Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

38 M. Pires and A. Sequeira

the existence of a relation between the sign of τ

w

at the points θ =0and2π,and

the orientation of the rotation.

A ﬁnal observation is related to the maximum values of the stream function.

Independently of the viscoelastic and viscosity parameters, the maximum values

dramatically increase in the neighborhood of the critical value q

var

. For the cases

where the maximum value of the exponent q is big enough, we observe that after

this peak, the maximum values decrease before stabilizing.

4.3.2. Inertial generalized Oldroyd-B ﬂows. In the previous subsection, we studied

the behavior of the generalized Oldroyd-B ﬂows in the absence of inertia (creeping

ﬂows). Our aim here is to consider the more general case of inertial ﬂows, and to

analyse the eﬀect of the Reynolds number in combination with the Weissenberg

number, the viscosity parameter η, the exponent q, and the curvature ratio δ.

We ﬁrst consider a pipe with a small curvature ratio (δ =0.001) in the case

of a constant viscosity (inertial Oldroyd-B ﬂuid). The secondary ﬂows exist and

the corresponding stream function and wall shear stress have globally the same

behavior as the creeping Oldroyd-B ﬂows. At this stage, the nature of the ﬂow is

qualitatively identical to that of a Newtonian ﬂuid.

In order to compare with the case of generalized creeping ﬂows, the viscosity

parameter η is set to 0.4, 0.5, 0.6 and as previously, several tests were performed for

diﬀerent values of the Weissenberg and the Reynolds numbers, with a continuation

in the exponent q.

One of the ﬁrst remarks is that the transition phenomenon observed in the

case of generalized creeping ﬂows does not hold, even for relatively small Reynolds

number (Re = 15). This fact is evident when η takes the values 0.4and0.5and

the behavior in these two cases is close to that of inertial generalized Newtonian

ﬂows.

Fixing η =0.4, and varying the Reynolds and the Weissenberg numbers,

the ﬂow is globally stable for q less than some critical value. From the contours

of the stream function and of the wall shear stress for the exponent |q|≤5, we

observe that the qualitative behavior is similar and independent of both inertia and

viscoelasticity. The only diﬀerence lies in the values and the magnitude of these

quantities, which clearly depend on the involved parameters. In the neighborhood

of a critical value of q, the behavior presents some changes and is no more uniform.

The rotation already observed initiates, and its orientation depend on Re and We.

Indeed, ﬁxing the Reynolds number to 15, we can see that for We =2,acounter-

clockwise rotation occurs for q

var

= −6 and that the contours remain stable till

we reach the maximum value for which the convergence is ensured. For We =3,

the stream function initiates a very slight counter-clockwise rotation at the same

value q

var

, but recovers the symmetry very quickly. Finally, for We =3,thesame

behavior is captured, but with a very slight clockwise rotation. The wall shear

stresses behave in an analogous way. In order to emphasize the role of the inertia,

we ﬁxed the Weissenberg number and increase the Reynolds number.

Flows of Generalized Oldroyd-B Fluids in Curved Pipes 39

In a second step, we consider the case η =0.5. For Re = 15, we can observe

that the Weissenberg number favorite the clockwise rotation (cf. Figure 7). Indeed,

when the Weissenberg number is small (even with larger exponent viscosity), the

contours remain symmetric. As We increases, the rotation initiates and holds ear-

lier. On the other hand, the Reynolds number does not seem to have a signiﬁcative

inﬂuence on the nature of the ﬂow. We implement several tests corresponding to

larger values of this characteristic parameter (Re =1, 15, 30, 50, 70) and did not

observe any signiﬁcative diﬀerence.

Table 3 summarizes the results of the maximum values of viscosity parame-

ter |q| with respect to the viscoelasticity (We) and the inertia (Re) parameters,

as a function of the curvature ratio (δ). For ﬁxed δ and We,thevalueof|q| de-

creases when Re increases. The same occurs if δ and Re are ﬁxed: the value of

|q| decreases when We increases. The curvature ratio associated with (Re)and

(We) has a strong inﬂuence on the convergence, since it can be shown that when

these parameters increase, the values of |q| decrease considerably and in some cases

convergence is not achieved.

Figure 7. Streamlines and wall shear stress for inertial generali-

zed Oldroyd-B ﬂows with η =0.5,Re= 15, We = 1 and diﬀerent

values of the |q| (δ =0.001).

The case η =0.6 is certainly the more surprising. For large values of the

Reynolds number and for the achieved viscosity exponents, the behavior seems to

be stable and no notable fact can be observed. The more interesting variations

were observed for relatively small Reynolds numbers. Fixing for example Re =15

and varying the Weissenberg number, we observe that the behavior is qualitatively

stable for small values of this parameter. For We = 3, 4 and 5, some new phenom-

enon initiates. The characteristics are very similar to those observed in the case

10−1

1

0−1

1

0−1

−1

0

1

−1

0

1

−1

0

1

−0.000557

−

0.000

509

−0.00046

−0.000412

−0.000364

−0.000315

−0.000267

−0.000218

−0.0

00

17

−0.00017

−0.000121

−7.29e−05

−2.44e−05

2.4e−05

7.25e−05

0.000121

0.000169

0.000218

0.000266

0.000315

0.000363

0.000412

0.00046

0.000509

0.000557

(a)

−0.00169

−0.00154

−

0.00

14

−0.00125

−0.0011

−0.000956

−0.00081

−0.000663

−

0.000663

−0.000663

−0.000517

−0.00037

−0.000224

−7.76e−05

6.88e−05

0.000215

0.000362

0.000508

0.000654

0.000801

0.000947

0.00109

0.00124

0.00139

0.00153

0.00168

(b)

−0.00129

−0.00118

−0.00107

−0.000956

−0.000844

−0.000731

−0.000618

−0.0005

06

−0.000506

−0.000393

−0.000281

−0.000168

−5.53e−05

−5.53e−

05

−5.53e−05

5.73e−05

0.00017

0.000283

0.000395

0.000508

0.00062

0.000733

337000.0

0.000846

0.000958

0.00107

0.00118

0.0013

(c)

1 2 3 4 5 6

−0.01

−0.005

0.005

0.01

t

w

q

q 4.80

q 4.25

q 4.20

q = −4.2

q = −4.25 q = −4.8

40 M. Pires and A. Sequeira

We 1 2 3 4 5

δ =0.001

Re =1 5.12 4.65 4.45 4.22 4.0

15 4.81 4.36 4.25 4.21 3.95

30 4.78 3.76 3.54 3.37 3.25

50 3.60 3.03 2.81 2.62 2.46

70 3.44 2.57 2.23 1.94 1.72

δ =0.1

Re =1 2.47 0.7 0.46 0.36 0.3

15 0.51 0.15 − − −

30 0.05 − − − −

50 − − − − −

δ =0.2

Re =1 1.26 0.53 0.36 − −

15 0.23 − − − −

30 − − − − −

Table 3. Maximum values of |q| with η =0.5.

of generalized creeping ﬂows during the phase transition: formation of boundary

layers and a pair of new vortices or strengthening of the new contours and weak-

ening of the original ones. However, in contrast with the creeping ﬂows, the new

state is not stable and at some level the inverse phenomenon initiates (Figure 8).

For this particular viscosity parameter, Table 4 shows the results of the max-

imum values of |q|, obtained for diﬀerent We and Re numbers,inthecaseof

δ =0.001. Comparing with Table 3 for the same curvature ratio, the same eﬀects

of viscoelasticity and inertia on |q| can be observed.

We 1 2 3 4 5

δ =0.001

Re =1 3.10 2.89 2.68 2.6 2.56

15 2.88 2.67 2.60 2.55 2.51

70 1.69 1.39 1.20 1.05 0.95

Table 4. Maximum values of |q| with η =0.6.

5. Conclusion

This paper is devoted to ﬁnite element simulations of ﬂows of incompressible vis-

coelastic non-Newtonian ﬂuids of Oldroyd-type through pipes of uniform circular

cross-section, and follows the work already published in [1] and [3] for generalized

Newtonian ﬂuids. We compare the quantitative and qualitative behavior of the

Flows of Generalized Oldroyd-B Fluids in Curved Pipes 41

Figure 8. Streamlines and wall shear stress for inertial genera-

lized Oldroyd-B ﬂows with η=0.6, Re=15 and We=5 (δ=0.001).

secondary streamlines and the wall shear stress for creeping and inertial generali-

zed Oldroyd-B ﬂows, performing computations for diﬀerent values of the Reynolds

number, the Weissenberg number, the curvature ratio and the non-dimensional

viscosity parameters involved in the governing equations.

In particular, we observe interesting viscosity eﬀects such that, for small

curvature ratio and within a certain range of viscosity parameters, the secondary

streamlines contours undergo a counter-clockwise rotation and lose symmetry. The

complexity of the ﬂow characteristics shown in the numerical tests suggest that

further theoretical analysis is needed to study the existence of more than one

solution and investigate the corresponding stability, for a range of appropriate

non-dimensional parameters.

More detailed discussion and numerical results can be found in [19] where

the generalized Newtonian ﬂows are obtained as a particular case of generalized

Oldroyd-B ﬂows, in the limit of vanish Weissenberg number (neglected viscoelastic

eﬀects). The numerical validation of the present results, using the perturbation

method [24] is a work in progress.

10−1 10−1 10−1 10−1

10−110−110−110−1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−1

0

1

−2.24e−05

−

2.05e−05

−1.85e−05

−1.66e

−05

−1.66e−

05

−1.46e−05

−1.27e−05

−1.07e−05

−8.77e−06

−8.77e−0

6

−6.82e−06

−6.82e−

06

−6.82e−06

−4.8

8e−

06

−4.88e−06

−2.93e−06

−9.77e−07

9.72

e−0

7

9.72e−07

9.72e−0

7

2.92e−06

4.87e−06

6.82e−06

8

.77e−0

6

8.77e−06

1.07

e−0

5

1.07e−05

1.27e−05

1.46e−05

1.66e−05

1.85e−05

1.8

5e−0

5

2.05e

−05

2.24e−05

(a)

−1.51e−05

−1.38e−05

−1.25e−05

−

1.12e−05

−1.12e−05

−

9.8

4e

−0

6

−9.84e−06

−9.8

4e−06

−9.84e−06

−8.53e−06

−7

.21

e−

06

−7.21e−06

−5.9e−06

−5

.9

e−06

−5.9e−06

−5.9e

−06

−5.9e−06

−4.58e−06

−3.27e−06

−

3.27e−06

−3.27e−06

−

3.27e−0

6

−1.95e−06

−6.4e−07

−6

.4e−0

7

−6.4e−07

−6.

4e−07

−6.4e−07

−6.4e−

07

−6. 4

e−

07

−6.4e−07

6.74e−07

1.99e−06

1.9

9e−06

6

0

−

e

3

.

3

3.3e−06

4.62e−06

5

.93

e−0

6

5.93e−06

7.25

e−06

7.25e−06

8.56e−06

8

.56

e−06

8.56e−06

9.87

e−06

9.87e−06

9.87e−

06

9.87e−06

9.87e−

06

1.12e−05

1.12e−

05

1.12e−05

1.25e−05

1.25e−

05

1

.25e

−05

1.25e−05

1.38e−05

1.51e−05

(b)

−3.03e−05

−2.76e−05

−2.5e−05

−2.24e−05

−

2.24e−05

−2.24e−05

−1.97e−05

−

1.97e−05

−1.97e−05

−1.7

1e−05

−1.71e−05

−1.44e−05

−1.44e−

05

−1.18e−05

−9.17e−06

−6.53e−

06

−6.53e−06

−6

.53e−06

−6.53e−06

−3.89e−06

−3.89e−0

6

−3.89e−06

−

3

.8

9e

−

0

6

−1.25e−06

−1.25

e

−

06

−1.25e−06

1.39e−06

1.3

9e

−0

6

1.39e−06

4.03e−06

4.03e

−06

6

0

−

e

3

0

.

4

4.03e−06

4.03

e−06

4.03e−06

6

0

−

e

7

6

.

6

6.6

7e−06

6.67e−06

6.67e−0

6

6.67e−06

9.31e−06

9.31e

−06

9.31e−06

1.19e−05

1.19e−

05

1.19e−05

1.46e−0

5

1.46e−05

1.72e−05

1.72

e−05

1.72e−05

1.99e−05

2

.25e−05

2.25e−05

2.51e−05

2.78e−

05

2.78e−05

3.04e−05

(c)

−4.92e−05

−4.49e−05

−4.07e−05

−3.64e−05

5

0

−

e

1

2

.

3

−

−3.21e−05

−2.78e−05

−

2.78e−05

−2.78e−05

−2.35e−05

−1.92e−05

−1.9

2e−05

−1

.92e−

05

−1.92e−

05

−1.49e−05

−1.06e−0

5

−1.06e−05

−1.06

e−05

−1.06e−05

−6.31e−06

−6

.3

1e−06

−6.31e−06

−2.02e−06

−2.02e−

06

2.27e−06

2.27e

−0

6

2.27e−06

2.27e

−0

6

2.27e−06

6.57e−

06

6.5

7e−06

6.57e−

06

6.57e−06

6.57e−

1.09e−05

1.09e−

05

1.52e−05

1.

52

e−

05

1.94e−05

1.94

e−

05

1.94e

−05

2

.37

e−05

2.37e−05

2.8e−05

3.23e−

05

3.23e−05

3.66e−05

4

.09e

−05

4.09e−05

4.52e−05

4.95e−05

4.9

5e−

05

(d)

−0.000235

−0.000214

−0.000194

−0.000173

−0.000153

−0.000132

−0.000111

−

0.000111

−9.09e−05

−7.04e−05

−

7.04e

−05

−7.04e−05

−

7.0

4e−0

5

−4.98e−05

−

4

.98

e−

0

5

−4.9

8e−05

−4.98e−05

−4.98

e−0

5

−2.93e−05

−

2.93

e

−0

5

−2.93e−05

−8.69e−06

−8.69

e−06

−8.69e−06

1.19e−05

1.19e−0

5

1.19e−05

3.24e−05

5.3e−05

7.36e−05

7.3

6e−05

7.36e−05

9.41e−05

9.41e−

05

9.41e−05

0.000115

0.000135

0.000156

0.00

0176

0.000176

0.000197

0.000217

0.000238

(e)

−0.000201

−0.000183

−0.000166

−0.000148

−0.000131

−0.000113

−9.54

e−05

−9.54e−

05

−

9.54

e−0

5

−9.54e−05

−7.78e−05

−7.78e−

05

−

7.78e−05

−7.78e−05

−6.03e−05

−6.0

3e−05

−6.03e−05

−4.27e−05

−

4.2

7e−05

−4.27e−05

−4.27e−

05

−2.51e−05

−7.58e−06

−

7

.5

8e−

0

6

−7.58e−06

9.99e−06

9.99e

−06

9.99e−06

2.76e−05

4.51e−05

4.51e

−05

4.51e−05

4.51e−

05

6.27e−05

8.02e−05

9.78e−05

9.78e−

05

9.78e−05

0.000115

0.000133

0.000151

0.000168

0.000186

0.000203

(f)

−0.000155

−0

.0

0

01

42

−0.000142

−0.000128

−0.000

115

−0.000115

−0.000101

−8.75e−05

−7.39e−05

−6.04e−05

−4

.68e−05

−4.68e−05

−3.32e−05

−3

.32e−05

−3.32e−05

−1.96e

−05

−1.96e−05

−1.96e−0

5

−1.96e−05

−6.04e−06

−6.04e−

06

−6.04e−

06

7.53e−06

2.11e−05

3.47e−05

4.83e−05

6.18e−05

7.54e−05

7.54

e−0

5

7.54e−05

8.9e−05

8.9e−0

5

0.0001

03

0.000103

0.0001

03

0.000116

0.0

0011

6

0.000116

0.00013

0.000143

0.000157

(g)

−0.000322

−0.000

29

2

−0.000262

−0.000232

−

0.000232

−0.000202

−0.000172

−0

.000172

−0.000142

−

0.0

00112

−0.0

00112

−8.22e−05

−5.22e−05

−

5.22e−05

−5.22e−05

−2.21e−05

−2.21e

−05

−2.21e−05

7.87e−06

3.79e−05

3.7

9e

−0

5

3.79e−05

6.79e−05

9.79e−05

0.000128

0.000158

0.0

00158

0.000188

0.000218

0.000248

0.000278

0.000308

(h)

2 3 4 5 6

q

−0.015

−0.01

−0.005

0.005

0.01

0.015

t

w

q 2.51

q 2.50

q 2.40

q 2.30

q 2.25

q 2.10

q 2.0

q 1.90

q 1.70

q 1.0

q 0.70

q 0.80

q 0.55

q 0.50

q 0.45

q 0.36

q = −0.36

q = −2

q = −2.25

q = −2.4

q = −2.5

q = −0.45

q = −0.5

q = −0.55

1

42 M. Pires and A. Sequeira

References

[1]N.Arada,M.PiresandA.Sequeira, Viscosity eﬀects on ﬂows of generalized New-

tonian ﬂuids through curved pipes, Computers and Mathematics with Applications,

53 (2007), 625–646.

[2] N. Arada, M. Pires and A. Sequeira, Numerical approximation of viscoelastic

Oldroyd-B ﬂows in curved pipes,RIMS,Kˆokyˆuroku Bessatsu, B1 (2007), 43–70.

[3]N.Arada,M.PiresandA.Sequeira,Numerical simulations of shear-thinning

Oldroyd-B ﬂuids in curved pipes,IASMETransactions,issue 6, Vol 2 (2005), 948–959.

[4] A.A. Berger, L. Talbot and L.-S. Yao, Flow in curved pipes, Ann. Rev. Fluid Mech.,

15 (1983), 461–512.

[5] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids,John

Wiley & Sons, New York (1987).

[6] S. Chien, S. Usami, R.J. Dellemback, M.I. Gregersen, Blood viscosity: inﬂuence of

erythrocyte deformation, Science 157 (3790), 827–829, 1967.

[7] S. Chien, S. Usami, R.J. Dellemback, M.I. Gregersen, Blood viscosity: inﬂuence of

erythrocyte aggregation, Science 157 (3790), 829–831, 1967.

[8] P. Daskopoulos and A.M. Lenhoﬀ, Flow in curved ducts: bifurcation structure for

stationary ducts, J. Fluid Mech., 203 (1989), 125–148.

[9] W.R. Dean, Note on the motion of ﬂuid in curved pipe, Philos. Mag., 20, 208 (1927).

[10] W.R. Dean, The streamline motion of ﬂuid in curved pipe, Philos. Mag., 30, 673

(1928).

[11] J. Eustice, Flow of water in curved pipes, Proc. R. Soc. Lond. A 84 (1910), 107–118.

[12] J. Eustice, Experiments of streamline motion in curved pipes, Proc. R. Soc. Lond. A

85 (1911), 119–131.

[13] Y. Fan, R.I. Tanner and N. Phan-Thien, Fully developed viscous and viscoelastic

ﬂows in curved pipes, J. Fluid Mech., 440 (2001), 327–357.

[14] J.H. Grindley and A.H. Gibson, On the frictional resistance to the ﬂow of air through

apipe, Proc. R. Soc. Lond. A 80 (1908), 114–139.

[15] D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer Verlag, New York

(1990).

[16] R. Keunings, A Survey of Computational Rheology, In: Proceedings of the XIII In-

ternational Congress on Rheology (D.M. Binding et al. ed.), British Soc. Rheol., 1

(2000), 7–14.

[17] R. Owens and T.N. Phillips, Computational Rheology, Imperial College Press, Lon-

don (2002).

[18] T.J. Pedley, Fluid Mechanics of Large Blood Vessels, Cambridge University Press,

Cambridge (1980).

[19] M. Pires, Mathematical and Numerical Analysis of Non-Newtonian Fluids in Curved

Pipes, PhD thesis, IST, Lisbon (2005)

[20] K.R. Rajagopal, Mechanics of non-Newtonian Fluids,Galdi,G.P.andJ.Neˇcas (eds.)

Recent Developments in Theoretical Fluid Mechanics, Pitman Research Notes in

Mathematics 291 (1993), Longman Scientiﬁc an Technical, 129–162.

[21] M. Renardy, Mathematical Analysis of Viscoelastic Flows, SIAM (2000).

Flows of Generalized Oldroyd-B Fluids in Curved Pipes 43

[22] M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical problems in Viscoelasticity,

Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman 35

(1987).

[23] A.M. Robertson, On viscous ﬂow in curved pipes of non-uniform cross section,Inter.

J. Numer. Meth. ﬂuid., 22 (1996), 771–798.

[24] A.M. Robertson and S.J. Muller, Flow of Oldroyd-B ﬂuids in curved pipes of circular

and annular cross-section, Int. J. Non-Linear Mechanics, 31 (1996), 1–20.

[25] A.M. Robertson, A. Sequeira and M. Kameneva, Hemorheology, Hemodynamical

Flows: Modeling, Analysis and Simulation, Series: Oberwolfach Seminars, G.P. Galdi,

R. Rannacher, A.M. Robertson, S. Turek, Birkh¨auser, 37 (2008), 63–120.

[26] W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, New York

(1978).

[27] W.Y. Soh and S.A. Berger, Fully developed ﬂow in a curved pipe of arbitrary curva-

ture ratio, Int. J. Numer. Meth. Fluid., 7 (1987), 733–755.

[28] G.B. Thurston, Viscoelasticity of human blood, Biophys. J., 12 (1972), 1205–1217.

[29] C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics,Encyclope-

diaofPhysics,(ed.S.Flugge),vol.III/3 (1965), Springer-Verlag.

[30] G.S. Williams, C.W. Hubbell and G.H. Fenkell, Experiments at Detroit, Mich. on the

eﬀect of curvature upon the ﬂow of water in pipes, Trans. ASCE 47 (1902), 1–196.

[31] Z.H. Yang and H.B. Keller, Multiple laminar ﬂows through curved pipes, Appl. Nu-

mer. Math., 2 (1986), 257–271.

Mar´ılia Pires

Centro de Investiga¸c˜ao em Matem´atica e Aplica¸c˜oes

Universidade de

´

Evora

Rua Rom˜ao Ramalho,

7000-671

´

Evora

Portugal

e-mail: marilia@uevora.pt

Ad´elia Sequeira

Centro de Matem´atica e Aplica¸c˜oes

Instituto Superior T´ecnico

Av. Rovisco Pais,

1049-001 Lisboa

Portugal

e-mail: adelia.sequeira@math.ist.utl.pt

Progress in Nonlinear Diﬀerential Equations

and Their Applications, Vol. 80, 45–55

c

 2011 Springer Basel AG

Remarks on Maximal Regularity

Pascal Auscher and Andreas Axelsson

In honour of H. Amann

Abstract. We prove weighted estimates for the maximal regularity operator.

Such estimates were motivated by boundary value problems. We take this

opportunity to study a class of weak solutions to the abstract Cauchy prob-

lem. We also give a new proof of maximal regularity for closed and maximal

accretive operators following from Kato’s inequality for fractional powers and

almost orthogonality arguments.

Mathematics Subject Classiﬁcation (2000). Primary 47D06; Secondary 35K90,

47A60.

Keywords. Maximal regularity, weighted estimates, abstract Cauchy problem,

Kato’s inequality, fractional powers, Cotlar’s lemma.

1. Weighted estimates for the maximal regularity operator

Assume −A is a densely deﬁned, closed linear operator, generating a bounded

analytic semigroup {e

−zA

, |arg z | <δ},0<δ<π/2, on a Hilbert space H.

Equivalently, A is sectorial of type ω(A)=π/2 − δ.LetD(A)denoteitsdomain.

The maximal regularity operator is deﬁned by the formula

M

+

f(t)=



t

0

Ae

−(t−s)A

f(s) ds.

This operator is associated to the forward abstract evolution equation

˙u(t)+Au(t)=f(t),t> 0; u(0) = 0

as for appropriate f, Au(t)=M

+

f(t). An estimate on M

+

f inthesamespace

as f gives therefore bounds on ˙u and Au separately. See Section 2.

The integral deﬁning M

+

f converges strongly in H for each t>0and

f ∈ L

2

(0, ∞; dt, D(A)). The estimate Ae

−(t−s)A

≤C(t − s)

−1

following from

the analyticity of the semigroup shows that the integral is singular if one only

assumes f(s) ∈H. The maximal regularity operator is an example of a singular

46 P. Auscher and A. Axelsson

integral operator with operator-valued kernel. The celebrated theorem by de Simon

[4] asserts

Theorem 1.1. Assume −A generates a bounded holomorphic semigroup in H.The

operator M

+

, initially deﬁned on L

2

(0, ∞; dt, D(A)), extends to a bounded operator

on L

2

(0, ∞; dt, H).

Motivated by boundary value problems for some second-order elliptic equa-

tions, we proved in [3] the following result.

Theorem 1.2. Assume −A generates a bounded holomorphic semigroup in

H and furthermore that A has bounded holomorphic functional calculus, then

M

+

, initially deﬁned on L

2

c

(0, ∞; dt, D(A)), extends to a bounded operator on

L

2

(0, ∞; t

β

dt, H) for all β ∈ (−∞, 1).

Here and in what follows the subscript

c

means with compact support.

The proof given there uses the operational calculus deﬁned in the thesis of

Albrecht [1]. It used as an assumption that A has bounded holomorphic functional

calculus as deﬁned by McIntosh [9]. Under this assumption estimates of integral op-

erators more general than the maximal regularity operator, with operator-kernels

deﬁned through functional calculus of A, were proved and gave other useful infor-

mation to understand also the case β = 1 needed for the boundary value problems.

However, not all generators of bounded analytic semigroups have a bounded holo-

morphic functional calculus (see [10], and Kunstmann and Weis [6, Section 11]

for a list of equivalent conditions.) So if we only consider the maximal regularity

operator, it is natural to ask whether one can drop the assumption on bounded

holomorphic functional calculus in Theorem 1.2. It is indeed the case and as we

shall see the proof is extremely simple assuming we know Theorem 1.1.

Theorem 1.3. Let −A be the generator of a bounded analytic semigroup on H.

Then M

+

, initially deﬁned on L

2

c

(0, ∞; dt, D(A)), extends to a bounded operator

on L

2

(0, ∞; t

β

dt, H) for all β ∈ (−∞, 1).

The subscript

c

means with compact support in (0, ∞). Set |||f(t)|||

2

=



∞

0

f(t)

2

dt

t

(we leave in the t-variable in the notation for convenience). As we

often use it, we recall the following simpliﬁed version of Schur’s lemma: if U(t, s),

s, t > 0, are bounded linear operators on H with bounds U(t, s)≤h(t/s)and

C =



∞

0

h(u)

du

u

< ∞,then

"



∞

0

U(t, s)f(s)

ds

s

"

≤ C |||f(s)|||.

Proof of Theorem 1.3. Let β<1. For β = 0, this is Theorem 1.1. Assume β =0

and set α = β/2. Observe that

M

+

f(t)

L

2

(t

β

dt,H)

= t

α

M

+

f(t)

L

2

(dt,H)

.

Remarks on Maximal Regularity 47

We have, with f

α

(s)=s

α

f(s),

t

α

M

+

f(t)=M

+

(f

α

)(t)+



t

0

Ae

−(t−s)A

(t

α

− s

α

)f(s) ds.

For the ﬁrst term apply Theorem 1.1. For the second, write

#



t

0

Ae

−(t−s)A

(t

α

−s

α

)f(s) ds

#

L

2

(dt,H)

=

"



∞

0

U(t, s)g(s)

ds

s

"

with g(s)=s

1/2+α

f(s)andU(t, s)=Ae

−(t−s)A

(t

α

−s

α

)s

1/2−α

t

1/2

for s<tand

0 otherwise. Since |||g(t)||| = f

L

2

(t

β

dt,H)

, it remains to estimate the norm of

U(t, s)onH.Wehave

U(t, s)≤C

|t

α

− s

α

|

|t − s|

s

1/2−α

t

1/2

,s<t.

It is easy to see that it is on the order of (s/t)

1/2−max(α,0)

as s<t. We conclude

by applying Schur’s lemma. 

Let

M

−

f(t)=



∞

t

Ae

−(s−t)A

f(s) ds.

This operator is associated to the backward abstract evolution equation

˙v(t) − Av(t)=f(t),t>0; v(∞)=0

as for appropriate f, Av(t)=−M

−

f(t).

Corollary 1.4. Assume that −A generates a bounded analytic semigroup on H.

Then M

−

, initially deﬁned on L

2

c

(0, ∞; dt, D(A)), extends to a bounded operator

on L

2

(0, ∞; t

β

dt, H) for all β ∈ (−1, ∞).

Proof. Observe that the adjoint of M

−

in L

2

(0,∞;t

β

dt,H) for the duality deﬁned

by L

2

(0,∞;dt,H)isM

+

in L

2

(0, ∞; t

−β

dt, H) associated to A

∗

and apply Theorem

1.3. 

We next show that the range of β is optimal in both results.

Theorem 1.5. For any non zero −A generating a bounded analytic semigroup on

H and β ≥ 1, M

+

is not bounded on L

2

(0, ∞; t

β

dt, H) and M

−

is not bounded

on L

2

(0, ∞; t

−β

dt, H).

Proof. It suﬃces to consider M

−

.SinceA =0,R(A), the closure of the range of

A, contains non zero elements. As

R(A)∩D(A) is dense in it, pick u ∈ R(A)∩D(A),

u =0,andsetf(t)=u for 1 ≤ t ≤ 2 and 0 elsewhere. Then f ∈ L

2

c

(0, ∞; dt, D(A))

and f ∈ L

2

(0, ∞; t

−β

dt, H)withf(t)

L

2

(0,∞;t

−β

dt,H)

= c

β

u < ∞.Fort<1,

one has

M

−

f(t)=(e

−(1−t)A

− e

−(2−t)A

)u,

which converges to (e

−A

− e

−2A

)u in H when t → 0.

Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift

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