Math. Ann. 315, 529–567 (1999) c Springer-Verlag 1999

Mathematische Annalen

Gain of regularity for semilinear Schr¨odinger equations Hiroyuki Chihara Received: 14 December 1998 Abstract. We discuss local existence and gain of regularity for semilinear Schr¨odinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi’s transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr¨odingertype equations. In particular, the sharp G˚arding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Mathematics Subject Classification (1991): 35Q55, 35B65, 35G25, 35S05

1 Introduction We are concerned with local existence and gain of regularity of solutions to the initial value problem for semilinear Schr¨odinger equations of the form ∂t u − i∆u = f (u, ∂u) in R × Rn , in Rn , u(0, x) = u0 (x)

(1) (2)

where u is a complex-valued and unknown function of (t, x) ∈ R × Rn , x = √ (x1 , . . . , xn ), i = −1, ∂t = ∂/∂t, ∂j = ∂/∂xj , ∂ = (∂1 , . . . , ∂n ), ∆ = ∂12 + · · · + ∂n2 , n is the spatial dimension, and the nonlinear term f (u, v) is a smooth function on R2 × R2n satisfying f (u, v) = O(|u|2 + |v|2 ) near (u, v) = 0.

(3)

The existence of time local solutions to (1)-(2) was studied in [1], [2], [3], [18] and [27]. The equation (1) does not necessarily allow the classical energy estimates of its solutions because the nonlinear term contains ∂u. So-called loss of derivatives occurs. To overcome this difficulty, smoothing effect of dispersivetype equations (see [5], [6], [7], [9], [12], [20], [21], [26], [30] and [35] for instance) effectively applied to (1). More precisely, the sharp smoothing estimate of eit∆ (see [27]) or the theory of Schr¨odinger-type equations (see [10], [11] and H. Chihara Department of Mathematical Sciences, Shinshu University, Matsumoto 390-8621, Japan (e-mail: [email protected])

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H. Chihara

[31, Lecture VII] for instance) makes (1)-(2) solvable locally in time. In fact we proved the local existence of smooth solutions to (1)-(2) by diagonalizing a 2×2 system for t [u, u] ¯ modulo bounded operators and applying Doi’s pseudodifferential operator discovered in [10] to it. See [2] and [3]. The drawback of these two works is that the initial data were required to be extremely smooth because we made strong use of pseudodifferential operators with smooth coefficients. We are interested in the gain of regularity associated with the spatial decay of the initial data as well. Such phenomena are generally observed in solutions to various dispersive-type equations. See [8], [13], [14], [15], [16], [17], [22], [23], [24], [34] and [36]. N. Hayashi, K. Nakamitsu and M. Tsutsumi ([15] and [16]), and S. Doi ([13]) studied this problem for (1) in which f (u, ∂u) was independent of ∂u and ∂ u. ¯ In [15] and [16] gauge invariance (see (5)) was assumed, and an operator J = (J1 , . . . , Jn ) defined by Jk u = xk u + 2it∂k u = ei|x|

2 /4t

2it∂k (e−i|x|

2 /4t

u),

q was effectively used, where |x| = x12 + · · · + xn2 . The operator J satisfies good commutation relations [∂t − i∆, J ] = 0 and [∂j , Jk ] = δj k , and acts on nonlinear terms with gauge invariance as if it were a usual differentiation ∂, where δj k = 1 if j = k, 0 otherwise. In [13] S. Doi developed their idea and made strong use of microlocal analysis and paradifferential calculus when the nonlinear term was a holomorphic function of u. Recently, in [17] N. Hayashi, P. I. Naumkin and P.-N. Pipolo studied this problem for (1) in one space dimension. Their nonlinear term f (u, ∂1 u) depends not only on (u, u) ¯ but also on (∂1 u, ∂1 u), ¯ and is gauge invariant. They saw (1) as a system for w ≡ t [J1k u]k=1,...,m1 , and applied a modified Doi’s operator to it, where m1 is a positive integer. It is very interesting to mention that they constructed the modified operator only from a multiplier and the Hilbert transformation, and that to eliminate the loss of derivatives, they obtained one kind of the ordinary G˚arding inequalities for singular integral operators of order one. Unfortunately, however, their results require the size of the initial data to be small. This seems to be natural, because they did not regard (1) as a system for t [w, w] ¯ though the nonlinear term contained ∂ u. ¯ There are two purposes in this paper. One is to improve the local existence theorems in [2] and [3] from the viewpoint of the smoothness of the initial data. Another is to observe the gain of regularity without restrictions on the size of the initial data and on the spatial dimension. To state our results, we recall several p function spaces and notation. Let m and l be real numbers. We set hxi = 1 + |x|2 and hDi = (1 − ∆)1/2 . H m,l is the set of all tempered distributions on Rn satisfying Z kukm,l =

l

Rn

m

1/2

|hxi hDi u(x)| dx 2

< +∞.

Gain of regularity for semilinear Schr¨odinger equations

531

In particular, we put H m = H m,0 , k·km = k·km,0 , L2 = H 0 and k·k = k·k0 . We often deal not only with scalar-valued functions but also with vector-valued ones, and we use the same notation of norms for them. In a similar way, (·, ·) denotes the inner product of scalar-valued or vector-valued L2 -functions. Any confusion will not occur. Let X be a Fr´echet space, and let I be an interval in R. C k (I ; X) denotes the set of all X-valued C k -functions on I for k = 0, 1, 2, . . . . Let H be a Hilbert space, and let p be an arbitrary exponent belonging to [1, ∞]. Lp (I ; H) denotes the set of all H-valued Lp -functions on I . For any real number s, [s] denotes the largest integer less than or equal to s. We now present our main results. Theorem 1 (Local existence for quadratic equations) Assume (3). Let m be an integer not less than [n/2] + 4. Then for any u0 ∈H m ∩H m−2,2 there exist a positive time T depending on ku0 k[n/2]+4 + ku0 k[n/2]+2,2 and a unique solution u to (1)-(2) belonging to C([−T , T ]; H m ∩H m−2,2 ). Theorem 2 (Local existence for cubic equations) Assume that f (u, v) is cubic, that is, f (u, v) = O(|u|3 + |v|3 ) near (u, v) = 0.

(4)

Let m be an integer not less than [n/2] + 4. Then for any u0 ∈H m there exist a positive time T depending on ku0 k[n/2]+4 and a unique solution u to (1)-(2) belonging to C([−T , T ]; H m ). Theorem 3 (Gain of regularity) Assume that f (u, v) is cubic and gauge invariant, that is, for any (u, v) ∈ C × Cn and for any θ ∈ R f (eiθ u, eiθ v) = eiθ f (u, v).

(5)

Let m be an integer not less than [n/2] + 4, and let m1 be a positive integer. Then for any u0 ∈H m,m1 there exist a positive time T depending on ku0 k[n/2]+4 and a unique solution u to (1)-(2) belonging to C([0, T ]; H m ). Moreover u satisfies hxi−|α| ∂ α u ∈ C([−T , T ] \ {0}; H m )

(6)

for |α| 6 m1 , where α = (α1 , . . . , αn ) ∈ {0, 1, 2, . . . }n , |α| = α1 + · · · + αn , and ∂ α = ∂1α1 · · · ∂nαn . Note that if f (u, v) is smooth, quadratic and gauge invariant, then f (u, v) is cubic. We would like to emphasize that the existence time T in Theorem 3 is independent of m1 . Therefore we can say that the solution to (1)-(2) gains regularity according to the decay of the initial data. Our idea of proof is basically the developed version of that of [2] and [3]. We see (1) as a system for t [J α u, J α u]|α|6m1 . For this reason, we study the L2 well-posedness for linear systems corresponding to nonlinear ones. To eliminate

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the loss of derivatives, we make use of block diagonalization and Doi’s operator. Our basic tools are pseudodifferential operators with nonsmooth coefficients. This paper is organized as follows. In Sect. 2 we introduce pseudodifferential operators with nonsmooth coefficients and prepare lemmas needed later. In addition Sect. 3 deals with the sharp G˚arding inequality for pseudodifferential operators of order one with C 2 -coefficients. In Sect. 4 we study well-posedness of linear systems. In Sects. 5, 6 and 7 we prove Theorems 1, 2 and 3 respectively. 2 Pseudodifferential operators with nonsmooth coefficients We here introduce classes of pseudodifferential operators whose coefficients have limited smoothness. Such an operator was originated by M. Nagase in [32]. Since then, the theory about it has advanced and has applied to studying nonlinear partial differential equations. See [37] and references therein. Definition 1 (Smooth symbols) Let m be a real number, and let ρ and δ be also real numbers satisfying 0 6 δ 6 ρ 6 1 and δ < 1. A C ∞ -function p(x, ξ ) on m if Rn × Rn is said to be a symbol belonging to a class Sρ,δ (α) |p(β) (x, ξ )| 6 Cαβ hξ im−ρ|α|+δ|β| (α) for all multi-indices α and β, where p(β) (x, ξ ) = ∂ β Dξα p(x, ξ ), Dξ = −i∂ξ , m m and ∂ξ = (∂ξ1 , . . . , ∂ξn ). Set S = S1,0 for short.

Definition 2 (Nonsmooth symbols) Let m be a real number, and let s be a nonnegative number. A function p(x, ξ ) on Rn × Rn is said to be a symbol belonging to a class B s S m if X kpkBs S m ,l = sup (hξ i−m+|α| kp (α) (·, ξ )kBs ) < +∞ |α|6l ξ ∈R

n

for all nonnegative integer l, where B s denotes the Banach space of all C [s] functions φ(x) on Rn satisfying X kφkBs = sup |∂ α φ(x)| |α|6[s] x∈R

+

X

n

sup

|α|=[s] x,y∈R x6=y

n

|∂ α φ(x) − ∂ α φ(y)| < +∞. |x − y|s−[s]

For the sake of convenience, we often use D = −i∂ below. If a symbol p(x, ξ ) is given, then a pseudodifferential operator P = p(x, D) is defined by ZZ 1 P u(x) = ei(x−y)·ξ p(x, ξ )u(y)dydξ n n (2π )n R ×R

Gain of regularity for semilinear Schr¨odinger equations

1 = (2π )n/2

Z

533

eix·ξ p(x, ξ )u(ξ ˆ )dξ

Rn

for u∈S, where x·ξ = x1 ξ1 + · · · + xn ξn , uˆ is the Fourier transform of u, and S denotes the Schwartz class on Rn . Conversely, if an operator P is given, then its symbol σ (P )(x, ξ ) is determined by σ (P )(x, ξ ) = e−ix·ξ P eix·ξ . Since the basic properties of pseudodifferential operators with smooth coefficients are well-known, we will make use of such operators freely. See [28]. In addition, we will often need the L2 -boundedness theorem for pseudodifferential operators with nonsmooth coefficients. Theorem 4 (M. Nagase [32, Theorem A]) Assume that p(x, ξ ) satisfies |p (α) (x, ξ )| 6 Cα hξ i−|α| , |p(α) (x, ξ ) − p (α) (y, ξ )| 6 Cα hξ i−|α|+τ |x − y|σ for |α| 6 n + 1 with 0 6 τ < σ 6 1. Then p(x, D) is L2 -bounded, that is, there exists a constant C1 depending only on n, σ and τ such that kp(x, D)uk 6 C1 C(p)kuk for any u∈L2 , where X C(p) =

sup (hξ i|α| |p (α) (x, ξ )|)

n |α|6n+1 x,ξ ∈R

+

X

sup

|α|6n+1 x,y,ξ ∈R x6=y

 n

|α|−τ

hξ i

 |p(α) (x, ξ ) − p (α) (y, ξ )| . |x − y|σ

M. Nagase proved Theorem 4 by the approximation of nonsmooth symbols by smooth ones. This is said to be symbol smoothing. We will observe that symbol smoothing is a strong method to deal with nonsmooth symbols. More precisely, we will prepare the fundamental theorem for algebra and the sharp G˚arding inequality of pseudodifferential operators defined by nonsmooth symbols of order zero or one, by using only symbol smoothing and well-known facts on smooth symbols. We now introduce symbol smoothing. Let p(x, ξ ) be a symbol belonging to Bs S m , and let ρ(x) ∈ S be a Friedrichs’ mollifier satisfying Z supp ρ ⊂ {|x| 6 1}, ρ(x) = ρ(−x) > 0, ρ(x)dx = 1. Rn

We put ρα,β (x) = x β ∂ α ρ(x) for short. Since ρ(x) is an even function, it follows that ρα,β (x) = (−1)|α+β| ρα,β (−x). We set Z p ] (x, ξ ) = ρ(y)p(x − hξ i−1/s y, ξ )dy Rn

534

H. Chihara

= hξ i

n/s

Z Rn

= hξ in/s

Z

Rn

ρ(hξ i1/s y)p(x − y, ξ )dy ρ(hξ i1/s (x − y))p(y, ξ )dy,

p b (x, ξ ) = p(x, ξ ) − p ] (x, ξ ). Then p(x, ξ ) is decomposed as p(x, ξ ) = p ] (x, ξ ) + p b (x, ξ ), and p ] (x, ξ ) and pb (x, ξ ) are the smooth principal part and the lower order term of p(x, ξ ) respectively. More precisely, the properties of symbol smoothing are the following. Lemma 1 Let m and s be real numbers satisfying 1 < s 6 2. Assume that p(x, ξ ) belongs to B s S m . Then for any multi-indices α and β (α)

|p ] (β) (x, ξ )| 6 Cαβ kpkB[s] S m ,|α| hξ im−|α|+(|β|−[s])+ /s , |p

b (α)

(x, ξ )| 6 Cα kpkBs S m ,|α| hξ i

(α) |pb (ej ) (x, ξ )|

|pb

(α)

6 Cα kpkBs S m ,|α| hξ i

(x, ξ ) − p b

(α)

m−1−|α|

,

m−(s−1)/s−|α|

(8) ,

−

(9)

(y, ξ )|

6 Cα kpkBs S m ,|α| hξ im−1+(s−1)/s−|α| |x − y|s−1 , (α) |pb (ej ) (x, ξ )

(7)

(10)

(α) p b (ej ) (y, ξ )|

6 Cα kpkBs S m ,|α| hξ im−|α| |x − y|s−1 ,

(11)

where τ+ = τ if τ > 0, 0 otherwise, and e1 , . . . , en are the fundamental vectors of Rn . Proof. We here show only (7) and (8). Let α and β be arbitrary multi-indices. We choose multi-indices β0 and β 0 satisfying β0 + β 0 = β, |β 0 | = (|β| − [s])+ . Applying ∂ β Dξα to p ] (x, ξ ) with the change of variables in the integral, we deduce (α)

p ] (β) (x, ξ ) β0

Z

ρ(y)p(β0 ) (x − hξ i−1/s y, ξ )dy   Z β0 α n/s 1/s ρ(hξ i (x − y))p(β0 ) (y, ξ )dy = ∂ Dξ hξ i

=∂

= ∂β

0

Dξα

Rn

X α1 +α2 +α3 =α

Rn

α! D α1 hξ in/s α1 !α2 !α3 ! ξ

Gain of regularity for semilinear Schr¨odinger equations

Z × =∂

(α3 ) Dξα2 ρ(hξ i1/s (x − y))p(β (y, ξ )dy 0)

Rn

X

β0

Z ×

α1 +α2 +α3 =α l≡|γ |6|α2 | α10 +···+αl0 =α2

l Y α0 α! α1 n/s Dξ hξ i hξ i−1/s Dξ j hξ i1/s α1 !α2 !α3 ! j =1

(α3 ) ργ ,γ (hξ i1/s (x − y))p(β (y, ξ )dy 0)

Rn

X

=

535

α1 +α2 +α3 =α l≡|γ |6|α2 | α10 +···+αl0 =α2 |β 0 |/s

l Y α0 α! α1 n/s hξ i−1/s Dξ j hξ i1/s Dξ hξ i α1 !α2 !α3 ! j =1

Z

× hξ i

0

Rn

(α3 ) (∂ β ργ ,γ )(hξ i1/s (x − y))p(β (y, ξ )dy. 0)

Then we obtain (7) since 0

supp[(∂ β ργ ,γ )(hξ i1/s (x − ·))] ⊂ {y ∈ Rn | |x − y| 6 hξ i−1/s }. We now show (8). The mean value theorem implies that Z b p (x, ξ ) = ρ(y){p(x, ξ ) − p(x − hξ i−1/s y, ξ )}dy Rn Z 1 XZ = hξ i−1/s ρ0,β (y) p(β) (x − θhξ i−1/s y, ξ )dθdy |β|=1 (n−1)/s

= hξ i

Rn

0

XZ

|β|=1

Z Rn

ρ0,β (hξ i

1/s

y)

1

p(β) (x − θy, ξ )dθdy.

0

Operating Dξα on the above and changing the variable in the integral, we have pb

(α)

(x, ξ ) X

=

α1 +α2 +α3 =α |β|=1 l≡|γ |6|α2 | α10 +···+αl0 =α2

l Y α0 α! Dξα1 hξ i(n−1)/s hξ i−1/s Dξ j hξ i1/s α1 !α2 !α3 ! j =1

Z

× =

Rn

Z

(ρ0,β )γ ,γ (hξ i1/s y)

X

α1 +α2 +α3 =α |β|=1 l≡|γ |6|α2 | 0 α1 +···+αl0 =α2

0

1

(α3 ) p(β) (x − θy, ξ )dθdy

α! hξ i−n/s Dξα1 hξ i(n−1)/s α1 !α2 !α3 !

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H. Chihara

×

l Y

α0

hξ i−1/s Dξ j hξ i1/s

j =1

Z

1

× 0

Z Rn

(ρ0,β )γ ,γ (y)

(α3 ) p(β) (x − θ hξ i−1/s y, ξ )dθdy,

(12)

where (ρ0,β )γR,γ (x) = x γ ∂ γ (x β ρ(x)). Note that (ρ0,β )γ ,γ (x) is an odd function and satisfies (ρ0,β )γ ,γ (y)dy = 0 provided that |β| = 1. Then (12) becomes pb =

(α)

(x, ξ ) X

α1 +α2 +α3 =α |β|=1 l≡|γ |6|α2 | α10 +···+αl0 =α2

α! hξ i−n/s Dξα1 hξ i(n−1)/s α1 !α2 !α3 !

Z l 0 Y −1/s αj 1/s × hξ i Dξ hξ i (ρ0,β )γ ,γ (y) Rn

j =1

Z

1

× 0

(α3 ) (α3 ) {p(β) (x − θ hξ i−1/s y, ξ ) − p(β) (x, ξ )}dθdy.

(13)

(α3 ) From the H¨older continuity of p(β) (x, ξ ) in x, it follows that (α3 ) (α3 ) (x − θ hξ i−1/s y, ξ ) − p(β) (x, ξ )| |p(β)

6 Cαβ kpkBs S m ,|α| hξ im−(s−1)/s−|α| |y|s−1 .

(14)

Combining (13) and (14), we obtain (8). The other estimates (9), (10) and (11) can be obtained by simple calculus similar to the above. t u The symbol smoothing naturally leads us to the fundamental theorem for algebra. Lemma 2 Let s be a real number greater than one. Assume that pj (x, ξ ) belongs to B s S j for j = 0, 1. Set q(x, ξ ) = p0 (x, ξ )p1 (x, ξ ), r(x, ξ ) = p1 (x, ξ ). Then p0 (x, D)p1 (x, D) ≡ p1 (x, D)p0 (x, D) ≡ q(x, D),

(15)

p1 (x, D)∗ ≡ r(x, D)

(16)

Gain of regularity for semilinear Schr¨odinger equations

537

modulo L2 -bounded operators, where p1 (x,D)∗ is the formal adjoint of p1 (x,D). More precisely, there exist a positive integer l and a positive constant C such that for any u∈L2 k(p0 (x, D)p1 (x, D) − q(x, D))uk 6 Ckp0 kBs S 0 ,l kp1 kBs S 1 ,l kuk, k(p1 (x, D)p0 (x, D) − q(x, D))uk 6 Ckp0 kBs S 0 ,l kp1 kBs S 1 ,l kuk, k(p1 (x, D) − r(x, D))uk 6 Ckp1 kBs S 1 ,l kuk. Note that the formula (15) immediately brings a commutator estimate k[p0 (x, D), p1 (x, D)]uk 6 Ckp0 kBs S 0 ,l kp1 kBs S 1 ,l kuk for any u∈L2 . This may be one kind of the extension of the Coifman-Meyer or the Kato-Ponce estimates of commutator of a multiplier belonging to the Lipschitz class and an operator of order one with constant coefficients. See [4] and [25]. Since both of p0 (x, D) and p1 (x, D) are operators with variable coefficients, we require slightly more smoothness of coefficients. It is also interesting to j mention that if pj (x, ξ ) ∈ B 1 Scl , then such a commutator estimate holds. See [37, Proposition 4.1.C]. Proof of Lemma 2 It follows from Theorem 4 and Lemma 1 that pj (x, ξ ) = pj] (x, ξ ) + pjb (x, ξ ), (α)

|pj] (β) (x, ξ )| 6 Cαβ kpj kBs S j ,|α| hξ ij −|α|+(|β|−1)+ /s , and that all of p1b (x, D), p0b (x, D)hDi, p0b (e1 ) (x, D), . . . , p0b (en ) (x, D) are L2 -bounded. We here remark that hDip0b (x, D) is also L2 -bounded. Indeed, using hξ i 6 1 + |ξ1 | + · · · + |ξn | 6 (n + 1)hξ i and the Plancherel theorem, we obtain khDip0b (x, D)uk 6 kp0b (x, D)uk +

n X

k∂j p0b (x, D)uk

j =1

6 kp0b (x, D)uk +

n X

kp0b (x, D)∂j uk +

j =1

6 (n + 1)kp0b (x, D)hDiuk +

n X kp0b (ej ) (x, D)uk j =1

n X j =1

kp0b (ej ) (x, D)uk.

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H. Chihara

Then we show that p1 (x, D)p0 (x, D) = (p1] (x, D) + p1b (x, D))(p0] (x, D) + p0b (x, D)) = p1] (x, D)p0] (x, D) + p1b (x, D)(p0] (x, D) + p1 (x, D)hDi−1 hDip0b (x, D) ≡ p1] (x, D)p0] (x, D) modulo L2 -bounded operators. Hence p1 (x, D)p0 (x, D) − p1] (x, D)p0] (x, D) j

is L2 -bounded. Since pj] (x, ξ ) ∈ S1,1/s and 1/s < 1, an asymptotic formula σ (p1] (x, D)p0] (x, D))(x, ξ ) ∼

X

p1]

(α)

] (x, ξ )p0(α) (x, ξ ).

α

holds. If |α| > 1, then p1] ple calculus, we have

(α)

−(s−1)(|α|−1)/s

] (x, ξ )p0(α) (x, ξ ) belongs to S1,1/s

. By sim-

p1] (x, ξ )p0] (x, ξ ) − q(x, ξ ) = p1 (x, ξ )p0b (x, ξ ) + p1b (x, ξ )p0] (x, ξ ). Note that the right hand side of the above fulfills the requirements of Theorem 4 with σ = s − 1 and τ = (s − 1)/s. Then it follows that p1] (x, D)p0] (x, D) − q(x, D) is L2 -bounded. This shows that p1 (x, D)p0 (x, D) − q(x, D) is L2 -bounded. In a similar way, we can show p0 (x, D)p1 (x, D) ≡ q(x, D) and (16) by symbol smoothing and an asymptotic formula. We here omit the rigorous proof. u t 3 The sharp G˚arding inequality In this section we show the sharp G˚arding inequality for systems of pseudodifferential operators of order one with B 2 -coefficients.

Gain of regularity for semilinear Schr¨odinger equations

539

Lemma 3 (The sharp G˚arding inequality) Let m be a positive integer. Assume that p(x, ξ ) = [pij (x, ξ )]i,j =1,...,m is an m × m matrix of symbols belonging to the class B 2 S 1 , and assume that there exists a nonnegative constant R such that p(x, ξ ) + t p(x, ξ ) > 0 for |ξ | > R. Then there exist a positive constant C1 and a positive integer l such that for any u ∈ (S)m Re (p(x, D)u, u) > −C1

m X

kpij kB2 S 1 ,l kuk2 .

(17)

i,j =1

One can find in [37, Proposition 2.4.A] that the Fefferman-Phong inequality proves the scalar-valued case (i.e. m = 1) and cannot apply to the matrix-valued case. We will here prove the general case only with symbol smoothing and the Friedrichs symmetrization. Without loss of generality, we may assume that p(x, ξ ) is hermitian and nonnegative for all (x, ξ ) ∈ Rn × Rn . From Lemma 1 and Theorem 4, it follows that p(x, ξ ) = p] (x, ξ ) + p b (x, ξ ), (α)

|pij] (β) (x, ξ )| 6 Cαβ kpij kB2 S 1 ,|α| hξ i1−|α|+(|β|−2)+ /2 and that p b (x, D) is an L2 -bounded operator. Moreover, p ] (x, ξ ) is also hermitian and nonnegative for all (x, ξ ) ∈ Rn × Rn since we chose a nonnegative mollifier. Then the proof of Lemma 3 is reduced to that of the following. Lemma 4 Assume that a matrix p(x, ξ ) = [pij (x, ξ )]i,j =1,...,m is hermitian and nonnegative, and satisfies 1−|α|+(|β|−2)+ /2 |pij (α) (β) (x, ξ )| 6 Cαβ hξ i

for all (x, ξ ) ∈ Rn × Rn . Then there exists a positive constant C1 such that for any u ∈ (S)m Re (p(x, D)u, u) > −C1 kuk2 . Lemma 4 is not a new result. In fact, in [29, Lemma 3.4] H. Kumano-go and M. Nagase proved it with m = 1. Their method of proof can also apply to the case of m > 2. They adopted the advanced idea of M. Nagase discovered in [33]. We will here attempt to present the primitive proof of Lemma 4 in the spirit of [28, Sect. 4 in Chapter 3].

540

H. Chihara

1 Proof of Lemma 4 The symbol p(x, ξ ) apparently belongs to S1,1/2 . To make (α) full use of the exponent (|β| − 2)+ /2 of the growth order of p(β) (x, ξ ) in ξ , we treat p(x, ξ ) as if it belonged to the class S 1 . We first introduce the Friedrichs part of p(x, ξ ). Take a cut-off function q(ζ ) ∈ S satisfying Z supp q ⊂ {|ζ | 6 1}, q(ζ ) = q(−ζ ), q(ζ )2 dζ = 1.

We set ρ(ζ ) = q(ζ )2 and F (ξ, ζ ) = q((ζ − ξ )hξ i−1/2 )hξ i−n/4 . A double symbol pF (ξ, x, η) defined by Z F (ξ, ζ )p(x, ζ )F (η, ζ )dζ, pF (ξ, x, η) = Rn

is said to be the Friedrichs part of p(x, ξ ). Its simplified symbol pF,L (x, ξ ) is determined by an oscillatory integral ZZ 1 pF,L (x, ξ ) = e−iy·η pF (ξ + η, x + y, ξ )dydη. (18) n n (2π )n R ×R Since p(x, ξ ) is hermitian and nonnegative, it follows from simple calculus that for any u ∈ (S)m ZZ t Re (PF u, u) = v(y, ζ )p(y, ζ )v(y, ζ )dydζ > 0, Rn ×Rn

where

ZZ 1 e−i(x−y)·ξ F (ξ, ζ )u(x)dxdξ, v(y, ζ ) = (2π )n Rn ×Rn ZZZZ 1 PF u(x) = (2π )2n Rn ×Rn ×Rn ×Rn i(x−y)·ξ +i(y−z)·η ×e p(ξ, y, η)u(z)dzdηdydξ ZZ 1 = ei(x−y)·ξ pF,L (x, ξ )u(y)dydξ. (2π )n Rn ×Rn

In view of the Calder´on-Vaillancourt theorem (see [28, Theorem 1.6 in Chapter 7] 0 . for instance), we have only to show that pF,L (x, ξ ) − p(x, ξ ) belongs to S1/2,1/2 0

(α,α ) In the same way, we set pF (β) (ξ, x, η) = ∂ β Dξα Dηα pF (ξ, x, η). We first observe 0

0

0

(α,α ) (ξ, x, η)| 6 Cαα0 β hξ i1−|α|/2 hξ ; ηi(|β|−2)+ /2 hξ i−|α |/2 , |pF (β)

where hξ ; ηi =

p 1 + |ξ |2 + |η|2 . To prove (19), we need the following.

(19)

Gain of regularity for semilinear Schr¨odinger equations

541

Lemma 5 (H. Kumano-go [28, Lemma 4.1 in Chapter 3]) For all multi-index α X ∂ξα F (ξ, ζ ) = hξ i−n/4 ψα,γ ,γ1 (ξ )qγ ,γ1 ((ζ − ξ )hξ i−1/2 ), |γ |6|α| γ1 6 γ

ψα,γ ,γ1 (ξ ) ∈ S −|α|+|γ −γ1 |/2 ⊂ S −|α|/2 , γ

where qγ ,γ1 (ζ ) = ζ γ1 ∂ζ q(ζ ). Using Lemma 5, we have 0

(α,α ) pF (β) (ξ, x, η) Z 0 = Dξα F (ξ, ζ )p(β) (x, ζ )Dηα F (η, ζ )dζ Rn X = ψα,γ ,γ1 (ξ )ψα0 ,γ 0 ,γ10 (η)hξ i−n/4 hηi−n/4 |γ |6|α| |γ 0 |6|α 0 | γ1 6 γ γ10 6γ 0

Z

×

Rn

qγ ,γ1 ((ζ − ξ )hξ i−1/2 )qγ 0 ,γ10 ((ζ − η)hηi−1/2 )p(β) (x, ζ )dζ.

For the sake of convenience, we set Dξ,η = {ζ ∈ Rn | |ζ − ξ | 6 hξ i1/2 , |ζ − η| 6 hηi1/2 }. Note that for any ζ ∈ Dξ,η hζ i 6 2 min{hξ i, hηi}, |Dξ,η | 6 (min{hξ i, hηi})n/2 . Then, from the properties of ψα,γ ,γ1 (ξ ) and ψα0 ,γ 0 ,γ10 (η), it follows that 0

(α,α ) |pF (β) (ξ, x, η)|

6 Chξ i

−n/4−|α|/2 −n/4−|α|/2

6 Chξ i

hηi

−n/4−|α 0 |/2

hηi

−n/4−|α 0 |/2

Z Dξ,η

hζ i1+(|β|−2)+ /2 dζ

(min{hξ i, hηi})n/2+1+(|β|−2)+ /2 .

This implies (19). 0 To prove pF,L (x, ξ ) − p(x, ξ ) ∈ S1/2,1/2 , we split it into three parts. Taylor’s formula gives pF (ξ + η, x + y, ξ ) = pF (ξ, x + y, ξ ) +

X |α|=1

pF (α,0) (ξ, x + y, ξ )(iη)α

542

H. Chihara

+

XZ |α|=2

1

(1 − θ )pF (α,0) (ξ + θη, x + y, ξ )dθ(iη)α .

0

Substitute this into (18). Note that (iη)α e−iy·η = (−∂y )α e−iy·η . Then, by the integration by parts, we deduce pF,L (x, ξ ) = pF (ξ, x, ξ ) n X XZ 1 (e ,0) + pF (ejj ) (ξ, x, ξ ) + (1 − θ)rα,θ (x, ξ )dθ, |α|=2

j =1

1 rα,θ (x, ξ ) = (2π )n

ZZ Rn ×Rn

0

e−iy·η pF (α,0) (α) (ξ + θη, x + y, ξ )dydη.

We will check that (e1 ,0) (en ,0) 0 (ξ, x, ξ ), . . . , pF (e (ξ, x, ξ ) ∈ S1/2,1/2 , pF (ξ, x, ξ ) − p(x, ξ ), pF (e n) 1) 0 and that {rα,θ (x, ξ )}06θ 61 is bounded in S1/2,1/2 for |α| = 2. 0 . Since ρ(ζ ) = We first show that a(x, ξ ) ≡ pF (ξ, x, ξ ) − p(x, ξ ) ∈ S1/2,1/2 R 2 q(ζ ) is an even function, ρ0,γ (ζ )dζ = 0 for |γ | = 1. Then, using Taylor’s formula, we have Z −n/2 ρ((ζ − ξ )hξ i−1/2 )(p(x, ζ ) − p(x, ξ ))dζ a(x, ξ ) = hξ i n ZR = hξ i−n/2 ρ((ζ − ξ )hξ i−1/2 ) Rn o n X × p(x, ζ ) − p(x, ξ ) − i p (γ ) (x, ξ )(ζ − ξ )γ dζ

= −hξ i−n/2 Z

XZ n |γ |=2 R

|γ |=1

ρ((ζ − ξ )hξ i−1/2 )

1

(1 − θ )p (γ ) (x, θ ζ + (1 − θ)ξ )(ζ − ξ )γ dθdζ 0 XZ = −hξ i−n/2+1 ρ0,γ ((ζ − ξ )hξ i−1/2 ) ×

Z ×

|γ |=2 1

Rn

(1 − θ )p (γ ) (x, θ ζ + (1 − θ)ξ )dθdζ.

0

Note that the domain of the above integral in ζ is contained in Eξ ≡ {ζ ∈ Rn | |ζ − ξ | 6 hξ i1/2 },

(20)

Gain of regularity for semilinear Schr¨odinger equations

543

and that |Eξ | 6 Chξ in/2 . It is easy to see that for ζ ∈ Eξ and for large ξ |θ ζ + (1 − θ )ξ | 6 θ|ζ − ξ | + |ξ | 6 hξ i1/2 + |ξ | 6 2hξ i, hξ i . 4 From these inequalities, it follows that there exists a positive constant M such that M −1 hξ i 6 hθ ζ + (1 − θ)ξ i 6 Mhξ i |θ ζ + (1 − θ )ξ | > |ξ | − θ |ζ − ξ | > |ξ | − hξ i1/2 >

for ξ ∈ Rn , ζ ∈ Eξ and θ ∈ [0, 1]. Then we have (γ +α)

|∂ β Dξα p (γ ) (x, θ ζ + (1 − θ )ξ )| 6 |p(β)

(x, θζ + (1 − θ)ξ )|

6 Cαβ hξ i−1−|α|+(|β|−2)+ /2

(21)

for all x, ξ ∈ Rn , ζ ∈ Eξ and θ ∈ [0, 1], and for all multi-indices α and β provided that |γ | = 2. On the other hand, Lemma 5 implies that ∂ξα ρ0,γ ((ζ − ξ )hξ i−1/2 ) X ψα,γ2 ,γ1 (ξ )(ρ0,γ )γ2 ,γ1 (ζ )((ζ − ξ )hξ i−1/2 ), =

(22)

|γ2 0|6|α| γ1 6γ2

ψα,γ2 ,γ1 (ξ ) ∈ S −|α|+|γ2 −γ1 |/2 ⊂ S −|α|/2 , γ

where (ρ0,γ )γ2 ,γ1 (ζ ) = ζ γ1 ∂ζ 2 ρ0,γ (ζ ). Applying (21) and (22) to (20), we deduce that for any multi-indices α and β (α) |a(β) (x, ξ )| 6 Cαβ hξ i−|α|/2+(|β|−2)+ /2 .

(23)

(e ,0)

0 In the same way, we show bj (x, ξ ) ≡ pF (ejj ) (x, ξ ) ∈ S1/2,1/2 . We put γ γ1 gγ ,γ1 (ζ ) = ζ ∂ζ q(ζ )q(ζ ) for short. It follows from Lemma 5 that Z ∂ξj F (ξ, ζ )F (ξ, ζ )p(ej ) (x, ζ )dζ bj (x, ξ ) = −i Rn X = hξ i−n/2 ψej ,γ ,γ1 (ξ )

Z ×

Rn

|γ |61 γ1 6 γ

gγ ,γ1 ((ζ − ξ )hξ i−1/2 )p(ej ) (x, ζ )dζ.

Note that if |γ | = 1 and γ1 = 0, then gγ ,γ1 (ζ ) is an odd function. Then, applying the mean value theorem to the last term of the right hand side of the above, we have X bj (x, ξ ) = hξ i−n/2 ψej ,γ ,γ (ξ ) |γ |61

544

H. Chihara

Z

gγ ,γ ((ζ − ξ )hξ i−1/2 )p(ej ) (x, ζ )dζ X + hξ i−(n−1)/2 ψej ,γ ,0 (ξ )

×

Rn

|γ |=|γ 0 |=1

Z ×

Rn

gγ ,γ 0 ((ζ − ξ )hξ i−1/2 )

Z

1

× 0

(γ 0 )

p(ej ) (x, θζ + (1 − θ)ξ )dθdζ.

(24)

We here remark that ( 1 (|γ | 6 1), ψej ,γ ,γ (ξ ) ∈ S −1 , p(ej ) (x, ξ ) ∈ S1,1/2 0 (γ ) 0 −1/2 ψej ,γ ,0 (ξ ) ∈ S , p(ej ) (x, ξ ) ∈ S1,1/2 (|γ | = |γ 0 | = 1). Evaluating (24) in the same way as the proof of (23), we show that for any multi-indices α and β −|α|/2+(|β|−1)+ /2 |bj (α) . (β) (x, ξ )| 6 Cαβ hξ i (γ ,0)

We set cγ (ξ, x, η) = pF (γ ) (ξ, x, η). It follows from (19) that 0

(γ +α,α 0 )

(α,α ) (ξ, x, η)| = |pF (γ +β) (ξ, x, η)| |cγ (β) 0

6 Chξ i1−|γ +α|/2 hξ ; ηi(|γ +β|−2)+ /2 hξ i−|α |/2 0

= Chξ i−|α|/2 hξ ; ηi|β|/2 hξ i−|α |/2 for any multi-indices α, α 0 and β provided that |γ | = 2. This implies that 0 {rγ ,θ }06θ 61 is bounded in S1/2,1/2 for |γ | = 2. For full details, see [28, Lemma 2.4 in Chapter 2]. t u 4 Linear systems with nonsmooth coefficients This section is devoted to studying the well-posedness of the initial value problem for 2l × 2l systems of Schr¨odinger-type equations of the form Lw = g(t, x) in (0, T ) × Rn , w(0, x) = w0 (x) in Rn ,

(25) (26)

where w is a C2l -valued and unknown function, g(t, x) and w0 (x) are given functions, T is a positive time, l is a positive integer, and the operator L is defined as follows: L = I2l ∂t − iE2l ∆ +

n X k=1

B k (t, x)∂k + C(t, x),

Gain of regularity for semilinear Schr¨odinger equations

545

Ip is the p × p identity matrix (p = 1, 2, 3 . . . ),   I 0 E2l = Il ⊕ [−Il ] = l , 0 −Il B k (t, x) = [bijk (t, x)]i,j =1,...,2l , and C(t, x) = [cij (t, x)]i,j =1,...,2l . As a concrete example, we have a system for t [[J α u]|α|6m1 , [J α u]|α|6m1 ] in mind. In [2] and [3] we studied the case of l = 1 by diagonalization and Doi’s method. In this case the matrix of symbols iE2 |ξ |2 + i

n X

B k (t, x)ξk + C(t, x)

k=1

has two distinct eigen-values provided that |ξ | is sufficiently large. Hence the corresponding operator can be diagonalized modulo L2 -bounded operators, and then the system is essentially reduced to a couple of single equations. Therefore we can solve the system by using the theory of linear Schr¨odinger-type equations. The basic idea of this theory is the transformation of unknown functions. Under an appropriate condition, there exists a transformation such that the commutator of it and the principal part of the equation eliminates the bad terms of order one. Doi’s method is to find a transformation so that the commutator is elliptic and dominates the bad terms. This elliptic operator represents the smoothing effect of Schr¨odinger equations. The essential role of the diagonalization of 2×2 systems is to keep the action of Doi’s operator on the original terms of order one from bringing other bad terms. We will here develop this idea and obtain the sufficient condition of H m -well-posedness of (25)-(26). More precisely, we will solve (25)-(26) by l×l-block diagonalization and Doi’s operator. The former is the iteration of the diagonalization of 2×2 subsystems and makes Doi’s operator to act well on (25). Lemma 6 Let m be a nonnegative integer. Assume that for i, j = 1, . . . , 2l and for k = 1, . . . , n bijk (t, x) ∈ C([0, T ]; B max{m,2} ) ∩ C 1 ([0, T ]; B 0 ), cij (t, x) ∈ C([0, T ]; B m ), and assume that there exists a nonnegative function φ(t, s) on [0, T ] × R such that φ(t, s) ∈ C([0, T ]; B 2 (R)), Z

+∞

sup

t∈[0,T ]

−∞

Z φ(t, s)ds + sup t∈[0,T ] τ ∈R

0

τ

∂t φ(t, s)ds < +∞,

546

H. Chihara l n X X

k (|bijk (t, x)| + |b(l+i)(l+j ) (t, x)|) 6 φ(t, xp )

(27)

k=1 i,j =1

for (t, x) ∈ [0, T ] × Rn and for p = 1, . . . , n. Then (25)-(26) is H m -well-posed, that is, for any w0 ∈(H m )2l and for any g∈L1 (0, T ; (H m )2l ) there exists a unique solution w to (25)-(26) belonging to C([0, T ]; (H m )2l ). For the sake of convenience, we denote the l×l block diagonal part of B k (t, x) by B k,diag (t, x), that is, B k,diag (t, x) = [bijk (t, x)]i,j =1,...,l ⊕ [bijk (t, x)]i,j =l+1,...,2l . To solve (25)-(26), we introduce pseudodifferential operators as follows: ˜ ˜ Λ0 (t) = I2l + i Λ(t), Λ(t) = I2l − i Λ(t), ˜ = Λ(t)

n

1X E2l (B k (t, x) − B k,diag (t, x))∂k (1 − ∆)−1 2 k=1 n

1X k =− (B (t, x) − B k,diag (t, x))E2l ∂k (1 − ∆)−1 , 2 k=1 K(t) = [Il k1 (t, x, D)] ⊕ [Il k10 (t, x, D)], K 0 (t) = [Il k10 (t, x, D)] ⊕ [Il k1 (t, x, D)], k1 (t, x, ξ ) = e−p(t,x,ξ ) , k10 (t, x, ξ ) = ep(t,x,ξ ) , n Z xj X φ(t, s)dsξj hξ i−1 . p(t, x, ξ ) = j =1

0

The block diagonalization will be accomplished by Λ(t) and Λ0 (t). On the other hand, K(t) should be said to be Doi’s operator and will eliminate the loss of derivatives. We will make use of them in a transformation w 7→ K(t)Λ(t)w. This is automorphic on (L2 )2l . Lemma 7 Under the same hypothesis as that of Lemma 6, there exists a positive constant Mt depending only on l n X X

k k (kbi(l+j ) (t, ·)kB0 + kb(l+i)j (t, ·)kB0 )

k=1 i,j =1

+ kφ(t, ·)kB1 +

Z

+∞ −∞

φ(t, s)ds,

such that for any t ∈ [0, T ] and for any w ∈ (L2 )2l Mt−1 kwk 6 kK(t)Λ(t)wk + kwk−1 6 Mt kwk.

(28)

Gain of regularity for semilinear Schr¨odinger equations

547

Proof. Let L be the set of all scalar and L2 -bounded operators, and let k·kL be the norm of L or L2l . It follows from Lemma 2 that there exists an operator R1 (t) belonging to C([0, T ]; L2l ) such that K 0 (t)K(t) = K 0 (t)K(t)hDihDi−1 = (I2l hDi + R1 (t))hDi−1 = I2l + R1 (t)hDi−1 ,   Z +∞ kR1 (t)kL 6 C kφ(t, ·)kB1 + φ(t, s)ds −∞

0

0

Multiplying K (t)K(t) by Λ (t) and Λ(t) from the left and the right respectively, we have −1 ˜ ˜ Λ0 (t)K 0 (t)K(t)Λ(t) = (I2l + i Λ(t))(I 2l + R1 (t)hDi )(I2l − i Λ(t)) = I2l + R2 (t),

˜ 2 + Λ0 (t)R1 (t)hDi−1 Λ(t) R2 (t) = Λ(t) −1 ˜ ˜ 2 + R1 (t)hDi−1 + i Λ(t)R = Λ(t) 1 (t)hDi −1 ˜ ˜ + Λ(t)R ˜ − iR1 (t)hDi−1 Λ(t) 1 (t)hDi Λ(t).

This implies the first inequality of (28). The second one follows from elementary calculus. t u To prove Lemma 6, we need energy inequalities. Lemma 8 Assume the same hypothesis as that of Lemma 6. Let L∗ be the formal adjoint of L. Then there exists a positive constant CT such that for any w ∈ C([0, T ]; (H 2 )2l )∩C 1 ([0, T ]; (L2 )2l ) and for any t ∈ [0, T ]



Z

t

kw(0)k +



kLw(τ )kdτ ,   Z T ∗ kw(t)k 6 CT kw(T )k + kL w(τ )kdτ . kw(t)k 6 CT

(29)

0

(30)

t

Proof. Note that L∗ is well-defined since the coefficients bijk of ∂k in L are continuously differentiable with respect to x. We here show only (29) since the proof of (30) is the same as that of (29). In view of (28), we will obtain the differential inequality for N (w(t)) ≡ kK(t)Λ(t)w(t)k + kw(t)k−1 .

548

H. Chihara

We set f = Lw. It is easy to see that f belongs to C([0, T ]; (L2 )2l ). We first obtain the energy estimate of kw(t)k−1 . Operating hDi−1 on Lw = f , we have I2l ∂t hDi−1 w − iE2l ∆hDi−1 w +

n X

hDi−1 ∂j (B j (t, x)w)

j =1 n X ∂B j hDi−1 (t, x)w + hDi−1 C(t, x)w = hDi−1 f (t, x). − ∂x j j =1

Then, by the usual energy estimates and (28), we deduce d kw(t)k−1 6 Cb,c kw(t)k + kf (t)k−1 dt 6 Cb,c Mt N (w(t)) + kf (t)k−1 ,

(31)

where Cb,c is a positive constant depending only on 2l n X X

sup kbijk (t)kB1 +

k=1 i,j =1 t∈[0,T ]

2l X

sup kcij (t)kB0 .

i,j =1 t∈[0,T ]

We will derive the energy inequality for kK(t)Λ(t)w(t)k step by step. We first diagonalize L by Λ(t) and Λ0 (t). Multiplying L by Λ(t) from the left, we have Λ(t)L = I2l ∂t Λ(t) − iΛ(t)E2l ∆ +

n X

B j (t, x)∂j Λ(t) + R3 (t),

(32)

j =1

R3 (t) = i

n X ∂ Λ˜ ˜ (t) − i Λ(t) B j (t, x)∂j ∂t j =1

+i

n X

˜ + Λ(t)C(t, x). B j (t, x)∂j Λ(t)

j =1

We observe the second term of the right hand side of (32) in detail. Using identities ˜ 2 and I2l = E2l2 , we have I2l = Λ0 (t)Λ(t) − Λ(t) − iΛ(t)E2l ∆ ˜ 2) = −iΛ(t)E2l ∆(Λ0 (t)Λ(t) − Λ(t) ˜ 2 = −iΛ(t)E2l ∆Λ0 (t)Λ(t) + iΛ(t)E2l ∆Λ(t) ˜ ˜ ˜ 2 = −i(I2l − i Λ(t))E 2l ∆(I2l + i Λ(t))Λ(t) + iΛ(t)E2l ∆Λ(t) ˜ ˜ = −iE2l ∆Λ(t) − Λ(t)E 2l ∆Λ(t) + E2l ∆Λ(t)Λ(t) + R4 (t)

Gain of regularity for semilinear Schr¨odinger equations

549

˜ ˜ = −iE2l ∆Λ(t) − Λ(t)E 2l ∆Λ(t) + E2l Λ(t)∆Λ(t) + R5 (t) ˜ ˜ = −iE2l ∆Λ(t) + (Λ(t)E 2l − E2l Λ(t))(1 − ∆)Λ(t) + R6 (t) = −iE2l ∆Λ(t) + R6 (t) n X − (B j (t, x) − B j,diag (t, x))∂j Λ(t)

(33)

j =1

˜ ˜ ˜ 2 R4 (t) = −i Λ(t)E 2l ∆Λ(t)Λ(t) + iΛ(t)E2l ∆Λ(t) , ! n X ∂ 2 Λ˜ ∂ Λ˜ (t) + 2 (t)∂j Λ(t), R5 (t) = R4 (t) + E2l ∂xj ∂xj2 j =1 ˜ ˜ R6 (t) = R5 (t) − (Λ(t)E 2l − E2l Λ(t))Λ(t). Substituting (33) into (32), we obtain the block diagonalization of L as follows: Λ(t)L = Ldiag Λ(t) + R7 (t), Ldiag = I2l ∂t − iE2l ∆ +

n X

(34)

B j,diag (t, x)∂j ,

j =1

R7 (t) = R3 (t) + R6 (t). After the block diagonalization, we apply K(t) to Λ(t)L. Multiplying Ldiag by K(t) from the left, we have K(t)Ldiag = I2l ∂t K(t) − +

n X

∂K (t) − iK(t)E2l ∆ ∂t

K(t)B j,diag (t, x)∂j .

(35)

j =1

We observe the third and the last terms of the right hand side of (35) in detail. It follows from simple calculus that the third term becomes − iK(t)E2l ∆ = −i[Il k1 (t, x, D)∆] ⊕ [−Il k10 (t, x, D)∆] = −i[Il ∆k1 (t, x, D)] ⊕ [−Il ∆k10 (t, x, D)] n X 0 [Il k1(ej ) (t, x, D)∂j ] ⊕ [−Il k1(e + 2i (t, x, D)∂j ] j) j =1

+i

n X j =1

0 [Il k1(2ej ) (t, x, D)] ⊕ [−Il k1(2e (t, x, D)] j)

550

H. Chihara

= −iE2l ∆K(t) + i

n X

E2l

j =1

+ 2i

n X j =1

∂ 2K (t) ∂xj2

0 [Il k1(ej ) (t, x, D)∂j ] ⊕ [−Il k1(e (t, x, D)∂j ]. j)

Note that 2ik1(ej ) (t, x, ξ )iξj = 2φ(t, xj )ξj2 hξ i−1 k1 (t, x, ξ ), 0 (t, x, ξ )iξj = 2φ(t, xj )ξj2 hξ i−1 k10 (t, x, ξ ). −2ik1(e j)

Then it follows from (15) that −iK(t)E2l ∆ = −iE2l ∆K(t) + R8 (t) n X I2l φ(t, xj )Dj2 hDi−1 K(t), +2

(36)

j =1 n X

n X ∂ 2K E2l 2 (t) + 2 I2l φ(t, xj ) R8 (t) = i ∂xj j =1 j =1

× [Il [k1 (t, x, D), Dj2 hDi−1 ]] ⊕ [Il [k10 (t, x, D), Dj2 hDi−1 ]]. On the other hand, using (15) again, we have n X

K(t)B p,diag (t, x)∂p

p=1

=

n X

p

[k1 (t, x, D)bij (t, x)∂p ]i,j =1,...,l

p=1 p

⊕ [k10 (t, x, D)bij (t, x)∂p ]i,j =l+1,...,2l =

n X

p

[bij (t, x)∂p k1 (t, x, D)]i,j =1,...,l

p=1 p

⊕ [bij (t, x)∂p k10 (t, x, D)]i,j =l+1,...,2l + R9 (t) =

n X

B p,diag (t, x)∂p K(t) + R9 (t),

p=1 n X p R9 (t) = [[k1 (t, x, D), bij (t, x)∂p ]]i,j =1,...,l p=1

(37)

Gain of regularity for semilinear Schr¨odinger equations

551

p

⊕ [[k10 (t, x, D), bij (t, x)∂p ]]i,j =l+1,...,2l . Multiplying Λ(t)L by K(t) from the left, and combining (35), (36) and (37), we obtain K(t)Λ(t)L = (I2l ∂t − iE2l ∆ + q(t, x, D))K(t)Λ(t) + R10 (t), q(t, x, ξ ) =

(38)

n X (2I2l φ(t, xj )ξj2 hξ i−1 + iB j,diag (t, x)ξj ), j =1

R10 (t) = K(t)R7 (t) + (R8 (t) + R9 (t))Λ(t) −

∂K (t). ∂t

Note that supt∈[0,T ] kR10 (t)kL depends only on 2l n X X

sup (kbijk (t)kB2 + k∂t bijk (t)kB0 )

k=1 i,j =1 t∈[0,T ]

+

2l X

sup kcij (t)kB0 + sup kφ(t, ·)kB2

i,j =1 t∈[0,T ]

Z

+ sup

t∈[0,T ]

t∈[0,T ]

+∞

−∞

Z φ(t, s)ds + sup t∈[0,T ] τ ∈R

τ

0

∂t φ(t, s)ds .

(39)

Our hypothesis (27) implies that q(t, x, ξ ) + t q(t, x, ξ ) > 0 for |ξ | > 1. Then, by using the modified system (38) and the sharp G˚arding inequality (17), we show that there exists a positive constant Cb,c,φ depending only on (39) such that K(t)Λ(t)w(t) is estimated as follows: d kK(t)Λ(t)w(t)k 6 Cb,c,φ N (w(t)) + kK(t)Λ(t)f (t)k. dt

(40)

Combining (31) and (40), we obtain d N (w(t)) 6 Cb,c,φ N (w(t)) + N(f (t)). dt This implies (29). We will conclude this section by proving Lemma 6.

t u

552

H. Chihara

Proof of Lemma 6 We adopt the induction on m = 0, 1, 2, . . . . The duality principle, together with the energy inequalities (29) and (30), shows the L2 case. See [19, Sect. 23.1] for details. Suppose that Lemma 6 holds for m = 0, 1, . . . , k −1, and suppose that w0 and f belong to (H k )2l and L1 (0, T ; (H k )2l ) respectively. We show the H k -well-posedness. Then, the hypothesis of the induction implies that there exists a unique solution w to (25)-(26) belonging to C([0, T ]; (H k−1 )2l ). We have only to show that ∂ α w belongs to C([0, T ]; (L2 )2l ) for |α| = k. Applying ∂ α to Lw = f , we have X

L∂ α w +

β<α |α−β|=1

X

=−

β<α |α−β|>2

−

n

X ∂ α−β B j α! (t, x)∂ β+ej w (α − β)!β! j =1 ∂x α−β n

X ∂ α−β B j α! (t, x)∂ β+ej w (α − β)!β! j =1 ∂x α−β

X

β<α |α−β|>1

∂ α−β C α! (t, x)∂ β w + ∂ α f. (α − β)!β! ∂x α−β

Note that the coefficients of ∂ β+ej w in the left hand side of the above equation belong to C([0, T ]; B 0 ), and that the right hand side belongs to L1 (0, T ; (L2 )2l ). Then the system for [∂ α w]|α|=k is L2 -well-posed. This implies that ∂ α w belongs t u to C([0, T ]; (L2 )2l ) for all |α| = k. 5 Proof of Theorem 1 In this section we prove Theorem 1 by parabolic regularization and uniform estimates. We have only to consider (1)-(2) in the forward direction of the time variable. Let ε be a positive parameter less than or equal to one. We first solve an auxiliary problem of the form ∂t uε − (i + ε)∆uε = f (uε , ∂uε ) in (0, +∞) × Rn , in Rn . uε (0, x) = u0 (x)

(41) (42)

Note that the initial data is independent of ε. Let {e(i+ε)t∆ }t >0 be a semigroup generated by ∂t − (i + ε)∆. It is easy to see that (41)-(42) is equivalent to an integral equation of the form Z t uε (t) = e(i+ε)t∆ u0 + e(i+ε)(t−τ )∆ f (uε (τ ), ∂uε (τ ))dτ. (43) 0 (i+ε)t∆

gains smoothness of order one, we can easily solve (43) by the Since e contraction principle.

Gain of regularity for semilinear Schr¨odinger equations

553

Lemma 9 Let m be an integer not less than [n/2] + 2. Then for any u0 ∈H m there exist a positive time Tε depending only on ε and on ku0 k[n/2]+2 , and a unique solution u to (41)-(42) belonging to C([0, Tε ]; H m ). Moreover, the map u0 ∈H m 7→ uε ∈ C([0, Tε ]; H m ) is continuous. We will prove Theorem 1 by three steps. We will first construct a local solution belonging to L∞ (0, T ; H m ∩H m−2,2 ). Secondly we will show the uniqueness of solution. Lastly we will recover the continuity of the unique solution in the time variable. In all of these steps we make full use of the energy estimates developed in Sect. 4. For this reason, we introduce the notation of differentiation of f (u, v). We put u = ξ0 + iζ0 , vj = ξj + iζj , ξ0 , . . . , ξn , ζ0 , . . . , ζn ∈ R. Differentiations ∂/∂u, ∂/∂ u, ¯ ∂/∂vj and ∂/∂ v¯j are defined by     ∂ 1 ∂ 1 ∂ ∂ ∂ ∂ , , = = −i +i ∂u 2 ∂ξ0 ∂ζ0 ∂ u¯ 2 ∂ξ0 ∂ζ0     ∂ 1 ∂ ∂ 1 ∂ ∂ ∂ = −i = +i , . ∂vj 2 ∂ξj ∂ζj ∂ v¯j 2 ∂ξj ∂ζj Construction of a local solution Let m be an integer not less than [n/2] + 4, and let uε be a solution to (41)-(42) belonging to C([0, Tε ]; H m ). Suppose that the initial data u0 = uε (0, ·) belongs to H m ∩H m−2,2 . Is is easy to see that uε belongs to C([−T , T ]; H m ∩H m−2,2 ). We shall show that there exists a positive time T depending only on ku0 k[n/2]+4 such that Tε > T for any ε, and {uε }ε∈(0,1] is a bounded sequence in L∞ (0, T ; H m ∩H m−2,2 ). Let α and β be multi-indices satisfying |α + β| 6 m and |β| 6 2. Operating x β ∂ α on (41), we have ε , ∂t (x β ∂ α uε ) − (i + ε)∆(x β ∂ α uε ) − hεαβ = fαβ

hεαβ =

(44)

n X ∂f ε (u , ∂uε )∂j (x β ∂ α uε ) ∂v j j =1

n X ∂f ε + (u , ∂uε )∂j (x β ∂ α uε ), ∂ v ¯ j j =1

ε = 2(i + ε) fαβ β α

n X ∂x β j =1

∂xj

ε

ε

∂ α+ej uε + (i + ε)

+ x ∂ f (u , ∂u ) −

n X ∂ 2xβ j =1

hεαβ .

∂xj2

∂ α uε

554

H. Chihara

We denote nondecreasing and nonnegative functions on [0, +∞) by the same notation A(·). If |α+β| 6 m − 1 and |β| 6 2, then we get usual energy estimates d dt

X

kx β ∂ α uε (t)k

|α+β|6m−1 |β|62



 6 Cm A  

 X

|α+β|6[n/2]+4 |β|62

 X kx β ∂ α uε (t)k kx β ∂ α uε (t)k. 

(45)

|α+β|6m |β|62

To evaluate the derivatives of the highest order, we make use of the energy estimates in the previous section. Let l be a positive integer defined by l=

(m + n − 1)! (m + n − 2)! n(n − 1) (m + n − 3)! +n + . m!(n − 1)! (m − 1)!(n − 1)! 2 (m − 2)!(n − 1)!

This is the number of couples of multi-indices (α, β) satisfying |α + β| = m and |β| 6 2. We set ε = t [Umε , Umε ], Umε = [x β ∂ α uε ]|α+β|=m,|β|62 , wm ε = t [Fmε , Fmε ], gm

ε Fmε = [fαβ ]|α+β|=m,|β|62 .

Then (44) becomes ε I2l ∂t wm

−

ε iE2l ∆wm

−

ε εI2l ∆wm

ε ε B ε,j (t, x)∂j wm = gm ,

(46)

j =1

" B ε,j (t, x) =

+

n X

# ε,j ε,j Il b1 (t, x) Il b2 (t, x) ε,j

ε,j

Il b2 (t, x) Il b1 (t, x)

,

∂f ε (u (t, x), ∂(uε (t, x)), ∂vj ∂f ε ε,j b2 (t, x) = − (u (t, x), ∂(uε (t, x)). ∂ v¯j ε,j

b1 (t, x) = −

Since m > [n/2] + 4 and uε ∈ C([0, Tε ]; H m ∩H m−2,2 ) ∩ C 1 ([0, Tε ]; H m−2 ∩H m−4,2 ), the Sobolev embedding H [n/2]+1 ⊂ B 0 shows that B ε,j (t, x) ∈ C([0, Tε ]; (B 2 )l×l ) ∩ C 1 ([0, Tε ]; (B 0 )l×l ), ε gm ∈ C([0, Tε ]; (L2 )l ).

(47)

Gain of regularity for semilinear Schr¨odinger equations

555

Following Sect. 4, we introduce pseudodifferential operators Λε (t) and K(t) with φ(t, s) = Mhsi−2 , where M is a positive constant and is independent of ε. We will determine the size of M later. We obtain the differential inequality for X ε Nmε (t) = kx β ∂ α uε (t)k + kK(t)Λε (t)wm (t)k. |α+β|6m−1 |β|62 ε (0) for short. We here Since Nmε (0) is independent of ε, we put R = N[n/2]+4 ? introduce a positive time Tε defined by ε Tε? = sup{T > 0 | N[n/2]+4 (t) 6 2R for t ∈ [0, T ]}.

From Lemma 7 and the Sobolev embedding, it follows that there exist a positive constant MR depending only on R and a positive constant Cm depending only on m such that for t ∈ [0, Tε? ] X Cm−1 MR−1 Nmε (t) 6 kx β ∂ α uε (t)k 6 Cm MR Nmε (t). |α+β|6m |β|62

Since f (u, v) is quadratic, ∂f/∂vj (uε , ∂uε ) is dominated as follows: ∂f ε ε (u (t, x), ∂u (t, x)) ∂v j 6 A(kuε (t)k[n/2]+3 )(|uε (t, x)| + |∂uε (t, x)|) 6 A(kuε (t)k[n/2]+3 )(|uε (t, x)| + |∂uε (t, x)|)hxi2 hxi−2 6 A(kuε (t)k[n/2]+3 )kuε (t)k[n/2]+2,2 hxi−2 6 CR hxi−2 , where CR is a positive constant depending only on R. Then we set M = 2nlCR . We here remark that K(t)Λε (t)I2l ∆ ≡ I2l ∆K(t)Λε (t) n X φ(t, xj )Dj2 hDi−1 K(t)Λε (t) + 2iE2l j =1

modulo L2 -bounded operators. Then, applying K(t)Λε (t) to (46), we have ε (I2l ∂t − iE2l ∆ − εI2l ∆ + q ε (t, x, D)) K(t)Λε (t)wm ε ε + R ε (t)wm = K(t)Λε (t)gm ,

556

H. Chihara

q ε (t, x, ξ ) = 2(I2l + iεE2l )

n X

Mhxj i−2 ξj2 hξ i−1

j =1

+i

n X

B ε,j,diag (t, x)ξj .

j =1

Note that q ε (t, x, ξ ) + t q ε (t, x, ξ ) n X Mhxj i−2 ξj2 hξ i−1 = 4I2l j =1

+i

n X

(B ε,j,diag (t, x) − t B ε,j,diag (t, x))ξj > 0

j =1

for |ξ | > 1, and that supt∈[0,Tε? ] kR ε (t)kL 6 CR . In the same way as (40), we obtain d ε kK(t)Λε (t)wm (t)k 6 Cm CR Nmε (t). dt

(48)

Combining (45) and (48), we have d ε N (t) 6 Cm CR Nmε (t), dt m

(49)

for t ∈ [0, Tε? ]. If m = [n/2] + 4, then (49) implies that ε (Tε? ) 6 R exp(C[n/2]+4 CR Tε? ). 2R = N[n/2]+4

This obtains Tε? > T ≡

log 2 . C[n/2]+4 CR

Clearly T depends only on ku0 k[n/2]+4 +ku0 k[n/2]+2,2 , and {uε }ε∈(0,1] is a bounded sequence in L∞ (0, T ; H m ∩H m−2,2 ). Then the standard compactness argument implies that there exist a subsequence {uε }ε∈(0,1] and a function u such that for any δ > 0 w∗

uε −→ u in L∞ (0, T ; H m ∩H m−2,2 ), m−δ ) uε −→ u in C([0, T ]; Hloc

as ε ↓ 0. It is easy to check that u is a solution to (1)-(2).

t u

Gain of regularity for semilinear Schr¨odinger equations

557

Uniqueness of solution Let u and u0 be solutions to (1)-(2) belonging to L∞ (0, T ; H [n/2]+4 ∩H [n/2]+2,2 ). Setting w = t [u − u0 , u¯ − u¯ 0 ], we have I2 ∂t w − iE2 ∆w +

n X

B j (t, x)∂j w + C(t, x)w = 0,

(50)

j =1

"

# j j b1 (t, x) b2 (t, x)



 c1 (t, x) c2 (t, x) B (t, x) = j , C(t, x) = , j c2 (t, x) c1 (t, x) b2 (t, x) b1 (t, x) j

j b1 (t, x)

Z =−

1

∂f (θ u + (1 − θ)u0 , θ∂u + (1 − θ)∂u0 )dθ, ∂vj

1

∂f (θ u + (1 − θ)u0 , θ∂u + (1 − θ)∂u0 )dθ, ∂ v¯j

1

∂f (θ u + (1 − θ)u0 , θ∂u + (1 − θ)∂u0 )dθ, ∂u ∂f (θ u + (1 − θ)u0 , θ∂u + (1 − θ)∂u0 )dθ. ∂ u¯

0

Z

j

b2 (t, x) = −

0

Z c1 (t, x) = −

0

Z c2 (t, x) = −

0

1

Since [n/2] + 1 > (n + 1)/2, and u and u0 belong to C([0, T ]; H n/2+3+1/4 ∩H n/2+1+1/4,2 ) ∩ C 1 ([0, T ]; H n/2+1+1/4 ), it follows from the Sobolev embedding that there exists a positive constant M such that B 1 (t, x), . . . , B n (t, x), C(t, x) ∈ C([0, T ]; (B 2 )2 ) ∩ C 1 ([0, T ]; (B 0 )2 ). |B 1 (t, x)|, . . . , |B n (t, x)| 6 Mhxi−2 . Then the initial value problem for the system (50) is L2 -well-posed. Hence w = 0 u t in C([0, T ]; (L2 )2 ) since w(0, x) = 0. Recovery of continuity Let m be an integer not less than [n/2] + 4, and let u be a unique solution to (1)-(2) belonging to L∞ (0, T ; H m ∩H m−2,2 ). We here set (m + n − 2)! (m + n − 3)! +n (m − 1)!(n − 1)! (m − 2)!(n − 1)! n(n − 1) (m + n − 4)! , + 2 (m − 3)!(n − 1)!

l=

w = t [U, U¯ ], U = [x β ∂ α u]|α+β|=m−1,|β|62 , g = t [F, F¯ ], F = [fαβ ]|α+β|=m−1,|β|62 ,

558

H. Chihara

fαβ = 2i

n X ∂x β j =1

−

n X j =1

∂xj

∂ α+ej u + i

n X ∂ 2xβ j =1

∂xj2

∂ α u + x β ∂ α f (u, ∂u)

n X ∂f ∂f β α (u, ∂u)∂j (x ∂ u) − (u, ∂u)∂j (x β ∂ α u). ∂vj ∂ v ¯ j j =1

Similarly, it follows that for any δ > 0 w, g ∈ L∞ (0, T ; (H 1 )2l ) ∩ C([0, T ]; (H 1−δ )2l ), and that w solves I2l ∂t w − iE2l ∆w +

n X

B j (t, x)∂j w = g,

(51)

j =1

" B j (t, x) =

# j j Il b1 (t, x) Il b2 (t, x) j

j

Il b2 (t, x) Il b1 (t, x)

,

∂f (u(t, x), ∂u(t, x)), ∂vj ∂f j b2 (t, x) = − (u(t, x), ∂u(t, x)). ∂ v¯j j

b1 (t, x) = −

Since [n/2] + 1 > (n + 1)/2 and u ∈ C([0, T ]; H n/2+3+1/4 ∩H n/2+1+1/4,2 ) ∩ C 1 ([0, T ]; H n/2+1+1/4 ), it follows from the Sobolev embedding that there exists a positive constant M such that B 1 (t, x), . . . , B n (t, x) ∈ C([0, T ]; (B 2 )2 ) ∩ C 1 ([0, T ]; (B 0 )2 ). |B 1 (t, x)|, . . . , |B n (t, x)| 6 Mhxi−2 . Then, by Lemma 6, we deduce that the initial value problem for the system (51) is H 1 -well-posed and that w belongs to C([0, T ]; (H 1 )2l ). This shows that u t u belongs to C([0, T ]; H m ∩H m−2,2 ).

Gain of regularity for semilinear Schr¨odinger equations

559

6 Proof of Theorem 2 This section deals with cubic nonlinear equations in the usual Sobolev space H m . The basic idea of proof of Theorem 2 is the same as that of Theorem 1. We will here choose the function φ(t, s) different from the previous one. Since the nonlinear term is cubic, we can construct φ(t, s) from the unknown function u. Such an idea arose in [3]. Proof of Theorem 2 We will here show only the existence of a solution. Let m be an integer greater than or equal to [n/2] + 4, and let uε be a unique solution to (41)-(42) belonging to C([0, Tε ]; H m ). Applying ∂ α to (41), we have ∂t (∂ α uε ) − (i + ε)∆(∂ α uε ) − hεα = fαε , hεα =

(52)

n X ∂f ε (u , ∂uε )∂j (∂ α uε ) ∂v j j =1

+

n X ∂f ε (u , ∂uε )∂j (∂ α u¯ ε ), ∂ v ¯ j j =1

fαε = ∂ α f (uε , ∂uε ) − hεα . The usual energy estimates give d ε ku (t)km−1 6 Cm A(kuε (t)k[n/2]+2 )kuε (t)km . dt

(53)

Let l = (m + n − 1)!/m!(n − 1)!, and let ε = t [Umε , Umε ], Umε = [∂ α uε ]|α|=m , wm ε gm = t [Fmε , Fmε ],

Fmε = [fαε ]|α|=m .

Then (52) becomes α α α I2l ∂t wm − iE2l ∆wm − εI2l ∆wm +

n X

ε ε B ε,j (t, x)∂j wm = gm ,

(54)

j =1

where B ε,j (t, x) is the same as (47). Let δ be a positive constant less than 1/8. We put xˆj = (x1 , . . . , xj −1 , xj +1 , . . . , xn ) for short. In the same way, we here introduce pseudodifferential operators Λε (t) and K ε (t) with ε

φ (t, s) = M

n Z X j =1

Rn−1

|hDi(n+1)/2+δ uε (t, s, xˆj )|2 d xˆj .

560

H. Chihara

The coefficients B ε,j (t, x) (j = 1, . . . , n) are dominated by φ ε (t, s). Indeed, the Sobolev embedding H (n−1)/2+δ (Rn−1 ) ⊂ B 0 (Rn−1 ) shows that for any j, k = 1, . . . , n Z ε,j |hDi(n+1)/2+δ uε (t, x)|2 d xˆk 6 φ(t, xk ) |B (t, x)| 6 C Rn−1

provided that M is sufficiently large. We now verify the regularity of φ ε (t, s) needed. The Sobolev embedding H 1/2+δ (R) ⊂ B 0 (R) implies that kφ ε (t, ·)kB2 6 Ckuε (t)k2n/2+3+2δ . Note that n/2 + 3 + 2δ < [n/2] + 4, since 0 < δ < 1/8. Then φ ε (t, s) belongs to C([0, Tε ]; B 2 (R)) provided that uε ∈ C([0, Tε ]; H n/2+3+2δ ). On the other hand, it is easy to see that Z +∞ φ ε (t, s)ds 6 Ckuε (t)k2(n+1)/2+δ . −∞

ε

Moreover, since u is a solution to (41), by the integration by parts and the Sobolev embedding, we deduce Z τ ε ∂t φ (t, s)ds 0

Z n X 6M 2 Re

τ

Rn−1

0

j =1

Z n X 6M 2 Re j =1

Z

xj

hDi

(n+1)/2+δ

ε

∂t u hDi

u¯ d xˆj ds

(n+1)/2+δ ε

Z Rn−1

0

(n+1)/2+δ ε

(i + ε) u¯ d xˆj ds

(n+1)/2+δ ε

× ∆hDi u hDi Z +C |hDi(n+1)/2+δ f (uε , ∂uε )hDi(n+1)/2+δ u¯ ε |dx 6C

n X

Rn

sup

j =1 τ ∈R

Z Rn−1

|hDi(n+1)/2+1+δ uε (t, τ, xˆj )|2 d xˆj

+ Ckuε (t)k2(n+1)/2+1+δ + kf (uε (t), ∂uε (t))k[n/2]+2 kuε (t)k[n/2]+2 6 A(kuε (t)k[n/2]+2 )kuε (t)k2[n/2]+3 . Then the condition on the regularity of φ ε (t, s) required in Lemma 6 is fulfilled. We here remark that whenever a solution u to the equation (1) with cubic nonlinearity belongs to L∞ (0, T ; H [n/2]+4 ), it is possible to apply the energy method in Sect. 4 to the nonlinear system.

Gain of regularity for semilinear Schr¨odinger equations

561

Applying K ε (t)Λε (t) to (54), we have ε ε (I2l ∂t − (iE2l + εI2l )∆ + q ε (t, x, D))wm = gm ,

q ε (t, x, ξ ) = 2(I2l + iεE2l )

n X

φ ε (t, xj )ξj2 hξ i−1 + i

j =1

n X

B ε,j (t, x)ξj .

j =1

We set ε (t)k + kuε (t)km−1 . Nmε (t) = kK ε (t)Λε (t)wm

Since q ε (t, x, ξ ) + t q ε (t, x, ξ ) > 0 for |ξ | > 1, we obtain d ε kK ε (t)Λε (t)wm (t)k 6 Cm A(kuε (t)kn/2+3+2δ )Nmε (t). dt

(55)

Combining (53) and (55), we have d ε Nm (t) 6 Cm A(kuε (t)kn/2+3+2δ )Nmε (t). dt Then, in the same way as the proof of Theorem 1, we show that there exist a positive time T depending on ku0 k[n/2]+4 and a solution u to (1)-(2) belonging to L∞ (0, T ; H m ). The uniqueness and the continuity in time of u can be proved by the same energy method. We here omit the rigorous proof. t u 7 Proof of Theorem 3 Finally we prove Theorem 3 by two steps. The first one is devoted to getting the L2 -estimates of [∂ α J β u]|α|6m,|β|6m1 . In the second step we obtain (6) from the L2 -estimates by elementary calculus related to the definition of J . Lemma 10 Let m be an integer greater than or equal to [n/2] + 4, and let m1 be a positive integer. Assume that f (u, v) satisfies (4) and (5). Then for any u0 ∈H m,m1 there exist a positive time T depending only on ku0 k[n/2]+4 and a unique solution u to (1)-(2) satisfying J β u ∈ C([−T , T ]; H m )

(56)

for |β| 6 m1 . We will first prove the case of m > [n/2] + 5 by the induction on m1 . We will next show the case of m = [n/2] + 4 by the approximation of a solution.

562

H. Chihara

Proof of the case of m > [n/2] + 5 Let m be an integer greater than or equal to [n/2] + 5, and let m1 be a positive integer. Suppose that u0 belongs to H m,m1 , and suppose that u is a unique solution to (1)-(2). Theorem 2 shows that (56) holds for m1 = 0. Suppose that (56) holds for |β| 6 m1 − 1. We show (56) for |β| = m1 . We first show that ∂ α J β u ∈ C([−T , T ]; L2 ) for |α| 6 m − 1 and |β| = m1 . Operating ∂ α J β on (1), we have ∂t (∂ α J β u) − i∆(∂ α J β u) − hαβ = fαβ , hαβ =

n X X

∂f α! 0 (u, ∂u)∂j (∂ α J β u) 0 0 (α − α )!α ! ∂vj

j =1 |α−α 0 |61 n X |β|

+ (−1)

X

j =1 |α−α 0 |61

+

(57)

∂f α! (u, ∂u)∂j (∂ α0 J β u) (α − α 0 )!α 0 ! ∂ v¯j

∂f ∂f (u, ∂u)(∂ α J β u) + (−1)|β| (u, ∂u)(∂ α J β u), ∂u ∂ u¯ fαβ = ∂ α J β f (u, ∂u) − hαβ .

Note that fαβ is a cubic term of [∂ α1 J β1 u]|α1 |6|α|+1,|β1 |6|β|−1 , [∂ α1 J β u]|α1 |6|α|−1 . We set lp =

(p + n − 1)! (m1 + n − 1)! , p!(n − 1)! m1 !(n − 1)!

wp = t [Up , Up ], Up = [∂ α J β u]|α|=p,|β|=m1 , gp = t [Fp , Fp ],

Fp = [fαβ ]|α|=p,|β|=m1 .

We show that wp belongs to C([−T , T ]; (L2 )2lp ) for p 6 m − 1. From (57), it follows that wp solves a 2lp × 2lp system of the form I2lp ∂t wp − iE2lp ∆wp +

n X

Bpj (t, x)∂j wp + Cp (t, x)wp = gp ,

(58)

j =1 j

where Bp (t, x) is a quadratic term of [∂ α u]|α|61 , Cp (t, x) is a quadratic term of [∂ α u]|α|62 , and gp depends only on [∂ α J β u]|α|6m,|β|6m1 −1 and [wq ]q 6p−1 . If wq belongs to C([0, T ]; (L2 )2lq ) for all q 6 p − 1, then gp belongs to C([0, T ]; (L2 )2lp ). Then Lemma 6, together with the induction on p, implies that the initial value problem for (58) is L2 -well-posed for p 6 m − 1. Hence wp belongs to C([−T , T ]; (L2 )2lp ) for p 6 m − 1.

Gain of regularity for semilinear Schr¨odinger equations

563

To complete the proof of the present case, we show that ∂ α J β u belongs to C([−T , T ]; H 1 ) for |α| = m − 1 and |β| = m1 . Operating ∂ α J β on (1), we have ∂t (∂ α J β u) − i∆(∂ α J β u) − h˜ αβ = f˜αβ , h˜ αβ

(59)

  β! 0 β−β 0 ∂f = (u, ∂u) ∂j (∂ α J β u) J 0 0 (β − β )!β ! ∂vj j =1 β 0 6β   n X X β! |β 0 | β−β 0 ∂f + J (−1) (u, ∂u) ∂j (∂ α J β 0 u) 0 0 (β − β )!β ! ∂ v¯j j =1 β 0 6β   n X X α! 0 ∂f 0 ∂ α−α + (u, ∂u) (∂ α +ej J β u) 0 α ! ∂v j j =1 |α−α 0 |=1   n X X α! 0 |β| α−α 0 ∂f ∂ (u, ∂u) (∂ α +ej J β u) + (−1) 0 α! ∂ v¯j 0 j =1 n X X

|α−α |=1

+

∂f ∂f (u, ∂u)(∂ α J β u) + (−1)|β| (u, ∂u)(∂ α J β u), ∂u ∂ u¯ α β ˜ fαβ = ∂ J f (u, ∂u) − h˜ αβ .

To see (59) as a system, we here introduce the notation as follows: l˜ =

m1 (m + n − 2)! X (p + n − 1)! , (m − 1)!(n − 1)! p=0 p!(n − 1)!

w˜ = t [U˜ , U˜ ], U˜ = [∂ α J β u]|α|=m−1,|β|6m1 , g˜ = t [F˜ , F˜ ], F˜ = [fαβ ]|α|=m−1,|β|6m1 . Then (59) becomes I2l˜w˜ − iE2l˜∆w˜ +

n X

˜ x)w˜ = g, B˜ j (t, x)∂j w˜ + C(t, ˜

(60)

j =1

where

˜ x) B˜ 1 (t, x), . . . , B˜ n (t, x), C(t,

are quadratic terms of functions belonging to C([−T , T ]; H [n/2]+3−ε ) ∩ C 1 ([−T , T ]; H [n/2]+1−ε ) ˜

for any positive number ε, and g˜ belongs to C([−T , T ]; (H 1 )2l ). Then Lemma 6 shows that the initial value problem for (60) is H 1 -well-posed and that w˜ belongs ˜ to C([−T , T ]; (H 1 )2l ). This completes the proof of the case of m > [n/2] + 5. t u

564

H. Chihara

Proof of the case of m = [n/2] + 4 Suppose that u0 ∈H [n/2]+4,m1 with a positive p p integer m1 . Take a sequence {u0 }p=1,2,... in the Schwartz class satisfying u0 →u0 p [n/2]+4,m1 [n/2]+4 in H as p → ∞. Since {u0 }p=1,2,... is bounded in H , by Lemma 10 with m > [n/2] + 5, we deduce that there exist a common and positive time T depending only on ku0 k[n/2]+4 and a unique solution up to (1) given an initial p data u0 such that up satisfies J β up ∈ C ∞ ([−T , T ]; H ∞ ) m for all multi-indices β, where H ∞ = ∩∞ m=1 H . We obtain the uniform estimates of {up }p=1,2,... . Operating ∂ α J β on the equation of up , we have p

p

∂t (∂ α J β up ) − i∆(∂ α J β up ) − hαβ = fαβ , p

hαβ =

n X X j =1 β 0 6β

(61)

β! (β − β 0 )!β 0 !

 ∂f p 0 p (u , ∂u ) ∂j (∂ α J β u) × J ∂vj n XX β! 0 + (−1)|β | 0 )!β 0 ! (β − β j =1 β 0 6β   0 ∂f × J β−β (up , ∂up ) ∂j (∂ α J β 0 u), ∂ v¯j 

β−β 0

p

p

fαβ = ∂ α J β f (up , ∂up ) − hαβ . We set

m

1 ([n/2] + n + 3)! X (k + n − 1)! l= , ([n/2] + 4)!(n − 1)! k=0 k!(n − 1)!

wp = t [U p , U p ], U p = [∂ α J β up ]|α|=[n/2]+4,|β|6m1 , p

g p = t [F p , F p ], F p = [fαβ ]|α|=[n/2]+4,|β|6m1 . Using (61) with |α| = [n/2] + 4 and |β| = m1 , we show that w p solves (I2l ∂t − iE2l ∆ +

n X

B p,j (t, x)∂j )w p = g p ,

(62)

j =1

where B p,j (t, x) is a quadratic term of [∂ α J β ]|α|61,|β|6m1 . In the same way, by using (61) and (62), we can obtain the energy estimates of X kwp (t)k + k∂ α J β up (t)k. |α|6[n/2]+3, |β|6m1

Gain of regularity for semilinear Schr¨odinger equations

565

p

Since {u0 }p=1,2,... is a bounded sequence in H [n/2]+4,m1 , {J β up }p=1,2,... is also bounded in C([−T , T ]; H [n/2]+4 ) for |β| 6 m1 . Then there exist a subsequence {up } and a function u such that for any δ > 0 w∗

J β up −→ J β u in L∞ (−T , T ; H [n/2]+4 ), [n/2]+4−δ

J β up −→ J β u in C([−T , T ]; Hloc

)

as p → ∞. Theorem 2 implies that u is a unique solution to (1)-(2). In a similar way, we can recover the continuity in the time variable of J β u. This completes the proof of Lemma 10. t u Finally we obtain (6) from (56). Lemma 11 Let s be an arbitrary real number, and let m1 be a positive integer. Assume that a function u on [−T , T ] × Rn satisfies J β u ∈ C([−T , T ]; H s ) for |β| 6 m1 . Then it follows that for |β| 6 m1 hxi−|β| ∂ β u ∈ C([−T , T ] \ {0}; H s ).

(63)

Proof. Note that the explicit formula of the hermitian polynomials, that is, e−aτ

2

[µ/2] X d µ aτ 2 µ! a µ−ν (2τ )µ−2ν e = µ dτ ν!(µ − 2ν)! ν=0

for τ ∈ R, a ∈ C and µ = 0, 1, 2 . . . . By elementary calculus, we have hxi−|β| ∂ β u = hxi−|β| ∂ β (ei|x| /4t (e−i|x| /4t u)) X β! 2 2 = hxi−|β| (e−i|x| /4t ∂ β ei|x| /4t ) (β − α)!α! 2

2

α 6β

∂ β−α (e−i|x| /4t u)) X β! α! (β − α)!α! (α − 2γ )!γ ! α 6β γ 6[α/2]  |α−γ |  |β−α| i (2x)α−2γ β−α 1 × J u, 4t 2it hxi|β|

× (e X =

i|x|2 /4t

2

where [α/2] = ([α1 /2], . . . , [αn /2]). This shows (63).

t u

566

H. Chihara

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