Finite speed of propagation for the thin-film equation in the spherical geometry
UDC 517.953 We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the s...
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2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507262025465856 |
|---|---|
| author | Taranets, R. M. Таранець, Р. М. |
| author_facet | Taranets, R. M. Таранець, Р. М. |
| author_sort | Taranets, R. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:56:42Z |
| description | UDC 517.953
We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution $u>0$ and $u=0.$
Using local entropy estimates, we also obtain the upper bound for the rate of the interface propagation. |
| first_indexed | 2026-03-24T02:06:31Z |
| format | Article |
| fulltext |
UDC 517.953
R. M. Taranets (Inst. Appl. Math. and Mech. Nat. Acad. Sci. Ukraine, Sloviansk)
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION
IN THE SPHERICAL GEOMETRY
СКIНЧЕННА ШВИДКIСТЬ ПОШИРЕННЯ ЗБУРЕНЬ
ДЛЯ РIВНЯННЯ ТЕЧIЇ ТОНКОЇ ПЛIВКИ ВЗДОВЖ КУЛI
We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical
surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free
boundary separating the regions, where the solution u > 0 and u = 0. Using local entropy estimates, we also obtain the
upper bound for the rate of the interface propagation.
Показано, що рiвняння тонких плiвок iз подвiйним виродженням, яке виникає з моделювання потоку в’язкого
покриття на сферичнiй поверхнi, має скiнченну швидкiсть поширення носiя невiд’ємного сильного розв’язку, а
отже, iснує iнтерфейс або вiльна межа, що роздiляє областi, де розв’язок u > 0 i u = 0. Крiм того, за допомогою
локальної ентропiйної оцiнки отримано оцiнку зверху для швидкостi поширення iнтерфейсу.
1. Introduction. In this paper, we study a particular case of the following doubly degenerate
fourth-order parabolic equation:
ut +
\Bigl[
un(1 - x2)
\bigl(
a - bx+ c
\bigl(
2u+
\bigl(
(1 - x2)ux
\bigr)
x
\bigr)
x
\bigr) \Bigr]
x
= 0 in QT , (1.1)
where u(x, t) represents the thickness of the thin film, the dimensionless parameters a, b and c
describe the effects of gravity, rotation and surface tension, QT = \Omega \times (0, T ), n > 0, T > 0, and
\Omega = ( - 1, 1). For n = 3 (no-slip regime) this equation describes the dynamics of a thin viscous liquid
film on the outer surface of a solid sphere. For n = 2 the classical Navier slip condition is recovered.
On the other hand, parameter range n \in (0, 2) (n \in (2, 3)) in the equation (1.1) corresponds to
strong (weak) wetting slip regime. More general dynamics of the liquid film for the case when the
draining of the film due to gravity was balanced by centrifugal forces arising from the rotation of the
sphere about a vertical axis and by capillary forces due to surface tension was considered in [11].
In addition, Marangoni effects due to temperature gradients were taken into account in [12]. The
spherical model without the surface tension and Marangoni effects was studied in [17, 18].
We are interested in time evolution of the support of nonnegative strong solutions to
ut +
\bigl(
(1 - x2)| u| n((1 - x2)ux)xx
\bigr)
x
= 0. (1.2)
Equation (1.2) is a particular case of (1.1) with a = b = 0 with an absence of the second-order
diffusion term. Existence of weak solutions for (1.2) in a weighted Sobolev space was shown in [13]
and existence of more regular nonnegative strong solutions of (1.2) was recently proved in [16].
Unlike the classical thin-film equation
ut +
\bigl(
| u| nuxxx
\bigr)
x
= 0, (1.3)
the qualitative behavior of solutions for double degenerate thin-film equation (1.2) is still not well
understood. Note that the model equation (1.3) describes the coating flow of a thin viscous film on a
c\bigcirc R. M. TARANETS, 2019
840 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 841
flat surface under the surface tension effect. Depending on the value of the parameter n, nonnegative
solutions of this equation posses some interesting properties. For example, in 1990, Bernis and
Friedman [2] defined and constructed nonnegative weak solutions of the equation (1.3) when n \geq 1,
and it was also shown that for n \geq 4, with a uniformly positive initial condition, there exists a
unique positive classical solution. Later on, in 1994, Bertozzi et al. [6] generalised this positivity
property for the case n \geq 7
2
. In 1995, Beretta et al. [1] proved the existence of nonnegative weak
solutions for the equation (1.3) if n > 0, and the existence of strong ones for 0 < n < 3. Also,
they could show that this positivity-preserving property holds at almost every time t in the case
n \geq 2. This positivity-preservation result was generalised for a cylindrical surface was obtained
in [7]. Furthermore, for n \geq 3
2
the solution’s support to (1.3) is nondecreasing in time, and the
support remains constant if n \geq 4. The existence (nonexistence) of compactly supported spreading
source type solution to (1.3) was demonstrated for 0 < n < 3 (n \geq 3) in [5]. One of interesting
qualitative properties of nonlinear parabolic thin-film equations is finite speed of support propagation
that is not the case when the parabolic equation is a linear one. This property was first shown in [3]
if 0 < n < 2, and in [4, 10] if 2 \leq n < 3 for nonnegative strong solutions of (1.3). A similar result
on a cylindrical surface was obtained in [8].
Our main result for the thin-film equation on the spherical surface is the finite speed of the
interface propagation in the special case of the strong slip regime n \in (1, 2). Proof of the finite speed
of propagation property is based on local entropy estimate and Stampacchia’s lemma. Moreover, we
obtain an upper bound the time evolution of the support as: \Gamma (t) \leq C0t
1
n+4 . This bound coincides
with the asymptotic behaviour of self-similar type solutions to (1.3) (see [5]).
2. Main result. We study the thin-film equation
ut +
\bigl(
(1 - x2)| u| n
\bigl(
(1 - x2)ux
\bigr)
xx
\bigr)
x
= 0 in QT (2.1)
with the no-flux boundary conditions
(1 - x2)ux = (1 - x2)
\bigl(
(1 - x2)ux
\bigr)
xx
= 0 at x = \pm 1, t > 0, (2.2)
and the initial condition
u(x, 0) = u0(x). (2.3)
Here n > 0, QT = \Omega \times (0, T ), \Omega := ( - 1, 1), and T > 0. Integrating the equation (2.1) by using
boundary conditions (2.2), we obtain the mass conservation property\int
\Omega
u(x, t)dx =
\int
\Omega
u0(x)dx =: M > 0. (2.4)
Consider initial data u0(x) \geq 0 for all x \in \=\Omega satisfying\int
\Omega
\bigl\{
u20(x) + (1 - x2)u20,x(x)
\bigr\}
dx < \infty . (2.5)
Definition 2.1 (weak solution). Let n > 0. A function u is a weak solution of the problem
(2.1) – (2.3) with initial data u0 satisfying (2.5) if u(x, t) has the properties
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
842 R. M. TARANETS
(1 - x2)
\beta
2 u \in C
\alpha
2
,\alpha
8
x,t ( \=QT ), 0 < \alpha < \beta \leq 2
n
,
ut \in L2
\bigl(
0, T ; (H1(\Omega ))\ast
\bigr)
, (1 - x2)
1
2ux \in L\infty \bigl(
0, T ;L2(\Omega )
\bigr)
,
(1 - x2)
1
2 | u|
n
2
\bigl(
(1 - x2)ux
\bigr)
xx
\in L2(P ),
and u satisfies (2.1) in weak sense:
T\int
0
\langle ut, \phi \rangle (H1)\ast ,H1 dt -
\int \int
P
(1 - x2)| u| n
\bigl(
(1 - x2)ux
\bigr)
xx
\phi xdxdt = 0
for all \phi \in L2
\bigl(
0, T ;H1(\Omega )
\bigr)
, where P := \=QT \setminus
\bigl\{
\{ u = 0\} \cup \{ t = 0\}
\bigr\}
,
(1 - x2)
1
2ux(., t) \rightarrow (1 - x2)
1
2u0,x(.) strongly in L2(\Omega ) as t \rightarrow 0,
and boundary conditions (2.2) hold at all points of the lateral boundary, where \{ u \not = 0\} .
Let us denote by
0 \leq G0(z) :=
\left\{
z2 - n - A2 - n
(n - 1)(n - 2)
- A1 - n
1 - n
(z - A) if n \not = 1, 2,
z \mathrm{l}\mathrm{n} z - z(\mathrm{l}\mathrm{n}A+ 1) +A if n = 1,
\mathrm{l}\mathrm{n}
\biggl(
A
z
\biggr)
+
z
A
- 1 if n = 2,
where A = 0 if n \in (1, 2) and A > 0 if else.
Theorem 2.1. Assume that n \geq 1 and initial data u0 satisfies
\int
\Omega
G0(u0) dx < +\infty , then
the problem (2.1) – (2.3) has a nonnegative weak solution, u, in the sense of Definition 2.1, such that
(1 - x2)ux \in L2
\bigl(
0, T ;H1(\Omega )
\bigr)
, (1 - x2)
\gamma
2 ux \in L2(QT ), \gamma \in (0, 1],
u \in L\infty \bigl(
0, T ;L2(\Omega )
\bigr)
, (1 - x2)
\mu
2 u \in L2(QT ), \mu \in ( - 1, \beta ].
The solution in the sense of Theorem 2.1 is called a strong solution. The existence of these
solutions was proved in [16]. Our aim is to establish the finite speed of propagation property for a
strong solution u of (2.1).
Theorem 2.2 (finite speed of propagation). Assume that 1 < n < 2, the initial data satisfies the
hypotheses of Theorem 2.1 and the support of the initial data satisfies \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(u0) \subset \Omega \setminus ( - r0, r0),
where \Omega = ( - 1, 1) and r0 \in (0, 1). Let u be the strong solution from Theorem 2.1. Then there exists
a time T \ast > 0 and a nondecreasing function \Gamma (t) \in C
\bigl(
[0, T \ast ]
\bigr)
, \Gamma (0) = 0 such that u has finite
speed propagation, i. e.,
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(u(\cdot , t)) \subseteq
\bigl[
- r0 + \Gamma (t), r0 - \Gamma (t)
\bigr]
\subset \Omega
for all t \in [0, T \ast ]. Moreover, \Gamma opt(t) = C0t
1
n+4 for all t \in [0, T \ast ].
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 843
3. Proof of Theorem 2.2. 3.1. Local entropy estimate.
Lemma 3.1. Assume that 1 < n < 2 and \nu > 1. Let \zeta \in C1,2
t,x (
\=QT ) such that its support
satisfies \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\zeta ) \subseteq \Omega and (\zeta 4)x = 0 on \partial \Omega . Then there exist positive constants C1, C2 are
independent of \Omega , such that for all T > 0 the strong solution u of Theorem 2.1 satisfies\int
\Omega
(1 - x2)\nu \zeta 4(x, T )G0(u) dx -
\int \int
QT
(1 - x2)\nu (\zeta 4)tG0(u) dxdt+
+
1
4
\int \int
QT
(1 - x2)\nu +2u2xx\zeta
4 dxdt \leq
\int
\Omega
(1 - x2)\nu \zeta 4(x, 0)G0(u0) dx+
+C1
\int \int
QT
(1 - x2)\nu u2x
\bigl[
\zeta 4 + \zeta 2\zeta 2x + \zeta 3| \zeta xx|
\bigr]
dxdt+
+C2
\int \int
QT
(1 - x2)\nu - 2u2
\bigl[
\zeta 4 + \zeta 4x + \zeta 2\zeta 2xx
\bigr]
dxdt. (3.1)
Proof. Equation (2.1) is doubly degenerate when u = 0 and x = \pm 1. Therefore, for any \varepsilon > 0
and \delta > 0 we consider two-parametric regularised equations
u\varepsilon \delta ,t +
\Bigl[
(1 - x2 + \delta )
\bigl(
| u\varepsilon \delta | n + \varepsilon
\bigr) \Bigl(
(1 - x2 + \delta )u\varepsilon \delta ,x
\Bigr)
xx
\Bigr]
x
= 0 in QT (3.2)
with boundary conditions
u\varepsilon \delta ,x =
\bigl(
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr)
xx
= 0 at x = \pm 1,
and initial data
u\varepsilon \delta (x, 0) = u0,\varepsilon \delta (x) \in C4+\gamma (\=\Omega ), \gamma > 0,
where
u0,\varepsilon \delta (x) \geq u0\delta (x) + \varepsilon \theta , \theta \in
\biggl(
0,
1
2(n - 1)
\biggr)
,
u0,\varepsilon \delta \rightarrow u0\delta strongly in H1(\Omega ) as \varepsilon \rightarrow 0,
(1 - x2 + \delta )
1
2u0x,\delta \rightarrow (1 - x2)
1
2u0,x strongly in L2(\Omega ) as \delta \rightarrow 0.
The parameters \varepsilon > 0 and \delta > 0 in (3.2) make the problem regular up to the boundary (i.e., uniformly
parabolic). The existence of a local in time solution of (3.2) is guaranteed by the classical Schauder
estimates (see [9]). Now suppose that u\varepsilon \delta is a solution of equation (3.2) and that it is continuously
differentiable with respect to the time variable and fourth order continuously differentiable with
respect to the spatial variable. For the full detailed proof of existence of strong solutions please refer
to [16].
Multiplying the equation (3.2) by \phi (x, t)G\prime
\varepsilon (u\varepsilon \delta ), integrating over \Omega , and then integrating by
parts yield
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
844 R. M. TARANETS
d
dt
\int
\Omega
\phi G\varepsilon (u\varepsilon \delta ) dx -
-
\int
\Omega
\phi tG\varepsilon (u\varepsilon \delta ) dx =
\int
\Omega
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
xx
\phi dx+
+
\int
\Omega
(1 - x2 + \delta )
\bigl(
| u\varepsilon \delta | n + \varepsilon
\bigr)
G\prime
\varepsilon (u\varepsilon \delta )
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
xx
\phi x dx =
= -
\int
\Omega
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr] 2
x
\phi dx -
\int
\Omega
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
x
\phi x dx -
-
\int
\Omega
\bigl[
(1 - x2 + \delta )(| u\varepsilon \delta | n + \varepsilon )G\prime
\varepsilon (u\varepsilon \delta )\phi x
\bigr]
x
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
x
dx =
= -
\int
\Omega
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr] 2
x
\phi dx+
1
2
\int
\Omega
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr] 2
\phi xx dx -
-
\int
\Omega
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
x
\bigl(
| u\varepsilon \delta | n + \varepsilon
\bigr)
G\prime
\varepsilon (u\varepsilon \delta )
\bigl(
(1 - x2 + \delta )\phi x
\bigr)
x
dx -
-
\int
\Omega
\bigl[
(1 - x2 + \delta )u\varepsilon \delta ,x
\bigr]
x
(1 - x2 + \delta )
\Bigl[ \bigl(
| u\varepsilon \delta | n + \varepsilon
\bigr)
G\prime
\varepsilon (u\varepsilon \delta )
\Bigr] \prime
u
u\varepsilon \delta ,x\phi xdx. (3.3)
Integrating (3.3) in time and taking the regularising parameter \varepsilon \rightarrow 0, by applying the Young
inequality and znG\prime
0(z) =
1
1 - n
z, we finally get
\int
\Omega
\phi G0(u\delta ) dx -
\int \int
QT
\phi tG0(u\delta ) dxdt+
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2
x
\phi dxdt \leq
\leq
\int
\Omega
\phi G0(u0,\delta ) dx+
1
2
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2
\phi xx dxdt -
- 1
1 - n
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr]
x
u\delta ((1 - x2 + \delta )\phi x)x dxdt -
- 1
1 - n
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr]
x
(1 - x2 + \delta )u\delta ,x\phi x dxdt \leq
\leq
\int
\Omega
\phi G0(u0,\delta ) dx+ \mu
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2
x
\phi dxdt+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 845
+
2 - n
2(1 - n)
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2
\phi xx dxdt+
1
4\mu (1 - n)2
\int \int
QT
u2\delta
\bigl(
(1 - x2 + \delta )\phi x
\bigr) 2
x
\phi
dxdt, (3.4)
where \mu > 0. Choosing \mu =
1
2
in (3.4), we arrive at\int
\Omega
\phi G0(u\delta ) dx -
-
\int \int
QT
\phi tG0(u\delta ) dxdt+
1
2
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2
x
\phi dxdt \leq
\leq
\int
\Omega
\phi G0(u0,\delta ) dx+
2 - n
2(1 - n)
\int \int
QT
\bigl[
(1 - x2 + \delta )u\delta ,x
\bigr] 2| \phi xx| dxdt+
+
1
2(1 - n)2
\int \int
QT
u2\delta
\bigl(
(1 - x2 + \delta )\phi x
\bigr) 2
x
\phi
dxdt. (3.5)
Letting \delta \rightarrow 0 in (3.5), we deduce that\int
\Omega
\phi (T )G0(u) dx -
\int \int
QT
\phi tG0(u) dxdt+
+
1
2
\int \int
QT
\bigl[
(1 - x2)ux
\bigr] 2
x
\phi dxdt \leq
\int
\Omega
\phi (0)G0(u0) dx+
+
2 - n
2(1 - n)
\int \int
QT
\bigl[
(1 - x2)ux
\bigr] 2| \phi xx| dxdt+
1
2(1 - n)2
\int \int
QT
u2
\bigl(
(1 - x2)\phi x
\bigr) 2
x
\phi
dxdt. (3.6)
Taking \phi (x, t) = (1 - x2)\nu \zeta 4(x, t) in (3.6) for \nu > 1, we have\int
\Omega
(1 - x2)\nu \zeta 4(T )G0(u) dx -
\int \int
QT
(1 - x2)\nu (\zeta 4)tG0(u) dxdt+
+
1
2
\int \int
QT
(1 - x2)\nu
\bigl[
(1 - x2)ux
\bigr] 2
x
\zeta 4 dxdt \leq
\int
\Omega
(1 - x2)\nu \zeta 4(0)G0(u0) dx+
+ \~C1
\int \int
QT
\bigl[
(1 - x2)ux
\bigr] 2\bigl[
(1 - x2)\nu - 2\zeta 4 + (1 - x2)\nu
\bigl(
\zeta 2\zeta 2x + \zeta 3| \zeta xx|
\bigr) \bigr]
dxdt+
+C2
\int \int
QT
u2
\bigl[
(1 - x2)\nu - 2\zeta 4 + (1 - x2)\nu +2\zeta 4x + (1 - x2)\nu +2\zeta 2\zeta 2xx
\bigr]
dxdt \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
846 R. M. TARANETS
\leq
\int
\Omega
(1 - x2)\nu \zeta 4(0)G0(u0) dx+ \~C1
\int \int
QT
(1 - x2)\nu u2x
\bigl[
\zeta 4 + \zeta 2\zeta 2x + \zeta 3| \zeta xx|
\bigr]
dxdt+
+C2
\int \int
QT
(1 - x2)\nu - 2u2
\bigl[
\zeta 4 + \zeta 4x + \zeta 2\zeta 2xx
\bigr]
dxdt,
where \~C1 =
2 - n
1 - n
\mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
5\nu , 2\nu (\nu + 1), 2(3 + 2\nu )
\bigr\}
, C2 =
16(\nu + 1)4
(1 - n)2
. From here, due to
1
2
\int \int
QT
(1 - x2)\nu
\bigl[
(1 - x2)ux
\bigr] 2
x
\zeta 4 dxdt =
1
2
\int \int
QT
(1 - x2)\nu +2u2xx\zeta
4 dxdt -
- 2
\int \int
QT
x(1 - x2)\nu +1uxuxx\zeta
4 dxdt+ 2
\int \int
QT
x2(1 - x2)\nu u2x\zeta
4 dxdt \geq
\geq 1
4
\int \int
QT
(1 - x2)\nu +2u2xx\zeta
4 dxdt - 2
\int \int
QT
(1 - x2)\nu u2x\zeta
4 dxdt,
we deduce (3.1) with C1 = \~C1 + 2.
Lemma 3.1 is proved.
3.2. Finite speed of propagation. For an arbitrary s > 0 and 0 < \delta \leq s we consider the families
of sets
\Omega (s) := \{ x \in \=\Omega : | x| \leq s\} , QT (s) = (0, T )\times \Omega (s), KT (s, \delta ) = QT (s) \setminus QT (s - \delta ).
We introduce a nonnegative cutoff function \eta (\tau ) from the space C2(\BbbR 1) with the following proper-
ties:
\eta (\tau ) =
\left\{
1 if \tau \leq 0,
- \tau 3(6\tau 2 - 15\tau + 10) + 1 if 0 < \tau < 1,
0 if \tau \geq 1.
Next, we introduce our main cut-off functions \eta s,\delta (x) \in C2(\=\Omega ) such that 0 \leq \eta s,\delta (x) \leq 1 for all
x \in \=\Omega and possess the following properties:
\eta s,\delta (x) = \eta
\biggl(
| x| - (s - \delta )
\delta
\biggr)
=
\left\{ 1, x \in \Omega (s - \delta ),
0, x \in \Omega \setminus \Omega (s),
| (\eta s,\delta )x| \leq
15
8\delta
, | (\eta s,\delta )xx| \leq
5(
\surd
3 - 1)
\delta 2
for all s > 0 and 0 < \delta \leq s. Choosing \zeta 4(x, t) = \eta s,\delta (x)e
- t
T in (3.1), we arrive at\int
\Omega (s - \delta )
(1 - x2)\nu u2 - n(T ) dx+
1
T
\int \int
QT (s - \delta )
(1 - x2)\nu u2 - n dxdt+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 847
+C
\int \int
QT (s - \delta )
(1 - x2)\nu +2u2xx dxdt \leq e
\int
\Omega (s)
(1 - x2)\nu u2 - n
0 (x) dx+
\leq C
\delta 2
\int \int
KT (s,\delta )
(1 - x2)\nu u2x dxdt+
C
\delta 4
\int \int
KT (s,\delta )
(1 - x2)\nu - 2u2 dxdt (3.7)
for all 0 < \delta \leq s. Here and throughout this proof, C denotes a positive constant independent of \Omega .
By (3.7) we deduce
(1 - (s - \delta )2)\nu
\int
\Omega (s - \delta )
u2 - n(T ) dx+
(1 - (s - \delta )2)\nu
T
\int \int
QT (s - \delta )
u2 - n dxdt+
+C(1 - (s - \delta )2)\nu
\int \int
QT (s - \delta )
(1 - x2)2u2xx dxdt \leq
C(1 - (s - \delta )2)\nu
\delta 2
\int \int
KT (s,\delta )
u2x dxdt+
+
C(1 - (s - \delta )2)\nu
\delta 4
\int \int
KT (s,\delta )
(1 - x2) - 2u2 dxdt,
whence \int
\Omega (s - \delta )
u2 - n(T ) dx+
1
T
\int \int
QT (s - \delta )
u2 - n dxdt+
+C(1 - r20)
2
\int \int
QT (s - \delta )
u2xx dxdt \leq
C
\delta 2
\int \int
KT (s,\delta )
u2x dxdt+
C(1 - r20)
- 2
\delta 4
\int \int
KT (s,\delta )
u2 dxdt =: R(s) (3.8)
for all 0 < \delta \leq s \leq r0. We apply Lemma A.1 in the region \Omega (s - \delta ) to a function v := u with
a = d = j = 2, b = 2 - n, k = 0 (or k = 1), N = 1, and \theta 1 =
n
8 - 3n
\Bigl(
or \theta 2 =
4 - n
8 - 3n
\Bigr)
.
Integrating the resulted inequalities with respect to time and taking into account (3.8), we arrive at
the following relations:
A(s - \delta ) \leq C(1 - r20)
- \alpha 1T \beta 1
\bigl(
R(s)
\bigr) 1+\kappa 1 + C T
\bigl(
R(s)
\bigr) 1+\kappa 3 , (3.9)
B(s - \delta ) \leq C(1 - r20)
- \alpha 2T \beta 2
\bigl(
R(s)
\bigr) 1+\kappa 2 + C T
\bigl(
R(s)
\bigr) 1+\kappa 3 , (3.10)
where
A(s) :=
\int \int
QT (s)
u2dxdt, B(s) :=
\int \int
QT (s - \delta )
u2xdxdt,
\alpha 1 =
4(n+ 4)
8 - 3n
, \alpha 2 =
4(6 - n)
8 - 3n
, \beta 1 =
4(2 - n)
8 - 3n
, \beta 2 =
2(2 - n)
8 - 3n
,
\kappa 1 =
4n
8 - 3n
, \kappa 2 =
2n
8 - 3n
, \kappa 3 =
n
2 - n
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
848 R. M. TARANETS
Since all integrals on the right-hand sides of (3.9), (3.10) vanish as T \rightarrow 0 and u \in L2(0, T ;
H1
\bigl(
- r0, r0)
\bigr)
, then for sufficiently small T we get
A(s - \delta ) \leq C3(1 - r20)
- \alpha 1T \beta 1
\bigl(
\delta - 4A(s) + \delta - 2B(s)
\bigr) 1+\kappa 1 , (3.11)
B(s - \delta ) \leq C4(1 - r20)
- \alpha 2T \beta 2
\bigl(
\delta - 4A(s) + \delta - 2B(s)
\bigr) 1+\kappa 2 , (3.12)
where C3, C4 are a positive constant depending on all known parameters and independent of \Omega . Let
us denote by
D(s) := A1+\kappa 2(s) +B1+\kappa 1(s), \kappa = (1 + \kappa 1)(1 + \kappa 2),
C5(T ) := 2\kappa - 1\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigl[
C3(1 - r20)
- \alpha 1T \beta 1
\bigr] 1+\kappa 2 ,
\bigl[
C4(1 - r20)
- \alpha 2T \beta 2
\bigr] 1+\kappa 1
\Bigr\}
.
Without loss of generality, we can define the function
\~D(s) = D(s) if s \in (0, r0] and \~D(s) = 0 if s > r0.
Then by (3.11), (3.12) we arrive at
\~D(s - \delta ) \leq C5(T )
\bigl(
\delta - 4\kappa \~D1+\kappa 1(s) + \delta - 2\kappa \~D1+\kappa 2(s)
\bigr)
(3.13)
for all s \in \BbbR + and \delta \in (0, r0]. Choosing
\delta (s) = \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigl[
4C5(T ) \~D
\kappa 1(s)
\bigr] 1
4\kappa ,
\bigl[
4C5(T ) \~D
\kappa 2(s)
\bigr] 1
2\kappa
\Bigr\}
in (3.13), we find that
\~D
\bigl(
s - \delta (s)
\bigr)
\leq 1
2
\~D(s),
whence it follows
\delta
\bigl(
s - \delta (s)
\bigr)
\leq \gamma \delta (s) \forall s \in \BbbR +, (3.14)
where \gamma = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
2 -
\kappa 1
4\kappa , 2 -
\kappa 2
2\kappa
\bigr\}
< 1. Applying Stampacchia’s lemma (see Lemma A.2) to (3.14), we
obtain
\delta (s) = 0 for all s \leq r0 -
\delta (r0)
1 - \gamma
.
Next, we will find the upper bound for \delta (r0). In view of Theorem 2.1, (1 - x2)
\nu - 2
2 u \in L2(QT ) and
(1 - x2)
\nu
2 ux \in L2(QT ) for any \nu > 1, then the right-hand side of (3.7) is bounded for all T > 0.
So, taking s = 2r0 and \delta = r0 in (3.9) and (3.10), we obtain \~D(r0) \leq C6C5(T ), whence
\delta (r0) \leq C7(1 - r20)
- 2(6 - n)
8 - 3n T
2 - n
8 - 3n .
This implies the upper bound for speed of propagation to solution support, i.e.,
\Gamma (T ) \leq r0 - C8T
2 - n
8 - 3n for all T \leq T \ast :=
\biggl(
r0
C8
\biggr) 8 - 3n
2 - n
(3.15)
for any r0 \in (0, 1), where C8 =
C7
1 - \gamma
(1 - r20)
- 2(6 - n)
8 - 3n .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 849
3.3. Exact upper bound for speed of propagation. In this subsection, we refine the estimate
(3.15). Throughout this subsection, C denotes a positive constant independent of \Omega . Applying
Lemma A.1 in the region \Omega (s) \setminus \Omega (s - \delta ) to a function v := u with a = d = j = 2, b = 1, k = 0
(or k = 1), N = 1, and \theta 1 =
1
5
\Bigl(
or \theta 2 =
3
5
\Bigr)
, and integrating the resulted inequalities with respect
to time, taking into account the mass conservation (2.4), we arrive at the following estimates:
\int \int
KT (s,\delta )
u2 dxdt \leq C T 1 - \theta 1M2(1 - \theta 1)
\left( \int \int
KT (s,\delta )
u2xx dxdt
\right)
\theta 1
+ C \delta - 1TM2, (3.16)
\int \int
KT (s,\delta )
u2x dxdt \leq C T 1 - \theta 2M2(1 - \theta 2)
\left( \int \int
KT (s,\delta )
u2xx dxdt
\right)
\theta 2
+ C \delta - 3TM2. (3.17)
Using (3.16), (3.17) and Young inequality, from (3.8) we find\int
\Omega (s - \delta )
u2 - n(T ) dx+
1
T
\int \int
QT (s - \delta )
u2 - n dxdt+ C(1 - r20)
2
\int \int
QT (s - \delta )
u2xx dxdt \leq
\leq \varepsilon (1 - r20)
2
\int \int
KT (s,\delta )
u2xx dxdt+ C\varepsilon \delta
- 5(1 - r20)
- 3TM2,
where \varepsilon > 0. Selecting \varepsilon \in (0,2 - 5) enough small and making standard iteration process, we get\int
\Omega (s - \delta )
u2 - n(T ) dx+
1
T
\int \int
QT (s - \delta )
u2 - n dxdt+
+C(1 - r20)
2
\int \int
QT (s - \delta )
u2xx dxdt \leq C \delta - 5(1 - r20)
- 3TM2. (3.18)
Taking s = 2\Gamma (T ) and \delta = \Gamma (T ) in (3.18), we obtain\int \int
QT (\Gamma (T ))
u2xx dxdt \leq C \Gamma - 5(T )(1 - r20)
- 5TM2,
whence, similar to (3.16) and (3.17), we have
A(\Gamma (T )) \leq C \Gamma - 1(T )(1 - r20)
- 1TM2,
B(\Gamma (T )) \leq C \Gamma - 3(T )(1 - r20)
- 3TM2.
Hence,
\delta (\Gamma (T )) \leq C\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
[\Gamma - \kappa 1(T )(1 - r20)
- (\kappa 1+\alpha 1)T \kappa 1+\beta 1M2\kappa 1 ]
1
4(1+\kappa 1) ,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
850 R. M. TARANETS
[\Gamma - 3\kappa 2(T )(1 - r20)
- (3\kappa 2+\alpha 2)T \kappa 2+\beta 2M2\kappa 2 ]
1
2(1+\kappa 2)
\biggr\}
=
= C9\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\Gamma - n
n+8 (T )T
2
n+8 ,\Gamma - 3n
8 - n (T )T
2
8 - n
\biggr\}
.
Thus, we get
\Gamma (T ) + C10\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\Gamma - n
n+8 (T )T
2
n+8 ,\Gamma - 3n
8 - n (T )T
2
8 - n
\Bigr\}
\leq r0, (3.19)
where C10 =
C9
1 - \gamma
. Now we use the following calculus result: let a > 0 and b > 0, then the
function f(x) = x+a x - b for all x \geq 0 has minimum at xmin = (ab)
1
1+b and f(xmin) =
1 + b
b
xmin.
Hence, minimizing the right-hand side, we obtain
\Gamma opt(T ) = C0T
1
n+4 for all T \leq T \ast .
Theorem 2.2 is proved.
Appendix A.
Lemma A.1 [14]. If \Omega \subset \BbbR N is a bounded domain with piecewise-smooth boundary, a > 1,
b \in (0, a), d > 1, and 0 \leq k < j, k, j \in \BbbN , then there exist positive constants d1 and d2 (d2 = 0
if \Omega is unbounded) depending only on \Omega , d, j, b, and N such that the following inequality is valid
for every v(x) \in W j,d(\Omega ) \cap Lb(\Omega ):\bigm\| \bigm\| \bigm\| Dkv
\bigm\| \bigm\| \bigm\|
La(\Omega )
\leq d1
\bigm\| \bigm\| Djv
\bigm\| \bigm\| \theta
Ld(\Omega )
\| v\| 1 - \theta
Lb(\Omega ) + d2 \| v\| Lb(\Omega ) ,
where \theta =
1
b
+
k
N
- 1
a
1
b
+
j
N
- 1
d
\in
\biggl[
k
j
, 1
\biggr)
. Note that if \Omega = B(0, R) \setminus B(0, r), where B(0, x) is ball with
the radius x and the origin at 0, then d2 = c(R - r) -
(a - b)N
ab
- k.
Lemma A.2 [15]. Assume that f(s) is nonnegative nondecreasing function satisfying the fol-
lowing inequality:
f(s - f(s)) \leq \varepsilon f(s) \forall s \leq s0,
where \varepsilon \in (0, 1). Then f(s) = 0 for all s \leq s0 -
f(s0)
1 - \varepsilon
.
References
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FINITE SPEED OF PROPAGATION FOR THE THIN-FILM EQUATION IN THE SPHERICAL GEOMETRY 851
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ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 6
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| id | umjimathkievua-article-1479 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:31Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fd/4a2bfeb2cfa69b347fa6e878b2a200fd.pdf |
| spelling | umjimathkievua-article-14792019-12-05T08:56:42Z Finite speed of propagation for the thin-film equation in the spherical geometry Скiнченна швидкiсть поширення збурень для рiвняння течiї тонкої плiвки вздовж кулi Taranets, R. M. Таранець, Р. М. UDC 517.953 We show that a double degenerate thin-film equation obtained in modeling of a flow of viscous coating on the spherical surface has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution $u>0$ and $u=0.$ Using local entropy estimates, we also obtain the upper bound for the rate of the interface propagation. УДК 517.953 Показано, що рівняння тонких плівок із подвійним виродженням, яке виникає з моделювання потоку в'язкого покриття на сферичній поверхні, має скінченну швидкість поширення носія невід'ємного сильного розв'язку, а отже, існує інтерфейс або вільна межа, що розділяє області, де розв'язок $u>0$ і $u=0.$ Крім того, за допомогою локальної ентропійної оцінки отримано оцінку зверху для швидкості поширення інтерфейсу. Institute of Mathematics, NAS of Ukraine 2019-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1479 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 6 (2019); 840-851 Український математичний журнал; Том 71 № 6 (2019); 840-851 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1479/463 Copyright (c) 2019 Taranets R. M. |
| spellingShingle | Taranets, R. M. Таранець, Р. М. Finite speed of propagation for the thin-film equation in the spherical geometry |
| title | Finite speed of propagation for the thin-film equation in the spherical geometry |
| title_alt | Скiнченна швидкiсть поширення збурень для рiвняння течiї тонкої плiвки вздовж кулi |
| title_full | Finite speed of propagation for the thin-film equation in the spherical geometry |
| title_fullStr | Finite speed of propagation for the thin-film equation in the spherical geometry |
| title_full_unstemmed | Finite speed of propagation for the thin-film equation in the spherical geometry |
| title_short | Finite speed of propagation for the thin-film equation in the spherical geometry |
| title_sort | finite speed of propagation for the thin-film equation in the spherical geometry |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1479 |
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