Math 557
Spring 2010
Assignment 2
Due Thursday March 11, 2010.
Do three of the following.
1. (Lorentz transform) Suppose u(t, x) is a solution of the wave equation in R3 ,
utt = c2 ∆u,
(1)
where c > 0 is the speed of light. Show that for any v ∈ (−c, c),
v(t, x) = u(t0 , x0 ),
t0 = γ(t −
vx1
),
c2
x0 = (γ(x1 − vt), x2 , x3 ),
γ = (1 −
v 2 −1/2
)
c2
is also a solution. You may assume c = 1 by rescaling.
2. For the wave equation in R3 ,
utt = ∆u,
u(0) = u0 ,
ut (0) = u1 ,
(2)
we have the Kirchhoff’s solution formula (see Evans [PDE, §2.4])
Z
1
u(t, x) =
tu1 (y) + u0 (y) + ∇u0 (y) · (y − x)dS(y).
|∂B(x, t)| ∂B(x,t)
Derive a decay estimate for ku(t, ·)kL∞ in terms of certain norms of u0 and u1 .
It need not be optimal.
3. Find the explicit formula of the traveling kink solution with speed −1 of the modified
KdV equation
ut + uxxx − (u3 )x = 0.
(3)
R
R
4. Let N (u) = 12 R3 |u|2 dx and E(u) = R3 12 |∇u|2 + 14 (Φ ∗ |u|2 )|u|2 dx where Φ ∈ Lp ∩
Lq (R3 ) and Φ < 0. Decide for which p and q the constrained minimization problem
inf
u∈H 1 (R3 ),N (u)=C
E(u)
may have a nontrivial minimizer. It need not be optimal.
(4)