32 I. Theory of Distributions
and
u−(t) = lim
(x,t)→(x(t),t)
u(x, t), x x(t).
Let u(x, t) be a distribution solution to a nonlinear equation
∂u
∂t
+

∂x
f(u) = 0,
i.e.,
R2
−u(x, t)
∂ϕ
∂t
f(u)
∂ϕ
∂x
dxdt = 0
for any ϕ(x, t) C0
∞(R2).
We assume that f
C∞(R1).
Prove
that the following condition holds:
dx
dt
=
f(u+(t)) f(u−(t))
u+(t) u−(t)
.
This condition is called the Rankine-Hugoniot condition.
11. Let f(x1) be the Cantor function on [0, 1], i.e., f(x1) is continuous,
nondecreasing, f(0) = 0, f(1) = 1, and f (x1) = 0 a.e.
Set f0(x1) = f(x1) on [0, 1] and f0(x1) = 0 for x1 that does not
belong to [0, 1].
Prove that the distribution derivative of f0(x1) is the following
functional:
df0
dx
(ϕ) = ϕ(1)
1
0
ϕ(x1)df(x1),
where
1
0
ϕ(x1)df(x1) is the Stieltjes integral.
12. Prove that there is no distribution on
R1
such that its restriction
to (0, +∞) is a regular functional in D (0, +∞) corresponding to
f(x1) = e
1
x2
1
.
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