Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
WEAK
DISCONTINUITY
WAVES
IN
ISOTROPIC
POROUS
MEDIA
FILLED
BY
A
FLUID
FLOW
∗
Alessio
Fam`
a
†
Liliana
Restuccia
‡
Dedicated
to
Professor
Constantin
Fetecau
on
the
occasion
of
his
80th
anniversary
DOI 10.56082/annalsarscimath.2026.4.265
Abstract
In
this
paper
the
propagation
of
weak
discontinuity
waves
is
in-
vestigated
in
a
special
isotropic
porous
medium
filled
by
a
fluid
flow.
To
this
aim
we
introduce
a
new
variable
related
to
the
surface
across
which
the
first
derivatives
of
the
solutions
of
the
governing
equations
undergo
a
jump.
Following
a
Boillat’s
methodology
for
quasi-linear
and
hyperbolic
systems
of
the
first
order,
we
have
investigated,
in
an
one-dimensional
case,
the
propagation
of
one
solution
into
an
uniform
unperturbed
state
in
the
first
approximation,
obtaining
the
Bernoulli’s
equation
governing
the
evolution
of
the
amplitude
of
the
weak
discon-
tinuity
wave,
the
wave
front,
the
velocity
along
the
characteristic
rays,
the
approximated
solution
and
the
critical
time.
The
results
obtained
can
be
applied
in
certain
technical
fields,
such
as
biology,
the
medical
sciences
and
geology.
Keywords:
weak
discontinuity
waves,
partial
differential
equations,
ap-
proximate
solutions,
non-equilibrium
thermodynamics,
porous
media
filled
by
a
fluid
flow.
∗
Accepted
for
publication
on
October
02,
2026
†
alessio.fama.04@gmail.com
,
University
of
Messina,
Department
of
Mathematical
and
Computer
Sciences,
Physical
Sciences
and
Earth
Sciences,
Viale
F.
Stagno
d’Alcontres,
Salita
Sperone
31,
98166
Messina,
Italy
‡
lrestuccia@unime.it
,
University
of
Messina,
Department
of
Mathematical
and
Com-
puter
Sciences,
Physical
Sciences
and
Earth
Sciences,
Viale
F.
Stagno
d’Alcontres,
Salita
Sperone
31,
98166
Messina,
Italy
265
Weak
discontinuity
waves
in
isotropic
porous
media
270
reach
the
respective
thermodynamic
equilibrium
values.
Equations
(9)
and
(12)
describe
the
evolution
of
the
porosity
field,
of
its
flux
and
in
their
right
hand
sides
the
source
terms
represent
the
contributions
of
the
fields
occur-
ring
inside
these
media.
Equations
(10)
and
(11)
are
the
anisotropic
Fick-
Nonnenmacher
equation
for
the
fluid
concentration
flux
and
the
anisotropic
generalized
Maxwell-Vernotte-Cattaneo
equation
for
the
heat
flux,
respec-
tively.
2.1
Evolution
equations
for
the
concentration
of
the
fluid
and
the
porosity
fields
and
their
fluxes
in
a
special
case
In
[9]
we
have
treated
the
special
case
of
perfect
isotropic
media,
where
their
geometric,
transport
and
thermal
properties
are
invariant
respect
to
all
rotations
and
inversions
of
the
frame
axes
(under
orthogonal
transforma-
tions).
We
have
assumed
the
media
into
consideration
are
at
rest
and
we
have
disregarded
some
cross
effects
among
the
fields
confining
us
only
to
the
porosity
and
the
fluid
concentration
fields
and
their
fluxes.
In
particular,
taking
into
account
equation
(2),
(10),
(9)
and
(12),
in
[9]
we
have
worked
out
the
following
system
of
equations
ρ
∂c
∂t
+
j
c
k,k
=
0
,
(13)
τ
j
c
∂j
c
i
∂t
=
−
j
c
i
+
α
c
r
,i
−
ρD
c
c
,i
,
(14)
∂r
∂t
+
V
k,k
=
0
,
(15)
τ
ν
∂
V
k
∂t
=
−V
k
−
D
ν
r
,k
+
α
ν
c
,k
,
(16)
with
α
c
and
α
ν
constant
coupling
coefficients,
τ
j
c
and
τ
ν
the
constant
re-
laxation
times
of
the
field
j
c
i
and
V
ijk
,
and
D
c
,
D
ν
the
constant
diffusion
coefficients
of
the
fluid
concentration
and
porosity
fields,
respectively.
Fur-
thermore,
in
(13)
the
mass
density
was
supposed
constant
and
in
equation
(15)
the
number
of
the
very
thin
porous
channels
was
assumed
locally
con-
served.
In
this
paper
we
suppose
that
in
(14)
and
(16)
the
two
coupling
coef-
ficients
α
c
and
α
ν
are
functions
of
at
least
class
C
1
of
the
type
α
c
=
α
c
(
r
)
and
α
ν
=
α
ν
(
c
),
and
we
investigate
the
weak
discontinuity
waves
of
the
PDEs
system
(13)-(16).