*}
  \mathbf{G} = \mathbf{G}(x) =- \nabla z =
  \big( 0, 0,\, {}- \genfrac{}{}{}1{\partial z}{\partial x_3}
  \big)
  = (0,0,-g) =- g\mathbf{e}\in \mathbb{R}^3 \,.
\end{equation*}
To simplify our notation,
we normalize the gravitational constant to one, $g=1$; hence,
$\mathbf{G} =- \nabla z =- \mathbf{e}\in \mathbb{R}^3$.
Thus, we obtain
%
\begin{equation*}
  \nabla\Phi(\theta) = \psi'(\theta)\, \nabla\theta - \mathbf{e}
\end{equation*}
%
which, after being inserted into Darcy's law \eqref{e:p-Darcy}, yields
%
\begin{equation}\label{eq:p-Darcy}
  \mathbf{v} =- K(\theta)\, \psi'(\theta)\, \nabla\theta
             + K(\theta)\mathbf{e}
  =- \nabla\varphi(\theta) + K(\theta)\mathbf{e}
    \in \mathbb{R}^3
\end{equation}
where
\begin{equation*}
  \varphi(\theta):=
  \int_0^{\theta} K(\vartheta)\, \psi'(\vartheta) \,\mathrm{d}\vartheta
  \quad\text{ for }\, \theta\in \mathbb{R} \,.
\end{equation*}
%
In general, the vector field $\mathbf{v}$ stands for the seepage flow
which, in our applications, will be proportional to the fluid velocity,
thus denoted by~$\mathbf{v}$.
It is reasonable to assume
$K(\vartheta) > 0$ and $\psi'(\vartheta) > 0$
(see  Bear \cite{Bear_1972}),
so that also $\varphi'(\vartheta) > 0$ holds.
As a consequence,
$\varphi: \mathbb{R}\to \mathbb{R}$
is a strictly monotone increasing, continuously differentiable function.

Beginning in the $1870$s,
many engineers concerned with fluid dynamics
(including the works in
 \cite{Forch_1886, Forch_1901, Forch_1906, Forch_1914, Forch_1930,
Kroeber_1884,
Leiben_1945-1, Leiben_1945-2, Leiben_1945-3,
Missbach:IV, Missbach:V, Missbach:VI, Missbach:VII,
Reynolds_1883,
Smreker_1878, Smreker_1879, Smreker_1881, Smreker_1914,
White_1935,Zhukov_1889,
Zunker_1920, Zunker_1930})
have discovered that, if the fluid flow is in {\it turbulent regime\/},
the linear Darcy law \eqref{e:p-Darcy}
does not provide the correct relationship between
the {\it pressure slope\/} (force),
\[
  \mathbf{F} =- \nabla\varphi(\theta) + K(\theta)\mathbf{e} \,,
\]
on the right\--hand side and
the {\it velocity\/}, $\mathbf{v}$,
on the left\--hand side of Darcy's law \eqref{e:p-Darcy}.
 Oscar Smreker
\cite[Eqs.\ (5)--(7), pp.\ 361--362]{Smreker_1879}
shows by rigorous calculations how linear Darcy's law leads
to a contradiction in a practical problem
(dug well, ``Schachtbrunnen'' in German).
Among several ``correction'' alternatives to Darcy's law,
 O.\ Smreker \cite{Smreker_1878, Smreker_1879, Smreker_1881}
suggested the following {\it power law\/}:
%
\begin{equation} \label{e:slope-Darcy}
\mathbf{F} =- K(\theta)\, \nabla\Phi(\theta)
  =- \nabla\varphi(\theta) + K(\theta)\mathbf{e}
\end{equation}
 is given by
\begin{equation} \label{e:p-Darcy_power}
 \mathbf{F} = \mathrm{const}\cdot |\mathbf{v}|^{p'-2} \mathbf{v}
  \quad\text{with some } p'> 2 \,,
\end{equation}
with the power $s = (p'-2) + 1 = p'-1$,
where the multiplicative constant is set to one,
$\mathrm{const} = 1$, for simplicity.
 Smreker's work \cite{Smreker_1878} suggests $p'-1 = 3/2$,
i.e., $p'= 2.5$, whereas
 Reynolds's measurements \cite{Reynolds_1883}
show $p'-1 = 1.723$.
A.~M.\ White \cite{White_1935}
proposed an analogous relation with $p'-1 = 1.8$.
All such corrections to Darcy's law allow only the power range
$1\leq p'-1\leq 2$.
Denoting by $p = p'/(p'-1)$ the conjugate exponent, i.e.,
$\frac{1}{p} + \frac{1}{p'} = 1$, we thus have to deal with the range
$3/2\leq p\leq 2$ and the velocity
%
\begin{equation}\label{e:p-Smreker}
  \mathbf{v} = |\mathbf{F}|^{p-2} \mathbf{F} \,.
\end{equation}
%
Inserting  \eqref{e:slope-Darcy} and \eqref{e:p-Smreker}
into the continuity equation \eqref{e:cont_eq}
we finally arrive at problem \eqref{e:Diaz},
where $b = \varphi_{-1}$ denotes the inverse function to $\varphi$
and $f\equiv g\equiv 0$.

We refer an interested reader to
 J.~I.\ D\'{\i}az and  F.\ de~Th\'elin
\cite{Diaz-Thelin}
for how to obtain problem \eqref{e:Diaz}
in a model dealing with {\it unsaturated flow\/}
(gas flow, typically).
There, $p = 3/2$.

It is now evident, that the {\em $p$-Laplace operator\/} $\Delta_p$,
\begin{equation}
\label{def:p-Laplace}
  \Delta_p u\equiv \operatorname{div}
  \left( |\nabla u|^{p-2} \nabla u\right) \,,
  \quad\text{ for }\, 1 < p < \infty \,,
\end{equation}
is created by the nonlinear power law \eqref{e:p-Darcy_power}
or, equivalently, by \eqref{e:p-Smreker}.
The continuity equation \eqref{e:cont_eq} is standard for both,
linear and nonlinear Darcy's laws.
This means that the origin of the $p$-Laplacian $\Delta_p$
is closely tied to \underline{\em who was the first\/} to plug
the power law \eqref{e:p-Smreker} into the continuity equation
\eqref{e:cont_eq} or at least into its stationary special case
$\operatorname{div} \mathbf{v} = 0$.
There seems to be a wide\--spread agreement in the literature that
the power law \eqref{e:p-Smreker} with $p = 5/3$
was suggested first by
 Oscar Smreker \cite{Smreker_1878}
\footnote[1]{
Ref.\ \cite{Smreker_2008} by  M.\ Bren\v{c}i\v{c} provides
a ``Short description of life and work of  Oskar Smreker''.}
in $1878$ in the equivalent form
\eqref{e:p-Darcy_power} with $p'= 2.5$.
A number of ``power laws'' (with a more general exponent $s = p'-1$)
by various authors followed afterwards.
We will discuss the most important ones in the following two sections.

In this context (``{\em Who was the first?\/}''),
we should mention the articles by
  Smreker \cite{Smreker_1881} from $1881$
(used also in his doctoral dissertation \cite{Smreker_1914} in $1914$)
and by
 N.~E.\ Zhukovskii \cite{Zhukov_1889} from $1889$
(reprinted in his collected works \cite{Zhukovskii} in $1937$),
in which they give the explicit formula for
the radially symmetric solution,
$u(x)\equiv u(|x|)$, of the so\--called
{\it $p$-harmonic equation\/},
$  \Delta_p u = 0 $,
for any $1 < p < \infty$, $p\neq N$,
%
\begin{equation}\label{e:p-harm}
  u(r) = C_0 + \mathrm{const}\cdot r^{1 - \mu}
 \quad\text{ for every }\, r = |x| > 0 \,,
\end{equation}
\[
\mu = \frac{N-1}{p-1}\geq 0, \quad  \mu\neq 1,
\]
see \cite[Eq.~(I), p.~36]{Smreker_1914}
and \cite[Eq.~(13), p.~19]{Zhukovskii}, respectively.
Here, $C_0\in \mathbb{R}$ is a constant;
$C_0 = u(0)$ if $\mu < 1$ and
$C_0 = u(+\infty):= \lim_{r\to +\infty} u(r)$ if $\mu > 1$.
Although this formula is valid in any dimension $N\geq 1$,
both engineers, in \cite{Smreker_1881, Zhukov_1889},
treat only the planar case ($N=2$) given by the hydro\-engineering model.
They had {\em never\/} written down the $p$-harmonic equation
($\Delta_p u = 0$)
explicitly throughout their entire articles
\cite{Smreker_1881, Zhukov_1889};
rather, they preferred to refer to
 Smreker's work in \cite{Smreker_1878} for the power law.
In fact, since every radially symmetric solution
$u(x)\equiv u(|x|)$ to the $p$-harmonic equation
$\Delta_p u = 0$ in the plane ($N=2$) satisfies
the stationary case of the continuity equation \eqref{e:cont_eq},
$\operatorname{div} \mathbf{v} = 0$, which is equivalent to
%
\begin{equation*}
  \Delta_p u(x) = r^{1-N}\cdot
  \frac{\mathrm{d}}{\mathrm{d}r}
  \left( r^{N-1}\, |u'(r)|^{p-2} u'(r)\right) = 0
  \quad\text{for every } r = |x| > 0 \,,
\end{equation*}
both,
 Smreker \cite{Smreker_1881} and
 Zhukovskii \cite{Zhukov_1889},
may have very easily used an alternative way
(e.g., the surface integral over a sphere)
to obtain in the plane ($N=2$),
%
\begin{equation*}
  r\, |u'(r)|^{p-2} u'(r) = \mathrm{const}
  \quad\text{ for all }\, r = |x| > 0 \,,\ x\in \mathbb{R}^2 \,,
\end{equation*}
%
whence \eqref{e:p-harm} follows with $N=2$ and $\mu = 1/(p-1) > 0$
(recall that $p\neq N=2$).


\section{Flow in a Channel or Porous Media}
\label{s:Begin}

The rapid development of hydrology in the late $18^\mathrm{th}$
and early $19^\mathrm{th}$ centuries required
new theoretical background and related new measurement techniques.
Much of this research, particularly by French engineers
closely connected with the famous Parisian engineering school
{\it \'Ecole des ponts et chauss\'ees\/},
was published in $1804$ in the monograph by one of its former directors,
baron  Gaspard Riche de Prony \cite{deProny}.
This book is a very comprehensive description of French research
on water flow through channels and large pipes.
Some studies treat also smaller (thinner) pipes and hoses
which, towards the end of the $19^\mathrm{th}$ century,
developed into research on filtration through soil, sand, and other
similar porous materials.
Mathematically, all models in this research are set
in space dimension one.
The spectrum of specialists involved in
the $18^\mathrm{th}$ century research begins with
{\it civil engineers\/}
(count  Pierre Louis George du Buat \cite{duBuat} and
 Pierre\--Simon Girard \cite{Girard}),
continues with
{\it theoretical engineers\/} and {\it applied mathematicians\/}
like  de Prony himself and
 Antoine de Ch\'ezy \cite{Chezy}, and ends up with
{\it mathematicians\/}
(marquess  Pierre\--Simon de Laplace \cite{deLaPlace}).
The author,  de Prony \cite{deProny},
describes and further develops the research findings of
his former teacher,  Antoine de Ch\'ezy \cite{Chezy},
published in $1775$ which contains also
his famous mathematical formula on the average flow velocity.
De Prony's book \cite{deProny} was further influenced by
the work of P.~L.~G.\ du Buat \cite{duBuat}
and  P.-S.\ Girard \cite{Girard}.
One of their most important discoveries was
the formula for the resistance force due to adhesion of the fluid
to the contact surface, cf.\
 G.~R.\ de Prony \cite[pp.\ 44, 58]{deProny}:

If $u$ stands for the average flow velocity,
then this resistance force, $\chi\delta s\, \phi(u)$,
is proportional to a polynomial function
$\phi = \phi(u)$ of degree one to three, where
$\chi$, $\delta$, and $s$ are some positive constants that describe
the adhesion to the contact surface, and
%
\begin{equation}
\label{e:deProny}
  \phi(u) = c + \alpha u + \beta u^2 + \gamma u^3
\end{equation}
%
with some non\-negative constants $c$, $\alpha$, $\beta$, and $\gamma$.
We refer to pages $44$ and $58$ of  de Prony's book
\cite{deProny}.
Calculation of these constants from available measurements
was a subject of strong theoretical and practical interest
to civil engineers working on the constructions of channels
and water pipelines throughout entire France
(\cite[pp.\ 65--90]{deProny}).

The transformation of the research interests
in water flow through channels and large pipes into research
on filtration through porous materials began
in mid\--$19^\mathrm{th}$ century in the work by
 Henry Darcy \cite{Darcy} in $1856$,
a French hydro\-engineer working in Dijon,
with his famous (linear) {\it Darcy law\/}, and by
 Jules Dupuit \cite{Dupuit},
another French engineer and economist, published in $1863$,
who, in contrast, works with  de Prony's quadratic law
\eqref{e:deProny} (where $\gamma = 0$)
for the dependence of the resistance force or pressure loss (difference)
on the average flow velocity, $u$.
While {\it Darcy's law\/} became quickly a very popular, simple tool
for calculating the dependence of force or pressure on the velocity $u$
for small absolute values of $u$,
 de Prony's quadratic law has turned out to fit
the filtration problems much more accurately also with higher velocities
required to filter a sufficient amount of liquid (water)
needed by a large urban community.
Towards the end of the $19^\mathrm{th}$ century,
several civil engineers throughout Western Europe have adopted
 de Prony's polynomial formula \eqref{e:deProny}
(typically quadratic or cubic)
in their investigation of fluid filtration phenomena;
 Oscar Smreker \cite{Smreker_1878, Smreker_1879}%
\footnote[2]{
In fact, the latter article, \cite{Smreker_1879},
was intended to be an introduction to the former one,
\cite{Smreker_1878}.
The temporal order of publication is publisher's mistake; see
publisher's remark at the end of the latter \cite{Smreker_1879}.}
(an Austrian\--born engineer based in the city of Mannheim, Germany,
 and active in several neighboring countries)
seems to be the first of them in $1878$--$1879$
(with another work \cite{Smreker_1881} in $1881$),
followed by  C.\ Kr\"{o}ber \cite{Kroeber_1884} in $1884$
and  Philipp Forchheimer
\cite{Forch_1886} in $1886$ and \cite{Forch_1901} in $1901$
(another Austrian engineer active also in Germany).
Especially  Forchheimer's latter article, \cite{Forch_1901},
became a landmark in non\-linear fluid dynamics.
Owing to  Forchheimer's
tremendous theoretical and practical activity in filtration problems,
which includes several lecture notes and comprehensive textbooks
\cite{Forch_1906, Forch_1914, Forch_1930},
 de Prony's and  Smreker's quadratic law
\eqref{e:deProny}, $\gamma = 0$,
in filtration theory is called
{\it Forchheimer's equation\/}.
We will stick to this terminology in the rest of this article
while keeping in mind earlier contributions by
 de Prony and  Smreker.
 Smreker's main merit is an early application of
 Forchheimer's quadratic formula \eqref{e:deProny}
in civil engineering, particularly in
the construction of a water supply system
to the Alsatian city of Strasbourg (France)
(\cite{Smreker_1878}, see the sketches following p.~128).
This engineering project plays the key role in  Smreker's works
\cite{Smreker_1878, Smreker_1879, Smreker_1881} mentioned above
(in $1878$--$1881$).
This work (and from his other articles to follow it)
is collected in his doctoral dissertation \cite{Smreker_1914}
(Dr.--Ing.) from $1914$ at the age of sixty.
By then he had designed and/or built numerous water supply systems
in various European cities:
Belgrade, Ljubljana, Lvow (Lemberg), Mannheim, Prague, Trieste,
Vilnius, etc.
Greater details on his achievements can be found in
 M.\ Bren\v{c}i\v{c}'s survey \cite{Smreker_2008}.

Nevertheless, it was  Oscar Smreker \cite{Smreker_1878} again
who has discovered that, at ``low'' velocity levels $v$,
neither the linear {\it Darcy law\/} nor the quadratic (or cubic)
{\it de Prony\--Forchheimer\/} law \eqref{e:deProny}
describes the relation between the pressure loss and the velocity $v$
accurately.
He suggested the following correction for the {\it (pressure) slope\/}
$h / \ell$,
%
\begin{equation}
\label{e:Smreker}
  \frac{h}{\ell} = \frac{v^2}{2g}\cdot \xi
  \quad\text{ where }\quad
  \xi = f(v) \quad\text{ for }\; v > 0 \,,
\end{equation}
%
with the gravitational constant (acceleration) $g$ given by
$g = 9.81$ $(m/s^2)$ and the function $\xi = f(v)$ taking
the ``hyperbolic'' form
%the ``parabolic'' form
%
\begin{equation}
\label{e:f(v):3/2}
  f(v) = \alpha + \frac{\beta}{\sqrt{v}}
  \quad\text{ for }\; v > 0
\end{equation}
%
with some positive constants $\alpha$ and $\beta$.
The (positive) quantities $h$ and $\ell$, respectively,
stand for the difference $h$ of water levels before and after
the (horizontal) filter of length $\ell$; cf.\
 Forchheimer \cite[Fig.~1, p.~1736]{Forch_1901} and
 Smreker \cite[pp.\ 358--360]{Smreker_1879}.
Formulas \eqref{e:Smreker} and \eqref{e:f(v):3/2}
yield a very special, but important case of
the famous {\it power law\/},
%
\begin{equation}
\label{e:power:3/2}
    \frac{h}{\ell}
  = \frac{v^{3/2}}{2g}\cdot \left( \alpha\, \sqrt{v} + \beta\right)
  \approx \frac{\beta}{2g}\cdot v^{3/2}
  \quad\text{ for }\; v > 0 \,,
\end{equation}
%
with the approximation by the power
$({\beta} / {2g})\cdot v^{3/2}$ being valid for small velocities $v>0$.
In his work \cite[p.~127]{Smreker_1878},
 Smreker suggests also a much more general relation, namely,
%
\begin{equation}
\label{e:f(v):series}
  \xi = f(v) = \alpha
  + \sum_{n=1}^{\infty} \beta_n\, v^{-1/n}
  \quad\text{ for }\; v > 0
\end{equation}
%
with some nonnegative constants $\alpha$ and $\beta_n$.
This is how the {\it power law\/}
%
\begin{equation}
\label{e:power}
  \frac{h}{\ell} = \mathrm{const}\cdot v^s
  \quad (1 < s < 2) \quad\text{ for }\; v > 0
\end{equation}
%
was discovered for the (pressure) slope $h / \ell$.
Starting with the articles \cite{Forch_1901, Smreker_1878},
the precise value of the constant $s\in (1,2)$ was the subject of
numerous measurements and theoretical investigations;
$s > 1$ shows the tendency to approach one ($s\searrow 1$).
Of course, the case $s=1$ renders (linear) {\it Darcy's law\/}.
The power law \eqref{e:power}
for soil permeability and {\em high\/} water velocity $v$
was confirmed in the experiments performed by
 F.\ Zunker \cite{Zunker_1920} in $1920$ with $s = 3/2$;
see also  Zunker's survey article \cite{Zunker_1930}.
He claims that Darcy's law is applicable to
{\em medium\/} water velocities $v$.
In Great Britain, the two nonlinear Darcy laws,
the quadratic law \eqref{e:deProny} (where $\gamma = 0$) and
the power law \eqref{e:power} (where $s = 1.723$),
appear for the first time in $1883$ in the work by
 Osborne Reynolds
\cite[Sect.~III, {\S}37, pp.\ 973--976]{Reynolds_1883}.
He considers very briefly also  Smreker's general problem
\eqref{e:Smreker} (cf.\ \cite[p.~119]{Smreker_1878}).
However, the relation of his research findings to those of
 O.\ Smreker \cite{Smreker_1878} is unclear%
\footnote[3]{
 Reynolds \cite{Reynolds_1883} seems to be unaware of
 Smreker's results in \cite{Smreker_1878}
published five years earlier.}.


It was not until mid\--$1930$s when
 Smreker's power law \eqref{e:power} was verified by
 Alois Anton Missbach
\cite{Missbach:IV} -- \cite{Missbach:VII}
in many laboratory experiments with sugar juice and water
penetrating a medium consisting of
tiny glass balls of constant diameter.
The final comparison of  Smreker's power law with
 A.\ Missbach's laboratory experiments were published
in the (now) famous article \cite{Missbach:VII}.
His experiments are so well\--documented in the series of articles
\cite{Missbach:IV} -- \cite{Missbach:VII}%
\footnote[4]{
Part~VI (Ref.\ \cite{Missbach:VI}) of  Missbach's work
appeared before Part~V (Ref.\ \cite{Missbach:V}).}
that many researchers in non\-linear fluid dynamics,
especially in the ``West''
(Americas, Australia, Europe, and New Zeeland),
consider A.\ Missbach's article \cite{Missbach:VII}
as the {\em verification\/} of
 Smreker's power law \eqref{e:power}.
For this reason, this power law is often called
{\it Missbach's equation\/} in Western literature
(or the {\it Darcy\--Missbach equation\/}
 in~\cite{QuinnCherryParker2011}).
We will use this terminology in the rest of this article,
although many authors from Russia, the mainland China, and Taiwan
prefer to attribute the power law to, e.g.,
the prolific Russian engineer
 S.~V.\ Izbash \cite{Izbash_1931, Izbash_1933}; see also
 S.~V.\ Izbash and  Kh.~Yu.\ Khaldre \cite{Izbash-Khaldre}
and  H.\ Watanabe \cite{Watanabe_1982}.
 A.\ Missbach's work \cite{Missbach:VII}
summarizes the results of a large research program sponsored by
several sugar refineries in Czechoslovakia in the early $1930$s
on efficient sugar juice filtration.
It is the final part (Part~VII) of the series of seven articles on
{\it Filtration ability of separated and saturated juices\/}
inspired by the scientific and industrial activities of
 Missbach's doctoral adviser, Jaroslav D\v{e}dek,
who himself also contributed to this article series (Part~III), cf.\
 J.\ D\v{e}dek and D.\ Ivan\v{c}enko \cite{Missbach:III}.
The findings of the research reported in
 Missbach's article \cite{Missbach:VII},
albeit obtained with penetrating water rather than sugar juice,
were immediately incorporated into industrial sugar production.
This article is written in two parallel originals, Czech and German.
Further details on his professional involvement with
the Czechoslovakian sugar producing industry will be provided
in Section~\ref{s:biogr_Missbach}.
A very practical application of
{\it Missbach's equation\/}
to non\-linear Darcy flow (also called {\it non\--Darcian flow\/})
is provided in
 P.~M.\ Quinn,  J.~A.\ Cherry, and
 B.~L.\ Parker \cite{QuinnCherryParker2011}.
This flow occurs in
high\--precision straddle packer tests conducted in boreholes
in a fractured dolostone aquifer using constant rate injection
step tests to identify the conditions of change
from Darcian to non\--Darcian flow.
An interesting comparison of
{\it Forchheimer's\/} and {\it Missbach's\/}
{\it equations\/},
\eqref{e:deProny} and \eqref{e:power}, respectively,
is available in the survey article by
 K.~P.\ Stark and R.~E.\ Volker \cite{Stark-Volker}
who, unfortunately, seem to be unaware of
 O.\ Smreker's pioneering work
\cite{Smreker_1878, Smreker_1879, Smreker_1881, Smreker_1914}.


\section{The Russian School}
\label{s:Russian}

Significant contributions to the filtration problem
in porous materials by Russian (or Soviet) engineers and scientists
began in early $1920$s by  N.~N.\ Pavlovskii \cite{Pavlovskii}
in a hand\--written monograph of $753$ pages.
It provides a very well\--written, up\--to\--date introduction
to hydraulics from a (mostly) theoretical point of view,
with plenty of valuable references to the literature.
In Russia, this time is characterized by massive industrialization
($1920$s and $1930$s).
In the first chapter,  Pavlovskii surveys constitutive laws
(Darcy's law, Forchheimer's quadratic and cubic laws,
 and the power law).
In the second chapter, he suggests a criterion based on
the Reynolds number to establish the validity range of
the linear Darcy law and the range where
a nonlinear law must be used instead.
According to
 V.~I.\ Aravin and  S.~N.\ Numerov \cite{AravinNumerov},
p.~4 and also p.~33 with a detailed explanation,
 Pavlovskii's work \cite{Pavlovskii}
is the first one to use Reynolds number for this purpose.
Despite of the fact that the monograph \cite{Pavlovskii}
thoroughly discusses various constitutive laws
in its first two chapters,
the partial differential equations used throughout the book
to study the seepage are only linear.

Serious interests in non\-linear (and non\--Newtonian) fluid dynamics
in the former Soviet Union began in early $1930$s
with the works by
 S.~V.\ Izbash \cite{Izbash_1931, Izbash_1933},
who has published the power law \eqref{e:power}
already in $1931$ in a monograph available only in Russian.
Decisive contributions to fluid dynamics were made by
 N.~E.\ Zhukovskii
(see his collected works \cite{Zhukovskii} from $1937$),
the most relevant for us being \cite{Zhukov_1889} from $1889$.
As we have already mentioned in Section~\ref{s:Filer_Probl},
he gives the explicit formula for the radially symmetric solution,
$u(x)\equiv u(|x|)$, of the $p$-harmonic equation,
$\Delta_p u = 0$,
see \cite[Eq.~(13), p.~19]{Zhukov_1889}.
In the same article, \cite{Zhukov_1889},
 Zhukovskii discusses applicability of various constitutive laws
to filtration of water through sandy soil known to that date, i.e.,
Darcy's, Kr\"{o}ber's, and Smreker's power\--type laws
\cite{Darcy, Kroeber_1884, Smreker_1878},
and compares them to scores of available experimental results.
For instance, he derives Laplace's equation by inserting
the (linear) Darcy law into the differential equation of continuity.
Using the Laplace equation he studies
several configurations of water wells scattered in the field
(standalone well, wells in a row, and wells on a circle).
For the standalone case,
he finds out that the discrepancy between theoretical predictions
from the formula based on the solution of Laplace's equation
and the reality (measured data) is too large.
To fix this problem, he suggests to use the velocity $v$
given by Kr\"{o}ber's and Smreker's power law
\cite{Kroeber_1884, Smreker_1878}
%
\begin{math}
  - \frac{\mathrm{d}u}{\mathrm{d}x} =
  i = \mathrm{const}\cdot v^{p'-1} \,,
\end{math}
%
$2 < p' < \infty$, cf.\ eq.~\eqref{e:p-Smreker},
to be plugged into the stationary case of
the continuity equation \eqref{e:cont_eq} as described above.
In particular, eq.~\eqref{e:p-Darcy_power}
plays the role of the constitutive law.

To the best of our knowledge,
all work on non\-linear (and non\--Newtonian) fluid dynamics
until $1940$, throughout the entire world,
treated only spatially one-dimen\-sional problems.
( Smreker's and  Zhukovskii's
 radially symmetric planar solution in
 \cite{Smreker_1881, Zhukov_1889}
 mentioned above is essentially one-dimensional.)
It was the Russian scientist
 S.~A. Christianovitch \cite{Christ}
who employed nonlinear constitutive laws
(Forchheimer's quadratic and cubic laws and Missbach's power law)
to derive nonlinear partial differential equations for
the seepage movement of underground water.
He restricts himself to the spatially two\--dimensional case.
In the case of the power law, he obtains the following equation
(re\--written in contemporary notation):
%
\begin{equation*}
  \Delta_p u\equiv \operatorname{div}
  \left( |\nabla u|^{p-2} \nabla u\right) = 0 \,,
\end{equation*}
%
for the unknown function $u = u(x,y)$.
Since he works in two space dimensions, he can use methods of
complex analysis and suggest analytical techniques
to obtain approximations of the solution to this equation
with the so\--called {\em $p$-Laplace operator\/} $\Delta_p$,
$1 < p < \infty$.
The common (linear) Laplace operator $\Delta$ is obtained for
the (linear) Darcy law ($p=2$).

Another notable person in the Russian hydraulic engineering school was
 L.~S. Leibenson who investigated seepage of oil and gas
in the oil and gas fields near the city of Baku
(now Azerbaijan, formerly Soviet Union).
Much of his research from the $1920$s and early $1930$s
was published not only in brief article form, but also as
a survey monograph \cite{Leiben_1934}.
His most important findings concern turbulent filtration of gas
in porous medium \cite{Leiben_1945-1, Leiben_1945-2}
(see also \cite{Leiben_1953}).
It was his article \cite{Leiben_1945-1}
where the {\it doubly non\-linear parabolic equation\/},
%
\begin{equation}
\label{eq:Leiben:1}
  \frac{\partial u^m}{\partial t} = c\, \Delta_p u
  \quad\text{ for }\; (x,y,z,t)\in \mathbb{R}^3\times (0,T) \,,
\end{equation}
%
with $m+1 = p = 3/2$, appeared for the first time.
Here, $u = u(x,y,z,t)$ is the unknown function of space and time,
and $c>0$ is some constant.
Thanks to $m = p-1$, eq.~\eqref{eq:Leiben:1} is called
{\it $(p-1)$-homogeneous\/}.
He used the separation of space and time variables,
%
\begin{equation*}
  u(x,y,z,t) = v(t)\, w(x,y,z) \,,
\end{equation*}
%
in order to obtain the following equation with the so\--called
{\it $1$-Laplacian\/},
%
\begin{equation}
\label{eq:1-Laplace}
    \operatorname{div} \genfrac{(}{)}{}0{\nabla w}{|\nabla w|}
  + A\, \sqrt{w} = 0 \,,
\end{equation}
%
where $w = w(x,y,z)$ is the unknown function of space and
$A > 0$ is a constant.
This article, \cite{Leiben_1945-1}, published in $1945$
seems to be the first one to derive and consider
a quasilinear {\it parabolic\/} (time\--dependent) problem,
eq.~\eqref{eq:Leiben:1},
with the {\em $p$-Laplace operator\/} $\Delta_p$ in space dimension three
(defined in eq.~\eqref{def:p-Laplace}),
albeit for $p = 3/2$ only.
For the $p$-harmonic equation, $\Delta_p u = 0$ with $p = 3/2$,
Leibenson \cite{Leiben_1945-1}
finds solutions in the spatially
one\--dimensional and radially symmetric cases.
In contrast,
 S.~A. Christianovitch \cite{Christ} (in $1940$)
treated only a quasilinear {\it elliptic\/} (stationary) problem,
$\Delta_p u = 0$, in two space dimensions, but for any
$1 < p < \infty$.

In his next work \cite{Leiben_1945-2},
immediately following \cite{Leiben_1945-1},
 L.~S.\ Leibenson
allows for a wider range of values of $p$, $3/2\leq p\leq 2$.
Also his doubly non\-linear parabolic equation \eqref{eq:Leiben:1}
becomes more general,
%
\begin{equation}
\label{eq:Leiben:2}
    \frac{\partial}{\partial t}\left( u^{\frac{1}{m+1}}\right)
  = c\, \Delta_p u
  \quad\text{ for }\; (x,y,z,t)\in \mathbb{R}^3\times (0,T) \,,
\end{equation}
%
with $m > 0$, which is no longer $(p-1)$-homogeneous.
This equation results from  Leibenson's studies
\cite{Leiben_1945-2} of filtration of
turbulent polytropic gas flow through porous medium;
$m > 0$ is called the {\it polytropic index\/} of the gas.
It is a direct generalization of an earlier work by
 L.~S. Leibenson \cite{Leiben_1929}
which still uses the linear Darcy law, whereas \cite{Leiben_1945-2}
uses  Smreker's power law%
\footnote[5]{
 Leibenson \cite{Leiben_1945-2} was apparently not aware of
 Missbach's work \cite{Missbach:VII}.}
\eqref{e:power}.
Practically all  Leibenson's results we have mentioned above
are very carefully collected and explained in his monograph
\cite{Leiben_1947} published in $1947$;
his scientific articles
\cite{Leiben_1929, Leiben_1945-1, Leiben_1945-2, Leiben_1945-3}
are reprinted in \cite{Leiben_1953}.

An important member of the Russian school was also
 P.~Ya.\ Polubarinova\--Kochina.
Her Russian monograph \cite{PolubarKochina} from $1952$
(translated into English in $1962$)
became quickly a widely used textbook by hydro\-geologists
all over the world.


\section{From Darcy's law to Forchheimer's equation\\
        (from linear to nonlinear diffusion)}
\label{s:Darcy-Forch}

Although fluid flow through channels, large pipes, and hoses
had occupied theoretical hydrologists since the $18^\mathrm{th}$ century
(see  de Prony's equation \eqref{e:deProny}),
fluid flow through porous media attracted major attention much later,
in mid\--$19^\mathrm{th}$ century.
We recall from Section~\ref{s:Begin}
the research on filtration through porous materials by
 Henry Darcy \cite{Darcy} in $1856$
(the linear {\it Darcy law\/})
and by Jules Dupuit \cite{Dupuit} in $1863$
(working with de Prony's quadratic law).
The idea of the quadratic law \eqref{e:deProny}
was picked up by  Ph.\ Forchheimer
who, in his ground\-breaking work \cite{Forch_1901},
developed applications of  de Prony's quadratic law
to filtration through porous materials (soil, in particular),
%
\begin{equation}
\label{e:Forch_quadr}
  i = av + bv^2 \,.
\end{equation}
Here, the quantity $i$ is the (negative) total piezometric head gradient,
$  i =- \frac{\mathrm{d}u}{\mathrm{d}x}$,
$v$ stands for the average seepage velocity,
and $a$ and $b$ are nonnegative constants determined by
the properties of the fluid and medium;
typically, $a > 0$ and $b > 0$.
His article \cite{Forch_1901}, published in 1901,
meant also the introduction of nonlinear diffusion
after several decades of intensive studies of
linear diffusion prompted by Darcy's law.
A number of workers have inferred that Forchheimer's equation has
sound physical backing apart from its attraction as
a relatively simple nonlinear expression.
We refer the reader to
 J.\ Bear,  D.\ Zaslavsky, and  S.\ Irmay
\cite{Irmay_1968}, for example,
who have derived the Forchheimer relation
by inferred arguments from the fundamental Navier\--Stokes equations
for the general case when inertia terms are considered; see also
 Irmay \cite{Irmay_1967}.
A few decades later, in 1930,
 Ph.\ Forchheimer \cite{Forch_1930}
extended his nonlinear Darcy law to
%
\begin{equation}
\label{e:Forch_m}
  i = av + bv^m \,,
\end{equation}
%
where $m$ is a constant typically taking values in the interval $(1,2]$,
i.e., $1 < m\leq 2$.

\begin{remark}
\begingroup \rm
From the point of view of Mathematical Physics,
relation \eqref{e:Forch_quadr}
means that if $a > 0$, then
the head gradient $i$ has nearly linear, nontrivial growth
%
\begin{equation}
\label{eq:sec:2:growth:i}
  i(v) - i(0) = i = a v \Big( 1 + \frac{b}{a} v \Big) \approx a v
\end{equation}
%
for low velocity $v$.
%
On one hand, this phenomenon was confirmed for
certain types of fluids and media from both
theoretical and experimental viewpoints, e.g., in the work of
 V.~I.\ Aravin and  S.~N.\ Numerov \cite{AravinNumerov},
 E. Lindquist \cite{Lindq_1933}, and
 J.~C.\ Ward \cite{Ward_1964}.
%
On the other hand, the nontrivial growth \eqref{eq:sec:2:growth:i}
($a>0$), which yields
%
\begin{equation*}
  v = v(i)
    =- \frac{a}{2b} + \sqrt{\big( \frac{a}{2b} \big)^2
                       + \frac{i}{b}}
    = \frac{a}{2b}
  \Big( {}- 1 + \sqrt{ 1 + \big(\frac{2b}{a}\big)^2 \frac{i}{b} }
  \Big) > 0
    \quad\text{ if also } b > 0 \,,
\end{equation*}
%
whence $v\approx i/a$ for $i\geq 0$ small,
does not occur for other types of fluids and media studied in
 M.\ Anandakrishnan and  G.~H.\ Varadarajulu
\cite{Ananda},
 C.~R.\ Dudgeon \cite{Dudgeon_1964},
 C.~R.\ Dudgeon and  C.~N. Yuen \cite{Dudgeon_1970},
 L. Escande \cite{Escande_1953},
 A.\ Missbach
\cite{Missbach:IV, Missbach:V, Missbach:VI, Missbach:VII},
 A.~K. Parkin \cite{Parkin_1962},
 A.~M.\ White \cite{White_1935}, and
 J.~K.\ Wilkins \cite{Wilkins_1955}.
\endgroup
\end{remark}


\section{Missbach's power law (nonlinear, power\--type diffusion)}
\label{s:Missbach}

In contrast with Forchheimer's approach to generalizing Darcy's law,
 Alois Missbach \cite{Missbach:VII} based his approach to
the porous medium problem on numerous {\em experimental results\/}
that became available in the $1930$s in
various rapidly developing industries,
such as sugar and petroleum (oil) production,
where certain types of fluids are filtered through special porous media.
Missbach's experiments were prompted by
theoretical and experimental results obtained much earlier by
 C.\ Kr\"{o}ber \cite{Kroeber_1884},
 O.\ Reynolds \cite{Reynolds_1883},
 O.\ Smreker  \cite{Smreker_1878}, and
 F.\ Zunker \cite{Zunker_1920}.
The experimental results obtained during the sugar beet campaign of 1935
in Czechoslovakia led  A.\ Missbach \cite{Missbach:VII}
to verifying the {\em power law relation\/}
%
\begin{equation}
\label{eq:Missbach}
  i = c\, v^m
\end{equation}
%
between the head gradient and the velocity,
$i$ and $v$, respectively,
published in $1937$.
The power $m$ typically takes values in the interval $(1,2)$.
A couple of years before Missbach's article appeared, in 1935,
 A.~M.\ White \cite{White_1935}
proposed an analogous relation with $m = 1.8$.
As a porous medium, Missbach used gravels, sands, and
packings of uniform spheres (e.g., tiny glass balls),
while in his starting experiments
\cite{Missbach:IV} -- \cite{Missbach:VI}
the fluid was represented by sugar juice of various sugar contents.
However, in his most important work for us, \cite{Missbach:VII},
he used water as the penetrating fluid
(Figure \ref{fig1} below).
He found out that the power $m$ stays in $(1,2)$ and tends to $1$
with the decreasing diameter of the spheres.
 C.~R.\ Dudgeon \cite{Dudgeon_1964}
carried out tests on coarse materials
serving as porous medium (gravels, sands, and packings of uniform spheres)
and confirmed that while the results followed closely
an expression of Missbach's form \eqref{eq:Missbach}
the values of $c$ and $m$ were not constant
for the particular material for all fluid flow conditions.
These and other experimental results have confirmed
Missbach's equation \eqref{eq:Missbach}.
A theoretical derivation of the special case of
Missbach's equation \eqref{eq:Missbach} for $m = 3/2$
has been given in
 E.\ Skjetne and  J.-L.\ Auriault \cite{Skjetne_1999}.
%
The authors of the present article have not been able to find
any reference concerned with a theoretical derivation of
Missbach's equation \eqref{eq:Missbach}
for an arbitrary power $m\in (1,2)$.
The article by  A.\ Brieghel\--M\"uller \cite{Brieghel-Muell}
thoroughly surveys almost all results concerning constitutive laws
for filtration known up to $1940$ and discusses their applicability
to filtration processes in sugar production.


Since experiments and measurements play a decisive role in
 A.\ Missbach's work \cite{Missbach:IV} -- \cite{Missbach:VII},
we provide a brief description of his apparatus.
 A.\ Missbach \cite{Missbach:VII} calls his
experimental laboratory equipment
``Apparatus for   testing the hydraulic conductivity
  (permeability, porosity)  through a layer of glass balls''.

\begin{figure}[htb]
  \centering
   \includegraphics[width=0.4\textwidth]{fig1}
  \caption{Apparatus for testing the hydraulic conductivity
           through a layer of glass balls.}
\label{fig1}
\end{figure}


Figure \ref{fig1}  is a scanned copy of the original figure from
 Missbach's work \cite{Missbach:VII},
p.~294, Obr.~1 (in the Czech edition) and
p.~424, Abb.~1 (in the German edition).
 Missbach \cite{Missbach:VII}
credits the use of tiny glass balls to
 Zunker \cite{Zunker_1920}.

Figure \ref{fig1} {\bf description\/}:
%
\begin{enumerate}
\item 
Glass tube with strong walls of internal diameter $45$~mm,
slightly longer than $200$~mm.
\item 
Lower sieve.
\item 
Upper sieve with a steel spring.
\item 
Connecting rubber hose with strong walls.
\item 
Tin funnel with a sieve insole.
\item 
Thin connection pipe for the differential water manometer.
\item 
Faucet for flow regulation.
\item 
Outlet for flow regulation.
\item 
Screw thread with an inserted filter cloth.
\item 
Trench for draining overflowing liquid.
\item  
Manometer.
\end{enumerate}
%

In contrast with earlier filtration experiments
(e.g.,  F.\ Zunker \cite{Zunker_1920, Zunker_1930})
which used a system of parallel capillary tubes
having undesirable side effects,
 A.\ Missbach \cite{Missbach:VII}
decided to construct an apparatus of a relatively large diameter
($45$~mm)
whose walls do not influence (obstruct, slow down)
the fluid flow through the layer of tiny glass balls.
He used glass balls of four (4) different sizes
(A, B, C, D; specified in \cite[Table~I]{Missbach:VII})
and varied both,
the thickness (height) of the layer of glass balls and
the pressure of the fluid penetrating through the layer.
The fluid used in this experiment was tap water,
carefully filtered, with no air bubbles and other ``pollutants''.
The filtered water was pumped through
the outlet for flow regulation $(8)$ from the bottom,
under the atmospheric pressure of up to $0.5$~atm,
then led to penetrate through the layer of glass balls upwards.
In order to guarantee a constant fluid flow velocity, $v$,
throughout the horizontal cross section of the glass tube,
a sieve insole $(2)$ is inserted into the glass tube.
The upper sieve with a steel spring $(3)$ prevents the glass balls
from being moved upwards by the penetrating fluid.
Finally, the overflowing liquid is drained into the trench $(10)$
and its volume is measured in a cylindrical vessel.

The thickness of the layer of glass balls,
the size of the balls (A, B, C, D),
the vertical pressure difference in the layer,
the flow velocity, and many other important measurements
are carefully recorded in
\cite[Tables II through~V]{Missbach:VII}.
These experiments provide evidence for
 Missbach's power law relation \eqref{eq:Missbach}.


\section{Comparison of the Forchheimer and Missbach equations\\
        (two different types of nonlinear diffusion)}
\label{s:Forch-Missbach}

Both, Forchheimer's and Missbach's models have been very useful
in a number of various situations.
Which of the two nonlinear models is better (i.e., more accurate)
depends strongly on the fluid properties and the velocity~$v$.
A brief comparison of the two models has been carried out e.g.\ in
 P.~M.\ Quinn,  J.~A.\ Cherry, and
 B.~L.\ Parker \cite{QuinnCherryParker2011},
 K.~P.\ Stark and  R.~E.\ Volker \cite{Stark-Volker},
and numerically in  R. E. Volker \cite{Volker}.
The experimental conditions in \cite{QuinnCherryParker2011}
seem to be slightly more favorable for Missbach's model.
We refer to  Figure~5 in
\cite[Chapt.~9, pp.\ 9--12]{QuinnCherryParker2011}
for a detailed comparison of the two models.
It is interesting to observe that the authors in
\cite[Chapt.~5, pp.\ 131--196]{Stark-Volker}
slightly favor Forchheimer's model for water penetrating
a porous medium {\it between two horizontal plates\/}
(see \cite[pp.\ 144, 185--186, and 196]{Stark-Volker}),
whereas
 A.\ Missbach \cite{Missbach:VII} obtains highly favorable results
for filtration of water through a porous medium
{\it in a vertical cylinder\/} described in the previous section
(with applications to filtration of saturated sugar juice).
Although the laminar flow regime often obeys the linear Darcy law,
it is always nonlinear in character.
Thus, Missbach's equation applies also to the laminar flow regime and
in the transition to a turbulent regime.


\section{Some basic analytic and numerical results for the $p$-Laplacian}
\label{s:res-p_Laplace}

A comprehensive survey on only basic analytic and numerical results
for the $p$-Laplacian would have to contain literally
hundreds of references.
As this is not the purpose of our present article,
we have decided to mention only a few ones.
Perhaps the very basic monograph on modern (nonlinear)
functional\--analytic methods for the $p$-Laplacian and
similar quasilinear partial differential operators is
the classical book by  J.-L.\ Lions \cite{Lions_69}.
Besides methods of Nonlinear Analysis it contains also
many applications to various mathematical models.
Among important topics are
the {\it global climate modelling\/} treated in
 J.-I.\ D\'{\i}az,  G.\ Hetzer, and  L.\ Tello
\cite{Diaz-Hetzer}
and {\it non\-linear fluid dynamics\/} in
 J.~I.\ D\'{\i}az and  F.\ de~Th\'elin
\cite{Diaz-Thelin}.

The spectrum of the (positive) $p$-Laplace operator
${}- \Delta_p$ on the Sobolev space
$W_0^{1,p}(\Omega)$
(that is, a monotone nonlinear operator with
 the zero Dirichlet boundary conditions)
has been an interesting open problem for decades,
with the exception of the first eigenvalue; see the monograph by
 S.\ Fu\v{c}\'{\i}k,  J.\ Ne\v{c}as,  J.\ Sou\v{c}ek,
and  V.\ Sou\v{c}ek \cite{FucikNSS}.
The Fredholm alternative at the first eigenvalue is studied in
 P.\ Dr\'abek,  P.\ Girg,  P.\ Tak\'a\v{c}, and
 M.\ Ulm \cite{Drab-Girg-Tak-Ulm}
in a bounded domain $\Omega\subset \mathbb{R}^N$ and in
 J.\ Benedikt,  P.\ Girg and  P.\ Tak\'a\v{c}
\cite{Ben-Girg-Tak}
in a bounded open interval $\Omega\subset \mathbb{R}^1$.
Bifurcations at the first eigenvalue are treated in
 P.\ Girg and  P.\ Tak\'a\v{c} \cite{Girg-Takac}.


\section{A short sketch of Missbach's biography}
\label{s:biogr_Missbach}

 A.\ Missbach (full name  Alois Anton Missbach)
was born on the 11$^\mathrm{th}$ of June, 1897
in Plenkovice near Znojmo, Moravia (present Czech Republic),
and baptized on June 13$^\mathrm{th}$, 1897.
According to the population statistics office (``matrika'')
in the town of Lib\'a\v{n} in Eastern Bohemia (Czech Republic),
A. Missbach had moved to Lib\'a\v{n} in 1923
and stayed there until July 26$^\mathrm{th}$, 1945.
He was employed as a technical engineer
from $1923$ through $1945$ in the sugar refinery in Lib\'a\v{n}
where he performed his research reported in
Refs.\ \cite{Missbach:IV} -- \cite{Missbach:VII}.
While working full time as an engineer (the second technical adjunct),
he defended his doctoral thesis on June 26$^\mathrm{th}$, 1936
at the Czech Technical University in Brno, Moravia.
He received the degree of Doctor of Technical Sciences (Dr. techn.).
His thesis advisor was the well\--known expert
in Chemistry and sugar production,
prof. Ing. Dr. techn. et Dr. agr. h.c. Jaroslav D\v{e}dek.

A. Missbach got married in 1928 in the famous Old Town Hall
in the historic center of Prague, then the capital of Czechoslovakia.
According to the statistics office in Lib\'a\v{n},
he moved out to Havra\v{n} near the town of Most
in Northwestern Bohemia (Czech Republic).
As far as we know from the municipal office of Havra\v{n},
several months later he moved to the nearby village of Lene\v{s}ice,
also near the town of Most.
He was the director of the sugar refinery in Havra\v{n} at least during
his stay there.
His last residence known to us was the town of Most starting
on August 12$^\mathrm{th}$, 1953.
Both sugar refineries, in Lib\'a\v{n} and Havra\v{n},
have been closed down several decades ago.


\subsection*{Acknowledgments}
The research of
Ji\v{r}\'{i} Benedikt, Petr Girg, and Luk\'{a}\v{s} Kotrla
was partially supported by
the Grant Agency of the Czech Republic (GA\v{C}R)
under Project No.\ 13-00863S.
%
Luk\'{a}\v{s} Kotrla was partially supported also by
the Ministry of Education, Youth, and Sports
(M\v{S}MT, Czech Republic)
under the program NPU I, Project No.\ LO1506 (PU\--NTIS).
%
Peter Tak\'{a}\v{c} was partially supported by
the German Research Society (D.F.G.) under Grants
No.\ TA 213/15-1 and TA 213/16-1.
%
We (all four authors) would like to thank the employees of
the population statistics offices (``matrika'')
in Lib\'{a}\v{n}, Havra\v{n}, and Lene\v{s}ice, and to
(Mrs.) Dr.\ Eva B\'{i}lkov\'{a} of the municipal archive
in the town of Ji\v{c}\'{i}n (Czech Republic)
for their kind help in providing us with a number of
interesting and useful facts about the work and life of
Dr.\--Ing.\ Alois Anton Missbach.
Our sincere thanks belong also to (Mr.) Ing.\ Daniel Fron\v{e}k,
the curator (``program director'') of the
\begingroup\sl
``Museum of Sugar, Alcohol Production, Sugar Beet Growing,
  and the Town of Dobrovice''
\endgroup
in Dobrovice, Eastern Bohemia (Czech Republic), and to
(Mr.) Mgr.\ Josef Ne\v{s}n\v{e}ra,
a facility officer and a museum guide,
for their (rather time\--consuming) help in searching for
traces about A.~Missbach's studies, work, and life throuhgout
the local museum library,
particularly Refs.\ \cite{Missbach:III} and
\cite{Missbach:IV} -- \cite{Missbach:VII} in both Czech and German.


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\emph{Teoreticheskoe issledovanie o dvizhenii podpochvennykh vod},
Zhurnal Russkogo fiziko\--khimicheskogo obshchestva,
\textbf{21}(1) (1889) (in Russian).
\hfil\break
[Theoretical investigation on the movement of subsurface water],
[The journal of the Russian Physical and Chemical Society],
\textbf{21} (1) (1889).
Reprinted in Ref.~\cite{Zhukovskii}, 9--33.

\bibitem{Zhukovskii}
N.~E. Zhukovskii,
{\sl ``Polnoe sobranie sochinenii''} [Collected Papers]
(Russian with English Summary), Vol.~\textbf{7},
Moscow\--Leningrad, 1937, pp.\ 9--33.

\bibitem{Zunker_1920}
F. Zunker,
\emph{Das allgemeine Grund\-wasser\-fliess\-gesetz},
     Journal f\"ur Gasbeleuchtung und Wasserversorgung,
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\bibitem{Zunker_1930}
F. Zunker,
\emph{Das Verhalten des Bodens zum Wasser}, in:
{\sl ``Handbuch der Bodenlehre:
     Die Physikalische Beschaffenheit des Bodens''}, Vol.~\textbf{6},
E.~Blanck, Ed., pp.\ 70--180.
Springer\--Verlag GmbH, Berlin\--Heidelberg, 1930.
  doi: 10.1007/978-3-662-02172-9.


\end{thebibliography}

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