Universität Basel
Prof. Dr. Enno Lenzmann
Nichtlineare Evolutionsgleichungen
23.03.2015
Problem Set No. 4
Deadline: Due on Wednesday, April 1st, 2015.
Problem 4.1. (16 Points). Let Φ : Rd Ñ R with Φ P L8 pRd q be given. We
consider the nonlinear Schrödinger equation of the form
˘
`
(1)
iBt u “ ´∆u ` Φ ˚ |u|2 u, up0, xq “ φpxq,
ş
where pf ˚ gqpxq “ Rd f px ´ yqgpyq dy denotes the convolution of functions on Rd .
(i) Prove that for every initial condition φ P L2 pRd q the equation (1) has a
unique solution u P C 0 pr0, T q; L2 pRd qq, provided that T “ T p}φ}L2 q ą 0 is
sufficiently small.
(ii) Prove that the solution upt, xq obtained in (i) exhibit conservation of L2 mass, i. e., we have }uptq}L2 “ }φ}L2 for t P r0, T q. Deduce that upt, xq can
be extended to all times t ě 0.
Hint: To show (i), you don’t need to use Strichartz estimates. In fact, it is enough
to run a Banach fixed point argument for the integral equation
żt
`
˘
uptq “ eit∆ φ ´ i eipt´sq∆ pΦ ˚ |upsq|2 qupsq ds
0
just using Young’s inequality }f ˚ g}Lr ď }f }Lp }g}Lq with 1 ` 1{r “ 1{p ` 1{q.
To prove (ii), use the regularization operator Jε “ p1 ´ ε∆q´1 from the lecture
notes.
Problem 4.2. (16 Points). Let d ě 1 and 0 ă α ă minpd, 4q. We consider
`
˘
(2)
iBt u “ ´∆u ´ |x|´α ˚ |u|2 u, up0, xq “ φpxq.
It is known that (2) is locally well-posed for initial data in H 1 pRd q, i. e., for each
φ P H 1 pRd q there exists a unique solution u P C 0 pr0, T q; H 1 pRd qq of (2) provided
that T “ T p}φ}H 1 q ą 0 is sufficiently small. Moreover, we have conservation of
L2 -mass and energy, i. e.,
}uptq}L2 “ const.,
ż
1
1
2
|∇uptq| dx ´
p|x|´α ˚ |u|2 q|u|2 dx “ const.
Eruptqs “
2 Rd
4 Rd
ż
(i) Let 0 ă α ă minp2, dq. Show that every solution u P C 0 pr0, T q; H 1 pRd qq
extends to all times t ě 0, i. e., we can take T “ `8.
(ii) Let 2 ď α ă minp4, dq. Show that there exists blowup solutions for (2).
More preciesly, assume that φ P H 1 pRd q with |x|φ P L2 pRd q and derive the
differential inequality
ż
d2
|x|2 |upt, xq|2 dx ď 16Erφs
dt2 Rd
for t P r0, T q. Conclude that upt, xq cannot exist for all times if Erφs ă 0.
1
2
Hint: To show (i), use the conservation laws and the Hardy-Littlewood-Sobolev
inequality
ˇ
ˇż ż
ˇ
ˇ
f pxqgpyq
ˇ
ˇ
ˇ d d |x ´ y|α dx dy ˇ ď Cα,d,p }f }Lp }g}Lr
R
R
with p, r ą 1, 0 ă α ă d, and 1{p ` α{d ` 1{r “ 2 together with the GagliardoNirenberg inequality
2σ`2´dσ
}f }2σ`2
ď Cd,σ }∇f }dσ
L2 }f }L2
L2σ`2
2
for d ě 3.
valid for f P H 1 pRd q with 0 ă σ ă 8 for d “ 1, 2 and 0 ă σ ă d´2
As for (ii), you can adapt the proof of Lemma 4.1 (in the lecture notes) with
W “ ´p|x|´α ˚ |u|2 q and recall that
ż
xu, riW, Asuy “ ´
px ¨ ∇W q|u|2 dx
Rd
For notational convenience, let us
ˇ drop the t-dependence. To evalute the right side,
B ˇ
W pλxq. Thus (where you don’t need to justify
we note that px ¨ ∇W qpxq “ Bλ
λ“1
the interchange of differentiation and integration) we have
ż
ż
d ˇˇ
´
px ¨ ∇W q|u|2 dx “ ´ ˇ
W pλxq|upxq|2 dx
dλ λ“1 Rd
Rd
ż ż
d ˇˇ
|upxq|2 |upyq|2
“´ ˇ
dx dy.
dλ λ“1 Rd Rd |λx ´ y|α
Use now a suitable substitution for double integral on the right side in order to
easily evaluate the derivative with respect to λ.