J. Pseudo-Differ. Oper. Appl. (2013) 4:113–143 DOI 10.1007/s11868-013-0068-y

Well-posedness of the Cauchy problem for p-evolution systems of pseudo-differential operators Alessia Ascanelli · Chiara Boiti

Received: 16 January 2013 / Revised: 12 March 2013 / Accepted: 18 March 2013 / Published online: 11 April 2013 © Springer Basel 2013

Abstract We study p-evolution pseudo-differential systems of the first order with coefficients in (t, x) and real characteristics. We find sufficient conditions for the well-posedness of the Cauchy problem in H ∞ . These conditions involve the behavior as x → ∞ of the coefficients, requiring some decay estimates to be satisfied. Keywords p-evolution equations · H ∞ well-posedness · Pseudo-differential operators Mathematics Subject Classification (2000)

35S10 · 35F40

1 Introduction and main results We consider, in [0, T ] × R, systems of pseudo-differential operators of the form ⎛ ⎜ L = Dt + ⎝

μ1 (t, x, Dx )

⎞ ..

.

⎟ ⎠ + R(t, x, Dx ),

(1.1)

μm (t, x, Dx ) where Dt stands for Dt · I , μk (t, x, Dx ), for 1 ≤ k ≤ m, are pseudo-differential operators with symbol in C([0, T ]; S p ), for a given p ≥ 2, and R(t, x, Dx ) is a

A. Ascanelli · C. Boiti (B) Dipartimento di Matematica ed Informatica, Università di Ferrara, Via Machiavelli n. 35, 44121 Ferrara, Italy e-mail: [email protected] A. Ascanelli e-mail: [email protected]

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matrix of pseudo-differential operators with symbol in C([0, T ]; S 0 ). Here D = 1i ∂, and S m is the classical class of symbols a(x, ξ ) defined by |∂ξα Dxβ a(x, ξ )| ≤ Cα,β,h ξ m−α h

∀α, β ∈ N, h ≥ 1,

 for some Cα,β,h > 0, with ξ h := h 2 + ξ 2 . System (1.1) will be called a p-evolution system of the first order. It is the model system to which scalar differential equations of order m will be reduced in the forthcoming paper [4]. For this reason we shall assume in the following that, for all 1 ≤ k ≤ m, ( p)

μk (t, x, Dx ) = μk (t, Dx ) +

p−1

( j)

μk (t, x, Dx )

(1.2)

j=1 ( j)

( j)

with symbols μk ∈ C([0, T ]; S j ) such that μk (t, x, ξ ) ∈ C for j = p and ( p)

μk (t, ξ ) ∈ R

∀(t, ξ ) ∈ [0, T ] × R,

(1.3)

according to the necessary condition of the Lax–Mizohata theorem for well-posedness in a neighborhood of zero of the Cauchy problem for scalar differential equations in Sobolev spaces (cf. [15]). ( j) ( p) When all the coefficients μk (and not only μk ) are real, well-posedness results for p ≥ 2-evolution equations are known (cf., for instance, [1–3,6]). In the case of complex coefficients, some unavoidable decay conditions in x are needed, as shown by [11]; this leads us to conditions (1.5)–(1.7) below. Well posedness of first order p-evolution differential equations with complex coefficients has been studied, for instance, in [12,13] for the case p = 2, [8] for p = 3, [5] for p ≥ 4. Second order equations with complex coefficients have been considered, for example, in [7–9], for p = 2, 3. Higher order equations with complex coefficients have been studied, for instance, in [16] for p = 2 and will be studied in [4] for p ≥ 2. In this paper we focus on p ≥ 2-evolution pseudo-differential systems of the first order. The main result of this paper, Theorem 1.1, together with Remark 4.1 will be crucial in [4]. We thus consider the operator (1.1)–(1.3) and assume that for all 1 ≤ k ≤ m the ( p) functions ∂ξ μk have all the same constant sign for every fixed ξ ∈ R and moreover ( p)

|∂ξ μk (t, ξ )| ≥ C p |ξ | p−1

∀t ∈ [0, T ], |ξ |  1,

(1.4)

for some C p > 0. Assume also that for all (t, x, ξ ) ∈ [0, T ] × R2 , 1 ≤ k ≤ m and α ∈ N: ( j)

j − p−1

( j)

j−1 p−1

|Im∂ξα μk (t, x, ξ )| ≤ Cα x

−

|Im∂ξα Dx μk (t, x, ξ )| ≤ Cα x

j−α

,

j = 1, . . . , p − 1

(1.5)

j−α

,

j = 2, . . . , p − 1

(1.6)

ξ h ξ h

Well-posedness of the Cauchy problem

115 − j−[β/2]

( j)

|Im∂ξα Dxβ μk (t, x, ξ )| ≤ Cα x p−1 ξ h ,

 β = 1, . . . , j − 1, j = 3, . . . , p − 1 2 j−α

(1.7)

for some Cα > 0, where [β/2] denotes the integer part of β/2 and · := ·1 . Under the above assumptions, we prove the following Theorem 1.1 Let L be a system of the form (1.1) satisfying (1.2)–(1.7). Then there exists a constant σ > 0 such that for every U ∈ C([0, T ]; H s+ p ) ∩ C 1 ([0, T ]; H s ) the following estimate holds: ⎛

|U (t, ·) |2s−σ ≤ Cs ⎝ |U (0, ·) |2s +

t

⎞

|LU (τ, ·) |2s dτ ⎠ ,

∀t ∈ [0, T ], (1.8)

0 2 for

msome Cs2 > 0, where for a given vector V = (V1 , · · · , Vm ) we denote |V |s :=

V

. j s j=1

The energy estimate (1.8) leads to H ∞ well-posedness of the Cauchy problem 

LU (t, x) = F(t, x) U (0, x) = G(x)

(t, x) ∈ [0, T ] × R x ∈R

(1.9)

with loss of σ derivatives. In order to prove Theorem 1.1 we have to consider first the scalar case, for a pseudo-differential operator P of the form P(t, x, Dt , Dx ) = Dt + a p (t, Dx ) +

p−1

a j (t, x, Dx )

(1.10)

j=0

with a j ∈ C([0, T ]; S j ), 0 ≤ j ≤ p, a p (t, ξ ) ∈ R ∀(t, ξ ) ∈ [0, T ] × R

(1.11)

and a j (t, x, ξ ) ∈ C ∀(t, x, ξ ) ∈ [0, T ] × R2 , 0 ≤ j ≤ p − 1. For the scalar operator (1.10) we prove the following: Theorem 1.2 Let us consider an operator of the form (1.10) satisfying (1.11) and |∂ξ a p (t, ξ )| ≥ C p |ξ | p−1

∀t ∈ [0, T ], |ξ |  1,

(1.12)

for some C p > 0. Assume that j − p−1

|Im∂ξα a j (t, x, ξ )| ≤ Cα x

−

|Im∂ξα Dx a j (t, x, ξ )| ≤ Cα x

j−1 p−1

j−α

, 1≤ j ≤ p−1

(1.13)

j−α

, 2≤ j ≤ p−1

(1.14)

ξ h ξ h

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A. Ascanelli, C. Boiti − j−[β/2] p−1

|Im∂ξα Dxβ a j (t, x, ξ )| ≤ Cα x

j−α

ξ h

, 1≤

 β ≤ j − 1, 3 ≤ j ≤ p − 1 2 (1.15)

for all (t, x, ξ ) ∈ [0, T ] × R2 and for some Cα > 0. Then, the Cauchy problem  (t, x) ∈ [0, T ] × R P(t, x, Dt , Dx )u(t, x) = f (t, x) u(0, x) = g(x) x ∈R

(1.16)

is well-posed in H ∞ (with loss of derivatives). More precisely, there exists a constant σ > 0 such that for all f ∈ C([0, T ]; H s ) and g ∈ H s there is a unique solution u ∈ C([0, T ]; H s−σ ) which satisfies the following energy estimate: ⎞ ⎛ t

u(t, ·) 2s−σ ≤ Cs ⎝ g 2s + f (τ, ·) 2s dτ ⎠ ∀t ∈ [0, T ], (1.17) 0

for some Cs > 0. p

Theorem 1.2 is a generalization of Theorem 1.1 of [5] where a p (t, Dx ) = a p (t)Dx j with a p ∈ C([0, T ]; R+ ), and a j (t, x, Dx ) = a j (t, x)Dx were differential operators with uniformly bounded complex valued coefficients. In particular, the assumption a p (t) ∈ R+ of [5] is here replaced by the assumption (1.12), cf. (3.35) in the proof of Theorem 1.2. Remark 1.3 Formula (1.17) states that a loss of derivatives appears in the solution of (1.16). In the following, it will be clear that the loss comes from (2.7), more precisely from (2.9). If condition (1.13) for j = p − 1 |Im∂ξα a p−1 (t, x, ξ )| ≤

C p−1−α ξ  x h

is substituted by the slightly stronger condition |Im∂ξα a p−1 (t, x, ξ )| ≤

C p−1−α ξ  x1+η h

for some η > 0, then, by defining

    x y ξ −1−η λ p−1 (x, ξ ) = M p−1 ω y ψ dy , p−1 h ξ  0

h

(cfr. (2.6)), our method gives well-posedness of (1.16) in Sobolev spaces without any loss of derivatives. The same considerations hold for formula (1.8), which shows a loss of derivatives in the energy estimate for systems of pseudo-differential p-evolution operators. The loss can be avoided by modifying the assumptions

Well-posedness of the Cauchy problem ( p−1)

|Im∂ξα μk

117

(t, x, ξ )| ≤ Cα x−1 ξ h

p−1−α

, 1≤k≤m

into ( p−1)

|Im∂ξα μk

(t, x, ξ )| ≤ Cα x−1−η ξ h

p−1−α

, 1≤k≤m

for some η > 0. 2 Preliminary results We need first to prove Theorem 1.2. To this aim, by the energy method we write i P = ∂t + ia p (t, Dx ) +

p−1

ia j (t, x, Dx ) =: ∂t + A(t, x, Dx )

(2.1)

j=0

and compute, for a solution u(t, x) of (1.16), d

u 20 = 2Re∂t u, u = 2Rei Pu, u − 2ReAu, u dt ≤ f 20 + u 20 − 2ReAu, u,

(2.2)

where · 0 and ·, · denote, respectively, the norm and the scalar product in L 2 (R). We would like to obtain an estimate from below for ReAu, u of the form ReAu, u ≥ −c u 20 for some c > 0, but such an estimate does not hold true, in general, since 2ReAu, u = (A + A∗ )u, u and A + A∗ is an operator with symbol in S p−1 (A∗ is the formal adjoint of A). To overcome the obstacle, throughout the paper we work as follows: (1) we construct an appropriate transformation that changes ∂t + A into ∂t + A , where A is an operator of the form A := (e )−1 Ae for some pseudo-differential operator ; (2) we use sharp-Gårding Theorem and Fefferman–Phong inequality to obtain the estimate from below ReA u, u ≥ −c u 20 for some c > 0; (3) we produce the energy estimate for the transformed equation (∂t + A )v = f ; by this, we obtain the energy estimate (1.17) for the equation Pu = f . This section is devoted to the construction of the transformation in (1) and to his main features. We look for a transformation of the form e (x,Dx ) , where (x, Dx ) is a pseudo-differential operator of symbol (x, ξ ) such that:

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(x, ξ ) is real valued; e ∈ S δ , δ > 0, so that e : H ∞ → H ∞ ; e is invertible; (e )−1 has principal part e− .

Then, in Sect. 3, we consider the Cauchy problem 

P v = f v(0, x) = g

(2.3)

for P := (e )−1 Pe , f := (e )−1 f and g := (e )−1 g. There we show that (2.3) is well-posed in Sobolev spaces; since well-posedness of (2.3) is equivalent to that of (1.16) for u(t, x) = e (x,Dx ) v(t, x), from the energy estimate for v we gain the desired energy estimate (1.17) for u which proves Theorem 1.2. In the energy estimate for u a loss of σ = 2δ derivatives will appear, due to the fact that the transformations e± are of positive order δ. Finally, in Sect. 4 we prove our main Theorem 1.1 by applying Theorem 1.2. Let us now construct the operator (x, Dx ) by describing its symbol. By assumption (1.12) we can fix ρ > 1 such that |∂ξ a p (t, ξ )| ≥ C p |ξ | p−1

∀t ∈ [0, T ], |ξ | ≥ ρ.

(2.4)

We thus define (x, ξ ) := λ p−1 (x, ξ ) + λ p−2 (x, ξ ) + · · · + λ1 (x, ξ )

(2.5)

with

    x y ξ − p−k y p−1 ψ , 1 ≤ k ≤ p − 1, dyξ −k+1 λ p−k (x, ξ ) := M p−k ω h p−1 h ξ  0

h

(2.6) where the constants M p−k > 0 will be chosen later on and ψ ∈ C0∞ (R) satisfy: 0 ≤ ψ(y) ≤ 1 ∀y ∈ R  1 1 |y| ≤ 2 ψ(y) = 0 |y| ≥ 1, while ω ∈ C ∞ (R) is of the form  0 ω(ξ ) = sgn(∂ξ a p (t, ξ ))

|ξ | ≤ 1 |ξ | ≥ ρ

with 0 ≤ ω(ξ ) ≤ 1 for all ξ ∈ R, where sgn(·) is the sign function.

Well-posedness of the Cauchy problem

119

The construction (2.5), (2.6) is similar to the one in [5]. In what follows we list some properties of the just constructed function , that will be used in Sect. 3 to prove Theorem 1.2; proofs of these properties heavily use the following immediate features of : p−1

• ψ(y/ξ h

) is zero outside   p−1 . E ψ := y ∈ R : y ≤ ξ h

• the derivatives ψ (k) (y/ξ h

p−1

E ψ

), k ≥ 1 are zero outside

 1 p−1 p−1 . := y ∈ R : ξ h ≤ y ≤ ξ h 2 

This is very useful to give estimates of the derivatives of (x, ξ ). Lemma 2.1 There exist positive constants C, δ and δα,β , independent on h, such that | (x, ξ )| ≤ C + δ logξ h |∂ξα Dxβ (x, ξ )|

≤

δα,β ξ −α h ,

(2.7) ∀α ∈ N, β ∈ N\{0}.

(2.8)

Remark 2.2 We remark that the positive constant δ in (2.7) is explicitly determined; this is very important since we are going to show that the loss of derivatives is exactly σ = 2δ. The precise value of δ is obtained in formula (2.11). Proof Direct computations give |λ p−1 (x, ξ )| ≤ M p−1 log 2 + M p−1 ( p − 1) logξ h , k−1 p−1 x p−1 ξ −k+1 |λ p−k (x, ξ )| ≤ M p−k χ E ψ (x) ≤ M p−k , h k−1

(2.9) (2.10)

p−1 for M p−k = M p−k k−1 , and χ E ψ the characteristic function of E ψ . Since

| (x, ξ )| ≤ |λ p−1 (x, ξ )| +

p−1

|λ p−k (x, ξ )|,

k=2

estimates (2.9) and (2.10) give (2.7) for δ = ( p − 1)M p−1

(2.11)

p−1 and C = M p−1 log 2 + k=2 M p−k . Now, with the aim to prove (2.8), we derive some useful estimates for the functions λ p−k  p − 1. We first give estimates of the derivatives of the functions  ,1 ≤ k ≤ p−1

ψ y/ξ h

and ω(ξ/ h). For β ≥ 1 we have:

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A. Ascanelli, C. Boiti

          x rq x   (q) r1 x Cq,r ψ · · · ∂ ∂  x x p−1 p−1 p−1   ξ h ξ h ξ h   r1 +···+rq =β  ri ∈N\{0} 

     x   β = ∂ x ψ p−1   ξ  h

≤ cβ x−β

(2.12) p−1

since we are in the region x ≤ ξ h

; similarly, for α ≥ 1:

     x   α ≤ cα ξ −α ; ∂ξ ψ p−1   ξ 

(2.13)

h

finally, for α ≥ 1 and β ≥ 1, by (2.12) and (2.13):      x   α β ≤ ∂ξ ∂x ψ p−1   ξ  h





cα0 ,...,αq

α0 +...αq =α

r1 +···+rq =β ri ∈N\{0}

      x   α0 (q) Cq,r ∂ξ ψ · p−1   ξ  h

    αq rq x   α1 r1 x · ∂ξ ∂x · · · ∂ ∂ x ξ p−1 p−1   ξ  ξ  h

h

≤ cα,β x−β ξ −α h .

(2.14)

As concerns ω(ξ/ h), it is constant for |ξ | ≥ hρ, and hence      γ ∂ ω ξ  =  ξ h 

     1 (γ ) ξ   ω  χ {|ξ |
≤ Cγ ξ h

∀γ ≥ 1,

(2.15)

where χ{|ξ | 0. In order to prove (2.8), let us first consider the case α = 0, β ≥ 1; for 1 ≤ k ≤ p −1 ∂xβ λ p−k (x, ξ )

     x ξ − p−k β−1 ∂ = M p−k ω x p−1 ψ ξ −k+1 h p−1 h x ξ h        x ξ − p−k β−1 p−1 ψ ∂x x = M p−k ω p−1 h ξ h  ⎤  β−1 β − 1  p−k x − ⎦ ξ −k+1 . + ∂xβ−1−β1 x p−1 ∂xβ1 ψ h p−1 β1 ξ  β1 =1 h

Well-posedness of the Cauchy problem

121

By (2.12) there exist positive constants cβ and Ck,β such that − p−k p−1 −β+1

|∂xβ λ p−k (x, ξ )| ≤ M p−k cβ x k−1

ξ −k+1 χ E ψ (x) h

−β

χ E ψ (x) ≤ Ck,β x−β ≤ Ck,β x p−1 ξ −k+1 h ≤ Ck,β ∀β ≥ 1, 1 ≤ k ≤ p − 1.

(2.16)

For the case α, β ≥ 1 we compute: ∂ξα ∂xβ λ p−k (x, ξ ) =

  α   ξ α α−γ ∂ξ ω h γ γ =0  ·



− p−k p−1

γ ∂ξ ∂xβ−1

M p−k x

ψ

x p−1

ξ h



 ξ −k+1 h

  α   α α−γ ξ ∂ J p−k,γ , := ω γ ξ h γ =0

where

J p−k,γ = M p−k

γ1 +γ2 =γ β1 +β2 =β−1 γ1 ·β2 >0

 ·

γ ∂ξ 1 ∂xβ2 ψ

   γ β − 1 β1 − p−k ∂x x p−1 γ1 β1 

x

γ

p−1 ξ h

− p−k +M p−k ∂xβ−1 x p−1

∂ξ 2 ξ −k+1 h  ·ψ

x



p−1

ξ h

γ

· ∂ξ ξ −k+1 h

is such that k−1

|J p−k,γ | ≤ Cγ ,β M p−k x p−1

−β

−γ −k+1

ξ h

−γ

χ E ψ (x) ≤ Cγ ,β M p−k ξ h

because of (2.14). Thus, by (2.15): k−1

|∂ξα ∂xβ λ p−k (x, ξ )| ≤ Cα,β M p−k x p−1 ≤ Cα,β ξ −α h

−β

ξ −α−k+1 χ E ψ (x) h

∀α, β ≥ 1.

(2.17)

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Summing up, estimates (2.16) and (2.17) give k−1

|∂ξα ∂xβ λ p−k (x, ξ )| ≤ Cα,β M p−k x p−1 ≤ δα,β ξ −α h

−β

ξ −α−k+1 χ E ψ (x) h

∀1 ≤ k ≤ p − 1, α, β ∈ N, β = 0, (2.18)  

that is (2.8) by construction (2.5).

In the sequel we shall need also the following Lemmas; for their proofs please refer to [5]. Lemma 2.3 Let (x, ξ ) satisfy (2.7) and (2.8). Then there exists h 0 ≥ 1 such that for h ≥ h 0 the operator e (x,Dx ) , with symbol e (x,ξ ) ∈ S δ , is invertible and (e )−1 = e− (I + R)

(2.19)

where I is the identity operator and R is of the form R = symbol

+∞

n=1 r

n

with principal

r−1 (x, ξ ) = ∂ξ (x, ξ )Dx (x, ξ ) ∈ S −1 . Remark 2.4 After the choice of h in Lemma 2.3 we get (2.18) also for β = 0, with constants depending also on the fixed h, i.e.: k−1

|∂ξα ∂xβ λ p−k (x, ξ )| ≤ Cα,β,h M p−k x p−1 ≤ δα,β,h ξ −α h

−β

ξ −α−k+1 χ E ψ (x) h

∀1 ≤ k ≤ p − 1, α, β ∈ N.

(2.20)

Indeed, for β = 0, α ≥ 1 and 1 ≤ k ≤ p − 1, by (2.15) we have: |∂ξα λ p−k (x, ξ )| ≤ Cα ξ −α h χ{|ξ |
(2.21) where

I p−k,γ

⎛ x ⎞   p−k y γ − ⎠. := M p−k ∂ξ ⎝ y p−1 ψ dyξ −k+1 h p−1 ξ h 0

In the case k = 1, from (2.9) we have that |I p−1,0 | ≤ C M p−1 (1 + δ logξ h )

(2.22)

Well-posedness of the Cauchy problem

123

and, for 0 < γ ≤ α, |I p−1,γ | x ≤ M p−1 0

≤ M p−1

     y 1  γ   dy  ∂ξ ψ p−1  y  ξ  h



Cq,r

r1 +···+rq =γ ri ∈N\{0}

≤ M p−1

    y 1  (q) y r  r1 y · · · ∂ξ q dy  ∂ξ ψ p−1 p−1 p−1   y ξ  ξ  ξ 

x

h

0

Cq,r

r1 +···+rq =γ ri ∈N\{0}

h

     y 1 y   (q) sup · ψ  p−1  y1+ R  ξ ( p−1) ξ 

x



h

h

0

( p−1)−γ

· χ E ψ (y) dy · ξ h

( p−1)−γ

≤ M p−1 cγ x− χ E ψ (x)ξ h −γ

≤ M p−1 cγ ξ h ,

(2.23)

since yε ψ (q) (y) is bounded for every ε > 0. Hence, substituting (2.22) and (2.23) in (2.21) we get |∂ξα λ p−1 (t, x)| ≤ Cα M p−1 ξ −α h (1 + logξ h )χ{|ξ |
(2.24)

and this is the reason why we take β = 0 in (2.8). Estimate (2.24) is of the type (2.20) once h is fixed. On the other hand, for 2 ≤ k ≤ p − 1, we clearly have that |∂ξα λ p−k (x, ξ )| ≤ Cα M p−k ξ −α h . Thus (2.18) holds also for β = 0. Lemma 2.5 Let (x, ξ ) satisfy (2.8) and h ≥ 1 be fixed large enough to get (2.19). Then ± (x,ξ ) |∂ξα e± (x,ξ ) | ≤ Cα ξ −α h e

|Dxβ e± (x,ξ ) |

−β ± (x,ξ )

≤ Cβ x

e

∀α ∈ N

(2.25)

∀β ∈ N.

(2.26)

Lemma 2.6 Let A(t, x, Dx ) be the operator in (2.1), satisfying (2.8), h ≥ h 0 and R as in (2.19). Then the operator A (t, x, Dx ) := (e (x,Dx ) )−1 A(t, x, Dx )e (x,Dx )

(2.27)

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A. Ascanelli, C. Boiti

can be written as A (t, x, Dx ) = e− (x,Dx ) A(t, x, Dx )e (x,Dx ) p−2 p−1−m 1 − (x,Dx ) n,m + e A (t, x, Dx )e (x,Dx ) + A0 (t, x, Dx ), m! m=0

n=1

(2.28) where A0 (t, x, Dx ) has symbol A0 (t, x, ξ ) ∈ S 0 and σ (An,m (t, x, Dx )) = ∂ξm r n (x, ξ )Dxm A(t, x, ξ ) ∈ S p−m−n .

(2.29)

Lemma 2.7 Let be defined by (2.5), with λ p−k satisfying (2.20), h fixed. Then, for m ≥ 1, e− Dxm e =

p−2

f −s (λ p−1 , . . . , λ p−s−1 ) + f − p+1 (λ p−1 , . . . , λ1 )

(2.30)

s=0

for some f − p+1 ∈ S − p+1 depending on λ p−1 , . . . , λ1 and f −s ∈ S −s depending only on λ p−1 , . . ., λ p−s−1 , and not on λ p−s , . . . , λ1 , such that |∂ξα ∂xβ f −s | ≤ Cα,β,s

ξ −s−α h x

p−1−s p−1 +β

∀α, β ≥ 0,

(2.31)

for some Cα,β,s > 0. We conclude this Section by recalling the sharp-Gårding Theorem and the Fefferman–Phong inequality, the two main tools we are going to use in proving Theorem 1.2, referring respectively to [14] and [10] for proofs. Theorem 2.8 (Sharp-Gårding) Let A(x, ξ ) ∈ S m with Re A(x, ξ ) ≥ 0. There exist pseudo-differential operators Q(x, Dx ) and R(x, Dx ) with symbols, respectively, Q(x, ξ ) ∈ S m and R(x, ξ ) ∈ S m−1 , such that A(x, Dx ) = Q(x, Dx ) + R(x, Dx ) ReQ(x, Dx )u, u ≥ 0

∀u ∈ H m R(x, ξ ) ∼ ψ1 (ξ )Dx A(x, ξ ) + ψα,β (ξ )∂ξα Dxβ A(x, ξ ),

(2.32)

α+β≥2

with ψ1 , ψα,β real valued functions, ψ1 ∈ S −1 and ψα,β ∈ S (α−β)/2 . As a consequence, there exists c > 0 such that it holds the well-known sharp-Gårding inequality ReA(x, Dx )u, u ≥ −c u 2(m−1)/2 .

(2.33)

Well-posedness of the Cauchy problem

125

Theorem 2.9 (Fefferman–Phong inequality) Let A(x, ξ ) ∈ S m with A(x, ξ ) ≥ 0. There exists c > 0 such that ReA(x, Dx )u, u ≥ −c u 2(m−2)/2 .

(2.34)

3 The scalar energy estimate Let (x, Dx ) be the operator constructed in (2.5), (2.6). Fix h ≥ 1 large enough so that the operator e is invertible, and (2.19) holds. As described in Sect. 2, we set A = (e )−1 Ae with A(t, x, Dx ) =

p

ia j (t, x, Dx )

j=0

and a p = a p (t, Dx ). To prove Theorem 1.2 we need an estimate of the form ReA v, v ≥ −c v 20

∀v(t, ·) ∈ H ∞

for some c > 0. Such an estimate will be obtained by choosing the constants M p−1 , . . . , M1 in a suitable way and by several applications of sharp-Garding and Fefferman–Phong inequalities. In what follows, we state and prove some useful lemmas. Then, we give the proof of Theorem 1.2. Throughout this section, we work with the more simple operator e− Ae ; then, at the end of the proof, we recover by Lemma 2.6 the full operator A = (e )−1 Ae . − Lemma 3.1 Let  us consider the operator e Ae . Its terms of order p − k, denoted by (e− Ae )ord( p−k) , satisfy for 1 ≤ k ≤ p − 1:

     p−k − p−k  Re(e− Ae )ord( p−k) (t, x, ξ ) ≤ C(M p−1 ,...,M p−k ) x p−1 ξ h

(3.1)

for a constant C(M p−1 ,...,M p−k ) > 0 depending only on M p−1 , . . . , M p−k and not on M p−k−1 , . . . , M1 . Proof We compute first   1 ∂ m A(t, x, ξ )Dxm e (x,ξ ) σ A(t, x, Dx )e (x,Dx ) = m! ξ m≥0

=

p p−1 m=0 j=m+1

 1 m ∂ξ ia j (t, x, ξ ) Dxm e (x,ξ ) + A¯ 0 m!

A¯ 0 ∈ S 0 . Then, for some A0 ∈ S 0 (which may change from one equality to the other) we have:

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A. Ascanelli, C. Boiti

σ (e− Ae )

⎛ p−1 p 1 α − α ∂ξ e Dx ⎝ = α! α≥0

=

m=0 j=m+1

p−1

j−1−m

p

m=0 j=m+1 α=0

=

p−1

p

m=0 j=m+1

+

p−2

p

⎞  m (x,ξ ) 1 m ∂ ia j (t, x, ξ ) Dx e + A¯ 0 ⎠ m! ξ

α   1 1 α − α β m+α−β (∂ξ e ) (∂ξm Dx (ia j (t, x, ξ )))(Dx e ) + A0 α! m! β β=0

1 − m m (e Dx e )(∂ξ (ia j (t, x, ξ ))) m! j−1−m α

m=0 j=m+1 α=1 β=0

  1 1 α β m+α−β (∂ξα e− )(∂ξm Dx (ia j (t, x, ξ )))(Dx e )+ A0 . α! m! β

(3.2) Put now

A I :=

p−1 p m=0 j=m+1

A I I :=

1 − m m (e Dx e )(∂ξ (ia j (t, x, ξ ))), m! j−1−m α

p−2 p

m=0 j=m+1 α=1 β=0

(3.3)

  1 1 α (∂ξα e− )(∂ξm Dxβ (ia j (t, x, ξ )))(Dxm+α−β e ). α! m! β

(3.4) We have σ (e− Ae ) = A I + A I I + A0 .

(3.5)

We consider first A I I , where α ≥ 1. In the case m + α − β ≥ 1, from (2.20) we get: |(∂ξα e− )(∂ξm Dxβ ia j )(Dxm+α−β e )|      p−1    p−1   λ    j−m  α −λ p−k   m+α−β  p−k ≤ c ξ h · ∂ξ e e  · ∂ x   k=1    k  =1 j−m

= c ξ h ·



α1 +···+α p−1 =α

γ1 +···+γ p−1 = m+α−β

p−1  α α! |∂ξ k e−λ p−k | α1 ! · · · α p−1 ! k=1

p−1 (m + α − β)!  γk  λ p−k  |∂x e | γ1 ! · · · γ p−1 !  k =1

Well-posedness of the Cauchy problem

127



≤ c

p−1 

α1 +···+α p−1 =α k,k  =1 γ1 +...+γ p−1 =m+α−β r1 +···+rqk =αk ; ri ,αk ≥1 s1 +···+s p  =γk  ; si ,γk  ≥1

k−1

x p−1

q

k M p−k

qk

α +qk (k−1)

ξ h k

k

p



k · M p−k 

x

k  −1 p−1 pk  −γk 

p  (k  −1) ξ h k

j−m

ξ h

(3.6)

for some c, c > 0.

p−1

p−1 Each term of (3.6) has order j − m − α − k=1 qk (k − 1) − k  =1 pk  (k  − 1) and decay in x of the form x

p−1

p−1  k=1 qk (k−1)+ k  =1 pk  (k −1) −m−α+β p−1

−

p−1

p−1  k=1 qk (k−1)− k  =1 pk  (k −1) p−1

j−m−α−

≤ x

since −( p − 1)(m + α − β) ≤ − j + m + α for m + α − β ≥ 1.

p−1

p−1 Note also that j − m − α − k=1 qk (k − 1) − k  =1 pk  (k  − 1) ≤ p − k − 1 and

p−1

p−1 j − m − α − k=1 qk (k − 1) − k  =1 pk  (k  − 1) ≤ p − k  − 1, so that whenever M p−k or M p−k  appear in (3.6), then the order is at most p − k − 1 and p − k  − 1 respectively. In the case m + α − β = 0, by (2.20) we have, for all 0 ≤ β ≤ j − 1 with 1 ≤ j ≤ p − 1: |Re[(∂ξα e− )(∂ξm Dxβ ia j )e ]| ≤ |∂ξα e− | · |Im∂ξm Dxβ a j |e α! = α ! · · · α p−1 ! α1 +···+α p−1 =α 1 ⎛ ⎞ ·

p−1  k=1

≤C

⎜ ⎜ ⎝

r1 +···+rqk =αk ri ,αk ≥1



p−1 

⎟ rq m β Cq,k |∂ξr1 λ p−k | · · · |∂ξ k λ p−k |⎟ ⎠ · |Im∂ξ Dx a j |

α1 +···+α p−1 k=1 r1 +···+rqk =αk =α ri ,αk ≥1

q

k−1

q

−αk −qk (k−1)

k M p−k x p−1 k ξ h

· |Im∂ξm Dxβ a j | (3.7)

for some C > 0. Now, for

⎧ 0 ⎪ ⎪ ⎨ 1 γ (β) = β  ⎪ ⎪ ⎩ 2

β=0 β=1

(3.8)

β≥2

and min{β + 1, 3} ≤ j ≤ p − 1 we have that (3.7) becomes, because of (1.13)–(1.15):

128

A. Ascanelli, C. Boiti

|Re[(∂ξα e− )(∂ξm ia j )e ]| ≤C





p−1 



α1 +···+α p−1 k=1 r1 +···+rqk =αk =α ri ,αk ≥1

q

k−1

−αk −qk (k−1)

q

k M p−k x p−1 k ξ h

· x

(β) − j−γ p−1

j−m

ξ h

.

(3.9) Each term of (3.9) is a symbol of order j − m − α − in x of the form: x

p−1 k=1 qk (k−1)− j+γ (β) p−1

−

≤ x

p−1 k=1 qk (k−1) p−1

j−m−α−

p−1 k=1

qk (k − 1) and has decay

if min{β + 1, 3} ≤ j ≤ p − 1,

since γ (β) ≤ β = α + m.

p−1 Here again j − m − α − k=1 qk (k − 1) ≤ p − k − 1 and hence M p−k appears in (3.9) only when the order is at most p − k − 1. Summing up, formulas (3.6) and (3.9) give that the terms of order p − k of A I I , denoted by A I I |ord( p−k) , satisfy: p−k   Re A I I |ord( p−k)  ≤ Cx− p−1 ξ  p−k h

(3.10)

for some C > 0. Moreover, Re A I I |ord( p−k) depends only on M p−1 , . . . , M p−k+1 and not on M p−k , . . . , M1 . We consider then AI =

p−1 p m=0 j=m+1

1 m (∂ (ia j ))(e− Dxm e ) m! ξ

p−1 k 1 m (∂ (ia p−k+m ))(e− Dxm e ) m! ξ k=0 m=0   p−1 k 1 m (∂ (ia p−k+m ))(e− Dxm e ) . = ia p + ia p−k + m! ξ

=

(3.11)

m=1

k=1

Note that Dx = Dx λ p−1 + Dx λ p−2 + · · · + Dx λ1 with Dx λ p−k ξ p−1 ∈ S p−k because of (2.16). Moreover, from Lemma 2.7 it follows that there exist f −s ∈ S −s , for 0 ≤ s ≤ p − 2, depending only on λ p−1 , …, λ p−s−1 , and f − p+1 ∈ S − p+1 such that, for f˜0 = (∂ξm a p−k+m ) f − p+1 ∈ S 0 , (∂ξm a p−k+m )(e− Dxm e ) =

p−2 s=0

f −s (λ p−1 , . . . , λ p−s−1 )∂ξm a p−k+m + f˜0 , (3.12)

Well-posedness of the Cauchy problem

129

and, from (2.31) for 0 ≤ s ≤ p − 2, | f −s ∂ξm a p−k+m | ≤

Cs p−1−s p−1

x

p−k−s

ξ h

≤

Cs x

p−k−s

p−k−s p−1

ξ h

∀k ≥ 1 (3.13)

for some Cs > 0. Rearranging the terms of the second addend of A I in (3.11) and putting together all terms of order p − k, we can thus write, because of (3.12), (3.13): A I = ia p +

p−1   ia p−k + i Dx λ p−k ∂ξ a p + B p−k + B˜ 0 , k=1

for some B˜ 0 ∈ S 0 and B p−k ∈ S p−k coming from (3.12) and of the form B p−k =

k

i f −(k−s) (λ p−1 , . . . , λ p−k+s−1 )

s=2

s

∂ξm a p−s+m , k = 1, . . . , p − 1.

m=1

(3.14) Notice that B p−k ∈ S p−k depends only on λ p−1 , . . . , λ p−k+1 and not on λ p−k , . . . , λ1 , and moreover |B p−k | ≤

Ck x

p−k p−1

p−k

ξ h

(3.15)

for some Ck > 0. Setting A0p−k := ia p−k + i Dx λ p−k ∂ξ a p

(3.16)

we write A I = ia p +

p−1

(A0p−k + B p−k ) + B˜ 0 .

(3.17)

k=1

Note that A0p−k , B p−k ∈ S p−k and, since Re(A0p−k ) = −Ima p−k + ∂x λ p−k ∂ξ a p , from (1.13) with j = p − k and α = 0, the first inequality in (2.16) with β = 1, and (3.15) we have − p−k p−1

|Re A0p−k | + |B p−k | ≤ Ck x

p−k

ξ h

(3.18)

for some Ck > 0. Moreover, A0p−k depends only on M p−k and B p−k depends only on M p−1 , . . . , M p−k+1 (and not on M p−k , . . . , M1 ) as a consequence of (3.14). Formulas (3.10) and (3.17)–(3.18) together give (3.1) because of (3.5). The proof is completed.  

130

A. Ascanelli, C. Boiti

 Lemma 3.2 Let us consider, for 1 ≤ k ≤ p − 3 the operator (e− Ae )ord( p−k) and define   R p−k = ψ1 (ξ )Dx (e− Ae )ord( p−k) + ψα,β (ξ )∂ξα Dxβ (e− Ae )ord( p−k) , α+β≥2

(3.19) with ψ1 , ψα,β as in Theorem 2.8. Denote by R p−k |ord( p−k−s) the terms of order p−k−s of R p−k , 1 ≤ s ≤ p − k − 1. Then:      p−k−s − p−k−s  Re(R p−k )ord( p−k−s) (t, x, ξ ) ≤ C(M p−1 ,...,M p−k−s ) x p−1 ξ h

(3.20)

for every 1 ≤ s ≤ p − k − 1 and for a positive constant C(M p−1 ,...,M p−k−s ) depending only on M p−1 , . . . , M p−k−s and not on M p−k−s−1 , . . . , M1 . Proof From (3.5), to estimate R p−k we need to give estimates of R( A I |ord( p−k) ) = ψ1 (ξ )Dx A I |ord( p−k) +

α+β≥2

ψα,β (ξ )∂ξα Dxβ A I |ord( p−k)

and R( A I I |ord( p−k) ) = ψ1 (ξ )Dx A I I |ord( p−k) +

α+β≥2

ψα,β (ξ )∂ξα Dxβ A I I |ord( p−k) .

We start by considering R( A I |ord( p−k) ) = R(A0p−k ) + R(B p−k ), because of (3.17) for A0p−k and B p−k defined respectively in (3.16) and (3.14). In computing

R(A0p−k ) = ψ1 Dx A0p−k +

α+β≥2

ψα,β ∂ξα Dxβ A0p−k

(3.21)

we find ψ1 Dx A0p−k = iψ1 Dx a p−k + i Dx2 λ p−k ψ1 ∂ξ a p ; by (1.14): |Re(ψ1 Dx A0p−k )| ≤ |Im Dx a p−k | · |ψ1 | ≤  ≤ ψ

since ψ1 ∈ S −1 and ξ h

p−k−1

/x

x p−1

ξ h p−k−1 p−1



C x

p−k−1 p−1

p−k−1 C  ξ h

x

p−k−1 p−1

p−k−1

ξ h

+ C 

is bounded on supp(1 − ψ).

(3.22)

Well-posedness of the Cauchy problem

131

We now look at ψα,β ∂ξα Dxβ A0p−k

α+β≥2

=



α+β≥2

=



α+β≥2

ψα,β ∂ξα Dxβ (ia p−k + i Dx λ p−k ∂ξ a p ) ψα,β i∂ξα Dxβ a p−k +







ψα,β

 α i∂ξα1 Dxβ+1 λ p−k · ∂ξα2 +1 a p . α1

α1 +α2 =α

α+β≥2

(3.23) β

Note that the first addend in (3.23) is ψα,β i∂ξα Dx a p−k ∈ S p−k−

considered at level p − k −

α+β $2

α+β 2

, so it has to be

α+β 2

if α + β is even, at level p − k − + 21 if α + β % 1 is odd, thus at level p − k + − α+β 2 + 2 . Looking also at its decay as x → ∞, we have by (1.14), (1.15), for p − k ≥ 3 and γ (β) defined by (3.8): p−k− α+β 2

|Re(ψα,β i∂ξα Dxβ a p−k )| ≤ ξ h

 ≤ Cψ

x

C x 

p−1 ξ h

p−k−γ (β) p−1

$ % 1 p−k+ − α+β 2 +2

ξ h x

$ % α+β p−k+ − 2 + 21

+ C

(3.24)

p−1

for some C  > 0, since



a+b 1 + − γ (b) ≥ − 2 2

∀a, b ≥ 0.

(3.25)

We remark that decay estimates of the form (3.24) are needed until level p − k − ≥ 21 , i.e.

 β ≤ p − k − 1, for p − k ≥ 3. (3.26) 0≤ 2

α+β 2

For the second addend of (3.23) by (2.20) we immediately get:     α α   α +1 β+1 1 2  i∂ξ Dx λ p−k · ∂ξ ψα,β a p   α 1   α +α =α α+β≥2

≤

1



2

Cα,β

α+β≥2

x

p−k p−1 +β

$ % 1 p−k+ − α+β 2 +2

≤ Cξ h

$ % 1 since β( p − 1) ≥ − α+β + 2 2 .

p−k− α+β 2

ξ h

−

x

$ % α+β p−k+ − 2 + 21 p−1

(3.27)

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A. Ascanelli, C. Boiti

Summing up, we have obtained, for the second addend of (3.21), that   % $ $ %   α+β p−k+ − 2 + 21 p−k+ − α+β + 21   2 − Re p−1 ψα,β ∂ξα Dxβ A0p−k  ≤ Cξ h x ψ + C   α+β≥2  for some C, C  > 0, because of (3.24) and (3.27). Note that only in (3.24) the assumptions (1.14), (1.15) are used. We have thus proved, looking also at (3.22), that R(A0p−k ) fulfills the decay estimate in (3.20) and, moreover, it depends only on M p−k and not on M j for j = p − k. We now estimate the other term R(B p−k ) =

=

s k

  R i f −(k−s) ∂ξm a p−s+m

s=2 m=1

⎡

k s

⎣ψ1 Dx (i f −(k−s) ∂ξm a p−s+m )

s=2 m=1

+

α+β≥2

α−β 2

for ψ1 ∈ S −1 , ψα,β ∈ S We have from (2.31):

⎤ ψα,β ∂ξα Dxβ (i f −(k−s) ∂ξm a p−s+m )⎦

(3.28)

and B p−k defined by (3.14).

|ψ1 Dx (i f −(k−s) ∂ξm a p−s+m )| ≤ |ψ1 (∂x f −(k−s) )∂ξm a p−s+m | + |ψ1 f −(k−s) ∂ξm ∂x a p−s+m )|   1 1 p−s −1 ≤ ξ h Ck−s + ξ h ξ −k+s p−1−k+s p−1−k+s h +1 x p−1 x p−1 Ck−s p−k−1 ≤ ξ h , p−k−1 x p−1 therefore, for each 2 ≤ s ≤ k, |ψ1 Dx (i f −(k−s) ∂ξm a p−s+m )|

 ≤ cψ

x p−1

ξ h



p−k−1

ξ h x

p−k−1 p−1

+ c

(3.29)

for some c, c > 0. For the second addend of (3.28) we write α+β≥2

=

ψα,β ∂ξα Dxβ (i f −(k−s) ∂ξm a p−s+m )

α+β≥2

ψα,β

β    α  α β    i(∂ξα Dxβ f −(k−s) )(∂ξα−α +m Dxβ−β a p−s+m ).   α β  

α =0 β =0

Well-posedness of the Cauchy problem

133 

β

β−β 



By (2.31) we have that ψα,β (∂ξα Dx f −(k−s) )(∂ξα−α +m Dx and 





a p−s+m ) ∈ S p−k−

α+β 2



|ψα,β (∂ξα Dxβ f −(k−s) )(∂ξα−α +m Dxβ−β a p−s+m )| ≤

Ck−s x

p−1−k+s  p−1 +β

Ck−s

≤ x

p−k− α+β 2

ξ h

% $ α+β p−k+ − 2 + 21

$ % 1 p−k+ − α+β 2 +2

ξ h

p−1

% $ 1 for some Ck−s > 0, since p −1−k +s ≥ p −k (being s ≥ 2) and β  ≥ − α+β + 2 2 . This, together with (3.29), means that R(B p−k ) satisfies the decay estimate in (3.20), independently of the conditions on the x-decay of the coefficients. Now we are going to estimate R( A I I |ord( p−k) ), where A I I is defined in (3.4). We have:   R (∂ξα e− )(∂ξm Dxβ (ia j (t, x, ξ )))(Dxm+α−β e ) $ % = ψ1 Dx (∂ξα e− )(∂ξm Dxβ (ia j (t, x, ξ )))(Dxm+α−β e ) $ %   + ψα  ,β  ∂ξα Dxβ (∂ξα e− )(∂ξm Dxβ (ia j (t, x, ξ )))(Dxm+α−β e ) α  +β  ≥2

(3.30) α  −β 

for ψ1 ∈ S −1 and ψα  ,β  ∈ S 2 . In order to avoid further computations analogous to those already made for the estimate of A I , we make some remarks. When the x-derivatives fall on m+α−β e ), the decay in x gets better because of Lemma 2.5, while the (∂ξα e− )(Dx β

level in ξ decreases. When the x-derivatives fall on ∂ξm Dx (ia j ) the assumptions (1.14) and (1.15) on the coefficients give a decay in x of order ( j − γ (β + 1))/( p − 1) in the first addend of (3.30), and of order ( j − γ (β + β  ))/( p − 1) in the second addend of (3.30), with γ the function defined in (3.8); at the same time we have that the level     in ξ decreases of 1 in the first addend of (3.30) and of α  − α −β = α +β in the 2 2 second addend of (3.30). Therefore the assumptions (1.14), (1.15) on the coefficients give that R( A I I |ord( p−k) ) satisfies the decay estimate in (3.20), since 

1 β − γ (β + 1) ≥ − − 1 + 2 2 

 + β α 1 β  −γ (β + β ) ≥ − − + 2 2 2 because of (3.25) with b = β + 1, a = 1 and b = β + β  , a = α  respectively.

(3.31) (3.32)

134

A. Ascanelli, C. Boiti

3.1 Proof of Theorem 1.2 The proof of Theorem 1.2 consists in choosing recursively positive constants M p−1 , . . . , M1 in such a way that  Re (e− Ae )ord( p−k) + C˜ ≥ 0

(3.33)

for some C˜ > 0, and applying the sharp-Gårding Theorem 2.8 to terms of order p − 2, p − 3, and so on, up to order 3, the Fefferman–Phong inequality to terms of order p − k = 2 and the sharp-Gårding inequality (2.33) to terms of order p − k = 1, finally obtaining that e− Ae = ia p (t, Dx ) +

p

Q p−s

s=1

with ReQ p−s v, v ≥ 0

∀v(t, ·) ∈ H p−s , s = 1, . . . , p − 3

ReQ p−s v, v ≥ −c v 20

∀v(t, ·) ∈ H p−s , s = p − 2, p − 1

Q0 ∈ S0. At the end of the proof we will show that the result holds not only for e− Ae , but also for the full operator (e )−1 Ae , finding a constant c > 0 such that ReA v, v ≥ −c v 20

∀v(t, ·) ∈ H ∞ .

From this, the thesis follows by standard energy arguments. Lemma 3.1 is fundamental to make these choices possible: it states that all terms of order p −k (1 ≤ k ≤ p −1) of the operator e− Ae have the “right decay at the right level”, in the sense that they satisfy (3.1); the fact that the constants C(M p−1 ,...,M p−k ) depend only on M p−1 , . . . , M p−k and not on M p−k−1 , . . . , M1 is very important in the following in the application of the sharp-Gårding Theorem, since we shall choose M p−1 , . . . , M1 step by step, and at each step (say “step p − k”) we need something which depends only on the already chosen M p−1 , . . . , M p−k+1 and on the new M p−k that we need to choose, and not on the constants M p−k−1 , . . . , M1 which will be chosen in the next steps. Lemma 3.2 states that not only the terms of order p − k of the operator e− Ae , but also remainder terms coming from an application of Theorem 2.8 have the “right decay at the right level” (formula (3.20)), with constants C(M p−1 ,...,M p−k−s ) depending only on M p−1 , . . . , M p−k−s and not on M p−k−s−1 , . . . , M1 ; this lets the recursive choice of the constants possible.

Well-posedness of the Cauchy problem

135

So, let us start with the proof. Choice of M p−1 . Let us define, with the notations of Lemma 3.1,  A p−k := (e− Ae )ord( p−k) = A I |ord( p−k) + A I I |ord( p−k) = A0p−k + B p−k + A I I |ord( p−k) ,

k = 1, . . . , p − 1.

(3.34)

We focus on the real part of A p−k . From (2.4), (1.13)–(1.15), (2.6) we have, for |ξ | ≥ hρ, Re A0p−k = −Ima p−k + ∂x λ p−k ∂ξ a p     ξ x − p−k p−1 x = M p−k ω ψ ∂ξ a p − Ima p−k ξ −k+1 h p−1 h ξ  h

≥

p−k − p−k C p M p−k x p−1 ξ h ψ − p−k p−1

−Cx ≥ψ

p−k

ξ h

· (C p M p−k

− p−k p−1

ψ − Cx

− p−k p−1

− C)x

p−k

ξ h

p−k ξ h

(1 − ψ)

− C 

(3.35)

for some C p , C  > 0 since ξ h /x is bounded on the support of (1 − ψ). Then, from (3.35), (3.18) and (3.10), for |ξ | ≥ hρ: p−1

Re A p−k = Re(A0p−k ) + Re(B p−k ) + Re( A I I |ord( p−k) ) − p−k p−1

≥ ψ(C p M p−k − C)x

p−k

ξ h

− p−k p−1

− C  − (Ck + C  )x

p−k

ξ h , (3.36)

where the constants C, C  , C  , Ck depend only on M p−1 , . . . , M p−k+1 and not on M p−k , . . . , M1 . In particular, for k = 1, Re A p−1 ≥ ψ(C p M p−1 − C − C1 − C  )x−1 ξ h

p−1

− C 

and we can thus choose M p−1 > 0 sufficiently large, so that Re A p−1 (t, x, ξ ) ≥ −C˜

∀(t, x, ξ ) ∈ [0, T ] × R2

for some C˜ > 0. Applying the sharp-Gårding Theorem 2.8 to A p−1 + C˜ we can thus find pseudo-differential operators Q p−1 (t, x, Dx ) and R˜ p−1 (t, x, Dx ) with symbols Q p−1 (t, x, ξ ) ∈ S p−1 and R˜ p−1 (t, x, ξ ) ∈ S p−2 such that

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A. Ascanelli, C. Boiti

A p−1 = Q p−1 + R˜ p−1 − C˜

(3.37)

ReQ p−1 v, v ≥ 0

∀(t, x) ∈ [0, T ] × R, ∀v(t, ·) ∈ H (R) α ψα,β (ξ )∂ξ Dxβ A p−1 (t, x, ξ ) R˜ p−1 (t, x, ξ ) ∼ ψ1 (ξ )Dx A p−1 (t, x, ξ ) + p−1

α+β≥2

with ψ1 ∈ S −1 , ψα,β ∈ S (α−β)/2 , ψ1 , ψα,β ∈ R. Therefore, the first application of the sharp-Gårding Theorem 2.8 gives, because of (3.5), (3.34) and (3.37): σ (e− Ae ) = ia p +

p−1

A p−k + A0 = ia p + A p−1 +

k=1

p−1

A p−k + A0

k=2

= ia p + Q p−1 +

p−1

  ( A I |ord( p−k) + A I I |ord( p−k) + R˜ p−1 

k=2

ord( p−k)

) + A0 (3.38)

for some A0 , A0 ∈ S 0 , where R˜ p−1 |ord( p−k) denotes the terms of order p − k of R˜ p−1 := R(A p−1 ). We have thus proved that it is possible to choose M p−1 > 0 such that σ (e− Ae ) = ia p (t, ξ ) + Q p−1 +

p−1

 (e− Ae )ord( p−k) + R˜ p−1 + A0 ,

k=2

(3.39) where Q p−1 (t, x, D) is a positive operator of order p − 1, R˜ p−1 is a remainder of order p − 2, and A0 (t, x, D) is an operator of order zero. Choice of M p−2 , . . . , M3 . To iterate this process, applying the sharp-Gårding Theorem 2.8 to terms of order p − 2, p − 3, and so on, up to order 3, we need to investigate the action of the sharp-Gårding Theorem on each term of the form  (e− Ae )ord( p−k) + S p−k , where S p−k denotes terms of order p − k coming from remainders of previous applications of the sharp-Gårding Theorem 2.8, for p − k ≥ 3. Lemma 3.2 says that remainders of terms of the form (e− Ae )ord( p−k) have “the right decay at the right level”, in the sense of (3.20); in what follows we show that also S p−k (and hence their remainders R(S p−k )) are sums of terms with “the right decay at the right level”. Then we apply the sharp-Gårding Theorem 2.8 to terms of order p − k, up to order p − k = 3.

Well-posedness of the Cauchy problem

137

To estimate S p−k and then R(S p−k ) we previously need to make some remarks. From (3.38) with R˜ p−1 = R(A p−1 ) we have σ (e− Ae ) = ia p + Q p−1 + R(A p−1 ) +

p−1

A p−k + A0

k=2

 = ia p + Q p−1 + A p−2 + R(A p−1 )ord( p−2) +

p−1

 (A p−k + R(A p−1 )ord( p−k) ) + A0 .

k=3

From (3.36) with k = 2 and Lemma 3.2 with k = 1, we can now choose M p−2 > 0 sufficiently large so that    Re A p−2 + R(A p−1 )ord( p−2) (t, x, ξ ) ≥ −C˜

∀(t, x, ξ ) ∈ [0, T ] × R2

for some C˜ > 0. Note that A p−2 depends on M p−1 and M p−2 , in the sense of (3.36), while R(A p−1 )|ord( p−2) depends only on the already chosen M p−1 . Thus, by the sharpGårding Theorem 2.8 there exist pseudo-differential operators Q p−2 and R˜ p−2 , with symbols in S p−2 and S p−3 respectively, such that ∀v(t, ·) ∈ H p−2 ReQ p−2 v, v ≥ 0  A p−2 + R(A p−1 )ord( p−2) = Q p−2 + R˜ p−2 , with   R˜ p−2 = R(A p−2 + R(A p−1 )ord( p−2) ) = R(A p−2 ) + R(R(A p−1 )ord( p−2) ), so that σ (e− Ae )  = ia p + Q p−1 + Q p−2 + R(A p−2 ) + R(R(A p−1 )ord( p−2) ) +

p−1

 (A p−k + R(A p−1 )ord( p−k) ) + A0

k=3

= ia p + Q p−1 + Q p−2   + A p−3 + R(A p−1 ) +

p−1  k=4

ord( p−3)

   + R(A p−2 )ord( p−3) + R 2 (A p−1 )ord( p−3)

 A p−k + R(A p−1 )ord( p−k)

   +R(A p−2 )ord( p−k) + R 2 (A p−1 )ord( p−k) + A0 .

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To proceed analogously for the terms of order p − 3, then p − 4 and so on up to order 3, we thus need to estimate, for p − k ≥ 3 and s ≥ 2: R s (A p−k ) = R s (A0p−k ) + R s (B p−k ) + R s ( A I I |ord( p−k) ). The arguments are analogous to those already made for the discussion of R(A0p−k ), R(B p−k ) and R( A I I |ord( p−k) ) in Lemma 3.2. Indeed, in the remainders of the sharpGårding Theorem 2.8 we have a first addend with some ψ˜ 1 ∈ S −1 and where some derivative Dx appears and a second addend with some ψα  ,β  ∈ S  β ∂ξα Dx

α  −β  2

and where

appear. some derivatives When the x-derivatives fall on λ p− j the decay in x gets better by (2.16), while the level in ξ decreases, so that we still have the “right decay”. When the x-derivatives fall on the coefficients then the assumptions (1.13)–(1.15)   (for α  = β  = 1 in still give the “right decay” since the level in ξ decreases of α +β 2 the first addend) and because of (3.31) and (3.32). Therefore, remainders coming from the sharp-Gårding Theorem 2.8 always have the “right decay”. This shows that we can apply again and again the sharp-Gårding Theorem 2.8 until we find pseudo-differential operators Q p−1 , Q p−2 , . . . , Q 3 of order p − 1, p − 2, . . . , 3 respectively and all positive definite, such that p−1

σ (e− Ae ) = ia p + Q p−1 + Q p−2 + · · · Q 3 +

(A p−k + S p−k ) + A˜ 0

k= p−2

for some A˜ 0 ∈ S 0 and S p−k coming from remainders of the sharp-Gårding theorem. Choice of M2 and M1 . Let us write A2 + S2 = T2 + i T2 with T2 = Re(A2 + S2 ) and T2 = Im(A2 + S2 ). As in the previous steps we choose M2 > 0 such that T2 = Re(A2 + S2 ) ≥ 0 (up to a constant that we can put in A˜ 0 ). Then, by the Fefferman–Phong inequality (2.34), we get that ReT2 v, v ≥ −c v 20 for some c > 0, without any remainder. On the other hand, we write i T2 =

i T2 + (i T2 )∗ i T  − (i T2 )∗ + 2 , 2 2

(3.40)

Well-posedness of the Cauchy problem

139

where ( Re

) i T2 − (i T2 )∗ u, u = 0, 2

(3.41)

while i T2 + (i T2 )∗ has a real principal part of order 1, has the “right decay” and does not depend on M1 . Therefore we can choose M1 > 0 sufficiently large so that  Re

i T2 + (i T2 )∗ + A1 + S1 2

 ≥0

and hence, by the sharp-Gårding inequality (2.33) for m = 1 , ( Re

 ) i T2 + (i T2 )∗ + A1 + S1 v, v ≥ −c v 20 . 2

(3.42)

By (3.40), (3.41) and (3.42) we finally get σ (e− Ae ) = ia p +

p−3

Q p−s + (A2 + S2 ) + (A1 + S1 ) + A˜ 0

s=1

with ReQ p−s v, v ≥ 0

∀v(t, ·) ∈ H p−s , s = 1, 2, . . . , p − 3

Re(A2 + S2 + A1 + S1 )v, v ≥ −c v 20

∀v(t, ·) ∈ H 2 .

Estimates for the operator A . We finally look at the full operator A in (2.27); by (2.28), (2.29) we notice that An,m is of the same kind of A with ∂ξm r n Dxm a j instead of a j . This implies that we have m more x-derivatives on a j , but the level in ξ decreases of −n − m < −m, so that we argue as for σ (e− Ae ) and find that also σ (e

−

A

n,m

e )=

p

Q n,m p−s

s=0

with Q n,m ∈ S 0 and 0 2 ReQ n,m p−s v, v ≥ −C n,m v 0

∀v(t, ·) ∈ H p−s 1 ≤ s ≤ p − 1

for some Cn,m > 0. Since every Q ∈ S 0 also satisfies ReQv, v ≥ −c v 20

∀v ∈ H 0

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A. Ascanelli, C. Boiti

for some c > 0, by Lemma 2.6 we finally have that ReA v, v ≥ −c v 20

∀v(t, ·) ∈ H ∞

(3.43)

for some c > 0, and hence if v ∈ C([0, T ]; L 2 ) is a solution of (2.3), by (2.2) with A instead of A we get that d

v 20 ≤ f 20 + v 20 − 2ReA v, v dt ≤ (2c + 1)( f 20 + v 20 ). By standard arguments we deduce that, for all s ∈ R, if v ∈ C([0, T ]; H s ), ⎛

v(t, ·) 2s

≤c

⎝

t

g 2s

+

⎞

f (τ, ·) 2s dτ ⎠

∀t ∈ [0, T ],

(3.44)

0

for some c > 0. Since e± ∈ S δ , for u = e v we finally have, from (3.44) with s − δ instead of s: ⎛

u 2s−2δ ≤ c1 v 2s−δ ≤ c2 ⎝ g 2s−δ + ⎛ ≤ c3 ⎝ g 2s +

t

⎞

t

⎞

f 2s−δ dτ ⎠

0

f 2s dτ ⎠

0

for some c1 , c1 , c3 > 0. This proves the existence of a solution u ∈ C([0, T ]; H ∞ (R)) of (1.16) which   satisfies (1.17) for σ = 2δ = 2( p − 1)M p−1 . Remark 3.3 For the choice of M p−1 , . . . , M3 we made use of the sharp-Gårding Theorem 2.8 obtaining, at each step, a new remainder given by (2.32). On the contrary, for the choice of M2 and M1 we made use of, respectively, the Fefferman–Phong inequality (2.34) and the sharp-Gårding inequality (2.33), where no new remainders appear. This lets us save some conditions on the coefficients a1 and a2 , for which we required, indeed, only conditions (1.13) and (1.13)–(1.14) respectively, in the statement of Theorem 1.2.

Well-posedness of the Cauchy problem

141

4 Energy estimate for systems: proof of Theorem 1.1 Let us now consider the operator L in (1.1) and the transformed operator L := (e )−1 Le , for defined by (2.5), (2.6): ⎛ ⎜ L = (e )−1 Dt e + (e )−1 ⎝ ⎛ ⎜ = Dt + ⎝

μ1

⎞ ..

⎟ −1 ⎠ e + (e ) Re

. μm

(e )−1 μ1 e

..

.

⎞ ⎟ ⎠ + R

(e )−1 μm e

with R (t, xξ ) ∈ S 0 . Setting ⎛ ⎜ A = ⎝

A1

⎞ ..

⎟ ⎠,

.

A j = i(e )−1 μ j e , 1 ≤ j ≤ m

Am we can thus write L = Dt − i A + R . As is Sect. 2 we substitute the Cauchy problem (1.9) by 

L V (t, x) = F (t, x) V (0, x) = G (x)

(t, x) ∈ [0, T ] × R x ∈R

(4.1)

for F = (e )−1 F and G = (e )−1 G. Proving the energy estimate for V we can then deduce the energy estimate for U = e V solution of (1.9). For a solution V of (4.1) we have: d

|V |20 = 2ReV  , V  = 2Rei F , V  − 2ReA V, V  − 2Rei R V, V  dt ≤ C( |Fλ |20 + |V |20 ) − 2ReA V, V  (4.2) for some C > 0, where for

given vectors U = (U1 , . . . , Um ) and V = (V1 , . . . , Vm ) we denote U, V  := mj=1 U j , V j . Note that every A j is of the same form as (2.27), so that by (3.43): ReA V, V  =

m j=1

ReA j V j , V j  ≥ −c

m j=1

V j 20 = −c |V |20 .

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A. Ascanelli, C. Boiti

Substituting in (4.2) we obtain, by standard arguments, the energy estimate for V ⎛

|V (t, ·) |2s ≤ C ⎝ |V (0) |2s +

t

⎞

|F (τ, ·) |2s dτ ⎠

0

for some C > 0, and hence the desired energy estimate for U = e V :

|U (t, ·) |2s−2δ = |e V |2s−2δ ≤ C1 |V |2s−δ ⎞ ⎛ t ≤ C2 ⎝ |V (0) |2s−δ + |F (τ, ·) |2s−δ dτ ⎠ 0

⎛ ≤

C3 ⎝ |U (0) |2s

t +

⎞

|F(τ, ·) |2s dτ ⎠

0

for some C1 , C2 , C3 > 0, since e ∈ S δ . This concludes the proof of Theorem 1.1.

 

Remark 4.1 If the remainder R in (1.1) is of order −n with n ≥ 2δ, δ in (2.11), then ( p) ( p) we can weaken the assumptions on the ∂ξ μk , avoiding the condition that “the ∂ξ μk have all the same constant sign for every fixed ξ ∈ R”. To obtain the energy estimate (1.8) in this more general case, it’s sufficient to substitute the transformation I e with ⎛ E := ⎝

⎞

e 1 ...

e m

⎠,

(4.3)

where, for 1 ≤ i ≤ m, i are of the form (2.5)-(2.6) with ωi (ξ/ h) instead of ω(ξ/ h), for some functions ωi ∈ C ∞ (R), 0 ≤ ωi ≤ 1 such that  ωi (ξ ) =

0 ( p) sgn(∂ξ μi (t, ξ ))

|ξ | ≤ 1 |ξ | ≥ ρ.

Note that every i satisfies (2.7) with the same constant δ. Then the transformed operator L = (E )−1 L E is of the form ⎛ ⎜ L = Dt + ⎝

(e 1 )−1 μ1 e 1

⎞ ..

.

⎟ ⎠ + R , (e m )−1 μm e m

Well-posedness of the Cauchy problem

143

where ⎛ ⎜ R = ⎝

(e 1 )−1

⎞ ..

.

⎛

⎟ ⎜ ⎠R⎝ (e m )−1

⎞

e 1 ..

.

⎟ ⎠ e m

has diagonal entries of order −n and the other entries of order not larger than −n+2δ ≤ 0 by the choice of n. Therefore R has order zero and we can proceed as in the proof of Theorem 1.1. This remark is useful for applications to higher order scalar differential equations with characteristic roots of possibly different sign at the same point (cf. [4]). References 1. Agliardi, R.: Cauchy problem for p-evolution equations. Bull. Sci. Math. 126(6), 435–444 (2002) 2. Agliardi, R.: Cauchy problem for evolution equations of Schrödinger type. J. Diff. Equ. 180(1), 89–98 (2002) 3. Agliardi, R., Zanghirati, L.: Cauchy problem for nonlinear p-evolution equations. Bull. Sci. Math. 133, 406–418 (2009) 4. Ascanelli, A., Boiti, C.: Cauchy problem for higher order p-evolution equations (2013, submitted) 5. Ascanelli, A., Boiti, C., Zanghirati, L.: Well-posedness of the Cauchy problem for p-evolution equations. J. Diff. Equ. 253, 2765–2795 (2012) 6. Ascanelli, A., Cicognani, M.: Schrödinger equations of higher order. Math. Nachr. 280(7), 717–727 (2007) 7. Ascanelli, A., Cicognani, M., Colombini, F.: The global Cauchy problem for a vibrating beam equation. J. Diff. Equ. 247, 1440–1451 (2009) 8. Cicognani, M., Colombini, F.: The Cauchy problem for p-evolution equations. Trans. Am. Math. Soc. 362(9), 4853–4869 (2010) 9. Cicognani, M., Reissig, M.: On Schrödinger type evolution equations with non-Lipschitz coefficients. Ann. Mat. Pura Appl. 190(4), 645–665 (2011) 10. Fefferman, C., Phong, D.H.: On positivity of pseudo-differential operators. Proc. Natl. Acad. Sci. USA 75(10), 4673–4674 (1978) 11. Ichinose, W.: Some remarks on the Cauchy problem for Schrödinger type equations. Osaka J. Math. 21, 565–581 (1984) 12. Ichinose, W.: Sufficient condition on H ∞ well-posedness for Schrödinger type equations. Commun. Partial Diff. Equ. 9(1), 33–48 (1984) 13. Kajitani, K., Baba, A.: The Cauchy problem for Schrödinger type equations. Bull. Sci. Math. 119, 459–473 (1995) 14. Kumano-Go, H.: Pseudo-Differential Operators. The MIT Press, London (1982) 15. Mizohata, S.: On the Cauchy problem. Notes and Reports in Mathematics in Science and Engineering, vol. 3. Academic Press, Orlando (1985) 16. Takeuchi, J.: Le problème de Cauchy pour certaines équations aux dérivées partielles du type de Schrödinger, IV. C. R. Acad. Sci. Paris Ser. I Math 312, 587–590 (1991)