Ockendon J., Howison S., Lacey A., Movchan A. Applied Partial Differential Equations

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268

PARABOLIC EQUATIONS

4. It is relatively easy to generalise (6.57) to account for transformations of the

dependent variable u. This is done in Exercise 6.21, where it is easily seen to

lead to similarity solutions of the form

u=t°F(

(6.65)

for any constant a. We recall that our fundamental `heat source' solution (6.30)

is of this form with a = -1/2.

An important example is that of the fundamental solution of the heat equation

in space dimension m > 1, which can be determined as one of these similarity

solutions. Spherical symmetry demands that

Ou_82u m-1Ou

Or2 +

r 8r'

where r = jxj and an obvious similarity solution is u = t°F(r/ft) = t°F(rl). The

value of a must be such that the total amount of heat is conserved, i.e.

f u(x, t) dx = t° f F (r/;)

dx = constant t°+"'/2

- mV t

is constant, and it is seen that a = -m/2. The equation for F is now

a2rF+(m 1+2)

d + 2F=0. (6.66)

Miraculously, there is an explicit solution F = constant

e-17 2/4 which satisfies the

regularity condition dF/dq (0) = 0. When we finally require fR" u dx = 1, we find

that

u(x, t) =

(4rt)-m/2e-r2/4t ,

in accordance with (6.35).

Another generalisation is the inclusion of a constant source term in the heat

equation, when again a similarity solution of the form (6.65) may be appropriate.

For example, the half-line problem

-6i=

in x > 0, (6.67)

with u = 0 on x = 0 and u = 0 at t = 0, models uniform heating in a half-

space with zero initial and boundary conditions. Now a similarity solution of the

form (6.65) is vital to give us enough freedom to allow group invariance: (6.67)

is invariant under the group x' = e'x, t' = e2At, u' = e2'u, and so we write

u = tF(q) and t = x/ f . This yields

d2F rl dF

drl2 +2d

-F+1=0 foul>0,

with F(0) = 0 and, since tF(x/t1/2) -+ 0 as t -+ 0, F(rl)/rl2 -i 0 as n -+ oo. It can

be shown that this is enough information for the solution to be written down in

terms of parabolic cylinder functions (these are defined in Exercise 5.40).

SIMILARITY SOLUTIONS AND GROUPS 269

*6.5.3 General remarks

Despite all the efforts that have been put into the development of a systematic the-

ory for similarity solutions, several puzzling aspects remain. The principal short-

coming of the group invariance approach is that, at the end of the day, someone

has to `spot' the group under which the differential equation is invariant; the ap-

proach has made the identification task easier than that of spotting the similarity

variables directly, but it has not automated it.

We wish to draw attention to two other aspects of the theory. The first is

the relationship between the invariance approach and that of separation of the

variables. The latter method has been freely used in the preceding chapters with

little or no comment, and in every case of its use the reader has had to 'eyeball'

the fact that separation was feasible. Group invariance sheds some light on this

situation for, if we return to the argument leading to (6.65) and in particular to

the calculation of Exercise 6.21, we soon see that the heat equation is invariant

under the group whose infinitesimal generator is

U=U a

+V8

+H -,

where one solution for U, V and W is

U = (Al + 2A2t)x + A3 + A4t,

V=A5+Alt+A2t2,

W =

(A6 -

1 1 1A2x21

A2t - 2Adx - 4

(6.68)

and A; are constants. (The general solution has an arbitrary solution of the heat

equation added to W.) Now it is easy to see that one possibility is

AeuB

so Y, = t and Y2 = ue-°t, where a = A6/A3, are both invariants for which

UY = 0. Hence there is a solution in which 12 = F(Y 2 ), i.e.

U = e' F(t),

which is just the result of separation of variables!

Although we do not advocate this approach to separation of variables in this

book, it is interesting to note that when we apply it to Laplace's equation, even

with the restriction that only the independent variables x and y are candidates for

transformation, we find that

where the infinitesimal generator U = U 8/8x + V 8/8y, and that U and V satisfy

the Cauchy-Riemann equations

270 PARABOLIC EQUATIONS

8x8y' Ox8y

(see Exercise 6.24). Hence, U and V are harmonic conjugates and we have recovered

the conformal invariance of §5.9.1.

This leads to our final observation. Clearly, the more terms that enter a partial

differential equation and its initial and boundary conditions, the less likely it is

to admit a group invariance. Hence, similarity solutions are often of most prac-

tical value as representations of the asymptotic behaviour of the solution in the

neighbourhood of certain interesting points in the space of independent variables

at which the initial and boundary conditions take a simple form. For example,

whenever we `switch on' a boundary value for the solution of a parabolic equation

that is different from its initial value, we expect the starting behaviour near this

boundary to be described by the similarity solution (6.45), where x is the coordi-

nate normal to the boundary. Such a result, if true, is of great value in ensuring

that discretised versions of the problem start off in the right way, because no Tay-

lor expansion exists near t = 0, x = 0. Equally, given any suitably localised initial

distribution of heat on the line -oo < x < oo, we expect the temperature to tend

to the similarity solution (6.34) as t -> oo. This is because, at a large distance from

the origin in the (x, t) plane, the initial data can be approximated by a multiple

of the delta function.tts

Such expectations are often correct but their proof is beyond the scope of this

text. We will, however, mention one fascinating aspect of `local similarity', which

can best be illustrated by an example of an elliptic Dirichlet problem. Suppose

V2U=0

in two-dimensional polar coordinates, with

9=0, 0<r<1,

r2,

6=a, 0<r<1,

f(9),

r=1, 0<9<a.

Assume, for the moment, that we are not clever enough to be able to solve the

problem exactly by separation of the variables, yet we still seek to find the be-

haviour of u near the `singular point' r = 0. A local similarity solution, obtained

in practice by separation of variables, is u oc r" sin nB, where, to satisfy the bound-

ary conditions, we need n = 2 and

r2 sin 29

sin 2a

(6.69)

However, this approximation clearly fails at a = it/2. All is revealed when we

consider the exact solution

1"81f f(x) is smooth and its integral is unity, then it is an easy exercise to show that

Iim,-+o(UE)f(x/c) = 6(x).

NONLINEAR EQUATIONS 271

n7rO

r2 00

sin 20

sin 2c,

+ E bnr" I° sin

(-f , (6.70)

n=1

where the b,, are Fourier sine series coefficients defined by

f(9) -

sin 29

JO (

nag

for

gin 2a =

n sin ` a J

0 < 6 < a.

n=1

The series (6.70) is valid for all a with 0 < a < a as long as a $ 7r/2, and,

indeed, can even be modified to work with a = it/2 (see Exercise 6.25). Hence, it

is clear that (6.69) is only the dominant term as r - 0 when a < 7r/2; for larger

values of a, this term is swamped by the first Fourier term blrl/11 sin(ir9/a),

which is still a similarity solution but one whose relevance is not obvious from a

cursory glance at the differential equation and boundary conditions near r = 0.

Indeed, its determination requires that we both solve an eigenvalue problem to

find the requisite power of r and then use global information to find b, . This is an

example of what is called second-kind similarity. It is of great practical importance

because it is often vital to know whether the local behaviour near a singularity of

a solution of a partial differential equation is controlled by local rather than global

considerations.

6.6

Nonlinear equations

6.6.1 Models

6.6.1.1

Semilinear equations

Reaction-diffusion equations are frequently semilinear and of the form

= V2u+ f(u,x,t).

(6.71)

Typically they appear as models in population dynamics, with the inhomogeneous

term depending upon a local population density, and in physical chemistry or

chemical engineering, where f varies with temperature and/or chemical concen-

tration in a reactor. The function f may be positive or negative to model say birth

or death, or exothermic or endothermic reactions, respectively.

If the problem is homogeneous and autonomous, f = f (u). For example, if u

is the concentration of some chemical undergoing an Nth-order chemical reaction,

f (u) = AuN. On the other hand, if u is the temperature in an exothermic reaction,

f is often of the form Ae"'(1+e0This is a resc.aling119 of the Arrhenius function

introduced in §5.11.1 and, since a is usually very small in practice, f is often

replaced by )eu, as in (5.159). Other autonomous models arise in situations where

diffusion is trying to spread out the dependent variable uniformly in space while

nonlinearity is trying to maintain it at a certain value. A famous example is the

"'The Arrhenius function of temperature T is e-R/RT, which can be written in the form

Aeu/(1+eu) by setting T = To(1 + cu), c = RTe/E and A = e-B/RTO, where To is a suitably

chosen reference temperature.

272 PARABOLIC EQUATIONS

Fisher equation for a population u in which f = u - u2, the first term representing

a linear birth rate and the second the limiting effect of the food supply, which

diminishes as u - 1. Another is the Cahn-Allen equation in which f = u - u3; u

is now the fraction of material that is undergoing a certain kind of phase change

from one stable phase u = -1 to another stable phase u = 1.

Often semilinear equations contain small parameters multiplying f or the

Laplacian in (6.71), and Exercise 6.26 indicates how such parameters can be ex-

ploited.

6.6.1.2

Quasilinear equations

Many problems in fluid dynamics lead to quasilinear rather than semilinear para-

bolic equations. A simple example concerns the flow of a compressible fluid through

a porous medium. Taking u to represent the fluid density, then Darcy's law (5.26),

which relates the velocity v to the pressure p by v = -(k/p)Vp, the equation of

conservation of mass, Ou/Ot + V (uv) = 0, and an equation of state, p = p(u),

can be combined to give

= V . (K(u)Vu),

where K(u) is proportional to udp/du. If we further specialise to the case of an

isothermal perfect gas, we have p oc u while, for an adiabatic situation in which

no heat is transferred to or from the gas, p oc uti with ry > 1. In either case the

model can be written as the porous medium equation

V. (u' Vu), (6.72)

where n is a positive constant.

This equation also arises in the study of thermal radiation, where energy is

transferred electromagnetically as well as by conduction or convection. Thus, at

any point, there is not only the absolute temperature T (x, t), but also an electro-

magnetic energy density Q which depends on the direction y through the point x,

so we write

Q = Q(x,Y, t),

1Y1=1;

(6.73)

electromagnetic energy propagates at the speed of light c in all directions y in an

isotropic material. In the absence of heat conduction, the key modelling assump-

tions are that the material emits electromagnetic energy at a rate /3T4 per unit

volume,120 where ,B is constant, and absorbs it at a rate proportional to Q. Thus,

since c is large, the two energy balances are

(6.74)

where the gradient is with respect to x, and

120The fourth power can be explained by the fact that the density of the energy emitted by

a `black body' as a function of the frequency w of the radiation is, by quantum statistics,

8ah(w/c)3/(ehW/kT - 1), where h is Planck's constant, k is Boltzmann's constant and T is

absolute temperature. Integrating over 0 < w < oo, we find that the total energy emitted is

proportional to T4.

NONLINEAR EQUATIONS

273

Q dS - 4rrl3T'

c-- =

(aQ -

8T4) dS = a 4 (6 75)

1=1

lYl=1

where p is the density, c is the specific heat and a is a constant. The presence

of the integral in (6.75) means that, as is the case with most radiative transfer

problems, the model is an integro-differential equation rather than a partial dif-

ferential equation. However, it can be approximated by a differential equation in

the so-called `optically thick' limit in which 0 and a are large and comparable in

an appropriate non-dimensionalisation. Then the dependence of Q on y is weak

and, substituting iteratively in (6.74), we can write

Q = a 1 Ta - a (Y . V)T' +

Ql (y . p)2T4 + ...1

. (6.76)

Now

T'dS=47rT4,

IYl=u

yl=t

lyl=t

and

(y. V)2T4 dS = 3

V2 (T4),

YI=u

so only the last term in (6.76) ultimately contributes to (6.75), which is simply

(6.72) with n = 3 after a trivial rescaling.121

The porous medium equation also models the thickness It of a viscous drop

spreading under gravity over a horizontal surface. as we can see by modifying the

derivation of the paint model of §1.1. The horizontal velocity is again proportional

to y(2h - y)G'p, but now p is nearly hydrostatic and thus approximately equal

to pg(h - y), so the horizontal components of Vp and Vh are, to leading order,

proportional to each other. Finally, the equation of conservation of mass gives

= V(hs`h)

with a suitable scaling of time.

A similar equation can model the horizontal spreading of highly fissured vol-

canos, which can be considered as shallow porous media through which magma

flows from below. The upper surface of the volcano moves normal to itself with

a velocity proportional to the rate of arrival of the magma. Again taking h to be

the height of the volcano surface above some horizontal datum, the magma pres-

sure is also approximately hydrostatic, so that the Darcy velocity of the magma

121 Unfortunately, radiative heat transfer in many processes such as glass manufacture is often

midway between optically thick and optically thin, but porous medium equation models are still

often used.

274 PARABOLIC EQUATIONS

is proportional simply to -Vh. The total horizontal flow rate is then proportional

to -hv'h and conservation of mass means that

8t =

V (hVh),

and we can anticipate solutions of this equation in which h has the topography of,

say, Mount Fuji.

Another example from fluid dynamics which leads to a quasilinear equation

different from the porous medium equation is Burgers' equation (2.50), which arises

in the theory of viscous one-dimensional gas dynamics; because of its importance,

its detailed derivation is given in Exercise 6.27.

We conclude our modelling discussion by quickly mentioning another famous

quasilinear equation which describes the two-dimensional steady flow of an incom-

pressible fluid flowing in a boundary layer on a wall. This is the Prandtl equation

for the stream function '(x, y) which, for the case of flow past a flat plate, takes

the form

Oi,

for 0 < y < oc, x > 0, (6.77)

8y 8x8y

8x 8yT 8y3

with

z/5=-y =0 aty=0

(6.78)

and

V) = y + O(1) at infinity. (6.79)

In this model the fluid velocity is, as usual, (80/8y, -Ski/8x) and some specifi-

cation of the behaviour at the `leading edge' x = y = 0 is necessary to close the

problem. The derivation of (6.77)-(6.79) is too complicated to describe here but

the basic idea will be given in Chapter 9. Suffice it to say that the left-hand side

of (6.77) models the inertia of the flow, the right-hand side models viscous forces,

(6.78) expresses the fact that the fluid adheres to the plate and (6.79) represents

the uniform flow outside the boundary layer. While (6.77) is not strictly speaking

parabolic, it can be made so by means of a partial hodograph transformation sim-

ilar to that described in §4.8.2. When we regard u = ft/8y as a function of x and

,y, the chain rule gives that

O2'

aye -

8lp'

8z1p _ 8u erg 8u

831

_ 8

8a8y

8x + 8x

8y3 - u T , 0

(u 8+p) '

and hence (6.77) becomes

8u_ 8 (Uau)

8x e0 00

which is just a special case of (6.72). However, in many applications it is easier to

work with (6.77) directly.

NONLINEAR EQUATIONS 275

6.6.2

Theoretical remarks

We have already remarked that almost all nonlinear partial differential equations,

be they hyperbolic, elliptic or parabolic, need to be treated on their own merits.

Concerning parabolic equations, a good rule is to start by asking the following

questions.

1. Are there steady solutions?

2. Are there spatially homogeneous solutions?

3. Are there any symmetries that can be exploited to find similarity solutions, as

in §6.5? (Questions 1 and 2 are really special cases of this.)

Other general questions which might also be posed are as follows.

4. Is the solution allowed to change sign? In all the examples above, except the

Burgers and the Cahn-Allen equations, the physical interpretation demands

that the dependent variable is non-negative.

5. Does the maximum principle apply? (When it does, it can be used to answer

the previous question.)

6. Is there degeneracy, i.e. is the equation `properly parabolic'? For example, the

porous medium equation, which can be written as

= u"v2u + nu"-11Qu12,

is not uniformly parabolic because as u -+ 0 the coefficient of the elliptic opera-

tor V2u vanishes. Indeed, regarding the right-hand side as u"-1(uV2u+nI Vuj2),

it is the gradient rather than the Laplacian of u that dominates when u be-

comes small, indicating that the equation looks more and more like a first-order

equation in this limit.

For degenerate problems it is possible, as we shall see later in this section and

again in Chapter 7, for there to be a fire boundary separating the support of u,

i.e. the region where u > 0, from the region where it vanishes. Of course, if such a

free boundary does occur, say for the porous medium equation, we cannot expect

the differential equation (6.72) to hold in its vicinity, because some derivatives

of u almost certainly fail to exist. Hence, as in Chapter 1, we need to consider

the possibility of defining a weak solution by integration in a suitably generalised

sense.

The next two sections are devoted to some answers to these questions for

equations of porous medium, reaction-diffusion and Burgers type.

6.6.3 Similarity solutions and travelling waves

Consider first the porous medium equation (6.72). It has no non-trivial spatially

homogeneous solutions, and steady asymmetric solutions in more than one space

dimension are hard to find, as we learned from the p-Laplacian problem of §5.11.1.

However, in one space dimension we can find travelling waves which describe a

free boundary separating a region in which u > 0, say t = x - V t < 0, from

one in which u =_ 0. Such a phenomenon would have been quite impossible for

276 PARABOLIC EQUATIONS

(a) (b)

Fig. 6.3 Diffusion from an initially localised source: (a) non-degenerate; (b) degenerate.

a non-degenerate parabolic equation; for example, we know from (6.32) that the

solution to a Cauchy problem for the heat equation is strictly positive for t > 0

for data that is positive only on a small interval and zero elsewhere (Fig. 6.3).

Writing u = F(£) in flu/8t = 8/8x (u"8u/8x) gives

d{( d)+Va =0,

so that one possibility is

r(nV(Vt-x))1l", x<Vt,

(6.80)

Note that, if n >, 1, u ceases to be differentiable at x = Vt and so the porous

medium equation certainly does not hold there. However, we can see that

ztms

(

n 8x

u")

(6.81)

which can be thought of as a Rankine-Hugoniot relation, as in §2.5. Indeed, this

equation expresses conservation of mass at x = Vt in the fluid dynamics inter-

pretations of §6.6.1.2. In any case, the way is open to construct a theory of weak

solutions of the porous medium equation by writing down the space-time integral

of (6.72), after multiplication by suitable test functions, and seeking functions u

that may be as badly behaved as (6.80), yet satisfy the integral identity. This is

precisely the procedure that we adopted in Chapter 1, and, not surprisingly, we

find that (6.81) is automatically satisfied by any weak solution. The analysis is

too intricate to describe here but it is interesting that no 'entropy-like' criterion

is now needed to ensure uniqueness.

Unexpected scenarios sometimes emerge when we seek travelling wave solutions

of semilinear equations. For the autonomous case of (6.71), flu/8t = V2u + f (U),

we encounter the ordinary differential equation

d2F

+ V

+ f (F) =

for which a phase-plane analysis can be carried out in terms of F and dF/dr;. For

the Fisher equation, where / = F(1-F), Exercise 6.28 reveals that travelling waves

NONLINEAR EQUATIONS 277

are possible in which the population F tends to zero ahead of the wave ({ -i +oo),

and to unity behind the wave ( -> -oo), but that V must exceed"' 2. The effect of

the nonlinearity is to legislate against the slow waves in the continuous `spectrum'

V > 0 that exists for the heat equation with zero temperature as C -+ +oo.

A contrast emerges when we proceed to the semilinear equations with a cubic

nonlinearity. When f = u(u - a) (1 - u) with 0 < a < 1/2, so that fo f (u) du >

0, the phase-plane in Exercise 6.29 reveals that there is a unique positive wave

speed for which u = F(t) connects the stable equilibrium points F(oo) = 0 and

F(-oo) = 1. However, when a = 1/2, which is effectively the Cahn-Allen equation,

fo f (u) du = 0. Then the unique wave speed is also zero and the only travelling

wave connecting u = 0 to u = 1 is the steady state.

We anticipated the structure of travelling wave solutions of Burgers' equation

after (2.50). In Exercise 2.18 we effectively showed that, if V > 0 and

-V d{ + F d{ =

a2dz

(6.83)

and F takes any prescribed values F± as

-+ ±oo, respectively, then

V = 2(F++F_).

Hence the wave speed can take any value, but we also recall that the restriction

F_ > F+ must be imposed. Note the importance of the quadratic nonlinearity in

(6.83). If a model ever led to Burgers' equation with a cubic nonlinearity, such as

20u

_ 02u

8t +

8x2'

we would find that travelling waves described by

- V d t + F2 dt =

d2ez

would not necessarily be constrained by the restriction F_ > F+. This can be

shown to allow the possibility of shock waves of expansion as well as compression,

which would have serious implications for the discussion in §2.5.2.

The existence of all these travelling waves can be viewed, perhaps perversely,

as resulting from the invariance of the relevant partial differential equations under

arbitrary translations of x and t. Greater invariance is possible for equations with

fewer terms, so let us now return to the porous medium equation (6.72). We just

consider one-dimensional problems for simplicity (although many of the arguments

12211 can be shown that the wave of minimum speed, 2, is the one that is observed as t -+ on,

because it has the best stability properties.