4. Scalar Conservation Law Equations- part III
We have been introduced to scalar conservation law equations in one space variable at a
fairly nontechnical level. In this section we will begin to consider some of the issues that
lead us to develop more sophisticated mathematical tools. We begin by reconsidering the
so called Lax entropy condition for selecting the physically relevant weak solution to a
problem for a scalar conservation law equation.
Parabolic Regularization
Consider the problem
/ t uÝx, tÞ + / x FÝuÞ = 0,
uÝx, 0Þ = u 0 ÝxÞ,
Ý4.1Þ
At a point where u L = uÝx ? , tÞ ® uÝx + , tÞ = u R , we have the condition that asserts that a
shock is the physically relevant solution if aÝu L Þ = F v Ýu L Þ > F v Ýu R Þ = aÝu R Þ. If we suppose
F”ÝuÞ > 0 for all u 5 R then aÝuÞ is increasing so aÝu L Þ > aÝu R Þ if and only if u L > u R ; i.e.,
the wave speed to the left of the discontinuity is greater than the wave speed to the right,
the fast wave ”crashes into” the slow wave.
An alternative dervation for this selection principle is obtained by considering the
perturbed problem,
/ t uÝx, tÞ + / x FÝuÞ = P / xx uÝx, tÞ,
uÝx, 0Þ = u 0 ÝxÞ.
Ý4.2Þ
This problem can be shown to have a unique smooth solution for all P > 0, and for arbitrary
initial data in L 2 ÝRÞ. The term / xx u has the physical interpretation of viscosity, hence the
term P / xx u suggests that a small amount of viscosity has been added to the fluid or gas
modelled by our original problem. If we can solve the perturbed problem and then let P ¸ 0,
hopefully the solution will tend to the physically relevant solution of the original zero
viscosity problem. The parabolic equation (4.2) is thus a regularized version of equation
(4.1) and drawing conclusions about the behavior of the solution to (4.1) by observing
behavior in the limit for solutions of (4.2) is referred to as the method of parabolic
regularization.
We look for a solution of the form uÝx, tÞ = g x ?P at for a 5 R and g 5 C 2 ÝRÞ to be
determined. Such a solution is called a travelling wave solution (TWS) since a solution of
this form represents the wave form gÝ6Þ propagating with speed a. Substituting this solution
into (4.2) leads to
i.e.,
and
? aP g v ÝsÞ + 1P d FÝgÝsÞÞ = P 12 g”ÝsÞ
ds
P
s = x ?P at ;
g”ÝsÞ = d FÝgÝsÞÞ ? a g v ÝsÞ
ds
g v ÝsÞ = FÝgÝsÞÞ ? a gÝsÞ + C.
It follows that
g v ÝsÞ | K?K = FÝgÝsÞÞ| K?K ? a gÝsÞ| K?K
v
g ÝsÞ ¸ 0 as | s| ¸ K, then it follows that
a=
Ý4.3Þ
and if we add the constraint that
FÝgÝKÞÞ ? FÝgÝ?KÞÞ
gÝKÞ ? gÝ?KÞ
1
Note that the speed of the TWS is consistent with a shock speed determined by the R-H
jump condition. To examine what happens between s = +K and s = ?K, write
g v ÝsÞ = FÝgÝsÞÞ ? a gÝsÞ + C and plot the two functions f 1 ÝgÞ = FÝgÞ + C and f 2 ÝgÞ = ag
versus g on the same axes.
We see from the picture that g v ÝsÞ = 0 at g = g 1 , g 2 where FÝgÞ + a = a g. It is also clear
from the picture that
i)
ii)
FÝgÞ + a < a g for g 1 < g < g 2 hence
g v ÝsÞ < 0 for g 1 < g < g 2 .
FÝgÞ + a > a g for g < g 1 or g > g 2 hence
Ý4.4Þ
g ÝsÞ > 0 for g < g 1 or g > g 2 .
v
These assertions imply that g 1 is a stable fixed point for gÝsÞ while g 2 is an unstable fixed
point for the differential equation for the unknown function, gÝsÞ. It is also evident from (4.4)
that the graph of gÝsÞ versus s must look like the following figure,
In the language of dynamical systems, there is a heteroclinic orbit joining the unstable fixed
point at g = g 2 to the stable fixed point at g = g 1 . Notice in particular that the stability of the
fixed points is determined by the relationship between the slopes of the two functions
2
f 1 ÝgÞ = FÝgÞ + C and f 2 ÝgÞ = ag at the critical points. Specifically,
f 1 v Ýg 2 Þ = F v Ýg 2 Þ > a = f 2 v Ýg 2 Þ
implies
g = g 2 is unstable
f 1 v Ýg 1 Þ = F v Ýg 1 Þ < a = f 2 v Ýg 1 Þ
implies
g = g 1 is stable.
F v Ýg 1 Þ < a < F v Ýg 2 Þ
Then the condition
Ý4.5Þ
u P Ýx, tÞ = g x ?P at
implies the existence of a TWS
for (4.2).
Now (4.2) tends to (4.1) as P tends to zero, and u P Ýx, 0Þ = g xP
u 0 ÝxÞ as P tends to zero, where
u 0 ÝxÞ =
g 2 if x < 0
Ý4.6Þ
.
g 1 if x > 0
tends to the limit
i.e.,
/ t u P Ýx, tÞ + / x FÝu P Þ = P / xx u P Ýx, tÞ,
¹
/ t uÝx, tÞ + / x FÝuÞ = 0,
u P Ýx, 0Þ = g xP .
¹
as P ¸ 0
uÝx, 0Þ = u 0 ÝxÞ,
It seems plausible then that as P ¸ 0, u P Ýx, tÞ tends (e.g. in some L p ? norm) to the solution
of (4.1) for u 0 given by (4.6).
In addition, since u P Ýx, tÞ satisfies the entropy condition (4.5) for all values of P > 0, it
follows that u = lim u P Ýx, tÞ must also satisfy (4.5). To see that u is the shock solution,
P¸0
consider the special case, FÝuÞ =
1
2
u 2 . In this case (4.3) becomes
g v ÝsÞ =
1
2
gÝsÞ 2 ? a gÝsÞ + C 0
=
1
2
ßg 2 ? 2a g + C 21 à
=
1
2
Ýg ? g 1 ÞÝg ? g 2 Þ,
g 1 = a ? a 2 + C 21 < g 2 = a + a 2 + C 21 .
It follows now that
gÝsÞ =
g 2 + g 1 e Ks
,
1 + e Ks
from which we can see :
for
K=
g2 ? g1
2
gÝsÞ ¸ g 2 as s ¸ ?K,
gÝsÞ ¸ g 1 as s ¸ +K,
gÝ0Þ = Ýg 1 + g 2 Þ/2
= a (wave speed = shock speed for Burger’s eqn)
Note also that plotting gÝ xP Þ versus x for various values of P > 0, the travelling wave
form g approaches the step discontinuity as P ¸ 0.
3
graph approaches step as eps tends to 0
Existence of a Weak Solution
Consider the initial value problem (4.1) where we suppose
i) u 0 5 L K ÝRÞ
ii) F 5 C 2 ÝRÞ with F”ÝuÞ > 0 for | u| ² || u 0 || K
Ý4.7Þ
Here L K ÝRÞ is the linear space of functions uÝxÞ for which there is a finite M > 0 such that
| uÝxÞ| ² M, almost everywhere. For u 5 L K ÝRÞ, the smallest such constant M is defined to
be || u|| K .
Under the conditions (4.7) it is possible to prove there exists a weak solution for (4.1).
The weak solution u = uÝx, tÞ has the following properties:
(a) | uÝx, tÞ| ² || u 0 || K for Ýx, tÞ 5 R 2+
(b) there exists a C > 0, depending only on u 0 and F such that
for all a > 0, and all Ýx, tÞ 5 R 2+ ,
uÝx + a, tÞ ? uÝx, tÞ
< C
a
t
(c) the solution depends continuously on the data in the sense that if u, v solve (4.1) for
initial data u 0 , v 0 respectively,
then for all x 1 < x 2 , and all t > 0
x +At
x
X x 2 | uÝx, tÞ ? vÝx, tÞ| dx ² X x 2?At | u 0 ÝxÞ ? v 0 ÝxÞ| dx
1
1
where A = max áaÝuÞ : | u| ² || u 0 || K â.
The proof of this theorem will not be given (cf shock waves and reaction diffusion
equations by Smoller, p266). The proof is constructive, showing that a difference equation
approximation to (4.1) is solvable and that the solution to the discrete problem converges in
an appropriate sense to the solution of (4.1). Although the details of this process are not
given, we can explain some of the implications of the conditions (a), (b), and (c).
Condition (c) asserts that the solution constructed in the proof is unique and depends
4
continuously on the data. However, it does not assert that there could not be another
solution to (4.1) which does not satisfy (c); i.e., although the solution whose existence is
implied by the proof has the properties indicated here, there is no guarantee at this point
that there is not another solution, obtained by some different algorithm, which does not
satisfy (c). Later we will state a theorem that rules out this possibility. Note also that (c)
implies that the solution has a finite speed of propagation.
Condition (b) says that if uÝx + a, tÞ ? uÝx, tÞ < 0 then the condition holds with any choice
of C (including zero). In this case the difference uÝx + a, tÞ ? uÝx, tÞ need not decay at t
increases. On the other hand, if uÝx + a, tÞ ? uÝx, tÞ > 0 then the condition asserts that there
is a constant C for which the estimate holds, and in this case the difference necessarily dies
out like t ?1 . As we have seen in the examples, for a convex FÝuÞ, the case
uÝx + a, tÞ ? uÝx, tÞ < 0 is associated with a shock while the case of a positive difference is
associated with a fan or rarefaction wave. Then the condition (b) could be used as a
selection principle instead of the previously given entropy condition. Moreover, when the
solution is a rarefaction wave, the condition implies that the solution is not only continuous
but is ”almost differentiable”. For example, it follows from (b) that the solution is a function
of bounded variation. To see this let b denote a constant such that b > C/t, and let
vÝx, tÞ = uÝx, tÞ ? bx. Then, for a > 0, (d) implies
vÝx + a, tÞ ? vÝx, tÞ = uÝx + a, tÞ ? uÝx, tÞ ? ba ² a C ? b
t
< 0,
which is to say, vÝx, tÞ is nonincreasing as a function of x. Then vÝx, tÞ is of bounded
variation as a function of x, and since this is also true of the function bx, it is also true of
uÝx, tÞ. Now it follows that uÝx, tÞ has, for each t, at most a countable number of jump
discontinuities as a function of x.
Uniqueness of the Weak Solution
Under the conditions (4.7) we can show that, up to a set of measure zero, there is at most
one solution for (4.1) which satisfies the condition (b) given above. Again, we cannot give
the proof since it is beyond the scope of this course, but we can indicate the idea of the
proof. Suppose that u and v are two weak solutions for (4.1). Then
X XR 2 ßu / t f + FÝuÞ / x fà dxdt + XR u 0 fÝx, 0Þ dx = 0 -f 5 C Kc ÝR 2 Þ
+
X XR 2 ßv / t f + FÝvÞ / x fà dxdt + XR u 0 fÝx, 0Þ dx = 0 -f 5 C Kc ÝR 2 Þ,
+
and
X XR 2 ßÝu ? vÞ / t f + ÝFÝuÞ ? FÝvÞÞ / x fà dxdt = 0
+
1
-f 5 C Kc ÝR 2 Þ.
d FÝs u + Ý1 ? sÞ vÞ ds = X 1 F v Ýs u + Ý1 ? sÞ vÞ ds Ýu ? vÞ
0
ds
Now
FÝuÞ ? FÝvÞ = X
hence
FÝuÝx, tÞÞ ? FÝvÝx, tÞÞ = aÝx, tÞ ÝuÝx, tÞ ? vÝx, tÞÞ
where
aÝx, tÞ = X F v Ýs uÝx, tÞ + Ý1 ? sÞ vÝx, tÞÞ ds .
Then
0
1
0
X XR 2 Ýu ? vÞ ß / t f + aÝx, tÞ / x fà dxdt = 0
+
-f 5 C Kc ÝR 2 Þ.
Now suppose for arbitrary d 5 C Kc ÝR 2 Þ we can find f 5 C Kc ÝR 2 Þ satisfying
5
/ t f + aÝx, tÞ / x f = d in R 2 .
Then it would follow that
X XR 2 Ýu ? vÞ d dxdt = 0
+
Ý4.8Þ
-d 5 C Kc ÝR 2 Þ
which would impy the uniqueness result. It only remains to show that for arbitrary
d 5 C Kc ÝR 2 Þ we can find a solution f for (4.8) that belongs to C Kc ÝR 2 Þ. This is the difficult
part of the proof and it is omitted. It is in this part of the proof that the condition (b) is
needed.
Decay of Entropy Solutions
In addition to existence and uniqueness results, it is possible to prove certain facts about
the long term behavior of weak solutions to scalar conservation law equations. These
results are the consequence of the entropy condition. Suppose first that uÝx, tÞ is a smooth
solution of (4.1) in the region áÝx, tÞ : x 5 IÝtÞ, t 5 ß0, Tàâ where IÝtÞ = Ýx 1 ÝtÞ, x 2 ÝtÞÞ and
x j ÝtÞ, j = 1, 2 are two base characteristics for (4.1).
Since u is constant along characteristics, it follows that the variation of u on IÝ0Þ in the
x-variable is the same as the variation of u on IÝTÞ = Ýx 1 ÝTÞ, x 2 ÝTÞÞ in the x-variable. That is,
varßuÝx, 0Þ, IÝ0Þà = sup > i | uÝx i , 0Þ ? uÝx i?1 , 0Þ|,
varßuÝx, TÞ, IÝTÞà = sup > i | uÝx i , TÞ ? uÝx i?1 , TÞ|,
where the sup is over all partitions of the interval, and since u is constant along
characteristics, these expressions are equal. Now suppose a shock develops at a point p of
IÝTÞ as the result of two shocks carrying conflicting information to this point.
6
Then
varßuÝx, TÞ, IÝTÞà = varßuÝx, TÞ, Ýx 1 ÝTÞ, y 1 ÝTÞÞà + varßuÝx, TÞ, Ýy 2 ÝTÞ, x 2 ÝTÞÞà
= varßuÝx, 0Þ, Ýx 1 Ý0Þ, y 1 Ý0ÞÞà + varßuÝx, 0Þ, Ýy 2 Ý0Þ, x 2 Ý0ÞÞà
< varßuÝx, 0Þ, IÝ0Þà.
This last inequality is the result of the fact that
varßuÝx, 0Þ, IÝ0Þà = varßuÝx, 0Þ, Ýx 1 Ý0Þ, y 1 Ý0ÞÞà + varßuÝx, 0Þ, Ýy 1 Ý0Þ, y 2 Ý0ÞÞà
+ varßuÝx, 0Þ, Ýy 2 Ý0Þ, x 2 Ý0ÞÞà
and it implies that the variation in u(x,t) in the x-variable is a decreasing function of time.
This observation can be made quantitatively precise in two different ways.
Suppose u = uÝx, tÞ solves (4.1). Then,
1.
if uÝx, 0Þ = u 0 ÝxÞ is periodic then u tends, uniformly in x at the rate t ?1 , to the
mean value of u 0 ÝxÞ taken over one period.
2.
if uÝx, 0Þ = u 0 ÝxÞ has compact support, then
a) || uÝ6, tÞ|| K ² C
t
b) || uÝ6, tÞ ? NÝ6, tÞ|| L 1 ÝRÞ ² C
t
We remark that while 2(a) implies that u tends to zero in the norm of L K ÝRÞ as t ¸ K, this
does not imply that u tends to zero in the norm of L 1 ÝRÞ. To see this suppose
FÝ0Þ = 0 and use the fact that uÝx, 0Þ = u 0 ÝxÞ has compact support to write,
d X uÝx, tÞ dx = X / t uÝx, tÞ dx = ? X / x ÝFÝuÝx, tÞÞÞ dx = ?FÝuÝx, tÞÞ| x=K = 0.
x=?K
R
R
dt R
i.e., since u 0 ÝxÞ has compact support and the solution has finite speed of propagation, the
derivative is zero. Of course the fact that the area under the graph of uÝx, tÞ vs x is constant
in time does not mean that the profiles do not change. The implication of 2(b) is that the
profile tends asymptotically to a shape called an ”N-wave”, defined as follows
NÝx, tÞ =
d = F”Ý0Þ,
1 Ý x ? aÞ if
d t
0
? pdt < x ? at <
qdt
otherwise
a = F v Ý0Þ, and p, q depend on u 0 ÝxÞ
7