MATH 220: FIRST ORDER SCALAR QUASILINEAR
EQUATIONS
We now consider the quasilinear equations; these have the form
(1)
a
(
x, y, u
)
u
x
+
b
(
x, y, u
)
u
y
=
c
(
x, y, u
)
,
with
a, b, c
at least
C
1
, given real valued functions. There is an immediate diFerence
between semilinear and quasilinear equations at this point: since
a
and
b
depend
on
u
, we cannot associate a vector ±eld on
R
2
to the equation: we need to work on
R
3
at least to account for the (
x, y, u
) dependence.
To achieve this, we proceed as follows. We consider the graph
S
of
u
in
R
2
×
R
,
given by
z
=
u
(
x, y
). If we know the solution
u
, we know the graph, and conversely
we can recover
u
if we ±nd its graph, a surface in
R
3
. So let
U
(
x, y, z
) =
u
(
x, y
)
−
z
;
we need to ±nd the 0-set of
U
. Now, substituting
U
into the left hand side of the
PDE we get
(
a
(
x, y, u
(
x, y
))
∂
x
+
b
(
x, y, u
(
x, y
))
∂
y
)
U
(
x, y, z
)
=
a
(
x, y, u
(
x, y
))
u
x
(
x, y
) +
b
(
x, y, u
(
x, y
))
u
y
(
x, y
)
.
If the PDE holds, this equals
c
(
x, y, u
(
x, y
)), which is in turn equal to
−
c
(
x, y, u
(
x, y
))
∂
z
U
as
∂
z
U
≡ −
1! Thus, our PDE implies that
(
a
(
x, y, u
(
x, y
))
∂
x
+
b
(
x, y, u
(
x, y
))
∂
y
+
c
(
x, y, u
(
x, y
))
∂
z
)
U
(
x, y, z
) = 0
for all
x, y, z
. This equation states that the directional derivative of
U
along the
vector ±eld (
a, b, c
) vanishes. Note that
∂
z
U
=
−
1, i.e. never vanishes, so in fact at
each point (
x, y, z
) the directional derivative can only vanish along a plane (and not
in every direction). But along any level set of
U
its directional derivative certainly
vanishes, so applied to the 0-level set, where
u
(
x, y
) =
z
, the PDE states that the
vector ±eld
W
(
x, y, z
) = (
a
(
x, y, z
)
, b
(
x, y, z
)
, c
(
x, y, z
))
is tangent to
S
. We can now rephrase our problem as follows: we want to ±nd
S
as a union of integral curves of
W
. As before, integral curves of
W
, where we
introduce a parameter
r
again, satisfy
∂x
∂s
(
r, s
) =
a
(
x
(
r, s
)
, y
(
r, s
)
, z
(
r, s
))
,
∂y
∂s
(
r, s
) =
b
(
x
(
r, s
)
, y
(
r, s
)
, z
(
r, s
))
,
∂z
∂s
(
r, s
) =
c
(
x
(
r, s
)
, y
(
r, s
)
, z
(
r, s
));
these are called the characteristic ODEs and integral curves are called the
charac-
teristics
, though perhaps it would be most appropriate to call them
lifted character-
istics
in view of our terminology for semilinear equations. Note that although the
integral curves are in
R
3
, hence there is a two-dimensional family of them, we only
care about the ones which will give
S
as their union, i.e. a one-dimensional family
1