Math 567: Assignment 4 (Due: Friday, Apr. 11)
1. Asymptotic stability in Hamiltonian systems
(a) Show that a critical point ~x0 ∈ Rn , ∇H(~x0 ) = 0, of a finite-dimensional Hamiltonian system ~x0 = J∇H(~x), cannot be asymptotically stable.
(b) On the other hand, show that every solution u(x, t) of the linear Schrödinger
equation in RnR (which is Hamiltonian) with u(x, 0) ∈ L2 (Rn ) tends to zero
locally in L2 : B |u(x, t)|2 dx → 0 as t → ∞ for any ball B ⊂ Rn (and so the
zero solution is asymptotically stable in this sense).
2. Euler-Lagrange equation for the best constant in the G-N-S inequality
Derive the Euler-Lagrange equation satisfied by a maximizer w(x) of the G-N-S
variational problem
R
4 dx
R2 |u(x)|
R
J(w) =
max
J(u),
J(u) := R
2
2
06=u∈H 1 (R2 )
R2 |∇u(x)| dx R2 |u(x)| dx
(by differentiating J(w + ξ) with respect to , where ξ(x) is an arbitrary smooth,
compactly supported function on R2 ) and show that a re-scaled version v(x) of such
a minimizer (assuming it exists) satisfies the soliton equation −∆v − |v|2 v + v = 0.
3. Sharp threshold for global existence for mass cirtical focusing NLS: Consider the 2D cubic focusing NLS:
iut + ∆u = −|u|2 u
(N LS3− ).
u(x, 0) = u0 (x) ∈ H 1 (R2 )
(a) Use the fact that the ground state soliton profile v(x) realizes the best constant
in the G-N-S inequality above, together with the Pohozhaev relations for v, to
show that solutions to (N LS3− ) with ku0 kL2 < kvkL2 are global.
(b) On the other hand, show that the pseudo-conformal transformation
i|x|2
x
1
−1/2 − 4(1−t)
u(x, t) 7→ ũ(x, t) := (1 − t)
e
u
,
1−t 1−t
1
maps smooth solutions of (N LS3− ) for t > 0 to solutions of (N LS3− ) for t < 1,
and apply this transformation to the soliton solution u(x, t) = v(x)eit to show
that there exist solutions of (N LS3− ) which blow up in finite time.
4. Non-compactness of the Strichartz ’embedding’ due to various ‘bubbles’:
Let φ(x) be a fixed non-zero function in Ḣ 1 (R3 ). In each case, show that the given
sequence uk (x)
• is bounded in Ḣ 1 (R3 )
R
• converges to 0 weakly in Ḣ 1 (R3 ) (that is, for any fixed f ∈ Ḣ 1 (R3 ), R3 ∇f (x) ·
∇uk (x)dx → 0 as k → ∞; hint: if it helps, you can argue that it suffices to
consider f smooth and compactly supported, by the density of such functions
in Ḣ 1 )
10
3
• eit∆ uk converges to 0 weakly in L10
t Lx (Rx × Rt )
10
3
• has no subsequence for which eit∆ uk converges in L10
t Lx (R × R):
(a) uk (x) = φ(x − xk ) with |xk | → ∞
−1/2
(b) uk (x) = λk
φ(x/λk ) with 0 < λk → 0 or → ∞
(c) uk (x) = eitk ∆ φ with tk → ±∞ (hint: the decay estimate keit∆ f kL∞ (R3 ) ≤
c|t|−3/2 kf kL1 (R3 ) in concert with an approximation argument might help)
5. Asymptotic non-interaction of bubbles: Let φ, ψ ∈ Ḣ 1 (R3 ). In each case, show
that
lim k∇uk k2L2 (R3 ) = k∇φk2L2 (R3 ) + k∇ψk2L2 (R3 )
k→∞
(a) uk (x) = φ(x − xk ) + ψ(x − yk ),
|xk − yk | → ∞
−1/2
−1/2
µk
λk
(b) uk (x) = λk φ λxk + µk ψ µxk ,
µ k + λk → ∞
−1/2
−1/2
µk
x−yk
λk
k
(c) uk (x) = λk φ x−x
+
µ
ψ
,
k
λk
µk
µk + λk +
(d) uk (x) = eitk ∆ φ + eisk ∆ φ,
|tk − sk | → ∞
2
|xk −yk |2
λk µ k
→ ∞.