Math 567: Assignment 1 (Due: Friday, Jan. 31)
1. Dispersion relations: find the dispersion relations and group velocities for each:
(a) Rossby waves: for ψ(x), x = (x1 , x2 ) ∈ R2 ,
∂
∂
∂
∆ψ + β
+c
ψ=0
∂t
∂x1
∂x1
(ψ is the ‘stream function’ of an atmospheric flow perturbing a constant-velocity
(c) equatorial (x1 -direction) flow, taking account (through constant β) of the
latitudinal (x2 -direction) variation in the Coriolis force due to earth’s rotation).
(b) (vacuum) Maxwell equations:
~ t = c2 ∇ × B,
~
E
~ t = −∇ × E,
~
B
~ =∇·B
~ = 0,
∇·E
~
~
for vector fields E(x,
t) ∈ R3 (electric), B(x,
t) ∈ R3 (magnetic), x ∈ R3
(c) Maxwell in an (uniaxial) anisotropic medium:
~ t = ∇ × B,
~ B
~ t = −∇ × E,
~ ∇ · (eE)
~ =∇·B
~ = 0, e = diag(α, 1, 1)
(eE)
2. Fundamental solution of the 1D transport equation: the dispersion relation
for ut + cux = 0 (n = 1) is h(ξ) = cξ. Show directly that the fundamental solution
Z ∞
1
1 h −ith(ξ) iˇ
(x) := lim
Φt (x) := √
e
eixξ e−itcξ e−|ξ| dξ = δ(x − ct)
→0+ 2π −∞
2π
in the ‘sense of distributions’. That is, for any function φ(x) which is smooth (C ∞ )
and compactly supported (vanishing outside a bounded interval), show
Z ∞
Z ∞
1
ixξ −itcξ −|ξ|
lim
φ(x)
e e
e
dξ dx = φ(ct).
→0+ −∞
2π −∞
3. Fundamental solution of the free Schrödinger equation: the dispersion relation for iut = ∆u (with x ∈ Rn ) is h(ξ) = |ξ|2 . Show directly that
Z
|x|2
1 h −ith(ξ) iˇ
1
1
2
2
i
Φt (x) :=
e
(x)
:=
lim
eix·ξ e−it|ξ| e−|ξ| dξ =
n
n e 4t
n
→0+ (2π)
(2π) 2
(4πit) 2
Rn
1
Hints: factor the integrand to reduce to the case n = 1. Then complete the square in
the exponent to turn the integrand into a (complex) Gaussian. Finally change variables (this is a complex change of variable, properly understood as a shift of contour
in the complex plane) to convert the integral into a standard Gaussian integral.
4. Fundamental solution of the wave equation: show that
ˇ
1
n=1
−n/2 sin(ct|ξ|)
2c χ[−ct,ct] (x)
Kt (x) := (2π)
(x) =
1
δ
(x) n = 3
c|ξ|
4πc2 t |x|=ct
indirectly – that is, by computing the Fourier transform of the right-hand side to get
the left-hand side. Use this to write solution formulas for the initial value problem
for the wave equation in dimensions n = 1 and n = 3.
5. Group velocity: if f (x) is a smooth function on Rn with compact support, and φ(x)
is a smooth (real-valued) function on Rn with only one critical point x0 (∇φ(x0 ) = 0)
in the support of f , the method of stationary phase yields the asymptotics:
Z
eitφ(x) f (x)dx = Ct−n/2 eitφ(x0 ) f (x0 ) + O(t−(n/2+1) ), t → ∞
Rn
(where C is a constant (depending on D2 φ(x0 ))). Recall the solution formula
Z
−n/2
u(x, t) = (2π)
ei(x·ξ−th(ξ)) û0 (ξ)dξ
Rn
for the dispersive PDE ut = −ih(−i∇x )u. Now, for a given velocity v ∈ R3 , use
the above to find the large t asymptotics of u(vt, t), the solution as observed while
moving out with velocity v. (You may assume h smooth and û0 smooth with compact
support). Determine the group velocity from your computation.
6. Dispersive estimates for the Airy equation: let
Z R
1 h −itξ3 iˇ
1
3
Φt (x) = √
lim
ei(xξ−tξ ) dξ
e
(x) =
2π R→∞ −R
2π
be the fundamental solution of the Airy equation.
(a) Show that Φt (x) = t−1/3 Φ1 (t−1/3 x).
(b) Show that Φ1 (x) is a bounded function over the interval x ∈ (−∞, −1]. Hint:
use the fact the phase φ = xξ − ξ 3 satisfies |φξ | ≥ 1 for all such x, and do an
RR
integration by parts on −R eiφ dξ to show it is bounded independently of R and
x ≤ −1. Note: you are basically doing the ‘method of non-stationary phase’.
(c) (Bonus): use more clever integration-by-parts arguments to show Φ1 (x) is
bounded over all x ∈ R.
(d) Given that Φ1 (x) is bounded, conclude |Φt (x)| ≤ Ct−1/3 , and hence for the solution u(x, t) = (Φt ∗ u0 )(x) of the initial value problem: |u(x, t)| ≤ Ct−1/3 ku0 kL1 .
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