STRICHARTZ ESTIMATES FOR THE SCHRÖDINGER
AND HEAT EQUATIONS PERTURBED WITH SINGULAR
AND TIME DEPENDENT POTENTIALS.
VITTORIA PIERFELICE
Abstract. In this paper, we prove Strichartz estimates for the Schrödinger
and heat equations perturbed with singular and time dependent potentials.
1. Introduction
In this work we study the following perturbations of Schrödinger and heat
equations
1
(1)
∂t u − ∆u + V u = 0,
u(0, x) = u0 (x),
i
(2)
∂t u − ∆u + V u = 0,
u(0, x) = u0 (x),
where ∆ is the n dimensional Laplacian and x ∈ Rn , n ≥ 3. Several works
have investigated these perturbed Cauchy Problems and the corresponding
Strichartz estimates; here in particular we are interested in the case of singular potentials V . The critical behavior for the potential is V ∼ |x|−2 . The
family of radial potentials
V (x) =
a
(n − 2)2
, n ≥ 2,
,
where
a
>
−
|x|2
4
is studied in the papers [15] and [4]. More precisely, in the first paper it is
proved that in the radial case, i.e. when the initial data are radially symmetric, the solution to the perturbed wave equation satisfies the generalized
space-time Strichartz estimates Lp Lq but not the dispersive estimate, as it
is shown by suitable counterexamples. Since their proof was based on estimates for the elliptic operator Pa := −∆ + |x|a 2 , the corresponding Strichartz
estimates hold also for the Schrödinger equation. In the second paper these
results are extended to general non radial initial data.
We also note that the heat and Schrödinger flows for the elliptic operator
have been studied in the theory of combustion (see [23]), and in quantum
mechanics (see [11]) respectively.
In this paper we treat several potentials for equations (1), (2), and our
goal is to prove Strichartz estimates for them. Our first result (Theorem 1)
concerns the Schrödinger equation; we can prove the full set of Strichartz
estimates in the case in which V = V (x) is a real-valued potential satisfying
the following assumption:
2n
(3)
kV kL( n2 ,∞) ≡ C0 <
,
Cs (n − 2)
1
2
where Cs is the universal Strichartz constant for the unperturbed equation.
Here with L(p,q) we denote the standard Lorentz spaces (and L(p,∞) is the
weak Lp space), see [3] for details about these spaces. We can deal also with
some cases in which V = V (t, x) is time dependent (Theorem 3 below). In
( n ,∞)
2
particular, we assume that V ∈ L∞
t Lx
that
is a real-valued potential such
kV (t, ·)kL( n2 ,∞) ≡ C0 .
(4)
is small enough.
As to the heat equation, our main result is Theorem 4. Also in this
case we are able to prove the Strichartz estimates, under the following assumptions: we take initial data u0 ∈ L1 ∩ L∞ , and we consider a rough
n
potential V (x) ∈ L( 2 ,∞) , which we split in positive and negative part
V (x) = V+ (x) − V− (x), V± ≥ 0, with
2n
.
(5)
kV− kL( n2 ,∞) ≡ C0 <
Cs (n − 2)
Notice that in the case of a non-negative potential we prove that the maximum principle holds, and hence also the stronger dispersive Lp −Lq estimates
are valid.
The paper is organized as follows. In section 2, we study the properties of
selfadjointness of the perturbed operator H = −∆ + V with a small singular
n
potential V = V (x) ∈ L( 2 ,∞) . In section 3 we deduce Strichartz estimates
for the solution of the perturbed Schrödinger equation making interpolation
into the endpoint estimate and energy estimate. The arguments of the previous sections can be extended to cover the case of a small, time dependent
potential V (t, x). Indeed, in section 4 we prove the existence of solution
to the perturbed Schrödinger equation with this potential, and we obtain
the full Strichartz estimates for it. We then do the same for the heat equation. Indeed, in section 5 we treat the heat equation perturbed by singular
potential and we deduce also for it the properties of decay of solution.
2. Selfadjointness of H = −∆ + V
In this section we check that the sum H = −∆ + V can be realized as
a selfadjoint operator on L2 by a standard Friedrichs extension. This will
allow us to consider the Schrödinger flow e−itH and the heat flow e−tH in
the following of the paper.
Consider the bilinear form
Z
(6) B(f, f ) = (∇f, ∇f )L2 (Rn ) +
V (x)|f (x)|2 dx, x ∈ Rn , n ≥ 3.
Rn
It is not difficult to see that
(7)
f →Vf
is a self adjoint operator with dense domain Ḣ 2 (Rn ). In this case we can
use the KLMN- theorem (see theorem 10.17 in [18]). Due to this theorem it
is sufficient to verify the estimate
¯
¯Z
Z
¯
¯
2
2
2
¯
¯
(8)
¯ n V (x)|f (x)| dx¯ ≤ a n |∇f (x)| dx − bkf kL2 (Rn ) ,
R
R
3
with a < 1. Indeed, the assumption (3) implies that
p
(9)
|V | ∈ L(n,∞) ,
so that, by the Hölder inequality for Lorentz spaces,
p
p
(10)
k |V |f kL2 ≤ Ck |V |kL(n,∞) kf kL(q,2) ≤ CC0 kf kL(q,2) ,
where
1 1
2n
1
= − , i.e. q =
.
q
2 n
n−2
Using the Sobolev embedding (see [3]) Ḣ 1 (Rn ) ,→ L(q,2) (Rn ), we get
(11)
and
(12)
kf kL(q,2) ≤ C1 kf kḢ 1
¯Z
¯
¯
¯
¯
p
¯
V (x)|f (x)| dx¯¯ ≤ k |V |f k2L2 (Rn ) ≤ C02 C 2 C12 k∇f k2L2 (Rn ) .
n
2
R
1
, where C is the constant
If C0 is such that CC0 C1 < 1 i.e. C0 < CC
1
from the Hölder inequality (for Lorentz spaces) and C1 is the constant from
Sobolev embedding, then we can conclude, using the KLMN theorem, that
there exists a self-adjoint operator H = −∆ + V such that
Z
2
V (x)|f (x)|2 dx.
(13)
((−∆ + V )f, f )L2 = k∇f kL2 +
Rn
3. Strichartz estimates for the Schrödinger flow e−itH
In this section we study the decay properties of the Schrödinger flow for
the operator H constructed above. More precisely, we can represent the
solution to the Schrödinger equation (1) as
(14)
u(t) = U (t)u0 , U (t) = e−itH .
Our starting point will be the following Strichartz estimate, essentially
proved in the paper [12]:
Proposition 1. Let n ≥ 3 and consider the Cauchy Problem for the Schrödinger
equation
(
1
i ∂t u − ∆u = F (t, x),
(15)
u(0, x) = 0, x ∈ Rn ,
then the following estimates hold:
(16)
kukLp L(q,2) ≤ CkF k
t
(17)
x
0
(q̃ 0 ,2)
Lp̃t Lx
,
kukLpt Lqx ≤ CkF kLp̃0 Lq̃0 ,
t
for all p, p̃ ∈ [2, ∞], and q, q̃ ∈ [2,
2n
n−2 ],
1
n
n
+
= ,
p 2q
4
x
such that
1
n
n
+
= .
p̃ 2q̃
4
Remark. Note that for the Schrödinger equation (p, q) = (2,
end-point Schrödinger-admissible for n ≥ 3.
2n
n−2 )
it is the
4
Proof. The second estimate (17) is the standard Strichartz estimate, proved
in [12]; notice that it follows from the stronger estimate (16) by embedding
of Lorentz spaces.
2n
is proved in section
Estimate (16) in the endpoint p = p̃ = 2, q = q̃ = n−2
6 of [12]. On the other hand, the point p = p̃ = ∞, q = q̃ = 2 reduces to the
standard conservation of energy since L(2,2) = L2 . Thus by interpolation we
obtain (16) in the dual case p = p̃, q = q̃. We conclude the proof applying
as usual the T T ∗ method.
¤
Our next step is to establish the end-point estimate for the perturbed
Schrödinger equation:
Proposition 2. Let n ≥ 3 and consider the Cauchy Problem
(
1
i ∂t u − ∆u + V u = F,
(18)
u(0, x) = 0, x ∈ Rn ,
where V = V (x) is a real-valued potential such that
2n
(19)
kV kL( n2 ,∞) ≡ C0 <
,
Cs (n − 2)
(here Cs is the constant appearing in the Strichartz estimates for the unperturbed equation). Then the following estimate holds
(20)
kukL2t Lqx ≤ CkF kLp̃0 Lq̃0 ,
x
t
where
2n
,
n−2
2n
] are such that
and p̃ ∈ [2, ∞], and q̃ ∈ [2, n−2
q=
n
n
1
+
= .
p̃ 2q̃
4
Proof. Indeed we can consider the solution u = u1 + u2 as the sum of solutions to following Cauchy problems
(
1
i ∂t u1 − ∆u1 = F,
(21)
u(0, x) = 0, x ∈ Rn , n ≥ 3,
and
(22)
(
1
i ∂t u 2
− ∆u2 = −V u,
u(0, x) = 0, x ∈ Rn , n ≥ 3.
For (21) we have the classical Schrödinger equation, such that
(23)
ku1 kL2 L(q,2) ≤ Cs kF k
t
x
0
(q̃ 0 ,2)
Lp̃t Lx
is satisfied for the Proposition 1 (see [12]).
Since for the Cauchy problem (22) we have
(24)
ku2 kL2 L(q,2) ≤ Cs kV uk
t
x
(q 0 ,2)
L2t Lx
,
we are in position to apply the Hölder estimate (see Theorem 3.5 in [14])
(25)
kV ukL(q0 ,2) ≤ C2 kV kL( n2 ,∞) kukL(q,2) ≤ C2 C0 kukL(q,2)
5
where
C2 = q ≡
2n
,
n−2
so if C0 is such that Cs C0 C2 < 1, i.e.
C0 <
2n
,
Cs (n − 2)
we see that from (23), (24) and (25) that
(26)
Cs
kF k p̃0 (q̃0 ,2) ,
Lt Lx
1 − C s C0 C2
kukL2 L(q,2) ≤
t
x
where
n
n
1
+
= .
p̃ 2q̃
4
So using the Theorem of Calderón (see Lemma 2.5 in [14])
µ ¶ 1 −1
d1 d1 d
(27)
kukL(p,d) ≤
kukL(p,d1 ) ,
p
for d > d1 , 1 < p < ∞, we get
(28)
kukL2t Lqx = kukL2 L(q,q)
t
x
and
(29)
kukLp̃0 Lq̃0 = kuk
t
x
µ ¶1−1
2 2 q
≤
kukL2 L(q,2)
t x
q
(q̃ 0 ,q̃ 0 )
0
Lp̃t Lx
≥ kuk
(q̃ 0 ,2)
0
Lp̃t Lx
,
so we arrive at
(30)
kukL2t Lqx ≤ CkF kLp̃0 Lq̃0 , q =
x
t
where
C=
and
µ
n−2
n
¶1 µ
n
2n
, n ≥ 3,
n−2
Cs
1 − C s C0 C2
¶
,
1
n
n
+
= .
p̃ 2q̃
4
¤
In the next step we consider the point p = ∞, q = 2:
Proposition 3. Let n ≥ 3 and consider the Cauchy Problem for the perturbed Schrödinger equation
(
1
i ∂t u − ∆u + V u = F,
(31)
u(0, x) = 0, x ∈ Rn ,
where V = V (x) is a real-valued potential such that
(32)
kV kL( n2 ,∞) < ∞.
Then the following estimate holds
(33)
0
0,
kukL∞
2 ≤ CkF k p
t Lx
L e ,Lqe
t
x
6
2n
where p̃ ∈ [2, ∞], and q̃ ∈ [2, n−2
] are such that
n
n
1
+
= .
p̃ 2q̃
4
Proof. Multipling the perturbed Schrödinger equation (31) by ū and taking
the Imaginary part of integral
¶
¶
µZ
µ Z
¶
µZ
¶
µZ
1
2
2
F ūdx ,
V |u| dx = Im
Im
|∇u| dx +Im
∂t u · ūdx +Im
i Rn
Rn
Rn
Rn
we notice that
Im
and
Im
thus we have
(34)
µZ
µZ
2
¶
=0
2
¶
= 0,
|∇u| dx
Rn
V |u| dx
Rn
µ Z
¶
¶
µZ
1
−Re
F ūdx .
∂t u · ūdx = Im
i Rn
Rn
The Cauchy-Scwhartz inequality implies
∂t ku(t)k2L2 ≤ kF kL2 kukL2 ,
(35)
and we obtain
ku(t)kL2 ≤
(36)
Z
t
0
kF kL2 dt
so we obtain the following estimate
kukL∞ L2 ≤ CkF kL1 L2 .
(37)
The estimate (20) leads to
(38)
kukL2 Lq ≤ CkF kL1 L2 , q =
2n
,
n−2
by duality we have also
(39)
kukL∞ L2 ≤ CkF kL2 Lq0 , q 0 =
2n
.
n+2
Interpolating between (37) and (39), we obtain
(40)
0
0,
kukL∞
2 ≤ CkF k p
t Lx
L e ,Lqe
t
where
x
1
n
n
+
= .
p̃ 2q̃
4
¤
We can now conclude the proof of the full Strichartz estimates for the
problem:
7
2n
Theorem 1. Let n ≥ 3, p, p̃ ∈ [2, ∞], and let q, q̃ ∈ [2, n−2
] be such that
n
n
1
n
n
1
+
= ,
+
= .
p 2q
4
p̃ 2q̃
4
Let V = V (x) be a real-valued potential such that
2n
(41)
kV kL( n2 ,∞) ≡ C0 <
,
Cs (n − 2)
where Cs is the universal Strichartz constant for the unperturbed equation.
Then the solution to the Cauchy Problem
(
1
i ∂t u − ∆u + V (x)u = F (t, x),
(42)
u(0, x) = f,
satisfies the estimates
(43)
kukLp (R
(q,2)
)
t ;Lx
+ kukC(Rt ;L2 ) ≤ CkF k
and
(44)
(e
q 0 ,2)
0
Lpe (Rt ;Lx
kukLp (Rt ;Lqx ) + kukC(Rt ;L2 ) ≤ CkF kLpe0 (R
qe0
t ;Lx )
)
+ Ckf kL2 ,
+ Ckf kL2 .
Proof. Assume first that f = 0. By interpolation between (20) and (33), we
get
(45)
kukLpt Lqx ≤ CkF kLpe0 ,Lqe0
t
x
for all (p, q), (p̃, q̃) as in the statement of the Theorem.
Assume now that F = 0 and f arbitrary. The previous estimate and the
T T ∗ argument of [9], yield the estimate
kukLpt Lqx ≤ Ckf kL2 .
(46)
Notice that the conservation of energy gives also
kukLpt Lqx + kukCt L2 ≤ Ckf kL2 .
(47)
Summing up we obtain (44). The proof of (43) is similar (see also the proof
of Proposition 1).
¤
If we start from the local Strichartz estimates instead of the global ones,
in a similar way we can prove the following
Theorem 2. Under the assumptions of Theorem 1 we have
(48)
kukLp ([0,T ];L(q,2) ) + kukC([0,T ];L2 ) ≤ CkF k
x
0
(e
q 0 ,2)
Lpe ([0,T ];Lx
)
+ Ckf kL2
for all T > 0 and with a constant C independent of T .
4. The case of time dependent potentials
The arguments of the previous sections can be extended to cover the
case of a small, time dependent potential V (t, x). Indeed, our method of
proof is based on a perturbation of the standard Strichartz estimates for
the Schrödinger and heat equations. However, we notice that in this case
we cannot use the standard theory of selfadjoint operators to study the
perturbed Hamiltonian H = −∆ + V (t, x). Thus in the following we shall
consider the problem of existence and of the decay of solutions.
Our first result is the following:
8
2n
Theorem 3. Let n ≥ 3, p, p̃ ∈ [2, ∞], and let q, q̃ ∈ [2, n−2
] be such that
1
n
n
+
= .
p̃ 2q̃
4
n
n
1
+
= ,
p 2q
4
Let V = V (t, x) be a real-valued potential such that
(49)
kV k
( n ,∞)
2
L∞
t Lx
≡ C0
0
0
is small enough. Then for any F (t, x) ∈ Lp̃ Lq̃ there exists a unique global
solution u(t, x) of the the Cauchy Problem
(
1
i ∂t u − ∆u + V (t, x)u = F (t, x),
(50)
u(0, x) = f.
which satisfies the estimates
(51)
kukLp (R
(q,2)
)
t ;Lx
+ kukC(Rt ;L2 ) ≤ CkF k
and
(52)
(e
q 0 ,2)
0
Lpe (Rt ;Lx
kukLp (Rt ;Lqx ) + kukC(Rt ;L2 ) ≤ CkF kLpe0 (R
qe0
t ;Lx )
)
+ Ckf kL2 ,
+ Ckf kL2 .
Analogous estimates hold on finite time intervals [0, T ] with constants independent of T .
Proof. The proof follows the lines of the proof of Theorem 1. We define Φ(v)
as the solution u of the linear problem
(
1
i ∂t u − ∆u = F (t, x) − V (t, x)v,
(53)
u(0, x) = f.
By Proposition 1 and [12] we have
kukL∞ L2 + kukL2 L(q,2) ≤ CkF − V vkL2 L(q0 ,2) + kf kL2
≤ kF kL2 L(q0 ,2) + kV vkL2 L(q0 ,2) + kf kL2 ,
2n
. Using the Hölder inequality for Lorentz spaces (see [14])
where q = n−2
and the assumption (49), we get
(54)
kukL∞ L2 + kukL2 L(q,2) ≤ CkF kL2 L(q0 ,2) + C0 kvkL2 L(q,2) + kf kL2 .
Thus Φ : v ∈ L2 L(q,2) 7→ u ∈ L2 L(q,2) ∩ L∞ L2 . We show now that Φ is a
contraction on the space L2 L(q,2) . Let v1 , v2 ∈ L2 L(q,2) such that Φ(vi ) =
ui , i = 1, 2; then we have
ku1 −u2 kL∞ L2 +ku1 −u2 kL2 L(q,2) ≤ kV (v1 −v2 )kL2 L(q0 ,2) ≤ C0 kv1 −v2 kL2 L(q,2) .
If C0 < 1 the map Φ is a contraction, and this implies that for any F ∈
0
L2 L(q ,2) and f ∈ L2 there exists a unique solution u(t, x) ∈ L2 L(q,2) ∩ L∞ L2
of the Cauchy problem (50).
In particular for all F ∈ Cc∞ and f ∈ L2 there exists a unique solution.
When F ∈ Cc∞ , we can proceed as in Proposition 2 and we can prove the
endpoint estimate
(55)
kukL2t Lqx ≤ CkF kLp̃0 Lq̃0 + kf kL2 ,
t
x
9
with
2n
,
n−2
2n
] are such that
and p̃ ∈ [2, ∞], and q̃ ∈ [2, n−2
q=
n
n
1
+
= .
p̃ 2q̃
4
The only difference in the proof is to replace (25) with the following Hölder
estimate
kV ukL2 L(q0 ,2) ≤ CkV kL∞ L( n2 ,∞) kukL2 L(q,2) ≤ CC0 kukL2 L(q,2) .
(56)
On the other hand, we can repeat the proof of Proposition 3 and we obtain
0
0 + kf kL2 ,
kukL∞
2 ≤ CkF k p
t Lx
L e ,Lqe
(57)
t
where
x
1
n
n
+
= .
p̃ 2q̃
4
Then by interpolation we obtain the full Strichartz estimates
(58)
kukLp (R
(q,2)
)
t ;Lx
+ kukC(Rt ;L2 ) ≤ CkF k
(e
q 0 ,2)
0
Lpe (Rt ;Lx
)
+ Ckf kL2
for all F ∈ Cc∞ .
Since we have proved that for all such F there exists a unique solution
0
0
u(t, x), by a density argument we easily obtain that for all F ∈ Lpte Lqxe there
n
= n4 .
exists a unique global solution u(t, x) ∈ Lpt Lqx , with p̃1 + 2q̃
¤
5. Heat equation perturbed with a singular potential
This section is devoted to a study of the perturbed heat equation. The
ideas of the preceding sections can be applied also in this case with some
modifications. The main difference is the role of the positive part V+ of
the potential V ; indeed, in order to prove the decay of the solution, weaker
assumptions on V+ are sufficient.
Our result is the following:
n
Theorem 4. Let n ≥ 3 and assume the potential V ∈ L( 2 ,∞) . Moreover,
assume that the negative part V− = −(V ∧ 0) satisfies
2n
.
(59)
kV− kL( n2 ,∞) ≡ C0 <
Cs (n − 2)
Then any solution to the following Cauchy problem
(
∂t u − ∆u + V (x)u = F (t, x),
(60)
u(0, x) = u0 ∈ L1 ∩ L∞ ,
satisfies the Strichartz estimate
(61)
kukLp (Rt ;Lqx ) + kukC(Rt ;L2 ) ≤ CkF kLpe0 (R
qe0
t ;Lx )
2n
] are such that
where p, p̃ ∈ [2, ∞], and q, q̃ ∈ [2, n−2
1
n
n
+
= ,
p 2q
4
1
n
n
+
= .
p̃ 2q̃
4
+ Cku0 kL2 .
10
We split the proof of Theorem 4 in several parts.
Proposition 4. Let n ≥ 3 and consider the following Cauchy problem
(
∂t u − ∆u + V (x)u = 0,
(62)
u(0, x) = u0 ≥ 0,
with initial data u0 ∈ L1 ∩ L∞ , and we assume that
(63)
V (x) ≥ 0
and
n
V ∈ L( 2 ,∞) .
Then there exists a unique solution to the Cauchy problem (62)
u(t, x) = e−tH0 u0
satisfying the maximum principle, i.e.
(64)
u ≥ 0.
Proof. Since we know that the maximum principle holds if the potential
is positive and V ∈ L∞ , we consider a sequence of truncated potentials
Vk = V ∧ k, k ≥ 1 so that Vk ∈ L∞ . We consider then the respectively
approximated Cauchy problem
(
∂t uk − ∆uk + Vk (x)uk = 0, k ≥ 1,
(65)
uk (0, x) = u0 , u0 ≥ 0,
and by maximum principle 0 ≤ uk+1 ≤ uk ≤ u0 . Since {uk } is a sequence
decreasing and u0 ∈ L1 ∩ L∞ , then by monotone convergence Theorem we
have that {uk } converge in strong sense to u(t, x)
(66)
u(t, x) = Lp − lim uk (t, x),
1 ≤ p < ∞.
k→∞
Now it suffices to prove that u(t, x) is a solution to (62), so we have that
0 ≤ u ≤ uk ≤ u0 . Thus since u(t, x) satisfies the Maximum principle (see
[13]), we have the uniqueness of the solution to (62).
Since u0 ∈ L1 ∩L∞ and {uk } is a sequence decreasing such that uk ≤ |u0 |,
by Theorem of Lebesgue we have the convergence uk → u in L1 .
As consequence we have following convergences in the distributional sense
D0 ∀k → ∞
uk → u,
∂t uk → ∂t u,
(67)
∆uk → ∆u.
Then it remains to prove that we have the following convergence
(68)
V k uk → V u
in the distributional sense. Indeed, we shall use the identity
(69)
Vk uk − V u = (Vk − V )uk + V (uk − u).
n
Consider the first term to (69) and since L( 2 ,∞) ⊂ L1loc we can take
V ∈ L1loc (Rn ),
that implies
Z
|V (x) − Vk (x)|dx → 0
K
∀k → ∞,
11
so that
Z
Z
|V (x)−Vk (x)|dx → 0,
|V (x)−Vk (x)||uk (t, x)|dx ≤ sup |uk (t, x)|
x∈Rn
K
∀k → ∞.
K
Thus the first term converges
(Vk − V )uk → 0 ∀k → ∞
in the distributional sense D 0 .
Now we are ready to estimate the second term to (69). We have
kV (uk − u)kL1 ≤ kV kL( n2 ,∞) kuk − ukL(q,1) ≤ C0 kuk − ukL(q,1)
(70)
where
1
q
= 1 − n2 , and using the real interpolation (see [14])
L(q,1) = (L1 , L∞ )(1− 1 ,1) ,
q
we have the following
2
(71)
n−2
n
kuk − ukL(q,1) ≤ kuk − ukLn1 kuk − ukL∞
.
Since {uk } is decreasing and uk ≤ u0 ∈ L1 ∩ L∞ , by monotone convergence
Theorem one obtains
kuk − ukL1 → 0,
and
kuk − ukL∞ → 0.
1
Thus V (uk − u) → 0 in L , and so it converges in distributional sense, i.e.
Vk uk − V u → 0.
This concludes the proof.
¤
Proposition 5. Let n ≥ 3 and assume that
(72)
V+ (x) ≥ 0,
n
V+ ∈ L( 2 ,∞) .
Then any solution to the Cauchy problem
(
∂t u − ∆u + V+ (x)u = 0,
(73)
u(0, x) = u0 ,
satisfies the dispersive estimate
C
n ku0 kL1 .
t2
Proof. Consider the Cauchy problem for the heat equation with the same
initial data to (73)
(
∂t ũ − ∆ũ = 0,
(75)
ũ(0, x) = u0 , u0 ≥ 0.
(74)
ku(t, ·)kL∞ ≤
The dispersive estimate (74) is valid for this problem.
Let w = ũ − u. Then w is a solution to the following Cauchy problem
(
∂t w − ∆w = V+ (x)u,
(76)
w(0, x) = 0.
12
n
Since 0 ≤ V+ ∈ L( 2 ,∞) we can apply it the previous Proposition and we
obtain that
u ≥ 0.
So applying one more the maximum principle for (76) we obtain
0 ≤ w = ũ − u.
Thus we have
0 ≤ u ≤ ũ
and the dispersive estimate
C
n ku0 kL1 ,
t2
follows. This concludes the proof of this Proposition.
ku(t, ·)kL∞ ≤
(77)
¤
Now we use the connection between self-adjointness and semibounded
quadratic form, extending the notion of ”closed” from operators to forms.
Lemma 1. Let n ≥ 3 and assume that
(78)
n
V+ ∈ L( 2 ,∞) .
V+ (x) ≥ 0,
Then the operator H0 = −∆ + V+ is self-adjoint in H 2 (Rn ).
Proof. Consider the quadratic form
(79)
B(f, f ) = (∇f, ∇f )L2 (Rn ) +
Z
Rn
V (x)|f (x)|2 dx, x ∈ Rn , n ≥ 3,
on the dense subspace H 1 (Rn ) of L2 (Rn ).
To prove this Lemma it suffices to apply the standard theory of symmetric
quadratic forms (see e.g. Theorem VIII.15 in the [17]). One can see easily
that B(f, f ) is a positive quadratic form, thus it remains to see that it is
closed in H 1 (Rn ), i.e. H 1 (Rn ) is complete under the norm
|||f |||2 := B(f, f ) + kf k2L2 .
(80)
Since V+ (x) ≥ 0 one obtains
(81)
|||f |||2 = k∇f k2L2 + (V+ f, f )L2 + kf k2L2 ≥ Ckf k2H 1 .
The assumption on the potential implies that
p
V+ ∈ L(n,∞) ,
(82)
so that, by the Hölder inequality for Lorentz spaces,
p
p
(83)
k V+ f kL2 ≤ Ck V+ kL(n,∞) kf kL(q,2) ≤ CC0 kf kL(q,2) ,
where
1
1 1
= − .
q
2 n
Using the Sobolev embedding (see [3]) Ḣ 1 (Rn ) ,→ L(q,2) (Rn ), we get
kf kL(q,2) ≤ C1 kf kḢ 1
(84)
and
(85)
(V+ f, f )L2
¯Z
¯
= ¯¯
¯
p
¯
V (x)|f (x)| dx¯¯ ≤ k V+ f k2L2 (Rn ) ≤ C̃kf k2Ḣ 1 (Rn ) ,
n
2
R
13
so that
|||f |||2 ≤ Ckf k2H 1 .
(86)
Thus we have the equivalence
(87)
|||f ||| ' kf kH 1 ,
and the conclusion follows at once.
¤
Remark. Since H0 = −∆ + V+ is a self-adjoint operator, we can represent
the solution to the Cauchy problem
(
∂t u − ∆u + V+ (x)u = 0,
(88)
u(0, x) = u0 ,
as
(89)
u(t) = U (t)u0 ,
U (t) = e−tH0 ,
and U (t) is a continuous semigroup in L2 and we have the energy inequality
kU (t)u0 kL2 ≤ ku0 kL2 .
(90)
Notice that interpolating the dispersive estimate (74) with the energy inequality we obtain Lp -decay estimates, and using the T T ∗ method of Ginibre
and Velo (see [9], [12]) it is possible obtain the full Strichartz space-time estimates
(91)
with
kukLp (Rt ;Lqx ) + kukC(Rt ;L2 ) ≤ CkF kLpe0 (R
1
n
n
+
= ,
p 2q
4
qe0
t ;Lx )
+ Cku0 kL2 ,
1
n
n
+
= .
p̃ 2q̃
4
Remark. Consider the following perturbed Cauchy problem
(
∂t u − H0 u + V− (x)u = F (t, x),
(92)
u(0, x) = u0 ,
n
where V− ∈ L( 2 ,∞) , kV− kL( n2 ,∞) ≤ C0 . Using the same argument of section
2 we show that the operator H = H0 − V− is selfadjoint, so the solution to
(92) is u(t, x) = e−tH u0 . Moreover, repeating the same steps of section 3, it
is not difficult to show the full Strichartz estimates for the heat flow e−tH
and this concludes the proof of Theorem 4.
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Vittoria Pierfelice: Università di Pisa, Dipartimento di Matematica, Via
Buonarroti 2, I-56127 Pisa, Italy
E-mail address: [email protected]