MATH 220: PROBLEM SET 3
DUE THURSDAY, OCTOBER 15, 2009
Problem 1.
Let
ψ
∈
C
(
R
) be given by
ψ
(
x
) =
0
,
x<
−
1
,
1 +
x,
−
1
<x<
0
,
1
−
x,
0
<x<
1
,
0
,
x>
1
,
so
ψ
≥
0,
ψ
(
x
) = 0 if

x
 ≥
1 and
integraltext
R
ψ
(
x
)
dx
= 1. Let
ψ
j
(
x
) =
jψ
(
jx
), so
ψ
j
(
x
) = 0
if

x
 ≥
1
/j
, and
integraltext
ψ
j
(
x
)
dx
= 1 for all
j
.
(1) Show that
ψ
j
→
δ
0
in
D
′
(
R
) (i.e., to be pedantic,
ι
ψ
j
→
δ
0
). (Hint: this is
essentially written up in the notes on distributions!)
(2) Let
φ
∈
C
∞
c
(
R
) be such that
φ
(
x
) = 1 for

x

<
1. Show that
{
ι
ψ
2
j
(
φ
)
}
∞
j
=1
is
not
a convergent sequence in
R
. Use this to conclude that
{
ι
ψ
2
j
}
∞
j
=1
does
not converge to any distribution.
(3) Show that there is no continuous extension of the map
Q
:
f
mapsto→
f
2
on
C
(
R
)
(so
Q
:
C
(
R
)
→
C
(
R
)) to
D
′
(
R
), i.e. there is no map
˜
Q
:
D
′
(
R
)
→ D
′
(
R
)
such that
•
˜
Q
(
ι
f
) =
ι
Qf
for every
f
∈
C
(
R
) and
•
u
j
→
u
in
D
′
(
R
) implies
˜
Qu
j
→
˜
Qu
in
D
′
(
R
).
Problem 2.
Consider the conservation law
u
t
+ (
f
(
u
))
x
= 0
, u
(
x,
0) =
φ
(
x
)
,
with
f
∈
C
2
(
R
). Let
v
=
f
′
(
u
). Show that if
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 '09
 Math, Differential Equations, Applied Mathematics, Equations, Partial Differential Equations, Partial differential equation, burger, conservation law

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