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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 "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 1-15. doi: 10.1007/3-7643-7385-7_1.  Google Scholar

[2]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157. doi: 10.3934/dcdss.2010.3.141.  Google Scholar

[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), 361-394. doi: 10.1619/fesi.47.361.  Google Scholar

[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), 737-744. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.  Google Scholar

[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), 51-89.  Google Scholar

[6]

T. G. Cowling, "Magnetohydrodynamics," Interscience Tracts on Physics and Astronomy, No. 4, Interscience Publishers, Inc., New York, 1957.  Google Scholar

[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), 545-583. doi: 10.1007/s00208-003-0493-y.  Google Scholar

[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), 653-669. doi: 10.1090/S0002-9947-08-04684-9.  Google Scholar

[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), viii+114.  Google Scholar

[10]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[11]

M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[12]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains, Arch. Math. (Basel), 91 (2008), 178-186. doi: 10.1007/s00013-008-2621-0.  Google Scholar

[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), 747-801. doi: 10.1007/s00208-005-0742-3.  Google Scholar

[14]

L. D. Landau and E. M. Lifschitz, "Lehrbuch der Theoretischen Physik ('Landau-Lifschitz'), Band VIII," Fourth edition, Elektrodynamik der Kontinua [Electrodynamics of continua], Translated from the second Russian edition by S. L. Drechsler, Translation edited by Gerd Lehmann, With a foreword by P. Ziesche and Lehmann, Akademie-Verlag, Berlin, 1985,  Google Scholar

[15]

A. McIntosh, Operators which have an $H_\infty$ functional calculus, in "Miniconference on Operator Theory and Partial Differential Equations" (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, (1986), 210-231.  Google Scholar

[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), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7.  Google Scholar

[17]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on $L_q$-spaces, Math. Z., 244 (2003), 651-688.  Google Scholar

[18]

R. T. Seeley, Complex powers of an elliptic operator, in "Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., (1967), 288-307.  Google Scholar

[19]

R. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math., 91 (1969), 889-920.  Google Scholar

[20]

R. Seeley, Norms and domains of the complex powers $A_Bz$, Amer. J. Math., 93 (1971), 299-309.  Google Scholar

[21]

J. A. Shercliff, "A Textbook of Magnetohydrodyamics," Pergamon Press, Oxford-New York-Paris, 1965.  Google Scholar

[22]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann., 319 (2001), 735-758. doi: 10.1007/PL00004457.  Google Scholar

[23]

Z. Yoshida and Y. Giga, On the Ohm-Navier-Stokes system in magnetohydrodynamics, J. Math. Phys., 24 (1983), 2860-2864. doi: 10.1063/1.525667.  Google Scholar

show all references

References:
[1]

H. Abels, Bounded imaginary powers and $H_\infty$-calculus of the Stokes operator in unbounded domains, in "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 1-15. doi: 10.1007/3-7643-7385-7_1.  Google Scholar

[2]

H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157. doi: 10.3934/dcdss.2010.3.141.  Google Scholar

[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), 361-394. doi: 10.1619/fesi.47.361.  Google Scholar

[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), 737-744. doi: 10.1002/(SICI)1099-1476(199706)20:9<737::AID-MMA863>3.3.CO;2-9.  Google Scholar

[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), 51-89.  Google Scholar

[6]

T. G. Cowling, "Magnetohydrodynamics," Interscience Tracts on Physics and Astronomy, No. 4, Interscience Publishers, Inc., New York, 1957.  Google Scholar

[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), 545-583. doi: 10.1007/s00208-003-0493-y.  Google Scholar

[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), 653-669. doi: 10.1090/S0002-9947-08-04684-9.  Google Scholar

[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), viii+114.  Google Scholar

[10]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[11]

M. Haase, "The Functional Calculus for Sectorial Operators," Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[12]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains, Arch. Math. (Basel), 91 (2008), 178-186. doi: 10.1007/s00013-008-2621-0.  Google Scholar

[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), 747-801. doi: 10.1007/s00208-005-0742-3.  Google Scholar

[14]

L. D. Landau and E. M. Lifschitz, "Lehrbuch der Theoretischen Physik ('Landau-Lifschitz'), Band VIII," Fourth edition, Elektrodynamik der Kontinua [Electrodynamics of continua], Translated from the second Russian edition by S. L. Drechsler, Translation edited by Gerd Lehmann, With a foreword by P. Ziesche and Lehmann, Akademie-Verlag, Berlin, 1985,  Google Scholar

[15]

A. McIntosh, Operators which have an $H_\infty$ functional calculus, in "Miniconference on Operator Theory and Partial Differential Equations" (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., 14, Austral. Nat. Univ., Canberra, (1986), 210-231.  Google Scholar

[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), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7.  Google Scholar

[17]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on $L_q$-spaces, Math. Z., 244 (2003), 651-688.  Google Scholar

[18]

R. T. Seeley, Complex powers of an elliptic operator, in "Singular Integrals" (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., (1967), 288-307.  Google Scholar

[19]

R. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math., 91 (1969), 889-920.  Google Scholar

[20]

R. Seeley, Norms and domains of the complex powers $A_Bz$, Amer. J. Math., 93 (1971), 299-309.  Google Scholar

[21]

J. A. Shercliff, "A Textbook of Magnetohydrodyamics," Pergamon Press, Oxford-New York-Paris, 1965.  Google Scholar

[22]

L. Weis, Operator-valued Fourier multiplier theorems and maximal $L_p$-regularity, Math. Ann., 319 (2001), 735-758. doi: 10.1007/PL00004457.  Google Scholar

[23]

Z. Yoshida and Y. Giga, On the Ohm-Navier-Stokes system in magnetohydrodynamics, J. Math. Phys., 24 (1983), 2860-2864. doi: 10.1063/1.525667.  Google Scholar

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