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Anomalous exponents and dipole solutions for the
thin lm equation
M. Bowen , J. Hulshof and J. R. King
Mathematical Institute
Division of Theoretical Mechanics
Leiden University
University of Nottingham
P.O. Box 9512
University Park
2300 RA Leiden
Nottingham NG7 2RD
The Netherlands
United Kingdom
1
1
2
1
2
Abstract
We investigate similarity solutions of the second kind (in that they feature an anomalous
exponent) for a fourth order degenerate diusion equation on the half-line x 0. These
self-similar solutions are termed dipole solutions and, using a combination of phase space
analysis and numerical simulations, we numerically construct trajectories representing these
solutions, at the same time obtaining broader insight into the nature of the four-dimensional
phase space. Additional asymptotic analysis provides further information concerning the
evolution to self-similarity.
1 Introduction
This paper builds on the results of [1], to which we refer for background and relevant references,
by studying further the asymptotic behaviour of the `thin lm' equation (see, for instance, [2])
@h = ? @ hn @ h
@t
@x @x
3
!
3
(1.1)
with n < 2 on the half-line x > 0, a problem noted by Barenblatt in [3]. For most of the
analysis, we shall impose on (1.1) the conditions
at x = 0 and x = s(t) (zero contact angle);
h = @h
@x = 0
(zero ux);
lim hn @ h = 0 at x = s(t)
x"s t @x
3
( )
3
(1.2)
with h = 0 for x s(t), and the (compactly supported) initial data
at t = 0 h = h0 (x); s = a;
(1.3)
in which h0 (x) = 0 for x a. The constraint n < 2 is needed for drainage of uid over the
edge x = 0 to be possible (in other words, to permit the ux
h @x x
J (t) = ?hn @
3
3
(1.4)
=0
to be positive) { see [1] for further details.
For 0 < n < 2 the asymptotic behaviour of (1.1)-(1.3) as t ! 1 is given by a similarity
solution of the second kind (see [3] for numerous other such examples), whereby
with (from (1.1))
h t?f (x=t ) as t ! 1 with x = O(t )
(1.5)
n + 4 = 1
(1.6)
and with (the `anomalous exponent') determined by a nonlinear eigenvalue problem specied
in [1] (the exception to this is the special case n = 1 in which (1.5) is a similarity solution of
the rst kind, with = 1=3, = 1=6, because the rst moment of h is conserved). Further
phase space results for this eigenvalue problem are given in Section 2 below and a detailed
numerical investigation, which in particular makes possible the calculation of the anomalous
exponents (n), is outlined in Section 3. One interesting feature of the numerical analysis is the
appearance of a second family of self-similar solutions in the range 0 n < 1=2. These solutions
have zero contact angle at both the xed and the free boundary, and are specied by imposing
optimal smoothness at the free boundary, characterised by the exponent 3=(n + 1) (which is
larger than the exponent 2). A similar family of solutions may be obtained for the Cauchy
problem (cf. [4]). In the n = 0 linear limit case this corresponds to having h = hx = hxx = 0
at the free boundary, so that h is a solution of the obstacle problem for ht + hxxx = 0, the
obstacle being h 0; see also the discussion at the end of Section 2 in [5]. Furthermore, we
expect these solutions to exist in the whole of the range ?1 < n < 1=2.
We anticipate that (1.5) also describes the asymptotic behaviour of (1.1)-(1.3) for nc <
n 0 for some nite nc, i.e. down to some (currently unknown) critical exponent. As in the
case of the Cauchy problem ([4],[5]), we expect that for n = nc
h e?t f (x=e?nct= ) as t ! 1 with x = O(e?nct= )
4
4
(1.7)
and for n < nc
h (tc ? t) f (x(tc ? t) ) as t ! t?c with x = O((tc ? t)? );
(1.8)
this being a second kind solution to (1.1)-(1.2), with ; > 0 satisfying ?n ? 4 = 1, which
represents extinction at some nite time t = tc (by contrast, the corresponding solution to the
Cauchy problem is of rst kind, with = = ?1=(n + 4)). We shall leave such issues for
n < 0 largely as open problems, though we do give results for the limit n ! ?1; in view of the
status of (1.2) as a combination of the xed boundary conditions of [6], for which nite time
extinction occurs for n < 0, and of the moving boundary conditions of the Cauchy problem
analysed in [4],[5], for which nc = ?4, we anticipate that the critical exponent nc for (1.1)-(1.3)
will satisfy ?4 < nc < 0.
In Section 4 we address three asymptotic regimes, namely n ! 1, n ! 2? and n ! ?1,
using formal methods. Consideration of the rst two arises from the rigorous results of [1] which
cover the existence of (rst kind) similarity solutions for n = 1 and n = 2; we show below how
properties of these solutions determine the evolution to second kind behaviour for neighbouring
values of n. The methods we adopt here are fairly widely applicable to problems involving
quantities which are almost conserved. The limit n ! ?1 is also instructive, in particular
because it illustrates the nite time extinction behaviour suggested above. We conclude in
Section 5 with some brief discussion of other possible boundary conditions.
2 The dynamical system
In this section we identify the appropriate similarity solution in (1.5) as corresponding to a
connection between two critical points of a dynamical system. We shall refer to this solution as
the dipole, and to the corresponding connection as the dipole connection. The dipole is given
by
h(x; t) = t? f (x=t ); n + 4 = 1;
(2.1)
where the prole f () 0 must solve
(f nf 000 )0 = f 0 + f; 0 < < 0 ;
(2.2)
and satisfy, see (1.2), the boundary conditions,
f (0) = f 0(0) = f ( ) = f 0( ) = f ( )nf 000( ) = 0:
0
0
0
0
(2.3)
Here 0 is positive and may be used as scaling parameter. We restrict our attention to 0 < n < 2.
In [7] we transformed (2.2) into a 4-dimensional quadratic system by setting
0
00
000
x = ff ; y = ff ; z = ff ; u = f ?n; t = log ;
2
3
4
(2.4)
where x and t are not to be confused with x and t in (1.1). The system for x, y, z , u as functions
of t reads
x_ = x(1 ? x) + y;
y_ = y(2 ? x) + z;
(2.5)
z_ = z(3 ? (n + 1)x) + u( + x);
u_ = u(4 ? nx):
This transformation maps positive solutions f of (2.2) to orbits of (2.5) in the half-space
H = fu > 0g. To identify the dipole connection we need the local structure near the critical
points, as determined in [7]. We recall, see [7], that the critical points are
(0; 0; 0; 0); (1; 0; 0; 0); P1 = (2; 2; 0; 0); P2 = ( n +3 1 ; ? ?6 + 3 n2 ; (6 n ? 3) (?23 + n) ; 0); (2.6)
(n + 1)
(n + 1)
and
(?2 + n) (n ? 4) ):
(2.7)
P3 = ( n4 ; ? 4 nn?2 16 ; (?16 + 8nn3) (n ? 4) ; (8 n + n32)
3
( n + 4 )
Orbits coming out of these critical points into H correspond to solutions f () dened on some
interval (0; 0 ) with 0 < 0 1. Near = 0 they behave like Ab where b is the x-coordinate
of the corresponding critical point. The constant A > 0 is arbitrary, except for P3 , for which
it follows easily from the u-coordinate (in the range of and n considered below there will be
no such solutions). In [7] the number of degrees of freedom for these solutions with powerlike
behaviour is determined from the linearisation of (2.5) around the corresponding critical point.
Relevant for the analysis in this paper are the points
P for 0 < n < 21 and P for 21 < n < 2;
1
2
(2.8)
3
corresponding to the exponents 2 and n+1
, which are the exponents of the dipole for respectively
0 < n < 21 and 12 < n < 2, see [1]. In both cases the relevant critical point has a 2-dimensional
unstable manifold, denoted by W u , with tangent space spanned by two eigenvectors, one with
nonzero u-component, as can easily be seen from the linearisations
0
?3 1 0
0
B
B
B
B
?2 0 1
0
B
B
B
B
0 0 1 ? 2n + 2
B
B
@
0 0
0
4 ?2n
0
1
BB
CC
BB
CC
BB
CC
at
P
and
BB
CC
BB ?
CC
[email protected]
A
1
n?5
n+1
?6+3 n2
0
0
n?1
n+1
1
0
0
0
n++3 n+1
0
0
n+4
n+1
2
n
( +1)
(6
1
n?3)(?2+n)
2
(n+1)
0
1
CC
CC
CC
CC at P :
CC
CA
2
In the light of the description above, we look for an orbit in H \ W u with the appropriate
large t behaviour. Let us rst review the possiblities. We assume, as will be supported by
the numerics, that orbits in H \ W u have simple !-limit sets, by which we mean singletons
consisting of a single critical point, possibly at innity.
Orbits in H \ W u cannot lie in the stable manifolds of P1 and P2 , because these stable
manifolds are contained in @H = fu = 0g. This observation also applies to the other critical
points with u = 0, provided n < 2. On the other hand, it is perfectly possible for an orbit in
H \ W u to connect to P3 , corresponding to solutions f having algebraic growth (exponent n4 ).
This point has linearisation, see [7],
0
BB
BB
BB
BB ?
BB ?
@
?
(
n?8
n
4 n?16
n2
16+8 n)(n?4)( n++3 )
n2 ( n+4 )
(8 n+32)(?2+n)(n?4)
n2 ( n+4 )
2
1
0
0
n?4
n
1
0
0
0
n+4 ? n+4
n
n
0
0
1
CC
CC
CC :
CC
CC
A
(2.9)
For n = 2 the eigenvalues are ?3; ?2; ?1; 0. The determinant of this matrix being 8(n +
4)(n ? 2)(n ? 4)n?3 , it follows that for n slightly smaller all eigenvalues are negative. Analysing
the characteristic polynomial we nd that there are no purely imaginary eigenvalues in the
parameter range under consideration, which is 0 < n < 2, 0 < n < 1. Thus P3 is an attractor
and we can expect all bounded orbits to approach this point as t ! 1, corresponding to
solutions
!
3
n
(2.10)
f 8(n + 4)(n ? 2)(n ? 4) n4 as ! 1:
Next we consider the possibilities for orbits to go to a point at innity. In [7] this was done
by means of the standard Poincare transformation. Here we use a slightly modied form of this
transformation, see also [8] and [1], and set
Y ;z = Z ;u = U ;
x= X
;
y
=
V
V
V
V
2
3
X + Y + Z + 2U = C;
2
3
2
(2.11)
2
where C is an arbitrary nonzero number. Note that we are restricted to U; V > 0. After a
change of coordinate for the independent variable, the new system reads
X_ = (?Y + (n ? 2)Z ? UZ + U (n ? 3))X ? ZY X + 2Y + (3U + 3Z )Y
?XU (1 + Z)V;
Y_ = Y X + ZX + (?2Y + ((2n ? 1)Z ? 2UZ + (2n ? 3)U )Y )X + 3ZU + 3Z
?2Y U (1 + Z)V;
Z_ = ((?n + 2)Z + U )X + (((?2n + 1)Z + 2U )Y ? 3Y Z + 3U ? 3ZU )X
?3Y Z + U (3U + 2Y + X ? 3Z )V;
U_ = (?n + 3)UX + ((?2n + 3)UY ? 3Y U + 3UZ ? 3U Z )X ? 3UY Z
+U (X ? 3ZU + 3Z + 2Y )V;
V_ = (X + (Y ? Y + (n + 1)Z ? UZ + Un)X ? Y Z )V
+(?ZU ? 3Z ? X ? 4U ? 2Y )V :
2
2
3
2
2
2
3
2
3
2
2
2
3
2
2
2
2
2
2
2
3
2
3
(2.12)
2
2
2
2
2
2
2
Modulo the scaling (a; b; c; d) ! (as; bs2 ; cs3 ; ds3 ) the critical points at innity (V = 0) with
U 0 and X 0 are
Q = (?1; 0; 0; 0); Q = (?1; 12 ; 0; 0); Q = (?1; ? 31 (n ? 2); ? 19 (2n ? 1)(n ? 2); 0); (2.13)
1
2
3
Q = (?1; ? 13 (n ? 3); ? 19 (2n ? 3)(n ? 3); ? 91 (2n ? 3)(n ? 3)) for 3=2 n 3;
4
and Q5 = (0; 0; 0; 1). The point Q1 has linearised matrix
0
BB 0
BB
BB 0
BB
BB 0
BB
BB 0
[email protected]
0
0
?1
1
n?3 0
0
0
0 ?2 + n ?
0
0
0
0 0
0
n?3 0
0
?1
1
CC
CC
CC
CC
CC
CC
CC
CA
(2.14)
(2.15)
and is an attractor (modulo the scaling), its eigenvalues being ?1; ?1; n ? 2; n ? 3 (ignoring in
here and what follows the zero eigenvalue due to the scaling invariance).
None of the other points is an attractor. In the numerics below we will see that orbits in
W u \ H typically select one of the two attractors P3 and Q1. Standard continuity arguments
imply that there must be an orbit in W u \ H which is not in the domain of attraction of P3
and Q1 . We refer to this orbit as the critical orbit, although we only have numerical evidence
of its uniqueness. Tuning we look for an -value for which the critical orbit corresponds to a
solution f satisfying (2.3). To this end, we will identify a critical point in whose 2-dimensional
stable (fast stable if n < 21 ) manifold the critical orbit must be contained for (2.3) to hold.
Generically we expect the critical orbit to connect to a critical point with a 3-dimensional
stable manifold. As we explain next there are two such points and it depends an the value of which of the two is selected by the critical orbit. The rst possibility is Q5 = (0; 0; 0; 1), which
has matrix
0
1
B 0 3 0 0 0 C
BB
BB 0
BB
BB 3 BB
BB
BB 0
@
0
p
p
p
and eigenvalues 0; 3 3 ; 3 (?1 ? i 3);
0 3 0
0 0 0
0 0 0
CC
0 C
CC
CC
3 C
CC :
C
0 C
CC
A
0
p
(2.16)
0 0 0
p (?1 + i 3). We note that this is the only point
3 3
3
2
2
with X = 0 and this suggests that x may be bounded. Modifying (2.11) to
x = X ; y = VY ; z = VZ ; u = VU ;
2
Y + Z + 2U = C;
2
3
(2.17)
2
we nd (modulo scaling) only one critical point at V = 0 with U 0, namely Q~ 5 = ( ? ; 0; 0; 1)
which corresponds to Q5 and has exactly the same eigenvalues. The point Q5 has a 3dimensional centre-stable manifold. We note that the zero eigenvalue is the only one with
an eigenvector having nonzero V -component and that all orbits coming from fV > 0g enter Q5
along this vector. These observations are based on a WKBJ-analysis giving solutions of (2.2)
with leading order term
(2.18)
f A? ;
and possible correction terms
p
!
p
3 1
3 1
B1 cos 3 23 An 3 ; B2 sin 3 23 An 3
1
1
!!
+2 ?1
3
!
2 1
exp ? 32 An 3 :
(2.19)
1
We note that the nonzero eigenvalues are zeros of the same cubic appearing in the WKBJanalysis. Numerically we nd that the critical orbit connects to Q5 for small .
The other three points, Q2 , Q3 and Q4 , are in direct correspondence to points discussed in
[7], Sections 3.2, 3.3, 3.5. Their eigenvalues are, up to a positive multiple,
?1; 2; 2n ? 1; 2n ? 3 for Q ;
(2.20)
p
0; ? (n + 1) ; ?3; ? 4 n ? 5 12+ 20 n ? 8 n for Q ;
(2.21)
p
3; ?n; ? 4 n ? 9 ?82n + 36 n ? 27 for Q :
(2.22)
For the last point the multiple changes sign as changes sign.
For < n < the point Q has a 2-dimensional stable manifold W s containing solutions
2
2
3
2
4
1
2
3
2
2
with the required interface behaviour: zero contact angle and ux, zero ux following from
z(t) = 1 f ()n?1 f 000() ! ;
(2.23)
u(t) and quadratic behaviour near 0 . Thus in this range, to construct the dipole, we need to tune
so as to make the critical orbit connect to Q2. For n > 32 the stable manifold is contained
in V = 0 and may be ignored (it will appear below that in this range we have to connect the
critical orbit to Q4 ). For 0 < n < 21 , however, it is 3-dimensional, but only the orbits coming
in along eigenvectors corresponding to the two most negative eigenvalues have (2.23). Thus we
must tune to make the critical orbit connect to Q2 along this fast stable manifold. Solutions
connecting to Q2 along the generic eigenvector (eigenvalue 2n ? 1, both directions allowed) do
not have zero ux.
The stable manifold of Q3 contains solutions f () which go to zero as ! 0 as a power3
function with exponent n+1
. For 21 < n < 2 its dimension is 3, and the critical orbit goes into
Q3 for too large. For 0 < n < 12 however, the dimension is 2, and Q3 takes over from Q2. In
this range we can also tune to make the critical orbit connect to Q3 . Solutions thus obtained
gain, rather than conserve, mass at the interface, but do represent the smoothest non-negative
solutions for this range of n.
Finally there is Q4 , which for 23 < n < 3 takes over the role of Q2 , having W s with dimension
2 and containing the zero ux, zero contact angle solutions (exponent n3 near 0 ), so that for
3
< n < 2 we tune to connect the critical orbit to Q4 . For n < 32 this point has U < 0 and is
2
not relevant here, while the point ?Q4 has a 3-dimensional stable manifold contained in V = 0
and is also irrelevant here.
3 Numerical construction of the trajectories
3.1 Preliminaries
In this section we outline the numerical approach used to construct the trajectories (outlined
in the previous section) between local equilibrium points and those at innity. The approach is
based on a shooting method program which follows trajectories emanating from the linearized
unstable manifold of the relevant local equilibrium point. These trajectories may either escape
to innity or approach some other local equilibrium point.
The program is based around NAG routine D02PAF and solves equations (2.5) using a
Runge-Kutta-Merson technique. For the work featured here, we always shoot from a twodimensional unstable manifold and the initial values for x; y; z and u come from choosing points
on a closed curve (x; y; z; u) = ?(), traced out by the (single-valued) function ?() dened by
? = Pc + (a cos + b sin ) ;
(3.1)
where a and b represent (suciently small) eigenvectors corresponding to the unstable eigenvalues of the relevant critical point Pc (namely P1 or P2 ) and 2 [0; 2), restricting attention
to u > 0; in practice we take jaj = jbj = d with d = 0:1.
Solutions of (2.5), which, in terms of (2.11) converge to one of the points Q1 , Q2 , Q3 , Q4 ,
escape to innity in nite t, having y ax2 , where a is the Y -coordinate of Qi in (2.13)-(2.14).
These solutions correspond (see [7]) to solutions f () with local interface behaviour
1
f A( ? ) 1?a as ! ?:
0
0
(3.2)
For clarity we include Table 1 which lists this correspondence and all possible behaviours,
including convergence to Q5 or P3 . The dimension of the stable manifold as n varies for
each equilibrium point is also included in Table 1. We note that Q5 corresponds to solutions
satisfying x ! ?= as the solution escapes to innity as t ! 1, whence (2.18). As explained
in Section 2 we look for connections to Q1 {Q5 and in particular, tuning and , to Qi 's with
dim(W s) = 2.
We recognise escape to innity in nite time with y ax2 by projecting the trajectories
onto the (log jxj, log jyj) plane, noting that points where a trajectory crosses an axis in the
(x; y) plane correspond to asymptotes in the log plot. As trajectories escape to innity in the
log plot we therefore expect them to have gradient 2 (corresponding to reaching Q2 , Q3 or Q4 ),
dim(W S )
Equilibrium point
Q
Q
Q
Q
Q
P
1
2
3
4
5
3
f -behaviour y ax n <
<n<
<n<2
f ( ? )
a=0
4
f ( ? )
a=
3
2
0
3
f ( ? ) n+1 a = ?n 2
3
3
3
f ( ? ) n a = ?n 1
2
2
f ? 3
f n4
4 (0 < n < 1)
2
0
0
0
1
2
3
2
3
2
1
1
2
2
2
0
1
2
3
3
3
Table 1: Possible f -behaviours and dimension of the stable manifold of the
corresponding critical point.
gradient less then 2 (corresponding to reaching Q1 ), or x ! 4=n or x ! ?= (corresponding
to P3 and Q5 , but these are better seen in a plot of x versus t). Additionally, if we extrapolate
a line with slope 2 we nd an intercept of log(a) (see Figure 2).
To nd the critical pair of values that corresponds to the dipole solution we perform numerical simulations varying and . For xed we rst determine the critical -value for which
the solution is nongeneric (i.e. not in the domain of attraction of P3 or Q1 ). This -value is
denoted by c(). We then vary to obtain a pair (c ; c (c )) for which the solution connects
to a critical point (Q2 or Q4 ) with dim(W s)=2. This c is the anomalous dipole exponent. For
n < 1=2 the situation is slightly dierent; see Section 2 and Section 3.4 below.
3.2 Behaviour for 1=2 < n < 3=2
In this range of n we look for the orbit connecting the two-dimensional unstable manifold of
P2 to the two-dimensional stable manifold of Q2 (Table 1 shows this point to relate to the
required quadratic behaviour for f local to the interface). The behaviour of orbits coming out
of P2 in this regime, described in terms of the (; ) parameter plane, is shown in the schematic
in Figure 1. The numbers which are not contained in parentheses represent the exponent in
the second column of Table 1. Numbers contained in braces denote the dimension of stable
manifolds.
Figure 2(a) shows the critical (c , c (c )) solution for n = 1:2. In this case c 0:30317.
1 {4}
3/(n+1) {3}
−α/β {3}
θ
2 {2}
4/n {4}
α
Figure 1: Schematic of the behaviour of trajectories in the (; ) plane for
1=2 < n < 3=2.
From the additional straight line superimposed on the graph we see that the gradient of the
solution in the far-eld is 2 and that the intercept of the straight line is given by log(1/2), as
desired. For the non-generic solution with > c we pick up the behaviour governed by Q3 as
shown in Figure 2(b) (cf. Figure 1). For (; c ()) with > c , by inspection of the x-t graph
we see that the solutions have x ! ?= as t ! 1 (as far as numerical accuracy will allow),
corresponding to Q5 . We note that it is a consequence of unavoidable numerical inaccuracies
that the solution eventually escapes to one of the generic behaviours (see Figure 2(c)). It is
thus dicult to recognise this solution in terms of the log x, log y plot, as the required solution
has x and y converging to a specic point - numerical errors take the solution away from this
point and the fact that the trajectory moves slowly in its neighbourhood is not apparent from
a plot involving x and y but not t.
Figure 3 shows the dipole connection for n = 1 found numerically and this is consistent
with the analysis in [1] (for n = 1 the rst moment of h is conserved which xes = 1=3),
helping to validate the numerical approach.
3.3 Behaviour for 3=2 < n < 2
For 3=2 < n < 2 we retain most of the structure that exists for 1=2 < n < 3=2, the only change
being that the stable manifold of Q2 becomes one-dimensional and that of Q4 two-dimensional
20
15
log|y|
log|y|
15
10
10
5
5
0
Intercept=log((2-n)/3)
0
Intercept=log(1/2)
-5
-5
-10
-10
-15
-15
Gradient=2
Gradient=2
-20
-20
-8
-6
-4
-2
0
2
4
6
8
10
-8
-6
-4
-2
0
2
4
6
8
log|x|
log|x|
c ; c (c ))
> c ; c ())
(a) (
(b) (
2
x
1.5
1
0.5
0
-0.5
-1
α/β=−0.4545
-1.5
-2
0
1
2
3
4
5
t
6
< c ; c ())
(c) (
Figure 2: Critical trajectories that reach innity for n = 1:2.
(cf. Section 2). Q4 takes over from Q2 and is the relevant equilibrium point for the dipole
solution in this range of n. The (; ) parameter plane illustrating the structure of the phase
space is shown in Figure 4, where we employ the same notation as discussed in Figure 1.
Figure 5(a) shows the critical trajectory representing the dipole solution for n = 1:8. We
see that the superimposed straight line has intercept log (3 ? n)=3 and gradient 2, as desired.
For completeness we also include Figures 5(b) and (c) to illustrate the behaviour of non-generic
solutions on each side of the dipole solution.
15
log|y|
10
5
0
-5
Intercept=log(1/2)
-10
-15
Gradient=2
-20
-10
-8
-6
-4
-2
0
2
4
6
8
log|x|
Figure 3: Graph of the (c ; c (c )) solution for n = 1 with c = 1:0=3:0 and
c(c) 1:5676027.
1 {4}
3/(n+1) {3}
−α/β {3}
θ
3/n {2}
4/n {4}
α
Figure 4: Schematic of the behaviour of trajectories in the (; ) plane for
3=2 < n < 2.
3.4 Behaviour for 0 < n < 1=2
In this regime, the behaviour is more complex in that we now have two possible critical connections. We can nd either a connection between the two-dimensional unstable manifold of P1 and
the two-dimensional stable manifold of Q3 , or we may also connect to the fast two-dimensional
stable manifold of Q2 . The situation is illustrated in Figure 6.
15
15
log|y|
log|y|
10
10
5
5
0
0
-5
Intercept=log((2-n)/3)
-5
-10
Intercept=log((3-n)/3)
-10
-15
Gradient=2
Gradient=2
-15
-20
-8
-6
-4
-2
0
2
4
6
8
-8
-6
-4
-2
0
2
log|x|
4
6
8
log|x|
c ; c (c ))
; c ()), > c
(a) (
(b) (
2
x
1.5
1
0.5
0
-0.5
-1
α/β=−0.4878
-1.5
-2
0
1
2
3
4
5
t
6
; c ()), < c
(c) (
Figure 5: Critical trajectories that reach innity for n = 1:8.
We comment at this point that such zero ux solutions with quadratic interface behaviour
are found by studying graphs of z against t for the non-generic solutions. The zero ux solution
has z ! 0 as t ! 1 in correspondence with Q2 while all other non-generic solutions have z
escaping to innity.
The solutions with interface behaviour f (0 ? )3=(n+1) as ! 0 are particularly
sensitive to numerical errors and it is dicult to locate these solutions by numerical study of
(2.5). This problem persists for all n < 1=2 and so we use a dierent numerical methodology in
1 {4}
Zero flux {2}
Nonzero flux {3}
−α/β {3}
3/(n+1) {2}
θ
4/n {4}
α
Figure 6: Schematic of the behaviour of trajectories in the (; ) plane for
0 < n < 1=2.
this case. We solve (2.2) directly by shooting from = 0 with initial conditions adapted from
[6], namely
f 2 + B3?2n as ! 0;
(3.3)
(with B being determined as part of the solution) and nd the value of 0 such that f (0 ) =
f 0(0 ) = 0 for xed . Varying then allows us to construct graphs of 0 against for various
n, see Figure 7. The critical value of that we seek is the minimum value for each curve.
Here the transition from f having a minimum to f having a maximum in 0 occurs. Thus
we also nd that f 00 (0 ) = 0, consistent only with Q3 . It is noteworthy that the branches
of solutions can be continued (and now correspond to solutions with sign changes), but fold
back. It is this borderline behaviour which causes the above noted numerical diculties; as is
to be expected, they become more severe as n ! 1=2. We determine the sign change branches
numerically only for n = 0, this being discussed below. Note that the form of the dynamical
system (2.5) makes it inconvenient to nd trajectories representing solutions with sign changes.
Moreover, numerous dierent branches of sign change solutions can be specied, depending on
how the local behaviour at a zero of h is prescribed; those given below for n = 0 are analytic
at a zero.
For n = 0 we may easily construct additional branches of solutions featuring sign changes
and having zero contact angle at their free boundary. It is the linearity of (2.2) that allows us
7.5
η0
7
n=0
6.5
n=0.1
n=0.2
6
n=0.3
5.5
n=0.4
5
4.5
0.42
0.44
0.46
0.48
0.5
0.52
0.54
α
0.56
Figure 7: Graph of 0 against for various n 2 [0; 1=2).
to continue the solutions through zero without diculty and nd further values of 0 for which
zero contact angle conditions hold at the interface. We proceed by xing and then separating
the initial condition (3.3) into two parts, denoting the solution of (2.2) with f = 2 + o(3 ) as
f1 and the solution with f 3 as f2. Values of 0 at which the Wronskian, W (f1(0 ); f2 (0 )),
becomes zero correspond to solutions of f (0 ) = f1 (0 ) + Bf2(0 ) = 0, with zero contact angle
at the interface. The rst zero of W corresponds to the positive solution discussed above and
B is specied from the requirement that f1(0 ) + Bf2(0 ) = 0. By counting zeros and varying
we then build up branches of solutions displayed as 0 against in Figure 8. Each fold
corresponds to an increase in the number of sign changes. We use numbers in braces to denote
the number of sign changes in the solution, so that f0g denotes the branch of non-negative
solutions.
3.5 The overall picture
By combining the approaches from the previous sub-sections we may construct the graph of c
against n, as shown in Figure 9. The 3=(n + 1) branch merges with c (n) as n ! 1=2? ; the
numerics become increasingly delicate as n approaches 1=2 and 2, resulting in the gaps in the
curves. It is noteworthy that the asymptotics about n = 2 indicate a rather rapid change in in that neighbourhood.
14
η0
{3}
13
12
11
{2}
10
{1}
9
8
7
6
{0}
5
4
0.4
0.5
0.6
0.7
0.8
0.9
α
1
Figure 8: Graph of 0 against for n = 0 illustrating branches of zero contact angle sign change solutions. Crosses indicate solutions with
f 00(0 ) = 0 and circles label solutions with f 000(0 ) = 0. Note that,
somewhat surprisingly, there appear to be no zero ux solutions
that contain a single interior zero, but there are two with three interior sign changes; moreover no f 00 (0 ) = 0 solution with a single
interior zero manifests itself.
0.6
0.55
2
0.5
0.45
3/(n+1)
0.4
α
0.35
0.3
0.25
asymptotics
0.2
β
0.15
0.1
0
0.5
1
1.5
2
n
Figure 9: Graph of exponents of the exceptional connections against n. The
(dashed) straight lines show the asymptotic results (4.9), (4.23)
derived below.
4 Asymptotic results
In this section we investigate the behaviour of the anomalous exponent and its corresponding
dipole solution in the limit cases n ! 1, n " 2 and n # ?1. This will be done by means of
formal asymptotics for the full partial dierential equation.
4.1 n ! 1
In this limit we can exploit the result from [1] that for n = 1 the asymptotic solution is of the
rst kind (due to conservation of the rst moment), with
h t? 31 f (x=t 61 ) as t ! 1:
(4.1)
We write n = 1 + " with j"j 1 and analyse the limit " ! 0 on the timescale = j"j ln t =
O(1) in which deviations from (4.1) are signicant. It is easy to show that, for ?1 < " < 1=2,
d Z s xhdx = 1 "(1 + ") Z s h"?1 @h 3 dx;
dt 0
2
@x
0
so writing
(4.2)
h = t? 31 F (; ); s = t 61 ( ); = x=t 16 ;
yields
d Z Fd = 1 (1 + ")sgn(")e? 13 sgn " Z F "? @F d:
d
2
@
Expanding in the form F F (; ) as " ! 0, it follows from (1.1) and (4.2) that
!
1 2F + @F = e? 13 sgn " @ F @ F ;
6
@
@
@
3
1
( )
0
0
(4.3)
0
0
0
implying that
( )
3
0
0
3
F = e 13 sgn " ( )f (=( ));
0
( )
4
where f ( ) represents the solution (4.1) for n = 1, which we specify uniquely via
f (1) = 0; f () > 0 for 0 < < 1:
(4.4)
The expression (4.3) then provides a solvability condition determining the scaling factor ( ),
whereby
Z1
Z 1 df 3 1
1 sgn(") 6 1
d
3
fd d e
( ) = 2 sgn(") f ?1 d de 3 sgn(") 6( ):
(4.5)
0
0
Since
we have
1 2 f = f d3 f ? f d2 f + 1 df 2 ;
6
d3 d2 2 d
Z
1
f ?1
df Z
3
1
d d = 2
so, dening the positive constants and by
0
Z
1
ddf
2
!
2
2
!
2
2
d; =
Z1
2
d ? 3 fd;
0
Z
1
fd;
(4.6)
( ) = exp 61 ? 32 sgn(") ;
(4.7)
=
equation (4.5) yields
0
ddf
0
2
0
0
for some constant 0 which is determined by matching into the timescale t = O(1) in which
the leading order problem is simply that for n = 1; 0 thus depends only on the rst moment
of the initial data. For = O(1) we thus have
Zs
0
1 xhdx exp ? 3 sgn(") as " ! 0:
6
0
Hence
6 3 1
1
1
1
3 ? 9 ? 1 "; 6 + 18 ? 2 "
We nd by computing f ( ) numerically that
as " ! 0:
1:5998 10? ; 3:9023 10?
4
4
(4.8)
(4.9)
(4.10)
so is decreasing near n = 1 and 3 > , as might be expected.
4.2 n ! 2?
The analysis here is similar, relying on the rst kind result for n = 2 that
h t? 61 f (x=t 61 ) as t ! 1;
the borderline case n = 2 lying in the mass preserving regime (see [1]). We write n = 2 ? with 0 < 1 and = ln t. The analysis of this case relies on the local result from [1] that
for 0 < < 3=2
! 3?1 3
(3
?
)
3
as x ! 0+ ;
(4.11)
h 3(3 ? 2) J (t)x
where J (t) is dened in (1.4), so that
d Z s hdx = ?J (t):
dt
(4.12)
0
In this case we need to write
h = t? 16 F (; );
s = t 16 ( );
= x=t 16 ; J = K ( )=t;
giving to leading order as ! 0+ that
(4.13)
!
1 F + @F0 = e 16 @ F 2 @ 3 F0 ;
6 0
@
@ 0 @3
from which it follows that
F = e? 121 ( )f (=( ));
(4.14)
2
0
where f ( ) gives the (mass preserving) similarity solution for n = 2, which we again specify via
(4.4). We thus have
df ? 121 (1 ? ) = f ddf ? 12 d
;
2
2
2
2
so that
df = p1 at = 0:
d
6
(4.15)
G (; ) = (3K ) 13 e 13 ? 181 ;
(4.16)
We now need the result that the local expression (4.11) also provides the leading order
solution as ! 0+ in the (exponentially small) inner region = O(1) with < 0, where
= ln , in which G(; ) = F (; )= is given to leading order by
0
0
writing K K0 ( ) as ! 0+ . The quantity F0 (; ), introduced above, represents the leading
order solution in the outer region = O(1) and matching with (4.16) as ! 0 thus requires
that
F0 (3K0 ) 13 e? 18 as ! 0+ ;
(4.17)
it then follows from (4.14)-(4.15) that
e? 121 = 6 32 3K ;
3
0
(4.18)
this being the rst of the two relations between ( ) and K0 ( ) that we require. The second
comes from (4.12)-(4.14), which yield
d e? 121 = ?K ;
d
3
where
=
Finally, (4.18)-(4.19) imply
Z
1
0
(4.19)
0
fd:
! !
2 12 1
1
= 0 exp 36 1 ? 3 3 ;
(4.20)
1 K = 3 exp ? 3 ;
62 3
6 2 3
3
0
0
(4.21)
where 0 = (0) is determined by the mass of the initial data, the leading order problem for
t = O(1) being that with n = 2. It follows from (4.21) that for = O(1)
Zs
0
1 hdx exp ? 3 3
0
(which, as is to be expected, decays with ) and that
!
2 12 2
1
1
6 + 36 1 + 3 3 ;
(4.22)
6 2 3
!
2 21 1
1
1
6 + 36 1 ? 3 3 as ! 0+ :
(4.23)
Numerically we nd that 5:2641 10?2 . Additionally decreases monotonically with n
and 3 < (2=3) 12 , 3 < 2 , with decreasing from 1=4 at n = 0 through 1=6 at n = 1 before
subsequently rising again to reach 1=6 at n = 2 (see Figure 10).
We note that the timescale = O(1) corresponds to exponentially large t, the rate of loss
of mass being extremely slow. This analysis therefore claries the apparently abrupt transition
noted in [1] between mass loss for n < 2 and mass conservation for n 2.
4.3 n ! ?1
Letting n ! ?1 we have in particular n < nc so that (1.8) applies. We note that nc may be
characterised by the fact that the rescaled ODE for f , namely
(f n f 000)0 = f 0 + f;
= ;
(4.24)
has a positive solution with f (0) = f 0 (0) = f (1) = f 0 (1) = lim
f n(1)f 000 (1) = 0 precisely for
n"1
= ?4=n.
α/β
0.3
β
0.2
2
0.1
2β−α
0
1.5
-0.1
1
β−α
-0.2
0.5
-0.3
-0.4
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0
0.5
1
n
1.5
2
n
(a)
(b)
Figure 10: Graphs comparing the values of (a) , 2 ? (rst moment),
? (mass) against n and (b) = against n.
The analysis partly follows that of [6]. We write n = ?1= and take the limit ! 0+ ; we
scale such that h0 (x0 ) = max h0 = 1 and assume h0 to be monotonic either side of x = x0 .
The initial evolution occurs over the exponentially short timescales t = O(1) < 1; we
have
h hI x =q?
0 < x < q? (t);
h h (x)
q?(t) < x < q (t);
h hI (s ? x) =(s ? q ) q (t) < x < s (t);
2
2
0
0
2
+
0
+
2
+
0
with hI (t) = t and where q? < x0 and q+ > x0 are dened by h0 (q ) = hI ; the leading order
interface position s0 (t) is then determined by conservation of mass, giving (recalling s(0) = a)
1 h (s ? q ) = Z a h (x)dx:
0
3 I 0 +
q+
When hI reaches one, each of q? and q+ coincide with x0 ; a slow phase of evolution then occurs
(cf.[6]), of which we omit details, leading on to the nal regime of exponentially short timescales
(tc ? t) = O(1) < 1 which are of most interest to us here since they govern the extinction
behaviour, whereby h 0 for t tc . Dening hm (t; ) to be the maximum of h(x; t), located
at x = q(t; ) say, the evolution proceeds as follows.
Writing hm h0m (t), q q0 (t) as ! 0, we have the outer solutions
0 < x < q (t)
h hm x =q
h hm (s ? x) =(s ? q ) q (t) < x < s (t):
(4.25)
x = q(t; ) + z; h = hm (1 ? )
(4.26)
0
0
The inner scalings
yield
2
2
0
0
2
0
0
0
2
0
dhm + h dq @ ? h ?
m dt @z
m
dt
0
0
so that
3
(
0
=
1)
@ e0 @ @z
@z
3
0
!
(4.27)
3
e0 @@z = ?dz + b + c;
2(?z)=q as z ! ?1;
(4.28)
= @
=
0
at
z
=
0
;
@z
2z=(s ? q ) as z ! +1;
where the quantities b, c and d are independent of , with
? hm? = dhdtm d; hm= dqdt b;
(4.29)
and c arises on integrating (4.27). The constraints in (4.28) correspond to six boundary conditions, so are in principle sucient to determine d(q ; s ), b(q ; s ) and c(q ; s ), as well as
(z; q ; s ). Two leading order ordinary dierential equations for the three unknowns hm ; q
and s are then provided by (4.29); the third follows from conservation of mass, which implies
3
0
0
3
0
0
0
0
0
3
0
(1
)
0
3
0
1
0
0
0
0
0
0
0
0
0
0
0
that
d Z s hdx = ? dq h + h? = @ h
dt q
dt
@x
3
1
3
!
x
q+
=
and hence from the second of (4.25) that
d 1 h0 (s ? q ) = ?h0 dq0 :
(4.30)
m dt
dt 3 m 0 0
Further information can be deduced from the scaling properties of (4.28). Writing q0 (t) =
(t)s0 (t), where 0 < < 1, together with z = s0Z , d = D=s40, b = B=s30 and c = C=s30, yields
e0 @@Z = ?D()Z + B () + C ();
2(?Z )= as Z ! ?1;
= @
@Z = 0 at Z = 0;
2Z=(1 ? ) as Z ! +1;
3
0
0
3
0
0
0
0
(4.31)
which determines D(), B (), C () and 0 (Z ; ); the boundary value problem (4.31) is the key
ingredient in governing the evolution. The system (4.29)-(4.30) can now be reduced at leading
order to
h0m = (tc ? t) ;
1 dq0 = () dh0m ;
q0 dt h0m dt
(4.32)
d 1 ? h q = ?h dq ;
m dt
dt 3 m
where we have written () = B ()=D() (the appearance of the small parameter in the
leading order expression hm is acceptable because the behaviour we are outlining takes place
on the timescale T = O(1), where T = ? ln(tc ? t), implying hm (T ) = exp(?T ); however, for
brevity we shall retain tc ? t throughout this description). As t ! t?c we have (t) ! c , a
0
0
0
0
0
constant, corresponding to the required self-similar behaviour, with, using (4.32),
q kc(tc ? t)? c ;
1 ? c (1 ? ( )) = ( );
c
c
3c
where the constant kc depends on the initial data. Hence c is determined by
0
(
)
(c ) = (1 ? c)=(1 + 2c)
(4.33)
where () is (in principle) known from (4.31). Unfortunately, it does not seem possible to
obtain c explicitly, but it follows from (4.31) that D is even and B is odd about ? 1=2 and
we anticipate that D > 0 and that B > 0 for < 1=2; this implies that 0 < c < 1=2. Since
we have (c ) < 1, we can conclude that we have nite time extinction, with the mass tending
to zero as t ! t?c . Finally, using ? 4 = we obtain
+ 4 (c); (c)
2
as ! 0+ :
(4.34)
5 Discussion
We have mainly been concerned in this paper with the one-dimensional thin lm equation (1.1)
for the zero contact angle conditions imposing zero ux at the free boundary (1.2). Numerous
generalisations are also amenable to the approaches we have adopted; the one to which we
restrict our discussion here replaces the zero contact angle conditions in (1.1) by zero pressure
ones (such a possibility was already noted in [1]), giving
@ h =0
h = @x
at x = 0 and x = s(t) (zero pressure);
(5.1)
@
h
n
= 0 at x = s(t)
(zero ux);
lim h
x"s t @x
The conditions at x = 0 in (5.1) are admissible if n < 1 and the conditions at x = s(t) if n < 2;
2
2
3
3
( )
these ranges can be extended if the second derivative boundary conditions are replaced by the
constraints that the x2 and (s ? x)2 terms be absent from the relevant local expansions { at
x = 0 the rst two terms are then of the form x and x3?n (so the constraint is admissible
for n < 2 but has unbounded second derivative if n > 1) and at x = s they are (s ? x) and
(s ? x)4?n (admissible for n < 3 but unbounded second derivative if n > 2).
Our main observation concerning (5.1) is that rst kind behaviour now occurs for n = 0
rather that n = 1 (as noted in [9], these are the values of n for which (1.1) admits conservation
laws other than conservation of mass). Indeed, for n = 0 we have
"
d Z s t xh(x; t)dx = ? x @ h ? @ h
dt
@x @x
( )
2
3
3
0
2
#s t
( )
;
(5.2)
0
so the rst moment is conserved when (5.1) holds, giving
h(x; t) t? 12 f (x=t 41 ) as t ! 1
(5.3)
as the relevant asymptotic behaviour (in fact, (5.3) is simply the x-derivative of the zero contact
angle source solution for n = 0 given in [4]); for other n a new second kind formulation is
required and a new critical exponent nc will arise.
Sign change scenarios provide other avenues for further investigation, leading to discrete
spectra of second kind exponents (cf. [10]). Figure 8 illustrates a particular case with n = 0;
for this exponent, a possible limit behaviour as the number of sign changes tends to innity
can be deduced from
"
#
d Z 1 x hdx = ? x @ h ? 2x @ h + 2 @h 1 ;
(5.4)
dt
@x
@x
@x
which implies that the second moment is conserved (so that = 3=4) for solutions to (1.1)
with n = 0 which decay in an oscillatory fashion as x ! 1 and satisfy zero contact angle
conditions at x = 0. Figure 8 indeed indicates that = 3=4 is a plausible limit for both crosses
2
0
2
3
2
3
2
0
and circles as the number of sign changes tends to innity; however, the third moment of h is
R
also conserved, as is 01 (x; t)hdx for any with t = xxxx, (0; t) = x (0; t) = 0 and with
not blowing up exponentially fast as x ! +1 (there being an innite number of suitable
polynomial solutions for ), so there is a family of other (more non-generic) limit solutions,
further illustrating the complexity of such formulations.
The above examples illustrate the variety of distinct prescriptions which are available for
(1.1); this is to be contrasted with the much more widely studied second order case
@h = @ hn @h @t @x @x
(5.5)
for which the only choice of boundary conditions relevant in the current context is
at x = 0
at x = s(t)
h = 0;
h = hn @h
@x = 0:
(5.6)
The higher order case (1.1) not suprisingly admits an innite number of analogs of (5.6), of
which (1.2) and (5.1) are arguably the most signicant mathematically, at least for n > 1=2.
Acknowledgements
JH and JRK gratefully acknowledge the support of the NWO/British Council and MB that of
the TMR network Nonlinear Parabolic Partial Dierential Equations: Methods and Applications (ERBFMRXCT980201). JRK is also grateful for the support of the Leverhulme Trust.
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