Math 567: Assignment 3 (Due: Friday, Mar. 21)
1. Hamiltonian structure for a Boussinesq system: For the following variant of
the Boussinesq system, for horizontal velocity v = v(x, t) and surface height h =
h(x, t), x ∈ R
ht = −[(1 + αh)v]x + βhxxt
vt = −[h + 12 αv 2 ]x + βvxxt
(α > 0, β > 0), find a Hamiltonian formulation, and identify the conserved energy
and momentum (conserved quantity arising from translation invariance). Hint: it
may help to employ the inverse operator (1 − βd2 /dx2 )−1 , which is easily seen to be
a nice (eg. bounded on L2 ) operator by taking the Fourier transform.
2. Hamiltonian structure for the BBM equation: Find a Hamiltonian formulation, as well as the conserved energy and momentum, for the ‘Benjamin-BonaMahoney’ equation for u(x, t), x ∈ R:
ut + uux − uxxt = 0.
3. Scaling and criticality: Determine the scaling-critical Sobolev index s (in terms of
p and n), and determine the ‘energy’ (i.e. the norm connected to the Hamiltonian)
criticality for each of:
(a) (gKdV) ut + up−1 ux + uxxx = 0
(b) (NLW) utt − ∆u ± up = 0
(x ∈ R)
(x ∈ Rn )
(c) (Navier-Stokes) ut + (u · ∇)u − ∆u = ∇p (x ∈ Rn , u ∈ Rn , p ∈ R, and the role
of ‘energy’ here is played by the L2 norm)
4. Global Existence for the Generalized KdV equation Consider the IVP for
gKdV in the ‘energy space’:
ut + up−1 ux + uxxx = 0,
u(x, 0) = u0 (x) ∈ H 1 (R).
The ‘standard argument’ gives a good (i.e. ‘subcritical) local existence theory in
H 1 for any 1 < p < ∞ (note all these exponents are H 1 sub-critical). Use the
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Gagliardo-Nirenberg-Sobolev inequality
Z
p−1 Z
p+3
Z
4
4
2
p+1
2
ux (x)dx
u (x)dx ≤ Cp
u (x)dx
R
R
R
together with conservation laws to show that the solution is global if
(a) p is L2 (‘mass’)-subcritical
(b) p is mass-critical and the mass is sufficiently small ( and identify this threshold
mass in terms of Cp )
5. Morawetz identities Show that if u(x, t) solves
(N LS3+ ) iut + ∆u = |u|2 u
with n = 3 (i.e. x ∈ R3 ), then
−
∂
|u|4 + 2(|∇u|2 − |(x̂ · ∇)u|2 )
Im (ūx̂ · ∇u) =
+∇·F
∂t
|x|
for some vector field F (here x̂ = x/|x|).
6. Decay estimate via ‘Pseudoconformal Energy’: The H 1 scattering theory for
the defocusing NLS (N LSp+ ) using the Morawetz identity which we outlined in class
requires p > 1 + 4/n (mass supercritical) and fails for lower powers. But scattering
can be recovered for some sub-critical powers by changing the space/norm. This
exercise gives an example of how a different identity/estimate can be used to obtain
some decay in the mass critical case p = 1 + 4/n. Take the defocusing cubic NLS in
two space dimensions:
iut + ∆u = |u|2 u,
u(x, 0) = u0 (x).
(a) Show that the ‘psuedoconformal energy’
Z 1
Epc (u, t) :=
|(x + 2it∇)u(x, t)|2 + t2 |u(x, t)|4 dx
2
R2
is conserved by solutions of the PDE (assume you are working with a smooth
solution which decays, along with its various derivatives, quickly as x → ∞).
(b) Thus derive the following decay estimate for the L4 norm:
Z
Z
1
|u(x, t)|4 dx ≤ 2
|xu0 (x)|2 dx.
t
2
2
R
R
Remark: this decay estimate can be used to show that a solution with u0 ∈ L2 and
xu0 ∈ L2 (extra spatial decay!) is global and scatters in L2 . It is MUCH harder
to prove this scattering without the extra xu0 ∈ L2 assumption, and indeed this is a
recent breakthrough development of just the last few years!
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