Maximal regularity in lp spaces for fractional lattice … Results Maximal regularity in l p spaces...

$Page 1: Maximal regularity in lp spaces for fractional lattice … Results Maximal regularity in l p spaces for fractional lattice models Marina Murillo Arcila Joint work with Carlos Lizama$
IntroductionResults

Maximal regularity in lp spaces for fractionallattice models

Marina Murillo ArcilaJoint work with Carlos Lizama

Universidad de Santiago de Chile

MSLMS, BCAM

March, 2016

Marina Murillo Arcila Maximal regularity in l

p

spaces for fractional lattice models 1/32

IntroductionResults

Contents

1 Introduction

2 ResultsMaximal `

p

-regularity for the linear modelPreliminariesThe general case: On UMD spacesThe Hilbert case

Existence of solutions for the nonlinear modelThe discrete time fractional Fisher equationThe discrete time fractional Nagumo equation


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IntroductionResults

Fractional calculus

Studies di↵erential operators of an arbitrary real order notonly integer order.

In contrast to ordinary derivative operators, fractionaloperators are non-local and incorporate memory e↵ects intomodelling.

They capture the memory and the heredity of the process. Itis an e↵ective tool for revealing phenomena in nature becausenature has memory.

Applications in science, engineering, and mathematics:viscoelasticity, electrical circuits, chemistry, neurology,di↵usion, control theory, statistics,....


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Some history

Fractional calculus on R : D↵

Authors: Leibniz, L’Hopital, Liouville, Grunwald, Letnikov,Riemann, Podlubny.

Fractional calculus on Z : �↵

Diaz and Osler (1974): A fractional di↵erence operator as aninfinite series.Gray and Zhang (1988): A fractional calculus for the discretenabla (backward) di↵erence operator.Miller and Ross (1989): A fractional sum via the solution of alinear di↵erence equation.Atici and Eloe (2007): The Riemann-Liouville like fractionaldi↵erence using the fractional sum of Miller and Ross.Anastassiou (2010): The Caputo like fractional di↵erence usingthe fractional sum from Miller and Ross.


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Definition

Lp

(R;X ) = {u : R ! X |RR ku(t)kp

X

dt < 1}

A continuous time fractional model: definition

D↵t

u(t, x) = Au(t, x) + G (u(t, x)), t 2 R, x 2 ⌦ ⇢ RN ,↵ 2 R+,

where A is a closed linear operator with domain D(A) defined on aBanach space of functions X , G : L

p

(R,X ) ! Lp

(R,X ) and D↵t

denotes a fractional type derivative.

Motivation

It models subdi↵usion (0 < ↵ < 1) and superdi↵usion phenomena(1 < ↵ < 2) when A = ��.


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Continuous time fractional model: state of the art

The existence and regularity of solutions when A is thegenerator of a C

0

-semigroup. Wang, Chen and Xiao (2012).

The existence, stability and uniqueness of solutions when A isa closed linear operator using methods from operator theory.Kovacs, Li and Lubich (2014, 2015).

What happens when we consider the discrete time version of thismodel?


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Definition

`p

(Z;X ) = {u : Z ! X |P

n2Zku(n)kp

X

< 1}

Model of interest: the discrete time fractional model

We are concerned with the study of existence and uniqueness ofsolutions in `

p

(Z;X ) for:

�↵u(n, x) = Au(n, x) + G (u(n, x)), n 2 Z, x 2 ⌦ ⇢ RN , ↵ 2 R+,

where A is a closed linear operator with domain D(A) on a Banachspace of functions X , G : `

p

(Z;X ) ! `p

(Z;X ) and �↵ denotesthe generalized Grunwald–Letnikov derivative.

This equation appears when dealing with numerical methods forsolving the continuous time model.


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Grunwald–Letnikov derivative: definition

The forward Euler operator � : `p

(Z;X ) ! `p

(Z;X ) isdefined by �f (n) := f (n + 1)� f (n), n 2 Z and for eachm 2 Z, we define �m := �m�1 ��.

Given ↵ > 0 the fractional sum of order ↵ is given by:(��↵f )(n) := (k↵ ⇤ f )(n) =

Pn

j=�1 k↵(n � j)f (j), n 2 Z,

where k↵(n) =

8>><

>>:

�(↵+ n)

�(↵)�(n + 1)n 2 Z

+

,

0 otherwise,and � is the Euler gamma function.

Given ↵ > 0 the Grunwald–Letnikov derivative of order ↵is defined by �↵f (n) = �m��(m�↵)f (n), n 2 Z,m = [↵] + 1.


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The discrete time model: state of the art

Methods to solve mixed di↵erence equations, i.e. discrete intime and continuous in space. (see e.g.: Bateman (1943)).

Kovacs, Li and Lubich (2014, 2015): existence and uniquenessof solutions of the discrete and continuous model when A isthe generator of a C

0

-semigroup.

Lizama (2015): existence and uniqueness of solutions when Ais a bounded operator and 0 < ↵ 2 using ↵-resolventsequence of bounded and linear operators.

No results were known when A is an unbounded closed linearoperator.


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Objective

Provide a characterization of the existence and uniqueness ofsolutions in `

p

(Z,X ) for any ↵ 2 R+ and any closed linearoperator A not necessarily bounded.

Maximal `p

-regularity

Let p 2 (1,1),↵ 2 R+ and A be a closed linear operator definedon a Banach space X . We say that

�↵u(n) = Au(n) + f (n), n 2 Z, (1)

has maximal `p

-regularity if for each f 2 `p

(Z;X ) there exists aunique solution u 2 `

p

(Z; [D(A)]) of (1).


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Work Plan1 Provide a characterization of maximal `

p

-regularity for thelinear model in UMD spaces (Theorem 1).

2 Provide a characterization of maximal `p

-regularity for thelinear model in Hilbert spaces (Corollary 1).

3 Prove the existence of solutions for the nonlinear model(Theorem 2).

4 Show the existence of solutions for two nonlinear equations:The discrete time fractional Fisher equation (Example 1) andthe discrete time fractional Nagumo equation (Example 2).


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Maximal `p

-regularity for the linear modelExistence of solutions for the nonlinear model

Theorem 1: Maximal `p

-regularity for the linear model

Let X be a UMD space, p 2 (1,1) and ↵ 2 R+. If{(1� e�it)↵}

t2T ⇢ ⇢(A) then the following assertions areequivalent:

(i) �↵u(n) = Au(n) + f (n), n 2 Z has maximal `p

-regularity;

(ii) M(t) := ((1� e�it)↵ � A)�1 is an `p

-multipler from X to[D(A)];

(iii) The set {M(t) : t 2 T} is R-bounded.

Tools

The concepts of R-boundedness and `p

-multipler.

The Discrete Time Fourier Transform in `p

spaces.

For the equivalence (ii) , (iii) Blunck’s theorem.


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Maximal `p


B(X ,Y ) := space of bounded linear operators from X into Yendowed with the uniform operator topology

R-boundedness

Let X ,Y Banach spaces. T ⇢ B(X ,Y ) is R-bounded if there is

a constant c � 0 such that

k(T1

x1

, ...,Tn

xn

)kR

ck(x1

, ..., xn

)kR

,

for all T1

, ...,Tn

2 T , x1

, ..., xn

2 X , n 2 N and

k(x1

, ..., xn

)kR

:=1

2n

X

✏j

2{�1,1}n

��nX

j=1

✏j

xj

��,

for x1

, ..., xn

2 X .


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Maximal `p


Definition

S(Z;X ) = {f : Z ! X : 8k 2 N0

: supn2Z|n|kkf (n)k < C

k

}C1((�⇡,⇡);X ) := {' : (�⇡,⇡) ! X , 2⇡-periodic :'|

[�⇡,⇡] infinitely di↵erentiable}C1((�⇡,⇡);R) = C1(�⇡,⇡) and S(Z;R) = S(Z).S(Z;X )0 := {T : S(Z) ! X : linear and continuous}

The Discrete time Fourier Transform (DTFT) in `p

(Z;X )

The DTFT (F) in S(Z;X )0 allows us to define F in `p

(Z;X ).Indeed, given f 2 `

p

(Z;X ) we can identify it with Tf

2 S(Z;X )0:

Tf

( ) := hTf

, i :=X

n2Zf (n) (n), 2 S(Z), then

hFTf

,'i = hTf

, 'i =X

n2Zf (n)'(n), ' 2 C1(�⇡,⇡).


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Maximal `p


Definition

Let M : T ! B(X ,Y ) be continuous. We say that M is a`p

-multiplier (from X to Y ) if there exists a bounded mappingT : `

p

(Z;X ) ! `p

(Z;Y ) such that

hF(Tf ),'i =X

n2Z(Tf )(n)'(n) =

X

n2Z' ·M(n)f (n) (2)

for all f 2 `p

(Z;X ) and all ' 2 C1((�⇡,⇡)). Here

' ·M(n) :=1

2⇡

Z ⇡

�⇡e int'(t)M(�t)dt, n 2 Z.


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Maximal `p


Blunck’s Theorem (2011)

Let p 2 (1,1) and X ,Y - UMD spaces. Let M : T ! B(X ;Y ) bea di↵erentiable function such that

�M(t), (1� e it)(1 + e it)M 0(t) : t 2 T

is R-bounded. Then there exists TM

: `p

(Z;X ) ! `p

(Z;Y ) abounded operator such that for each f 2 F�1(L1(T;X )), we haveTM

f 2 S(Z;Y ) and

\(TM

f )(t) = M(t)bf (t), for all t 2 T. (3)

Remark

Note that (3) is equivalent to write (TM

f )(n) =P

j2Z

⇣ R ⇡�⇡ M(t)e i(n�j)tdt

⌘f (j) ⌘ (M ? f )(n), n 2 Z.


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Maximal `p


The converse of Blunck’s theorem (2011)

Let p 2 (1,1) and X ,Y be Banach spaces. Let M : T ! B(X ;Y )be an operator valued function. Suppose there is a boundedoperator T

M

: lp

(Z;X ) ! lp

(Z;Y ) such that

\(TM

f )(t) = M(t)bf (t), for all t 2 T.

Then {M(t) : t 2 T} is R-bounded.


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Maximal `p


Theorem 1: Maximal `p

- regularity for the linear model

Let X be a UMD space, p 2 (1,1) and ↵ 2 R+

. If{(1� e�it)↵}

t2T ⇢ ⇢(A) then the following assertions areequivalent:

(i) �↵u(n) = Au(n) + f (n), n 2 Z has maximal `p

-regularity;

(ii) M(t) := ((1� e�it)↵ � A)�1 is an `p

-multipler from X to[D(A)];

(iii) The set {M(t) : t 2 T} is R-bounded.


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Maximal `p


Sketch of the proof: (iii) ) (ii)

We first observe the following identity, valid for all f 2 `p

(Z;X )and ' 2 C1(�⇡,⇡)

X

n2Z(M ? f )(n)'(n) =

X

n2Z' ·M(n)f (n),

(iii) ) (ii) We prove that {(1� e it)(1 + e it)M 0(t) : t 2 T} is R-bounded and by Blunck’s Theorem there is a bounded operatorTM

such that :X

n2Z(T

M

f )(n)'(n) =X

n2Z(M ? f )(n)'(n) =

X

n2Z' ·M(n)f (n).


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Maximal `p


Sketch of the proof: (ii) ) (iii)

(ii) ) (iii) Since M(t) is an `p

-multipler there exists a boundedoperator T such that for all f 2 `

p

(Z;X ) and ' 2 C1(�⇡,⇡):

hF(Tf ),'i = hTf , 'i = h' ·M, f i = hM ? f , 'i = hF(M ? f ),'i

By the converse of Blunck’s theorem M(t) is R- bounded.


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Maximal `p


Sketch of the proof: (i) ) (ii)

(i) ) (ii) Given f 2 `p

(Z;X ) there exists a unique solutionuf

2 `p

(Z; [D(A)]) for

�↵uf

(n) = Auf

(n) + f (n), n 2 Z.

Denote by T↵ : `p

(Z;X ) ! `p

(Z; [D(A)]) the operator defined byT↵f = u

f

. By the closed graph theorem T↵ is bounded and T↵

satisfies the definition that M(t) is an `p

-multipler.


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Maximal `p


Sketch of the proof: (ii) ) (i)

(ii) ) (i) This implication is very technical and we have toconstruct the solution u of the equation. To prove the uniqueness,we take u : Z ! D(A) a solution of the equation with f ⌘ 0 andwe prove u ⌘ 0.


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Maximal `p


Corollary 1: Maximal `p

- regularity in Hilbert spaces

Let H be a Hilbert space and ↵ 2 R+. Suppose

{(1� e�it)↵}t2T ⇢ ⇢(A).

The following assertions are equivalent:(i) For all f 2 `

p

(Z;H) there exists a unique u 2 `p

(Z;H) suchthat u(n) 2 D(A) for all n 2 Z and

�↵u(n) = Au(n) + f (n), n 2 Z.

(ii) We have

supt2T

��⇣(1� e�it)↵ � A

⌘�1

�� < 1.


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IntroductionResults

Maximal `p


We study the existence of solutions in `p

(Z;X ) for :

�↵u(n) = Au(n) + G (u)(n) + ⇢f (n) (4)

⇢ 2 R+, f 2 `p

(Z,X ) and G : `p

(Z;X ) ! `p

(Z;X ).

Theorem 2: Existence of solutions for the nonlinear model

Let X be a UMD space, 1 < p < 1 and ↵ 2 R+. Assume that

{(1� e�is)↵}t2T ⇢ ⇢(A).

Suppose that

(i) The set {((1� e�it)↵ � A)�1}t2T is R-bounded.

(ii) G (0) = 0 is continuously Frechet di↵erentiable at u = 0 andG 0(0) = 0,

then there exists ⇢⇤ > 0 such that equation (4) is solvable for each⇢ 2 [0, ⇢⇤), with solution u = u⇢ 2 `

p

(Z;X ).


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Maximal `p


Sketch of the proof

Define L(u)(n) = �↵u(n)� Au(n) with domain`p

(Z; [D(A)]). L is an isomorphism.

By (i) the equation �↵u(n)� Au(n) = f (n) has maximal `p

-regularity and then L is surjective.

Given ⇢ 2 (0, 1), we define: H[u, ⇢] = �Lu + G (u) + ⇢f . By(ii) H[0, 0] = 0, H is continuously di↵erentiable at (0, 0) andH1

(0,0) = �L which is invertible.

By the Implicit function theorem there exists ⇢⇤ such thatfor all ⇢ 2 [0, ⇢⇤) there u = u⇢ 2 `

p

(Z;X ) such that

�↵u(n) = Au(n) + G (u(n)) + ⇢f (n).


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Maximal `p


Example 1: The discrete time fractional Fisher equation

d�↵u(n, x) = ux

(n, x)� ku(n, x)(1� u(n, x)) + ✏f (n, x), (5)

where n 2 Z, x 2 R, k and d are positive numbers, ✏ > 0 andf 2 `

p

(Z,X ) is an external force.

When ↵ = 2 and ✏ = 0 the unidimensional discrete Fisher’sequation.

A discrete di↵erential equation could behave di↵erently fromthe corresponding reaction–di↵usion equation (Keener, 1987).


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Maximal `p


Example 1: The existence of solutions

We consider X = L2(R) and let 1 < ↵ < 3.

Our Equation can be modeled as a nonlinear one takingAu = 1

d

u0 � k

d

u and G (u) = k

d

u2 =) Hypothesis (ii) ofTheorem 2 for G is verified.

Since Bu = u0 generates a contraction semigroup on L2(R):

k(�� A)�1k = k(�+k

d� 1

dB)�1k 1

<(�) + k

d

.

We only need to check that supt2T k((1� e�it)↵ � A)�1k is

bounded.


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Maximal `p


Example 1: The set (1� e�it)2

-4 -3,2 -2,4 -1,6 -0,8 0 0,8 1,6 2,4 3,2 4 4,8 5,6

-4

-3,2

-2,4

-1,6

-0,8

0,8

1,6

2,4

3,2

4

4,8

Figure: (1� e�it)2, t 2 T.


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Maximal `p


Example 1: The minimum of the set <[(1� e�it)↵]

Let !(↵) =: mint2T<[(1� e�it)↵] =

(2 + 2 cos(2⇡( 1

↵+1

)))↵/2 cos(↵ arctan(� sin(2⇡( 1

↵+1

))

1+cos(2⇡( 1

↵+1

))

)).

Figure: !(↵), 0 < ↵ < 3.5


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Maximal `p


Example 1: The conclusion

For all d such that 0 < d < �k

!(↵) we have

supt2T

��⇣(1� e�it)↵ � A

⌘�1

�� < 1,

and there exists ✏⇤ > 0 such that our equation admits a solutionu✏ 2 `

p

(Z, L2(R)) for all 0 ✏ < ✏⇤.


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Maximal `p


Example 2: The discrete time fractional Nagumo equation

d�↵u(n, x) = ux

(n, x)+u(n, x)(u(n, x)�a)(1�u(n, x))+✏f (n, x),

n 2 Z, x 2 R, d > 0, 0 < a < 1

2

, 1 < ↵ < 3, ✏ > 0 andf 2 `

p

(Z, L2(R)).When ↵ = 2 and ✏ = 0 it corresponds to the discrete Nagumoequation. It models the spread of genetic traits (Aronson,1975) and the propagation of nerves pulses in a nerve axon.


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Maximal `p


Example 2: The existence of solutions

Let Au = 1

d

u0 � a

d

u and G (u) = � 1

d

u3 + (1 + a

d

)u2 =)Hypothesis (ii) of Theorem 2 for G is verified.

As in Example 1,

k(�� A)�1k = k(�+a

d� 1

dB)�1k 1

<(�) + a

d

,

Then for all d such that,

0 < d <�a

!(↵)

the set {(1� e�it)↵}t2T is contained in ⇢(A) and there exists

✏⇤ > 0 such that the non-homogeneous fractional Nagumoequation admits a solution u✏ 2 `

p

(Z, L2(R)) for all0 ✏ < ✏⇤.


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