PDE LECTURE NOTES, MATH 237AB
1
1.
Some Examples
Example 1.1
(Tra
ﬃ
c Equation)
.
Consider cars travelling on a straight road, i.e.
R
and let
u
(
t, x
)
denote the density of cars on the road at time
t
and space
x
and
v
(
t, x
)
be the velocity of the cars at
(
t, x
)
.
Then for
J
= [
a, b
]
⊂
R
, N
J
(
t
) :=
R
b
a
u
(
t, x
)
dx
is the number of cars in the set
J
at time
t.
We must have
Z
b
a
˙
u
(
t, x
)
dx
=
˙
N
J
(
t
) =
u
(
t, a
)
v
(
t, a
)
−
u
(
t, b
)
v
(
t, b
)
=
−
Z
b
a
∂
∂x
[
u
(
t, x
)
v
(
t, x
)]
dx.
Since this holds for all intervals
[
a, b
]
,
we must have
˙
u
(
t, x
)
dx
=
−
∂
∂x
[
u
(
t, x
)
v
(
t, x
)]
.
To make life more interesting, we may imagine that
v
(
t, x
) =
−
F
(
u
(
t, x
)
, u
x
(
t, x
))
,
in which case we get an equation of the form
∂
∂t
u
=
∂
∂x
G
(
u, u
x
)
where
G
(
u, u
x
) =
−
u
(
t, x
)
F
(
u
(
t, x
)
, u
x
(
t, x
))
.
A simple model might be that there is a constant maximum speed,
v
m
and maxi
mum density
u
m
,
and the tra
ﬃ
c interpolates linearly between
0
(when
u
=
u
m
)
to
v
m
when
(
u
= 0)
,
i.e.
v
=
v
m
(1
−
u/u
m
)
in which case we get
∂
∂t
u
=
−
v
m
∂
∂x
(
u
(1
−
u/u
m
))
.
Example 1.2
(Burger’s Equation)
.
Suppose we have a stream of particles travelling
on
R
,
each of which has its own constant velocity and let
u
(
t, x
)
denote the velocity
of the particle at
x
at time
t.
Let
x
(
t
)
denote the trajectory of the particle which
is at
x
0
at time
t
0
.
We have
C
=
˙
x
(
t
) =
u
(
t, x
(
t
))
.
Di
ff
erentiating this equation in
t
at
t
=
t
0
implies
0 = [
u
t
(
t, x
(
t
)) +
u
x
(
t, x
(
t
))
˙
x
(
t
)]

t
=
t
0
=
u
t
(
t
0
, x
0
) +
u
x
(
t
0
, x
0
)
u
(
t
0
, x
0
)
which leads to Burger’s equation
0 =
u
t
+
u u
x
.