International Journal of Modern Nonlinear Theory and Application, 2013, 2, 161-163
http://dx.doi.org/10.4236/ijmnta.2013.23021 Published Online September 2013 (http://www.scirp.org/journal/ijmnta)
On the Geometric Blow-Up Mechanism to Scalar
Conservation Laws
Shikuan Mao, Yongqin Liu
School of Mathematics and Physics, North China Ele ct ric Power Uni versity, Beijing, China
Email: shikuanmao@ncepu.edu.cn, yqliu2@ncepu.edu.cn
Received July 4, 2013; revised August 3, 2013; accepted August 13, 2013
Copyright © 2013 Shikuan Mao, Yongqin Liu. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The focus of this article is on the geometric mechanism for the blow-up of solutions to the initial value problem for
scalar conservation laws. We prove that the sufficient and necessary condition of blow-up is the formation of character-
istics envelope. Whether the solution blows up or not relates to the topology stru cture of a set dominated by in itial data.
At last we take Burger’s equation as an example to verify our main theorem.
Keywords: Blow-Up; Conservation Laws; Characteristics Envelope
1. Introduction
In this short article we consider the blow-up phenomena
to scalar conservation laws. We are interested in the
blow-up mechanism of the following initial value prob-
lem

 
0
0, 0,,
0,, ,
tx
ufut x
uxuxx
 

(1.1)
where , without loss of generality, we can
assume that , i.e., we aim at the classical
solution to the Equa tion (1.1).

2
fC
0
uH

2
By a routine proof we know that the so lu tion to (1.1 ) is
the limit of the corresponding viscous approximate solu-
tions as the viscosity coefficient tends to zero, herein we
omit the proof since it is not our aim. In view of the
maximum principle, we know that the -norm of the
solution is controlled by the initial data, i.e., the solution
itself can never be infinity at any time. On the other hand,
what we concern about is the classical solutions to (1.1).
So the way blow-up happens can only be when the first
spatial derivative of the solution becomes infinite at
some finite time.
L
As is well known, the solution to (1.1) blows up once
the characteristics intersect (see [1-4]). However, the
reverse is not true. In fact, what is needed for the
blow-up to happen is the formation of the envelopes. In
order to give an explicit description to the geometric
blow-up mechanics for scalar conservation laws, we
write this article.
In the remaining part we will study in what case the
solution blows up and at th e sa me time the ch aracteristics
intersect, and in what case the solution blows up while no
characteristics intersect simultaneously. In the last sec-
tion we take the Burger’s equation as an example to test
and verify our main theorems by constructing certain
initial data.
2. Main Theorems and Proof
Let us first introduce the characteristics issuing from y
 



d,,,,0,
d
0, ,.
tyf uttyty
t
yyy



,
(2.1)
From Equation (2.1) we get that


0
,,uttyuy
, (2.2)
and that



00
,1
ytytf u yu y
 
 . (2.3)
From Equation (2.2) we have that
 
0
,, ,
xy
u ttytyuy

.
(2.4)
Denote
,
x
ty
y and consider the family of lines
(2.1) with as parameter,


0,xytfuy y
.
(2.5)
In view of the knowledge of the differential geometry,
C
opyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
162
we know that the envelope of the family of lines (2.5), i.e.
characteristics envelope is the following curve with
as a parameter, y



0
0
0
10
.
tfu uy
xytfuy
 


,
0
(2.6)
From (2.3), (2.4) and (2.6), we have the following re-
sults.
Theorem 2.1. The solution to (1.1) blows up at
time if and only if there exists some such that
y, or equivalently the characteristics envelope
forms at time .
u
t

,ty y
t
From the previous analysis, the proof of Theorem 2.1
is obvious. Moreover, under some stronger assumptions,
we have more detailed results than Theorem 2.1, i.e. the
following Theorem 2.2.
If



00
min
y
f
uyuy

exists, then the concrete
relation between blow-up and characteristics is deter-
mined by the topology structure of the following set





00
00
:;
min 0
y
Ax fuxux
fuyuy
 




We denote 0
A
by the set of all the inner points of
A
.
Then we have the following t h eorem
Theorem 2.2. Assume that


00
min
y
f
uyuy

 

exists and
A
is defined as above. There exist the fol-
lowing three cases:
,A then the solution to Equation (1.1) exists
globally. u
0
,AA ,
1
then the solution to (1.1) blows
up, while no characteristics intersect when blow-up hap-
pens.
u
0,A then the solution u to (1.1 ) blo w s up and at
the same time the characteristics intersect.
Proof. 1) Since , and , we have
A
0, x

2
0
uH



00
fuxux
 .
It yields that , so

,
yty


0
,,
x
ut tyuy
,
i.e. the solution to Equation (1.1) exists globally.
2) Since , there exists . Let
0
,AA 
0
yA




0
1
0000
tfuy uy
 
 , then . From
(2.4) and Theorem 2.1,

00
,
yty

0
00
0
,,ty 
x
ut , so the so-
lution blows up and 0
t is the maximal existence
time of the smooth solution.
u
Now we prove that no characteristics intersect at time
0 by contradiction. Suppose that there are two charac-
teristics issuing from 1, , where , such that
. Since
ty
2
y1
yy







1
1
00000 00
min ,
y
tfuy uyfuyuy


 
 

we have that
0,
yty
0, and this results in
001
,,,ty y
12
. Then we have that ,ty yy

01
, ,y yy 2
,0
yty
, i.e.







000000
0
1,,
12
.
uyuyfuy uyyyy
t
 

It yields that 0
0
y
A
0
A. This contradicts with the as-
sumption that
. So the characteristics do not in-
tersect when the blow-up happens.
3) Since 0
A
, choose 0
0
y
A. Let



0
1
0000
tfuy uy
 
 , then

00
,
yty
0
. From
(2.4) and Theorem 2.1,

00
0
,,ty 
x
ut , so the so-
lution blows up and 0 is the maximal existence
time of the smooth solution.
ut
Now we prov e that the characteristics intersect at time
0. Since t0
0
y
A , then there exist 12
,yy
such
that
0
12 A,yy , i.e.


000000
0,, .
21
f
uyuyfuy uyyyy
 

It yields that
0
f
uy
is a line segment on
12
,
y
y and
000 1
,,0,,
yy
tytyy yy


2
..
So,
0001
,,,,tytyy yy


2
,
i.e. the characteristics intersect.
Remark 1). The solution to Equation (1.1) exists glob-
ally if
0
f
uy
is a nondecreasing function of
y
.
2) The solution blows up first along the character-
istics which issue from the points where
u
0
f
uy
decreases fastest.
3) If the set of the points where

0
f
uy
decreases
fastest contain an inner point, some characteristics inter-
sect and the solution blows up. Otherwise, the character-
istics do not intersect while the solution still blows up by
reason of the formation of the envelope so long as
0
f
uy
decrease in a neighborhood of some point.
3. Burger’s Equation
In this part, we take Burger’s equation as an example to
testify Theorem 2.2. The initial value problem for Bur-
ger’s equation is the following
 
0
0, 0,,
0,, ,
tx
uuut x
uxuxx
 

(3.1)
which is equivalent to (1.1) with

2
f
uu. The corre-
sponding set
A
is as follows with the assumption that
0
min
yuy
exists,
2
,

01 02
,ty ty

Copyright © 2013 SciRes. IJMNTA
S. K. MAO, Y. Q. LIU
Copyright © 2013 SciRes. IJMNTA
163
0.
 

00
;min
y
Ax uxuy

 
From Theorem 2.2, we know that if there is some
point on which 0 is negative, then the solution to
equation (3.1) blows up along the characteristics which
issues from the point. And if the picture of
u

0,uyy
u
restricted on the set of the points on which 0 is nega-
tive does not contain any line segments, then no charac-
teristics intersect, but the solution to Equation (3.1) still
blows up along these characteristics issuing from those
points on whi ch is negative.
0
Now we give certain initial data for Burger’s equation
corresponding to the case 2) in Theorem 2.2. We take u0
as follows
u
 

2
02
2
0, 2,
2, 21,
2, 11,
2, 12.
x
xx
ux xx
xx



Then we have that
 

0
0, 2,
22,2
2,1 1,
22,1 2.
x
xx
ux xx
xx



1,
It is easy to see that and

2
0,uH
2.
 
00
min 1
xux u

The characteristics issuing from
0
x
can be described as

000
,
x
xtux
1. taking
0 we get that 1,x
x
t By the computation simi-
lar to the proof of Theorem 2.2, we know that the maxi-
mal existence time of the so lution is

1
00
1
min ,
2
x
tux
 
and
3
1,
22
x
u.
How-
ever, there is no characteristics intersecting at time 1
2
t
when blow-up happens.
Next we give certain initial data for Burger’s equation
corresponding to the case 3) in Theorem 2.2. Assume
that u0 is as follows,




2
2
0
2
2
0, 2,
2, 21,
2,11,
32,1 2,
32,2 4,
5, 45,
0, 5.
x
xx
xx
ux xx
xx
xx
x



 
 
 
,




0
0, 2,
22,21
2,1 1,
2,1 2,
23,2 4,
25, 45,
0, 5.
x
xx
xx
ux x
xx
xx
x



 

 
It is easy to see that and

2
0,uH
001
min
xuxux

Then we have that
2,

for any
11, 2.x The equa-
tion for the characteristics issuing from 0
x
is
000
,
x
xtux taking
01
1, 2xx , we get that
32
11
.
x
xt x By the computation similar to the
proof of Theorem 2.2, we know that the maximal exis-
tence time of the solution is

1
00
1
min ,
2
x
tux
 
and
3
1,
22
x
u.
 Moreover at time 1,
2
t we have
that


111
3
132,1,2,
22
xxxx  i.e., the char-
acteristics issuing from the points in [1,2] intersect at the
point
3
1,
22 in the
,tx-plane as blow-up happens.
4. Acknowledgements
The authors would give thanks to Professor Weike Wang
for the helpful discussion. The first author is partially
supported by the National Natural Science Found ation of
China (Grant No. 11201142) and by the Fundamental
Research Funds for the Central Universities (Grant no.
11QL40). The second autho r is partially supported by the
National Natural Science Foundation of China (Gran t No.
11201144).
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