to (3.4). Also, we shall see that a naive use of the conservation laws for mass and energy in general leads to restrictions on |Ω| or{γj}dj=1. We shall show in a second step how to overcome this problems using a coordinate-change. Assertion (2) of our main Theorem is then proved in Section 3.4 and we finally collect some concluding remarks in Chapter 5.
The proof is an adaptation of classical arguments, based on a contraction mapping (via Duhamel’s formula (3.8)) and Strichartz estimates for the linear (unitary) group S(t) = eitH generated by the Hamiltonian (3.9). In our case, Strichartz estimates can be obtained by following the approach of Yajima [85]. As defined, the Hamiltonian is not of the form necessary for the results in [85] to hold. However, the Hamiltonian can be rewritten as
H = 1
2(i∇+A(x))2+U(x)−1
2A(x)2, where A(x) = Ω∧x.
Indeed it holds that
(i∇+ Ω∧x)2 =−∆ + (Ω∧x)2+ 2i(Ω∧x)· ∇+i(∇ ·A)
=−∆ + (Ω∧x)2+ 2iΩ·(x∧ ∇).
Since the rotation B(x) =∇ ∧A(x) = 2Ω is constant and the potential U(x)−A(x)2/2 is subquadratic, the results in [85] imply that there exist finite, positive, constantsC and δ such that
kS(t)ϕkL∞ ≤ C
|t|d/2kϕkL1, for |t|< δ.
In particular it follows that the Strichartz estimates forH are analogous to those found in the well-known case of NLS with quadratic potentials [17], i.e. the rotation term does not influence the dispersive behavior (locally in time). These Strichartz estimates have been proven under general circumstances in [46]. Recall that p0 denotes the conjugate H¨older coefficient ofp, i.e. 1/p+ 1/p0 = 1.
Lemma 3.3.3 (Strichartz estimates). The unitary group S(t) = eitH with Hamiltonian H defined in (3.9) satisfies a local in time Strichartz estimate. There exist TS > 0 and constants Cq, Cq,˜q such that
kS(t)ϕkLp(0,TS;Lq(Rd)) ≤CqkϕkL2(Rd) for all ϕ∈L2(Rd) and
Z t 0
S(t−s)F(s)ds
Lp(0,TS;Lq(Rd))
≤Cq,˜qkFkLp˜0
(0,TS;Lq˜0(Rd))
if (p, q) and (˜p,q)˜ are admissible coefficients, i.e. 2 ≤q <2d/(d−2) and 2/p=δ(q) :=
d(1/2−1/q) and likewise for (˜p,q).˜
Since the semigroup (S(t))t∈R⊂L(L2(Rd)) is continuous in time, the Strichartz estimates also yield the following continuity property.
Remark 3.3.4. Under the conditions of Lemma 3.3.3, it holds that S(t)ϕ∈C([0, TS];L2(Rd)) and
Z t 0
S(t−s)F(s)ds∈C([0, TS];L2(Rd)),
which will later allow us to prove continuity of time of fixed points of Duhamel’s for- mula (3.8).
The existence of a local in-time solution is then standard and we repeat it here for the reader’s convenience.
Proof of Lemma 3.3.2. The idea is to write the nonlinear Schr¨odinger equation as the solution of a fixed point equation, using Duhamel’s formula:
ψ(t) = S(t)ψ0−iλ Z t
0
S(t−s) |ψ|2σ(s)ψ(s) ds
=: Φ(ψ)(t).
Let us define parameters
q = 2σ+ 2, p= 4σ+ 4
dσ , k = 2σ(2σ+ 2) 2−(d−2)σ. We will presently show that Φ as a maps the space
XT :=
ψ ∈L∞(0, T; Σ) :ψ, xψ,∇ψ ∈Lp(0, T;Lq(Rd))
onto itself. Indeed, settingR:=kψ0kΣ, we shall establish that Φ is a contraction mapping in
XT,R:=
ψ ∈XT :kψkL∞(0,T;Σ)≤2R,
kxψkLp(0,T;Lq),k∇ψkLp(0,T;Lq) ≤2CqR , for all T small enough. We equip the space XT,R with a metric
d(ψ,ψ) =e kψ−ψke L∞
t L2x +kψ −ψke Lp
tLqx. Then (XT ,R, d) forms a complete metric space, cf. [19].
In order to proceed, we calculate the commutators [∇, S(t)] and [x, S(t)]. We find that [∇, H] =− 1
2[∇,∆] + [∇, U] +i[∇,Ω·(x∧ ∇)]
=∇U +i[∇, x·(∇ ∧Ω)]
=∇U +i∇ ∧Ω
by the well-known formulaa·(b∧c) = det(a, b, c) = (a∧b)·cfor three-dimensional vectors a, b, c. Similar calculations yield
[x, H] =∇ −iΩ∧x.
Since
i∂t[∇, S(t)] = [∇, HS(t)] =H[∇, S(t)] + [∇, H]S(t), we deduce that
(3.15) ∇Φ(ψ)(t) =S(t)∇ψ0−iλ Z t
0
S(t−τ)∇ |ψ(τ)|2σψ(τ) dτ
−i Z t
0
S(t−τ) (∇U−iΩ∧ ∇) Φ(ψ)(τ)dτ, and
xΦ(ψ)(t) = S(t)∇ψ0−iλ Z t
0
S(t−τ)∇ |ψ(τ)|2σψ(τ) dτ
−i Z t
0
S(t−τ) (∇ −iΩ∧x) Φ(ψ)(τ)dτ.
SinceT < TS, the Strichartz estimates of Lemma 3.3.3 yield (3.16) k∇Φ(ψ)kLp
tLqx ≤Cqkψ0kΣ+|λ|Cq,q
∇ |ψ(τ)|2σψ(τ)
Lpt0Lqx0
+Cq,2
∇UΦ(ψ)
L1tL2x +Cq,2
Ω∧ ∇Φ(ψ) L1tL2x. Since
1 q0 = 2σ
q +1
q and 1
p0 = 2σ k +1
p, H¨older’s inequality yields
∇ |ψ|2σψ
Lpt0Lqx0 ≤(2σ+ 1)
|ψ|2σ∇ψ Lpt0Lqx0
≤(2σ+ 1) ψ
2σ LktLqx
∇ψ LptLqx.
Furthermore, it holds that ψ
2σ
LktLqx ≤T2σk ψ
2σ k
L∞t Lqx ≤T2σk ψ
2σ L∞t Hx1
where we used the Gagliardo-Nirenberg inequality in the last estimate. Recall that the Gagliardo-Nirenberg inequality implies
kϕkLqx ≤Ckϕk1−δ(q)L2
x k∇ϕkδ(q)L2 x
for all ϕ∈H1(Rd), where δ(q) is defined in the statement of Lemma 3.3.3. Furthermore we can bound |∇U| ≤Cx. Thus estimate (3.16) yields
k∇Φ(ψ)kLp
tLqx ≤Cqkψ0kΣ+CT2σk ψ
2σ
L∞t Σxk∇ψkLp
tLqx
+CT
xΦ(ψ)
L∞t L2x+CT
∇Φ(ψ) L∞t L2x. Similarly we obtain that
kΦ(ψ)kLp
tLqx ≤Cqkψ0kΣ+CT2σk ψ
2σ
L∞t ΣxkψkLp
tLqx
and
kxΦ(ψ)kLp
tLqx ≤Cqkψ0kΣ+CT2σk ψ
2σ
L∞t ΣxkxψkLp
tLqx
+CT
∇UΦ(ψ)
L∞t L2x+CT
∇Φ(ψ) L∞t L2x. Since (∞,2) is an admissible pair, the same estimates hold for the L∞t L2x-norm, with Cq replaced by 1. If we choose T sufficiently small, this shows that Φ indeed constitutes a mapping Φ :XT,R→XT,R. In order to show that Φ :XT ,R →XT ,R is also a contraction, we take two functions ψ,ψe∈XT ,R and estimate the difference Φ(ψ)−Φ(ψ). In the samee way as above, we obtain that
Φ(ψ)−Φ(ψ)e Lp
tLqx =
Z t 0
S(t−s) |ψ(s)|2σψ(s)− |ψ(s)|e 2σψ(s)e ds
LptLqx
≤
|ψ|2σψ− |ψ|e2σψe Lpt0Lqx0, The point-wise inequality
|ψ|2σψ− |ψ|e2σψe
≤2σ |ψ|2σ +|ψ)|e 2σ
|ψ−ψ|e
and the same H¨older and Gagliardo-Nirenberg estimates used before then yield Φ(ψ)−Φ(ψ)e
LptLqx +
Φ(ψ)−Φ(ψ)e L∞t L2x
≤C kψk2σLk
tLqx +kψke 2σLk tLqx
kψ−ψke Lp
tLqx
≤CT2σk kψk2σL∞
t Hx1 +kψke 2σL∞
t Hx1
kψ−ψke Lp
tLqx.
Thus, choosing T ≤ TS small enough, there exists a fixed point ψ ∈ XT,R satisfying Φ(ψ) = ψ. Remark 3.3.4 applied to the fixed point equations, e.g. (3.15), yields ψ ∈ C([0, T]; Σ). Hence ψ is indeed a mild solution to the Schr¨odinger equation (3.4). The conservation laws (3.13) follow from straightforward calculations in combination with a standard density argument and reversibility (see e.g. [19]). Finally, in order to prove the blow-up alternative we first notice that the above local existence argument can be iterated as long askψ(t)kΣstays bounded. Thus we obtain a maximal solutionψ ∈C([0, Tmax); Σ) with Tmax>0. Assume thatTmax<+∞. In view of mass conservation, it follows that
t→Tlimmax kxψ(t)kL2 +k∇ψ(t)kL2
= +∞.
Furthermore we compute
(3.17) d
dtkxψ(t)k2L2 = 2Im Z
Rd
xψ(t)∇ψ(t)dx≤ kxψ(t)k2L2 +k∇ψ(t)k2L2.
Thus, as long ask∇ψ(t)kL2 is bounded, Gronwall’s inequality yields a bound onkxψ(t)kL2 as well. The only obstruction to global existence is therefore given by the possible un- boundedness of k∇ψ(t)kL2 in [0, T].
Remark 3.3.5. For quadratic potentials of the form (3.2), Strichartz estimates can be obtained explicitly by invoking a generalization of Mehler’s formula for the kernel ofS(t), c.f. [18]. Indeed, by making the following ansatz
S(t)ψ0(x) =
d
Y
j=1
(2πiµj(t))−1/2 Z d
R
ei2F(t,x,y)ψ0(y)dy,
where µj(t) ∈ R+ and F(t, x, y) is a general quadratic form in x and y with (yet to be determined) time-dependent coefficients. Substituting this into the linear Schr¨odinger equation yields a coupled system of differential equations for these coefficients. Solving this system, however, is in general rather tedious. This approach is therefore only feasible under some simplifying assumptions, such as Ω = 0 [16, 17], orU(x) = γ22|x|2 with|Ω|=γ as it is done in [37, 38].
In view, of (3.14), we immediately conclude the following important corollary.
Corollary 3.3.6. If U(x) is such that (Ω·L)U = 0, then we also have conservation of angular momentum, i.e. LΩ(t) =LΩ(0), and in addition it holds
(3.18) E0(t)≡ Z
Rd
1
2|∇ψ|2+U(x)|ψ|2+ λ
σ+ 1|ψ|2σ+2dx=E0(0).
Thus, in the case of axially symmetric potentials U(x), there are in fact two conserved energy functionals corresponding to (3.4).
With a local existence result in hand, we can ask about global existence. In order to infer T = +∞, one usually invokes the conservation of mass and energy (3.13). The problem is, that due to the appearance of the angular momentum rotation term, the energy EΩ(t) has no definite sign even ifU ≥0 andλ ≥0 (defocusing nonlinearity). A possible strategy to overcome this problem is to rewrite the linear Hamiltonian as
(3.19) H =−1
2∆−Ω·L+U(x) = 1
2(−i∇ −A(x))2+U(x)−|Ω|2 2 r2,
where A(x) = Ω ∧ x and r = |x ∧ Ω|/|Ω| denotes the radial distance perpenticular to Ω. Note that A(x) can be considered as the vector potential corresponding to a constant magnetic field B = ∇ ∧A = 2Ω. The corresponding “magnetic derivative”
DA:=−i(∇+A(x)) is known to satisfy, cf. [19, Chapter 7]:
k∇|ψ|kL2 ≤ kDAψkL2 ≤ k∇ψkL2 +kxψkL2.
It can therefore be used to control the nonlinear potential energy∝ kψkL2σ+2 via Gagliardo- Nirenberg type inequalities. If in addition, U(x) is given by (3.2) with |Ω| ≤ γ we infer that U(x)− |Ω|22r2 ≥ 0. In this case, the linear part of the energy is seen to be a sum of non-negative terms, and global existence can be concluded as in the case of NLS with quadratic confinement [17]. However, it seems impossible to extend this approach to sit- uations in which |Ω|> γ, even if λ >0. In order to do so, we shall follow a different idea, which invokes a time-dependent change of coordinates.
Proof of Assertion (1) of Theorem 3.2.3. We start with the L2-subcritical case, i.e. 0 <
σ < 2/d which follows by standard arguments. Recall that in the proof of Lemma 3.3.2 we showed that
kψkLp(0,T;Lq)∩L∞(0,T;L2) ≤Ckψ0kL2 +Ckψk2σLk(0,T;Lq), where
q= 2σ+ 2, p= 4σ+ 4
dσ , k = 2σ(2σ+ 2) 2−(d−2)σ.
Moreover, we showed that
kxψkLp(0,T;Lq)∩L∞(0,T;L2)+k∇ψkLp(0,T;Lq)∩L∞(0,T;L2)
≤Ckψ0kΣ+Ckψk2σLk(0,T;Lq) kxψkLp(0,T;Lq)+k∇ψkLp(0,T;Lq) + +CT kxψkL∞(0,T;L2)+k∇ψkL∞(0,T;L2)
. Ifσ <2/d, it holds that 1/p <1/k and thus
kψkLk(0,T;Lq)≤T1/k−1/pkψkLp(0,T;Lq).
If we choose T = T∗ > 0 small enough, we can absorb all terms on the right hand side except the term involving ψ0 and obtain a bound on kψkL∞(0,T∗;Σ). Since we can shift the time interval [0, T∗] by an arbitrary amount of time, in the same way we can get a uniform bound on kψkL∞(0,T∗;Σ) for every interval of length |I| ≤ T∗. Thus, by splitting any arbitrarily large time interval [0, T] into sufficiently small sub-intervals {In}Nn=1 such that |In| ≤ T∗ and iterating the bound kψkLp(In;Lq)∩L∞(I;L2) ≤ C∗, we infer kψkLp(0,T;Lq)∩L∞(0,T;L2) ≤ C where C < +∞ depends on the value of T. From here, we proceed as before to obtain a uniform bound for the left hand side for smallT∗ and thus by iteration for arbitrary time intervals [0, T].
Next we consider the case of an L2-supercritical nonlinearity σ > 2/d. In this case, the iterative argument given above breaks down. The basic idea is to use a change of coordinates in order to bring equation (3.4) into a more suitable form. For the sake of notation we shall only consider the case d = 3 in the following. We first note that by using the skew-symmetric matrix
Θ :=
0 Ω3 −Ω2
−Ω3 0 Ω1 Ω2 −Ω1 0
,
the wedge product with the angular momentum can be written as Ω∧x=−Θ·x.
Then the matrix-exponential
X(t, x) := eΘt·x
defines a rotation of the vector x ∈ R3 around the axis Ω by an angle of −|Ω|t. Its time-derivative can be calculated as
(3.20) ∂tX(t, x) = Θ·X(t, x) =−Ω∧X(t, x).
Denoting the wave function in rotated coordinates via
(3.21) ψ(t, x) =e ψ(t, X(t, x)),
we conclude from (3.20) that
i∂tψ(t, x) =e i∂tψ(t, X(t, x))−i(Ω∧X(t, x))· ∇ψ(t, X(t, x)).
Rewriting −i(Ω∧X)· ∇=−iΩ·(X∧ ∇) = Ω·L, we arrive at i∂tψe=−1
2∆ψe+λ|ψ|e2σψe+U(X(t, x))ψ,e
where we have also used the fact that the Laplace operator is invariant with respect to rotations, i.e.
∆Xψ(t, X(t, x)) = ∆xψ(t, X(t, x)).
Dropping all the tildes and denoting W(t, x) = U(X(t, x)), we conclude that up to a change of coordinates, equation (3.4) is equivalent to the following NLS with time- dependent potential
(3.22) i∂tψ =−1
2∆ψ+λ|ψ|2σψ+W(t, x)ψ.
Note that W(t, x) is smooth w.r.t. t∈Rand sub-quadratic w.r.t. x∈R3 with the same (uniform) constantsC(k) as given in Assumption 3 for U(x). Moreover, if U(x) is axially symmetric, i.e. (Ω·L)U(x) = 0, equation (3.20) implies that
∂tW(t, x) =−Ω∧X(t, x)· ∇U(X(t, x)) = −i(Ω·L)U(X(t, x)) = 0,
and hence W(t, x) =W(0, x)≡U(x). The energy corresponding to the transformed NLS (3.22) is given by
EW(t) :=
Z 1
2|∇ψ(t, x)|2+λ|ψ(t, x)|2σ+2+W(t, x)|ψ(t, x)|2 dx.
However, since the potentialW(t, x) in general is time-dependent, the EW(t) is no longer a conserved quantity. Rather, we obtain that
(3.23) d
dtEW(t) = Z
∂tW(t, x)|ψ(t, x)|2 dx.
Nevertheless it is not hard to prove Assertion (1)(ii) of Theorem 3.2.3: In view of the blow-up alternative, stated in Lemma 3.3.2, it suffices to show k∇ψ(t)kL2 <+∞, for all
T >0. To this end, we first estimate 1
2k∇ψ(t)k2L2 ≤EW(t) + Z
W(t, x)|ψ(t, x)|2 dx
≤EW(t) +Ckxψ(t)k2L2,
under the assumption that λ > 0. Integrating equation (3.23) and having in mind that W(t, x) is sub-quadratic inx, we obtain that
k∇ψ(t)k2L2 ≤EW(0) + Z t
0
d
dsEW(s)ds+Ckxψ(t)k2L2
(3.24)
≤C0
1 +kxψ(t)k2L2 + Z t
0
kxψ(s)k2L2 ds
.
Recalling inequality (3.17), we infer d
dtkxψ(t)k2L2 +kxψ(t)k2L2 ≤C0
1 +kxψ(t)k2L2 + Z t
0
kxψ(s)k2L2 ds
,
which by Gronwall’s inequality yields an uniform bound onkxψ(t)kL2 for every time inter- val [0, T]. With this in hand, we can boundk∇ψ(t)kL2 by simply using inequality (3.24) once more.
Remark 3.3.7. In particular, for Ω = (0,0, ω)> and U(x) given by (3.2), we explicitly find
W(t, x) = 1 2
γ12cos2(ωt) +γ22sin2(ωt) x21 + γ21sin2(ωt) +γ22cos2(ωt)
x22+ sin(2ωt) γ12−γ22
x1x2+γ32x23 . Clearly,W = 12(γ12x21+γ22x22+γ32x23) in the axially symmetric case γ12 =γ22.