STRICHARTZ ESTIMATES FOR THE SCHR ... - Semantic Scholar
¨ STRICHARTZ ESTIMATES FOR THE SCHRODINGER AND HEAT EQUATIONS PERTURBED WITH SINGULAR AND TIME DEPENDENT POTENTIALS. VITTORIA PIERFELICE
Abstract. In this paper, we prove Strichartz estimates for the Schr¨ odinger and heat equations perturbed with singular and time dependent potentials.
1. Introduction In this work we study the following perturbations of Schr¨odinger 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 Schr¨odinger equation. In the second paper these results are extended to general non radial initial data. We also note that the heat and Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 H˙ 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 H¨older 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]) H˙ 1 (Rn ) ,→ L(q,2) (Rn ), we get (11) and (12)
kf kL(q,2) ≤ C1 kf kH˙ 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 H¨older 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
¨ dinger flow e−itH 3. Strichartz estimates for the Schro In this section we study the decay properties of the Schr¨odinger flow for the operator H constructed above. More precisely, we can represent the solution to the Schr¨odinger 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 Schr¨ odinger 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
(˜ q 0 ,2)
Lpt˜ Lx
,
kukLpt Lqx ≤ CkF kLp˜0 Lq˜0 , t
for all p, p˜ ∈ [2, ∞], and q, q˜ ∈ [2,
2n n−2 ],
1 n n + = , p 2q 4
x
such that 1 n n + = . p˜ 2˜ q 4
Remark. Note that for the Schr¨odinger equation (p, q) = (2, end-point Schr¨odinger-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 = p˜ = 2, q = q˜ = n−2 6 of [12]. On the other hand, the point p = p˜ = ∞, q = q˜ = 2 reduces to the standard conservation of energy since L(2,2) = L2 . Thus by interpolation we obtain (16) in the dual case p = p˜, q = q˜. We conclude the proof applying as usual the T T ∗ method. ¤ Our next step is to establish the end-point estimate for the perturbed Schr¨odinger 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 kLp˜0 Lq˜0 , x
t
where
2n , n−2 2n ] are such that and p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 q=
n n 1 + = . p˜ 2˜ q 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 Schr¨odinger equation, such that (23)
ku1 kL2 L(q,2) ≤ Cs kF k t
x
0
(˜ q 0 ,2)
Lpt˜ 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 H¨older 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
d1 , 1 < p < ∞, we get (28)
kukL2t Lqx = kukL2 L(q,q) t
x
and (29)
kukLp˜0 Lq˜0 = kuk t
x
µ ¶1−1 2 2 q ≤ kukL2 L(q,2) t x q (˜ q 0 ,˜ q0 )
0
Lpt˜ Lx
≥ kuk
(˜ q 0 ,2)
0
Lpt˜ Lx
,
so we arrive at (30)
kukL2t Lqx ≤ CkF kLp˜0 Lq˜0 , 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 + = . p˜ 2˜ q 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 Schr¨ odinger 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 p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 ] are such that
n n 1 + = . p˜ 2˜ q 4 Proof. Multipling the perturbed Schr¨odinger equation (31) by u ¯ and taking the Imaginary part of integral ¶ ¶ µZ µ Z ¶ µZ ¶ µZ 1 2 2 Fu ¯dx , V |u| dx = Im Im |∇u| dx +Im ∂t u · u ¯dx +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 Fu ¯dx . ∂t u · u ¯dx = 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 + = . p˜ 2˜ q 4 ¤ We can now conclude the proof of the full Strichartz estimates for the problem:
7 2n Theorem 1. Let n ≥ 3, p, p˜ ∈ [2, ∞], and let q, q˜ ∈ [2, n−2 ] be such that
n n 1 n n 1 + = , + = . p 2q 4 p˜ 2˜ q 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), (˜ p, q˜) 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 Schr¨odinger 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, p˜ ∈ [2, ∞], and let q, q˜ ∈ [2, n−2 ] be such that
1 n n + = . p˜ 2˜ q 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) ∈ Lp˜ Lq˜ 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 H¨older 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 kLp˜0 Lq˜0 + kf kL2 , t
x
9
with
2n , n−2 2n ] are such that and p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 q=
n n 1 + = . p˜ 2˜ q 4 The only difference in the proof is to replace (25) with the following H¨older 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 + = . p˜ 2˜ q 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 n exists a unique global solution u(t, x) ∈ Lpt Lqx , with p1˜ + 2˜ q = 4. ¤ 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, p˜ ∈ [2, ∞], and q, q˜ ∈ [2, n−2
1 n n + = , p 2q 4
1 n n + = . p˜ 2˜ q 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 u ˜ − ∆˜ u = 0, (75) u ˜(0, x) = u0 , u0 ≥ 0. (74)
ku(t, ·)kL∞ ≤
The dispersive estimate (74) is valid for this problem. Let w = u ˜ − 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=u ˜ − u. Thus we have 0≤u≤u ˜ 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 H¨older 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]) H˙ 1 (Rn ) ,→ L(q,2) (Rn ), we get kf kL(q,2) ≤ C1 kf kH˙ 1
(84) and (85)
(V+ f, f )L2
¯Z ¯ = ¯¯
¯ p ¯ ˜ k2˙ 1 n , V (x)|f (x)| dx¯¯ ≤ k V+ f k2L2 (Rn ) ≤ Ckf H (R ) 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 + = . p˜ 2˜ q 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. References ∞
[1] Beals M.Optimal L decay for solutions to the wave equation with a potential. Comm. Partial Diff. Equations 1994 8, 1319-1369 [2] Beals M.; Strauss W. Lp estimates for the wave equation with a potential. Comm. Partial Diff. Equations 1993 18 , 1365 - 1397. [3] Bergh J.; L¨ ofstr¨ om J. Interpolation spaces. Springer, Berlin, Heidelberg, New York, 1976. [4] Burq N.; Planchon F; Stalker J.; Tahvilard-Zadeh S. Strichartz estimates for the Wave and Schr¨ odinger Equations with the Inverse-Square Potential. Preprint, 2002. [5] Cuccagna S. On the wave equation with a potential. Comm. Partial Diff. Equations, 2000 25, 1549-1565. [6] D’Ancona P., Georgiev V., Kubo H. Weighted decay estimates for the wave equation. J. Differential Equations 2001 177(1), 146-208.
14
[7] Georgiev V.; Visciglia N. Decay estimates for the wave equation with potential. Preprint 2002. [8] Georgiev V., Ivanov A. Existence and mapping properties of wave operator for the Schr¨ odinger equation with singular potential”. Preprint 2003. [9] Ginibre J., Velo G. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 1995133(1), 50-68. [10] Jensen A.; Nakamura S. Mapping properties of functions of Schr¨ odinger operators between Lp -spaces and Besov spaces. Spectral and scattering theory and applications. 1994, volume 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 187-209. [11] Kalf H.; Schmincke U.-W.; Walter J.; W¨ ust R. On the spectral theory of Schr¨ odinger and Dirac operators with strongly singular potentials. In Spectral theory and differential equations.(Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), 182-226. Lecture Notes in Math., Vol. 448. Springer, Berlin, 1975. [12] Keel M.; Tao T. Endpoint Strichartz estimates. Amer. J. Math. 1998 120(5), 955-980. [13] Murray H. Protter, Hans F. Weinberger Maximum Principles in Differential Equations. [14] O’Neil R. Convolution operator and Lp,q -spaces. Duke Math. J. 1963 it 30, 129 - 142. [15] Planchon F.; Stalker J.; Tahvildar-Zadeh Sh. Lp estimates for the wave equation with the inverse-square potential will appear in Descrete and Cont. Dyn. Systems. [16] Planchon F.; Stalker J.; Shadi Tahvildar-Zadeh A. Dispersive estimate for the wave equation with the inverse-square potential. Preprint 2001. [17] Reed M.; Simon B. Methods of Modern Mathematical Physics vol. I Functional Analysis, Academic Press, New York, 1975. [18] Reed M.; Simon B. Methods of Modern Mathematical Physics vol. II Fourier Analysis and Self Adjointness, 1975 Academic Press, New York, San Francisco , London . [19] Rodianski I.; Schlag W. Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials. Preprint 2002. [20] Simon B. Schr¨ odinger semigroups. Bulletin AMS 1982 7, 447-526. [21] Strichartz R. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 1977 44, 705-714. [22] Triebel H., Interpolation theory, functional spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [23] Vazquez J.; Zuazua E. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal., 2000173(1), 103-153. [24] Visciglia N. Semilinear wave equation with time - dependent potential. Methods Applied Science. [25] Visciglia N. About the Strichartz estimate and the dispersive estimate. Comptes Rendus de l’Acc. Bulgare 2002 55. [26] Visciglia N., PhD thesis, Pisa 2003. ` di Pisa, Dipartimento di Matematica, Via Vittoria Pierfelice: Universita Buonarroti 2, I-56127 Pisa, Italy E-mail address: [email protected]
Abstract. In this paper, we prove Strichartz estimates for the Schr¨ odinger and heat equations perturbed with singular and time dependent potentials.
1. Introduction In this work we study the following perturbations of Schr¨odinger 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 Schr¨odinger equation. In the second paper these results are extended to general non radial initial data. We also note that the heat and Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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 H˙ 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 H¨older 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]) H˙ 1 (Rn ) ,→ L(q,2) (Rn ), we get (11) and (12)
kf kL(q,2) ≤ C1 kf kH˙ 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 H¨older 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
¨ dinger flow e−itH 3. Strichartz estimates for the Schro In this section we study the decay properties of the Schr¨odinger flow for the operator H constructed above. More precisely, we can represent the solution to the Schr¨odinger 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 Schr¨ odinger 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
(˜ q 0 ,2)
Lpt˜ Lx
,
kukLpt Lqx ≤ CkF kLp˜0 Lq˜0 , t
for all p, p˜ ∈ [2, ∞], and q, q˜ ∈ [2,
2n n−2 ],
1 n n + = , p 2q 4
x
such that 1 n n + = . p˜ 2˜ q 4
Remark. Note that for the Schr¨odinger equation (p, q) = (2, end-point Schr¨odinger-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 = p˜ = 2, q = q˜ = n−2 6 of [12]. On the other hand, the point p = p˜ = ∞, q = q˜ = 2 reduces to the standard conservation of energy since L(2,2) = L2 . Thus by interpolation we obtain (16) in the dual case p = p˜, q = q˜. We conclude the proof applying as usual the T T ∗ method. ¤ Our next step is to establish the end-point estimate for the perturbed Schr¨odinger 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 kLp˜0 Lq˜0 , x
t
where
2n , n−2 2n ] are such that and p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 q=
n n 1 + = . p˜ 2˜ q 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 Schr¨odinger equation, such that (23)
ku1 kL2 L(q,2) ≤ Cs kF k t
x
0
(˜ q 0 ,2)
Lpt˜ 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 H¨older 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
d1 , 1 < p < ∞, we get (28)
kukL2t Lqx = kukL2 L(q,q) t
x
and (29)
kukLp˜0 Lq˜0 = kuk t
x
µ ¶1−1 2 2 q ≤ kukL2 L(q,2) t x q (˜ q 0 ,˜ q0 )
0
Lpt˜ Lx
≥ kuk
(˜ q 0 ,2)
0
Lpt˜ Lx
,
so we arrive at (30)
kukL2t Lqx ≤ CkF kLp˜0 Lq˜0 , 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 + = . p˜ 2˜ q 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 Schr¨ odinger 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 p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 ] are such that
n n 1 + = . p˜ 2˜ q 4 Proof. Multipling the perturbed Schr¨odinger equation (31) by u ¯ and taking the Imaginary part of integral ¶ ¶ µZ µ Z ¶ µZ ¶ µZ 1 2 2 Fu ¯dx , V |u| dx = Im Im |∇u| dx +Im ∂t u · u ¯dx +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 Fu ¯dx . ∂t u · u ¯dx = 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 + = . p˜ 2˜ q 4 ¤ We can now conclude the proof of the full Strichartz estimates for the problem:
7 2n Theorem 1. Let n ≥ 3, p, p˜ ∈ [2, ∞], and let q, q˜ ∈ [2, n−2 ] be such that
n n 1 n n 1 + = , + = . p 2q 4 p˜ 2˜ q 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), (˜ p, q˜) 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 Schr¨odinger 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, p˜ ∈ [2, ∞], and let q, q˜ ∈ [2, n−2 ] be such that
1 n n + = . p˜ 2˜ q 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) ∈ Lp˜ Lq˜ 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 H¨older 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 kLp˜0 Lq˜0 + kf kL2 , t
x
9
with
2n , n−2 2n ] are such that and p˜ ∈ [2, ∞], and q˜ ∈ [2, n−2 q=
n n 1 + = . p˜ 2˜ q 4 The only difference in the proof is to replace (25) with the following H¨older 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 + = . p˜ 2˜ q 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 n exists a unique global solution u(t, x) ∈ Lpt Lqx , with p1˜ + 2˜ q = 4. ¤ 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, p˜ ∈ [2, ∞], and q, q˜ ∈ [2, n−2
1 n n + = , p 2q 4
1 n n + = . p˜ 2˜ q 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 u ˜ − ∆˜ u = 0, (75) u ˜(0, x) = u0 , u0 ≥ 0. (74)
ku(t, ·)kL∞ ≤
The dispersive estimate (74) is valid for this problem. Let w = u ˜ − 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=u ˜ − u. Thus we have 0≤u≤u ˜ 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 H¨older 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]) H˙ 1 (Rn ) ,→ L(q,2) (Rn ), we get kf kL(q,2) ≤ C1 kf kH˙ 1
(84) and (85)
(V+ f, f )L2
¯Z ¯ = ¯¯
¯ p ¯ ˜ k2˙ 1 n , V (x)|f (x)| dx¯¯ ≤ k V+ f k2L2 (Rn ) ≤ Ckf H (R ) 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 + = . p˜ 2˜ q 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. References ∞
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[7] Georgiev V.; Visciglia N. Decay estimates for the wave equation with potential. Preprint 2002. [8] Georgiev V., Ivanov A. Existence and mapping properties of wave operator for the Schr¨ odinger equation with singular potential”. Preprint 2003. [9] Ginibre J., Velo G. Generalized Strichartz inequalities for the wave equation. J. Funct. Anal., 1995133(1), 50-68. [10] Jensen A.; Nakamura S. Mapping properties of functions of Schr¨ odinger operators between Lp -spaces and Besov spaces. Spectral and scattering theory and applications. 1994, volume 23 of Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 187-209. [11] Kalf H.; Schmincke U.-W.; Walter J.; W¨ ust R. On the spectral theory of Schr¨ odinger and Dirac operators with strongly singular potentials. In Spectral theory and differential equations.(Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), 182-226. Lecture Notes in Math., Vol. 448. Springer, Berlin, 1975. [12] Keel M.; Tao T. Endpoint Strichartz estimates. Amer. J. Math. 1998 120(5), 955-980. [13] Murray H. Protter, Hans F. Weinberger Maximum Principles in Differential Equations. [14] O’Neil R. Convolution operator and Lp,q -spaces. Duke Math. J. 1963 it 30, 129 - 142. [15] Planchon F.; Stalker J.; Tahvildar-Zadeh Sh. Lp estimates for the wave equation with the inverse-square potential will appear in Descrete and Cont. Dyn. Systems. [16] Planchon F.; Stalker J.; Shadi Tahvildar-Zadeh A. Dispersive estimate for the wave equation with the inverse-square potential. Preprint 2001. [17] Reed M.; Simon B. Methods of Modern Mathematical Physics vol. I Functional Analysis, Academic Press, New York, 1975. [18] Reed M.; Simon B. Methods of Modern Mathematical Physics vol. II Fourier Analysis and Self Adjointness, 1975 Academic Press, New York, San Francisco , London . [19] Rodianski I.; Schlag W. Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials. Preprint 2002. [20] Simon B. Schr¨ odinger semigroups. Bulletin AMS 1982 7, 447-526. [21] Strichartz R. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 1977 44, 705-714. [22] Triebel H., Interpolation theory, functional spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. [23] Vazquez J.; Zuazua E. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal., 2000173(1), 103-153. [24] Visciglia N. Semilinear wave equation with time - dependent potential. Methods Applied Science. [25] Visciglia N. About the Strichartz estimate and the dispersive estimate. Comptes Rendus de l’Acc. Bulgare 2002 55. [26] Visciglia N., PhD thesis, Pisa 2003. ` di Pisa, Dipartimento di Matematica, Via Vittoria Pierfelice: Universita Buonarroti 2, I-56127 Pisa, Italy E-mail address: [email protected]
