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October  2013, 6(5): 1259-1275. doi: 10.3934/dcdss.2013.6.1259

$H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions

1. 

TU Darmstadt, FB Mathematik, Schlossgartenstr 7, D-64289 Darmstadt, Germany, Germany, Germany

Received  January 2012 Revised  February 2012 Published  March 2013

In this paper we prove that the $L^p$ realisation of a system of Laplace operators subjected to mixed first and zero order boundary conditions admits a bounded $H^{\infty}$-calculus. Furthermore, we apply this result to the Magnetohydrodynamic equation with perfectly conducting wall condition.
Citation: Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259
References:
[1]

H. Abels, Bounded imaginary powers and $H_\infty$-calculus of the Stokes operator in unbounded domains,, in, 64 (2005), 1. doi: 10.1007/3-7643-7385-7_1.

[2]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141. doi: 10.3934/dcdss.2010.3.141.

[3]

T. Akiyama, H. Kasai, Y. Shibata and M. Tsutsumi, On a resolvent estimate of a system of Laplace operators with perfect wall condition,, Funkcial. Ekvac., 47 (2004), 361. doi: 10.1619/fesi.47.361.

[4]

J. Bolik and W. von Wahl, Estimating $\nablau$ in terms of div $u$, curl $u$ either $(\nu,u)$ or $\nu \times u$ and the topology,, Math. Methods Appl. Sci., 20 (1997), 737. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.

[5]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^{\infty}$ functional calculus,, J. Austral. Math. Soc. Ser. A, 60 (1996), 51.

[6]

T. G. Cowling, "Magnetohydrodynamics,", Interscience Tracts on Physics and Astronomy, (1957).

[7]

R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni, New thoughts on old results of R. T. Seeley,, Math. Ann., 328 (2004), 545. doi: 10.1007/s00208-003-0493-y.

[8]

E. Dintelmann, M. Geissert and M. Hieber, Strong $L^p$-solutions to the Navier-Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity,, Trans. Amer. Math. Soc., 361 (2009), 653. doi: 10.1090/S0002-9947-08-04684-9.

[9]

R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003).

[10]

G. Dore and A. Venni, On the closedness of the sum of two closed operators,, Math. Z., 196 (1987), 189. doi: 10.1007/BF01163654.

[11]

M. Haase, "The Functional Calculus for Sectorial Operators,", Operator Theory: Advances and Applications, 169 (2006). doi: 10.1007/3-7643-7698-8.

[12]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains,, Arch. Math. (Basel), 91 (2008), 178. doi: 10.1007/s00013-008-2621-0.

[13]

N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators,, Math. Ann., 336 (2006), 747. doi: 10.1007/s00208-005-0742-3.

[14]

L. D. Landau and E. M. Lifschitz, "Lehrbuch der Theoretischen Physik ('Landau-Lifschitz'), Band VIII,", Fourth edition, (1985).

[15]

A. McIntosh, Operators which have an $H_\infty$ functional calculus,, in, 14 (1986), 210.

[16]

M. Mitrea and S. Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds,, Trans. Amer. Math. Soc., 361 (2009), 3125. doi: 10.1090/S0002-9947-08-04827-7.

[17]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on $L_q$-spaces,, Math. Z., 244 (2003), 651.

[18]

R. T. Seeley, Complex powers of an elliptic operator,, in, (1967), 288.

[19]

R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. Math., 91 (1969), 889.

[20]

R. Seeley, Norms and domains of the complex powers $A_Bz$,, Amer. J. Math., 93 (1971), 299.

[21]

J. A. Shercliff, "A Textbook of Magnetohydrodyamics,", Pergamon Press, (1965).

[22]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity,, Math. Ann., 319 (2001), 735. doi: 10.1007/PL00004457.

[23]

Z. Yoshida and Y. Giga, On the Ohm-Navier-Stokes system in magnetohydrodynamics,, J. Math. Phys., 24 (1983), 2860. doi: 10.1063/1.525667.

show all references

References:
[1]

H. Abels, Bounded imaginary powers and $H_\infty$-calculus of the Stokes operator in unbounded domains,, in, 64 (2005), 1. doi: 10.1007/3-7643-7385-7_1.

[2]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141. doi: 10.3934/dcdss.2010.3.141.

[3]

T. Akiyama, H. Kasai, Y. Shibata and M. Tsutsumi, On a resolvent estimate of a system of Laplace operators with perfect wall condition,, Funkcial. Ekvac., 47 (2004), 361. doi: 10.1619/fesi.47.361.

[4]

J. Bolik and W. von Wahl, Estimating $\nablau$ in terms of div $u$, curl $u$ either $(\nu,u)$ or $\nu \times u$ and the topology,, Math. Methods Appl. Sci., 20 (1997), 737. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.

[5]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^{\infty}$ functional calculus,, J. Austral. Math. Soc. Ser. A, 60 (1996), 51.

[6]

T. G. Cowling, "Magnetohydrodynamics,", Interscience Tracts on Physics and Astronomy, (1957).

[7]

R. Denk, G. Dore, M. Hieber, J. Prüss and A. Venni, New thoughts on old results of R. T. Seeley,, Math. Ann., 328 (2004), 545. doi: 10.1007/s00208-003-0493-y.

[8]

E. Dintelmann, M. Geissert and M. Hieber, Strong $L^p$-solutions to the Navier-Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity,, Trans. Amer. Math. Soc., 361 (2009), 653. doi: 10.1090/S0002-9947-08-04684-9.

[9]

R. Denk, M. Hieber and J. Prüss, $\mathcal R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type,, Mem. Amer. Math. Soc., 166 (2003).

[10]

G. Dore and A. Venni, On the closedness of the sum of two closed operators,, Math. Z., 196 (1987), 189. doi: 10.1007/BF01163654.

[11]

M. Haase, "The Functional Calculus for Sectorial Operators,", Operator Theory: Advances and Applications, 169 (2006). doi: 10.1007/3-7643-7698-8.

[12]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains,, Arch. Math. (Basel), 91 (2008), 178. doi: 10.1007/s00013-008-2621-0.

[13]

N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators,, Math. Ann., 336 (2006), 747. doi: 10.1007/s00208-005-0742-3.

[14]

L. D. Landau and E. M. Lifschitz, "Lehrbuch der Theoretischen Physik ('Landau-Lifschitz'), Band VIII,", Fourth edition, (1985).

[15]

A. McIntosh, Operators which have an $H_\infty$ functional calculus,, in, 14 (1986), 210.

[16]

M. Mitrea and S. Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds,, Trans. Amer. Math. Soc., 361 (2009), 3125. doi: 10.1090/S0002-9947-08-04827-7.

[17]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on $L_q$-spaces,, Math. Z., 244 (2003), 651.

[18]

R. T. Seeley, Complex powers of an elliptic operator,, in, (1967), 288.

[19]

R. Seeley, The resolvent of an elliptic boundary problem,, Amer. J. Math., 91 (1969), 889.

[20]

R. Seeley, Norms and domains of the complex powers $A_Bz$,, Amer. J. Math., 93 (1971), 299.

[21]

J. A. Shercliff, "A Textbook of Magnetohydrodyamics,", Pergamon Press, (1965).

[22]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity,, Math. Ann., 319 (2001), 735. doi: 10.1007/PL00004457.

[23]

Z. Yoshida and Y. Giga, On the Ohm-Navier-Stokes system in magnetohydrodynamics,, J. Math. Phys., 24 (1983), 2860. doi: 10.1063/1.525667.

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