On convergence rate estimates for approximations of solution operators of linear non-autonomous evolution equations

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solution operators of linear non-autonomous evolution equations

Hagen Neidhardt, Artur Stephan, Valentin Zagrebnov

To cite this version:

Hagen Neidhardt, Artur Stephan, Valentin Zagrebnov. On convergence rate estimates for approxi-

mations of solution operators of linear non-autonomous evolution equations. Nanosystems: Physics,

Chemistry, Mathematics, St. Petersburg National Research University of Information Technologies,

Mechanics and Optics., 2017, 8 (2), pp.202 - 215. �10.17586/2220-8054-2017-8-2-202-215�. �hal-

01418846�

On convergence rate estimates for approximations of solution operators of linear non-autonomous evolution

equations

1

Hagen Neidhardt,

Artur Stephan,

Valentin A. Zagrebnov

1

WIAS Berlin

Mohrenstr. 39, D-10117 Berlin, Germany

2

Humboldt Universit¨ at zu Berlin Institut f¨ ur Mathematik

Unter den Linden 6, D-10099 Berlin, Germany

3

Aix Marseille Universit´ e, CNRS, Centrale Marseille, I2M UMR 7373, F-13453 Marseille, France

[email protected], [email protected], [email protected]

PACS 02.30.Sa,02.30.Tb,02.60.Cb

We improve some recent convergence rate estimates for approximations of solution operators of linear non- autonomous evolution equations. The approximation results from the Trotter product formula which is proved to converge in the operator-norm and its convergence can be estimated. The result is applied to a diffusion perturbed by a time-dependent potential.

Keywords: Trotter product formula, evolution equations, operator-norm topology, approximation, conver- gence rate; operator splitting

Classification: Primary 34G10, 47D06, 34K30; Secondary 47A55..

1. Introduction

The theory of equations of evolution plays an important role in various areas of pure and applied mathematics, physics and other natural sciences, [4, 10, 12]. We focus on a non-autonomous linear Cauchy problem of the form

˙

u(t) = −(A + B(t))u(t), u(s) = u

∈ X, 0 < s 6 t 6 T, (1.1)

where {A + B (t), dom(A) ∩ dom(B(t))}

is a family of closed linear operators on the sepa-

rable Banach space X, I = [0, T ] ⊂ R . Let I

= (0, T ]. The solution operator {U (t, s)}

,

i.e. u(t) = U (t, s)u

solves (1.1) in some sense, can be obtained using the Howland-Evans

approach. The main idea of this approach is to reformulate the non-autonomous problem

(1.1) on X to an autonomous Cauchy problem on the Banach space L

(I , X) of p-summable

functions on I with values in X. The solution of the autonomous and the non-autonomous

Cauchy Problem correspond to each other, and therefore it is equivalent to solve the one or

the other equation. Once, the solution is obtained, the problem of a good approximation

appears. The Trotter product formula [13] or [2, Theorem 3.5.8] provides an approximation

in the strong-topology. In practice, a convergence in the operator-norm is more useful, espe-

cially, if the convergence can be estimated. Then, for example, in spite of the initial values,

the smallness of the steps can be calculated such that the error rate of the approximation is

lower than a given accuracy.

Going to analyze a linear non-autonomous Cauchy problem of the form (1.1) the aim is to find the so-called ”solution operator” or propagator {U (t, s)}

, ∆ = {(t, s) ∈ I

× I

: 0 < s 6 t 6 T }, I

= (0, T ], of the Cauchy problem (1.1), which has the property that u(t) = U (t, s)u

for (t, s) ∈ ∆ is a “solution” of the initial Cauchy problem (1.1) for an appropriate set of initial data u

. By definition, a propagator {U (t, s)}

is a strongly continuous operator-valued function U (·, ·) : ∆ → B(X) satisfying

U(t, t) = I for t ∈ I

, U(t, r)U (r, s) = U (t, s) for t, r, s ∈ I

with s 6 r 6 t, kUk

:= sup

(t,s)∈∆

kU (t, s)k

< ∞.

Our goal is to find an approximation {U

(t, s)}

, n ∈ N , of the solution operator {U(t, s)}

in the operator norm with an explicit convergence rate estimate. Such con- vergence rate estimates were already found by Ichinose and Tamura for positive self-adjoint operators [3]. Recently, in [6] an error estimate was proved, where the underlying space is a Banach space. In [6] the main technical tool to get such an approximation was the Trotter product formula. It was verified in [6] that under the assumptions made there the Trotter product formula converges not only in the strong topology but actually in the operator norm.

In a second step this result was carried over to the solution operator.

Following the ideas of [6] we improve the convergence rate estimate of [6] under assumptions on the involved operators A and B(t) which are straightforward generalizations of those of [3] to the Banach space. Despite this straightforward generalization we were unable to reproduce the strong convergence rate of [3] for the Banach space. However, with respect to the Trotter product formula we get a slightly stronger convergence rate estimate than in [1].

2. Preliminaries and Assumptions 2.1. Preliminaries

Throughout the paper we are dealing with a separable Banach space (X, k · k

). For an linear operator A : dom(A) ⊂ X → X, we define the resolvent by R(λ, A) := (A − λ)

: X → dom(A). A family {T (t)}

of bounded linear operators on a Banach space X is called a strongly continuous (one-parameter) semigroup if it satisfies the functional equation

T (0) = I, T (t + s) = T (t)T (s), t, s > 0,

and the orbit maps [0, ∞) 3 t 7→ T (t)x are continuous for every x ∈ X. In the following we simply call them semigroups. For a given semigroup we define its generator by

Ax := lim

h&0

1

h (x − T (h)x) with domain

dom(A) := {x ∈ X : lim

h&0

1

h (x − T (h)x) exists}.

Note that the definition differs from the standard one by the sign minus. It is well-known that the generator of a semigroup is a closed and densely defined linear operator which uniquely determines the semigroup (see e.g. [2, Theorem I.1.4]). For a given generator A we will write T (τ ) = e

, τ > 0.

For any semigroup {T (t)}

there are constants M

, γ

, such that it holds kT (t)k 6

M

e

for all t > 0. Such semigroups are called of class G(M

, γ

) and we write A ∈

G(M

, γ

). If γ

6 0, {T (t)}

is called a bounded semigroup. If kT (t)k 6 1, the semigroup is called contractive.

For any semigroup we can construct a bounded semigroup by adding some constant ν > γ

to its generator: The operator ˜ A := A + ν generates a semigroup { T ˜ (t)}

with k T ˜ (t)k 6 M

. It is known that for a generator A ∈ G (M

, γ

), the open half plane {z ∈ C : Re(z) < γ

} is contained in the resolvent set %(A) of A and the estimate kR(λ, A)k 6

MA

Re(λ)−γ_A

holds. If ˜ A = A + ν, then the open half-plane {z ∈ C : Re(z) < γ

− ν} is contained in the resolvent set of ˜ A.

The semigroup {T (t)}

on X is called a bounded holomorphic semigroup if its generator A satisfies ran(T (t)) ⊂ dom(A) for all t > 0 and sup

ktAT (t)k < ∞. It is well- known, that in this case the semigroup {T (t)}

can be extended holomorphically to a sector {z ∈ C : | arg(z)| < δ} ∪ {0} ⊂ C of angle δ > 0. For generators A of bounded holomorphic semigroups with 0 ∈ %(A) one can define fractional powers A

. Then, for α ∈ (0, 1), it holds dom(A) ⊂ dom(A

) ⊂ dom(A

) = X. In the following we need the well-known estimate for generators of a bounded holomorphic semigroup:

sup

t>0

kt

A

T (t)k = M

< ∞. (2.1) 2.2. Assumptions

Below we made the following assumptions with respect to the operator A and the family {B (t)}

.

Assumption 2.1.

(A1) The operator A is a generator of a bounded holomorphic semigroup of class G(M

, 0) and 0 ∈ %(A). Let {B(t)}

be a family of generators on X belonging to the same class G(M

, β). The function I 3 t 7→ (B (t) + ξ)

x ∈ X is strongly measurable for any x ∈ X and any ξ > β.

(A2) There is an α ∈ (

, 1) such that for a.e. t ∈ I it holds that dom(A

) ⊂ dom(B(t)) and dom((A

)

) ⊂ dom(B (t)

). Moreover, it holds

C

:= ess sup

t∈I

kB(t)A

k

< ∞ and C

:= ess sup

t∈I

kB(t)

(A

)

k

< ∞, (2.2) where A

and B(t)

denote the adjoint operators of A and B(t), respectively.

(A3) There is a constant L > 0 such that estimate

kA

(B (t) − B(s))A

k

6 L|t − s|, holds for a.e. t, s ∈ I.

Remark 2.2.

(a) In [6] the assumptions are slightly weaker. It is assumed that the domains satisfy dom(A

) ⊂ dom(B(t)

).

(b) The assumption 0 ∈ %(A) is just for simplicity. Otherwise, the generator A can be shifted by a constant η > 0. One can prove that the domain of the fractional power of A does not change either.

(c) In [3] both operators A and B(t) are assumed to be positive selfadjoint operators on a separable Hilbert space. The assumptions made in [3] yield that Assumption 2.1 is valid.

(d) The assumptions above imply that for a.e. t ∈ I the operator B(t) is infinitesimally small with respect to A. Indeed, fix t ∈ I and assuming (A1), (A2) we conclude

dom(A + η) = dom(A) ⊂ dom(A

) ⊂ dom(B(t))

for η > 0 and hence

kB (t)(A + η)

k

6 kB(t)A

k

· kA

(A + η)

k

6 C

C

η

. Therefore for any x ∈ dom(A) ⊂ dom(B(t)), we get

kB (t)xk

6 C

C

η

· k(A + η)xk

6 C

C

η

1

η kAxk

+ kxk

.

The relative bound can be chosen arbitrarily small by shifting η > 0. In particular, using standard perturbation results ( [5, Corollary IX.2.5]), we conclude that A + B (t) is the generator of a holomorphic semigroup, i.e. problem (1.1) is a parabolic evolution equation.

3. Solving strategy

Let us describe briefly our solving strategy. Details can be found in [6]. The approach to finding the solution operator {U (t, s)}

of (1.1) leads to a perturbation or extension problem for linear operators. It can be used in very general settings and is quite flexible.

The usual idea can be described as follows: The non-autonomous Cauchy problem in X can be reformulated as an autonomous Cauchy problem in a new Banach space L

(I, X ), p ∈ [1, ∞), of p-summable functions on the interval I with values in the Banach space X.

An operator family {C(t)}

on X induces an multiplication operator C on L

(I, X ) defined by

(Cf )(t) := C(t)f (t), dom(C) :=

f ∈ L

(I, X ) : f(t) ∈ dom(C(t)) for a.e. t ∈ I I 3 t 7→ C(t)f(t) ∈ L

(I, X )

.

Theorem 3.1 ( [6, Theorem 2.8]). Let {C(t)}

be a family of generators on X such that for almost all t ∈ I it holds that C(t) ∈ G(M, β) for some M > 1 and β ∈ R . If the function I 3 t 7→ (C(t) + ξ)

x ∈ X is strongly measurable for ξ > β, x ∈ X, then the induced multiplication operator C is a generator in L

(J , X ) and its semigroup is given by

(e

f )(t) = e

f(t), f ∈ L6P (I, X ), for a.e t ∈ I. In particular, for the operator-norms we get

ke

k

= ess sup

t∈I

ke

k

. So the generators C(t) and C belong to the same class.

In particular in our case, the operator family {B (t)}

induces the generator B and A induces trivially the generator A on L

(I, X ). Assuming (A1) and (A2) it turns out that the operators BA

and A

B are bounded on L

(I, X ) and it holds that kBA

k

6 C

and kA

Bk

6 C

.

Let us introduce the operator D

:= ∂

on L

(I, X ) defined by

D

f(t) := ∂

f (t), dom(D

) := {f ∈ W

([0, T ], X ) : f(0) = 0}.

Then, D

is a generator of class G(1, 0) of the right-shift semigroup {S(τ )}

that has the form

(e

f )(t) = (S(τ)f )(t) := f(t − τ)χ

(t − τ), f ∈ L

(I, X ), a.e. t ∈ I.

We note that the generator D

has empty spectrum since the semigroup {S(τ )}

is nilpotent and therefore the integral R

0

e

S(τ)f dτ exists for any λ ∈ C and for any f ∈ L

(I, X ).

Let us look at the operator sum D

and A. Since A is time-independent, the operators A and D

commute, and, hence, also their semigroups commute. So, the operator family {e

e

}

defines a semigroup on L

(I, X ). Its generator is denoted by K

. It is closure of the operator sum D

+ A, i.e. K

= D

+ A. We note that all the generators K

, A, A belong to the same class.

Remark 3.2. By assumption (A1) the operator A generates a holomorphic semigroup. Note that the operator K

is not a generator of a holomorphic semigroup. Indeed, if we have

(e

f)(t) = (e

e

f )(t) = e

f(t − τ )χ

(t − τ), f ∈ L

(I, X ).

Since the right-hand side is zero for τ > t, the semigroup can not be extended to the complex plane.

Now, look at the operator sum

K e = D

+ A + B, dom( K) = dom(D e

) ∩ dom(A) ∩ dom(B). (3.1) In [6], the following theorem is proved.

Theorem 3.3 ( [6, Theorems 4.3 and 4.4]). Assume (A1) and (A2). Then, the operator closure K e =: K is a generator on L

(I, X ), and it holds

K = K

+ B, dom(K) = dom(K

) ∩ dom(B). (3.2) Moreover, it is an evolution generator, i.e. its semigroup defines the unique solution operator U(t, s) of problem (1.1) by

(e

f )(t) = (U (τ )f )(t) = U (t, t − τ )χ

(t − τ)f (t − τ), τ > 0, t ∈ I.

We note that for the proof it is not necessary that the operators B(t) are generators. After proving the existence of a unique solution, we want to approximate the solution operator U(t, s). This will be done by proving an operator-norm convergence for the Trotter product formula for K = K

+ B.

4. Stability

Proving the Trotter product formula, it is important to establish stability conditions.

Notice that stability is satisfied if the contractivity of the involved semigroups is assumed which might be too strong in applications. There are many stability conditions known for evolution equations. In particular, the Kato-stability is of interest, cf. [7, Definition 4.1], which is equivalent to a renormalizability conditions of the underlying Banach space, cf. [7].

We note that our following stability condition is weaker than Kato-stability.

Definition 4.1. Let A be a generator and let {B (t)}

be a family of generators in X. The family {B (t)}

is called A-stable if there is a constant M > 0 such that

ess sup

(t,s)∈∆

n←

Y

j=1

G

(t, s; n)

6 M

holds for any n ∈ N where G

(t, s; n) := e

e

, j = 0, 1, 2, . . . , n, and the

product is ordered increasingly in j from the right to the left.

Let us introduce the notion

T (τ) = e

e

, τ > 0.

Lemma 4.2 ( [6, Lemma 5.8]). If the operator family {B(t)}

is A-stable, then kT τ

n

k

6 M for any m ∈ N , n ∈ N and τ > 0. In particular, we have

kT (τ )

k

6 M for any m ∈ N and τ > 0.

5. Convergence in the operator-norm topology

Theorem 3.3 leads to the problem, how the semigroup of K can be approximated in terms of the semigroups generated by D

, A and B. The classical Trotter product formula gives an approximation in the strong topology. In this section, we establish an approximation in the operator-norm topology on L

(I, X ). This is done in several steps. This approximation in L

(I, X ) can be used to prove an convergence rate estimate in X for the propagators.

5.1. Technical Lemmata

In this section, we state and prove all technical lemmas that we used to prove the convergence and estimate of the Trotter product formula in the operator-norm in L

(I, X ).

For simplicity in notation, we set T (τ ) := e

e

, τ > 0. Note that T (τ ) = 0 for τ > T . Similarly, e

= 0 for τ > T .

Lemma 5.1. Let the assumptions (A1) and (A2) be satisfied.

(i) Then dom(K

) ⊂ dom(A

) and there is a constant Λ

> 0 such that kA

e

k

6 Λ

τ

(5.1)

holds for τ > 0.

(ii) If {B(t)}

is A-stable, then there is a constant Π

> 0 such that the estimates

k(T (τ ) − e

)A

k

6 Π

τ (5.2)

kA

(T (τ) − e

)k

6 Π

τ (5.3)

are valid for τ > 0.

(iii) If {B(t)}

is A-stable, then there is a constant Y

> 0 such that the estimate kT (τ)

A

k

6 Y

τ

+ 1 (kτ )

, τ > 0, k ∈ N . (5.4) holds for τ > 0.

Proof. (i)-(ii) The assertions dom(K

) ⊆ dom(A

) as well as (5.1) and (5.2) follow from

Lemma 7.3, Lemma 7.4 and Lemma 7.6 of [6]. To prove (5.3) one has slightly to modify the

second part of the proof of Lemma 7.6 of [6].

(iii) For kτ > T we have T (τ )

= 0. Hence, one has to prove the estimate (5.4) only for kτ 6 T . In fact, using Lemma 4.2, we get kT (τ )

k 6 M , τ ∈ [0, ∞). Hence,

kT (τ )

A

fk 6 k(T (τ )

− e

)A

f k + ke

A

f k 6 k

k−1

X

j=0

T (τ )

(e

− I)e

A

fk + ke

A

f k

6 M

k−1

X

j=0

Z

0

dσke

BA

k kA

e

fk + ke

A

fk, where we have used I − e

= R

0

Be

dσ. Moreover, from (2.1) we get kA

e

f k 6 M

((j + 1)τ )

kf k and kA

e

f k 6 M

(kτ )

kf k for τ > 0. Hence, using α >

, we get

kT (τ )

A

fk 6 M M

M

C

τ τ

k−1

X

j=0

1

(j + 1)

kfk + M

(kτ )

kf k 6 M M

M

C

ζ(2α)

τ

kfk + M

(kτ )

kfk for τ ∈ I , where ζ(β) := P

j=1 1

j^β

, β > 1, is the Riemann ζ-function and we have set M

:= sup

ke

k. Using that T (τ )

= 0 for τ k > T we find

kT (τ )

A

fk 6 M M

M

C

ζ(2α)

τ

kfk + M

(kτ )

kfk, f ∈ dom(A),

for τ > 0. Taking the supremum over the unit ball in dom(A) we prove (5.4).

Lemma 5.2. Let the assumptions (A1), (A2) and (A3) be satisfied. Then there is a constant Z

> 0 such that

kA

(T (τ ) − e

)A

k

6 Z

τ

, τ > 0. (5.5) Proof. Let f ∈ dom(K

) = dom(K). We have

d

dσ T (σ)e

f = d

dσ e

e

e

f

= − e

Be

e

f − e

e

K

e

f + e

e

Ke

f

= − e

Be

e

f + e

e

Be

f

=e

{e

Bf − Be

}e

f, which yields

T (τ )f − e

f = Z

0

d

dσ T (σ)e

f dσ = Z

0

e

{e

B − Be

}e

f dσ.

(5.6)

Now, we have the following identity e

e

B − Be

e

f

= (e

− I){e

B − Be

}(e

− e

)f+

+ (e

− I){e

B − Be

}e

f +

+ {e

B − Be

}(e

− e

)f + {e

B − Be

}e

f.

which yields for f = A

g

A

e

e

B − Be

e

A

g =

= A

(e

− I){e

B − Be

}(e

− e

)A

g+

+ A

(e

− I){e

B − Be

}A

e

g+

+ A

{e

B − Be

}(e

− e

)A

g+

+ A

{(e

− e

)B − B(e

− e

)}e

A

g+

+ A

(e

B − Be

)A

e

g.

(5.7)

In the following, we estimate the five terms separately.

At first we use the fact that A and K

commute and conclude that (e

− e

)A

g =

Z

0

e

BA

e

gdr.

Thus, for the first term we get

A

(e

− I ){e

B − Be

}(e

− e

)A

g

= − Z

0

A

Be

dr [e

, B]A

Z

0

A

e

BA

e

gdr.

where

[e

, B]f := {e

B − Be

}f, f ∈ dom(K

), τ > 0.

Using Lemma 5.1, we obtain the estimate

kA

(e

− I ){e

B − Be

}(e

− e

)A

gk 6 σ 2C

C

Λ

M

M

Z

0

1

(τ − σ − r)

dr kgk 6 σ(τ − σ)

2C

C

Λ

M

M

1 − α kgk

(5.8)

for σ ∈ [0, τ ] and τ > 0. For the second term, we get the estimate

kA

(e

− I){e

B − Be

}A

e

gk 6 σ 2C

C

M

M

kgk. (5.9) for σ ∈ [0, τ ] and τ > 0. Since we have

e

− e

h = Z

0

e

Be

hdr, h ∈ dom(K

), one obtains for the third term the estimate

kA

{e

B − Be

}(e

− e

)A

gk 6 (τ − σ) 2C

C

M

M

kgk (5.10)

for σ ∈ [0, τ ] and τ > 0. Moreover, using e

− e

h = −

Z

0

e

Ae

hdr, we get for the fourth term

A

{(e

− e

)B − B(e

− e

)}e

A

g = −

Z

0

A

e

e

drBA

+ A

B Z

0

e

A

e

dr

e

g, which yields the estimate

kA

{(e

− e

)B − B(e

− e

)}e

A

gk 6 C

M

M

Z

0

1

r

dr kgk + C

M

M

Z

0

1

r

dr kgk

= (C

+ C

) M

M

α σ

kgk

(5.11)

for σ ∈ [0, τ ] and τ > 0. To estimate the fifth term, we note that

(e

B − Be

)f = e

B (·)f (·) − Bχ

(· − σ)f (· − σ) =

= χ

(· − σ)B(· − σ)f(· − σ) − B (·)χ

(· − σ)f(· − σ) =

= χ

(· − σ){B(· − σ) − B(·)}f (· − σ), and therefore

kA

(e

B − Be

)e

A

gk

6 M

kA

{e

B − Be

}A

gk 6 ess sup

t∈I

kA

{B(t − σ) − B (t)}A

k

kgk 6 Lσkgk.

(5.12)

for σ ∈ [0, τ ] and τ > 0. From (5.7) we find the estimate kA

e

e

B − Be

e

A

gk

6 kA

(e

− I){e

B − Be

}(e

− e

)A

gk + kA

(e

− I){e

B − Be

}A

e

gk

+ kA

{e

B − Be

}(e

− e

)A

gk

+ kA

{(e

− e

)B − B(e

− e

)}e

A

gk + kA

(e

B − Be

)A

e

gk.

for σ ∈ [0, τ ] and τ > 0. Taking into account (5.8), (5.9), (5.10), (5.11) and (5.12) we find kA

e

e

B − Be

e

A

gk 6 n

σ(τ − σ)

2C

C

Λ

M

M

1 − α + σ 2C

C

M

M

+ (τ − σ) 2C

C

M

M

+ σ

(C

+ C

) M

M

α + σ L o

kgk

for σ ∈ [0, τ ] and τ > 0. Setting Z

:= 2C

C

Λ

M

M

1 − α , Z

:= 2C

C

M

M

+ L Z

:= 2C

C

M

M

, Z

:= (C

+ C

) M

M

α we obtain

kA

e

e

B − Be

e

A

gk 6 n

Z

σ(τ − σ)

+ Z

σ + Z

(τ − σ) + Z

σ

o

kgk (5.13)

From (5.6) we derive the representation A

(T (τ) − e

)A

g

= Z

0

A

e

{e

B − Be

}e

A

g dσ.

which yields the estimate

kA

(T (τ ) − e

)A

gk 6

Z

0

kA

e

{e

B − Be

}e

A

gk dσ.

Inserting (5.13) into this estimate and using Z

0

σ(τ − σ)

dσ = τ

Z

0

x(1 − x)

dx = τ

Γ(1 − α) Γ(2 − α) , we find the estimate

kA

(T (τ) − e

)A

gk 6 Z

Γ(1 − α)

Γ(2 − α) τ

+ Z

+ Z

2 τ

+ Z

1 + α τ

for τ > 0. We have

kA

(T (τ) − e

)A

gk 6

Z

Γ(1 − α)

Γ(2 − α) τ

+ Z

+ Z

2 τ

+ Z

τ

for τ > 0. Since T (τ ) = 0 and e

= 0 for τ > T we finally obtain

kA

(T (τ) − e

)A

g k 6

Z

Γ(1 − α)

Γ(2 − α) T

+ Z

+ Z

2 T

+ Z

τ

which proves the lemma.

Lemma 5.3. Let α ∈ [0, 1). Then the estimates

n−1

X

m=1

1

m

6 n

1 − α and

n−1

X

m=1

1

(n − m)

m

6 2

1 − α n

(5.14) are valid for n = 2, 3, . . . .

Proof. The function f (x) = x

, x > 0, is decreasing. Hence

n−1

X

m=1

1 m

6

Z

0

1

x

dx 6 (n − 1)

1 − α 6 n

1 − α

for n = 2, 3, . . . . Further, we have

n−1

X

m=1

1

(n − m)

m

6 2 1 n

n−1

X

m=1

1

m

6 2 1 n

n

1 − α = 2

1 − α n

,

and the claim follows.

5.2. The Trotter product formula in operator-norm topology

Now, we are able to prove and estimate operator-norm convergence of the Trotter product formula.

Theorem 5.4. Let the assumptions (A1), (A2) and (A3) be satisfied. If the family of generators {B(t)}

is A-stable, then there is a constant C

> 0 (depending on α ∈ (

, 1) and on the compact interval I) such that

k(e

e

)

− e

k

6 C

1

n

(5.15)

for τ > 0 and n = 2, 3, . . ..

Proof. Let T (σ) := e

e

and U (σ) := e

, σ > 0. Then the following identity holds T (σ)

− U (σ)

=

n−1

X

m=0

T (σ)

(T (σ) − U (σ))U (σ)

= T (σ)

(T (σ) − U (σ)) + (T (σ) − U(σ))U (σ)

+ +

n−2

X

m=1

T (σ)

(T (σ) − U (σ))U (σ)

= T (σ)

A

A

(T (σ) − U(σ)) + (T (σ) − U (σ))A

A

U (σ)

+ +

n−2

X

m=1

T (σ)

A

A

(T (σ) − U (σ))A

A

U (σ)

. which yields the estimate

kT (σ)

− U (σ)

k

6 kT (σ)

A

k kA

(T (σ) − U (σ))k + k(T (σ) − U (σ))A

k kA

U(σ)

k+

+

n−2

X

m=1

kT (σ)

A

k kA

(T (σ) − U(σ))A

k kA

U(σ)

k.

From Lemma 5.1 we get the estimates kT (σ)

A

k 6 Y

σ

+ 1 ((n − 1)σ)

, n > 2, as well as

kA

(T (σ) − U (σ))k 6 Π

σ and k(T (σ) − U (σ))A

k 6 Π

σ for σ ∈ (0, τ ]. Hence

kT (σ)

A

k kA

(T (σ) − U (σ))k 6 Π

Y

σ

σ

+ 1 (n − 1)

and

k(T (σ) − U (σ))A

k kA

U (σ)

k 6 Π

Λ

(n − 1)

σ

where we have used (5.1). Since

kA

(T (σ) − e

)A

k

6 Z

σ

, τ ∈ [0, τ

), by Lemma 5.2 we have

kT (σ)

A

k kA

(T (σ) − U (σ))A

k kA

U (σ)

k 6 Y

σ

+ 1

((n − m − 1)σ)

Z

σ

Λ

1 (σm)

6 Y

Z

Λ

σ

1

m

+ σ

1

(n − m − 1)

m

Using Lemma 5.3 we get

n−2

X

m=1

kT (σ)

A

k kA

(T (σ) − U (σ))A

k kA

U (σ)

k

6 Z

Λ

Y

σ

n−2

X

m=1

1

m

+ Z

Λ

Y

σ

n−2

X

m=1

1

(n − m − 1)

m

6 Z

Λ

Y

1 − α n

σ

+ 2n

σ

. Summing up we get the estimate

kT (σ)

− U (σ)

k 6 Π

Y

σ

σ

+ 1 (n − 1)

+ Π

Λ

(n − 1)

σ

+ Z

Λ

Y

1 − α n

σ

+ 2Z

Λ

Y

1 − α n

σ

. Setting σ := τ /n, we obtain

kT (τ /n)

− U (τ /n)

k 6 Π

Λ

T

(n − 1)

+ Π

Λ

n − 1 + Π

Λ

T

(n − 1) + Z

Λ

Y

T

1 − α

1

n

+ 2Z

Λ

Y

T

1 − α

1 n

. for τ > 0 and n = 2, 3, . . .. Hence there is a constant C

> 0 such that (5.15) holds.

Remark 5.5. It is worth saying that the result only depends on the domains of the operators A and B(t) and not on the concrete form of them.

5.3. Norm convergence for propagators

Let us investigate the consequences of Theorem 5.4 for the approximation of the solution operator {U (t, s)}

. We have

({(e

e

)

−e

}g)(t) = {U

(t, t − τ) − U (t, t − τ )}χ

(t − τ )g(s − τ).

for (t, t − τ) ∈ ∆ and g ∈ L

(I, X ) where U

(t, s) :=

−→

Y

n

j=1

e

n )

e

, (t, s) ∈ ∆.

Let us introduce the left-shift semigroup on L

(I, X )

(L(τ)f )(t) := χ

(t + τ )f(t + τ ), f ∈ L

(I, X ).

Theorem 5.6. Let the assumptions (A1), (A2) and (A3) be satisfied. If the family of generators {B(t)}

is A-stable, then there is a constant C

> 0

ess sup

(t,s)∈∆

kU

(t, s) − U(t, s)k

6 C

n

, n = 2, 3, . . . , (5.16) where the constant C

coincides with that one of Theorem 5.4.

Proof. We set

S

(t, s) := U

(t, s) − U (t, s), (t, s) ∈ ∆, n ∈ N , and

S

(τ ) := L(τ ){(e

e

)

− e

} : L

(I, X ) → L

(I, X ).

for τ > 0 and n = 2, 3, . . . . We have

(S

(τ )g)(t) = S

(t + τ, t)χ

(t + τ )g(t), t ∈ I

, g ∈ L

(I, X ).

Hence, for any τ ∈ I and n ∈ N , the operator S

(τ ) is a multiplication operator on L

(I , X) induced by the family {S

(· + τ, ·)χ

(· + τ)}

of bounded operators. Applying [6, Lemma

?], we conclude for τ > 0

k(e

e

)

− e

k

> kV (τ){(e

e

)

− e

}k

= kS

(τ )k

= ess sup

t∈I0

kS

(t + τ, t)χ

(t + τ)k

=

= ess sup

t∈I0

k{U

(t + τ, s) − U (t + τ, s)}χ

(t + τ)k

=

= ess sup

t∈(0,T−τ]

kU

(t + τ, s) − U(t + τ, s)k

. Taking into account Theorem 5.4 we find

ess sup

t∈(0,T−τ]

kU

(t + τ, t) − U (t + τ, t)k

6 C

n

, τ > 0, n ∈ 2, 3, . . . ,

which yields (5.16).

Remark 5.7.

(i) Ichinose and Tamura proved in [3] a convergence rate of O(

) assuming that the operators A and B (t) are positive and self-adjoint. In [6], the authors proved the convergence rate of O(n

) for any β ∈ (α, 1) assuming dom(A

) ⊂ dom(B (t)

).

(ii) Moreover, we note that indeed, the estimates (5.15) and (5.16) are equivalent.

(iii) We note that a priori the operator family {U

(t, s)}

do not define a propagator since the co-cycle equation is in general not satisfied. But one can check that

U

(t, s) = U

t, s + k

n (t − s)

U

s + k

n (t − s), s

,

is satisfied for 0 < s 6 t 6 T , n ∈ N and any k ∈ {0, 1, . . . , n}.

6. Example: Diffusion equation perturbed by a time-dependent potential We investigate the diffusion equation perturbed by a time-dependent potential. On the Banach space X = L

(Ω), where Ω ⊂ R

is a bounded domain with C

-boundary (d > 2) and q ∈ (1, ∞), the equation reads

˙

u(t) = ∆u(t) − B(t)u(t), u(s) = u

∈ L

(Ω), t, s ∈ I

. (6.1)

∆ denotes the Laplace operator on L

(Ω) with Dirichlet boundary conditions defined on

∆ : dom(∆) = H

(Ω) ∩ H ˚

(Ω) → L

(Ω).

It turns out that −∆ is the generator of a holomorphic contraction semigroup on L

(Ω) (cf. [9, Theorem 7.3.5/6]). B (t) denotes a time-dependent scalar-valued multiplication operator given by

(B(t)f )(x) = V (t, x)f(x), dom(B(t)) = {f ∈ L

(Ω) : V (t, x)f(x) ∈ L

(Ω)}, where

V : I × Ω → C , V (t, ·) ∈ L

(Ω).

For α ∈ (0, 1), the fractional power of −∆ are defined on the domain (−∆)

: ˚ H

(Ω) → L

(Ω).

Note, that for 2α <

, it holds that ˚ H

(Ω) = H

(Ω). The adjoint operator of ∆ is denoted by ∆

and it is defined on the domain dom(∆

) = H

(Ω) ∩ H ˚

(Ω) ⊂ L

(Ω), where

1

q

+

= 1. It is well-known that also the operator (−∆

) generates a bounded holomorphic semigroup. The operators B(t) are scalar-valued and hence B (t)

= B(t) : dom(B(t)) ⊂ L

(Ω) → L

(Ω). Moreover, one can show that K

= D

+ A, i.e. the operator sum D

+ A is already closed.

Now, we are going to verify the assumptions (A1) - (A3) in order to approximate the solution of (6.1). This means, we determine the required regularity of V (t, ·) ∈ L

(Ω) to ensure dom((−∆)

) ⊂ dom(B (t)) and dom((−∆

)

) ⊂ dom(B (t)

), i.e

H

(Ω), H

(Ω) ⊂ dom(B(t)). (6.2) Using Sobolev embeddings, one obtains the general embedding

H

(Ω) ⊂ L

(Ω) for

( γ

∈ [γ

,

d sγ1 d

s−γ1

], if γ

∈ (1,

)

γ

∈ [γ

, ∞), if γ

∈ [

, ∞) . (6.3) For our case (6.2), we obtain H

(Ω) ⊂ L

(Ω) and H

(Ω) ⊂ L

(Ω), for some constants r, ρ ∈ (1, ∞]. Hence, it suffices to ensure L

(Ω), L

(Ω) ⊂ dom(B(t)). The parameters r, ρ define ˜ r, ρ ˜ via

1 r + 1

˜ r = 1

q , 1 ρ + 1

˜ ρ = 1

q

(6.4)

and since the operator B(t) is a multiplication operator defined by V (t, ·), the regularity of V (t, ·) has to be at least % := max{˜ r, ρ}. ˜

Moreover, let

F (t) := (−∆)

B(t)(−∆)

: L

(Ω) → H ˚

(Ω) ⊂ L

(Ω).

We have to ensure Lipschitz-continuity for the operator-valued function t 7→ F (t). So, let f ∈ L

(Ω) and g ∈ L

(Ω). Define ˜ f = ∆

f ∈ H ˚

(Ω) ⊂ L

(Ω) and ˜ g = (∆

)

g = (∆

)

g ∈ H ˚

(Ω) ⊂ L

(Ω). Then, we have for t ∈ I

hF (t)f, gi = h(−∆)

B(t)(−∆)

f, gi = h(−∆)

f, B(t)

(−∆

)

gi = h f , B(t) ˜

˜ gi.

The boundedness of h f , B(t) ˜

gi ˜ can be ensured by V (t, ·) ∈ L

(Ω), where τ ∈ (1, ∞) is defined via

1 r + 1

τ + 1

ρ = 1. (6.5)

For fixed d > 2 and α ∈ (

, 1), the following table turns out for the parameters ˜ r, ρ, τ: ˜ q ∈ (1,

) q ∈ [

, ∞)

q

∈ (1,

) ˜ r ∈ [

, ∞], ρ ˜ ∈ [

, ∞], r ˜ ∈ (q, ∞], ρ ˜ ∈ [

, ∞], τ ∈ [

, ∞] τ ∈ [

, ∞]

q

∈ [

, ∞) r ˜ ∈ [

, ∞], ρ ˜ ∈ (q

, 2α, ∞], r ˜ ∈ (q, ∞], ρ ˜ ∈ (q

, ∞],

τ ∈ [

, ∞] τ ∈ (1, ∞]

We remark that since we have r > q, it holds that τ 6 ρ ˜ and hence, τ 6 % = max{˜ r, ρ}. To guarantee, that the operators ˜ B(t) are generators, we assume that the poten- tial V (t, x) is positive, i.e.

Re(V (t, x)) > 0, for a.e. (t, x) ∈ I × Ω

Then, for any t ∈ I the operator V (t, x) is a generator of a contraction semigroup on X = L

(Ω) (cf. [2, Theorem I.4.11-12]). In particular, the operator family B(t) is A-stable.

Theorem 6.1. Let Ω ⊂ R

be a bounded domain with C

-boundary, let q ∈ (1, ∞) and let α ∈ (

, 1). Let B(t)f = V (t, ·)f define a scalar valued multiplication operator on L

(Ω) with

V ∈ L

(I, L

(Ω)) ∩ C

(I, L

(Ω)),

where % = max{˜ r, ρ} ˜ and r, ˜ ρ, τ ˜ is chosen from the above table. Moreover, let Re(V (t, x)) > 0 for t ∈ I and for a.e. x ∈ Ω.

Then, the evolution problem (6.1) has a unique solution operator U (t, s) which can be approximated in operator-norm by

sup

||U

(t, s) − U (t, s)||

= O(n

), where

U

(t, s) =

−→

Y

n

j=1

e

e

. (6.6) Proof. Using relations (6.4), (6.5) and Sobolev embeddings (6.3), it is easy to see that the required inclusions hold. The claim follows, using Theorem 3.3 and Theorem 5.6. The

“ess sup” becomes a “sup”, since the solution operator and the approximating operator are

continuous.

Remark 6.2.

(i) In [11], the existence of a solution operator for equation (6.1) is shown assuming weaker

regularity in space and time for the potential. We assumed uniform boundedness of the

function t 7→ ||B(t)(−∆)

||

, which is indeed too strong but important for the consider-

ations.

(ii) We focused on domains, which are bounded and have C

-boundaries. Our considerations can be extended to other domains, too.

(iii) Although the approximating propagator {U

(t, s)}

defined in (6.6) looks elabo- rate, it has a simple structure. The semigroup of the Laplace operator on L

( R

) is given by the Gauss-Weierstrass semigroup (see for example [2, Chapter 2.13]) defined via

(e

u)(x) = (T (t)u)(x) = (4πt)

Z

R^d

e

2

4t

u(y)dy.

The terms e

are scalar valued and can be easily computed.

Acknowledgements

The preparation of the paper was supported by the European Research Council via ERC-2010-AdG no 267802 (“Analysis of Multiscale Systems Driven by Functionals”).

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