Abstract
Generalized Stokes operators \(A_S\) arise as linearizations of various models for non-Newtonian fluid flows. Here, it is proved that such operators in fairly general settings of domains and boundary conditions admit a bounded \({\mathcal H}^\infty \)-calculus in the framework of solenoidal \(L_q\)-spaces, \(1<q<\infty \), with \({\mathcal H}^\infty \)-angle less than \(\pi /2\). As a consequence, their fractional power spaces can be identified as complex interpolation spaces, and Amann’s theory of extra-interpolation scales becomes available in its full strength.
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References
H. Amann, Linear and Quasilinear Parabolic Problems I. Monographs in Mathematics 89, Birkhäuser, 1995.
D. Bothe, M. Köhne, and J. Prüss. On a class of energy preserving boundary conditions for incompressible Newtonian flows. SIAM J. Math. Anal. 45, 3768–3822 (2013).
D. Bothe and J. Prüss. \(L_p\)-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39, 379–421 (2007).
R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old results of R.T. Seeley. Math. Ann. 328, 545–583 (2004).
R. Denk, M. Hieber, J. Prüss, \({\cal{R}}\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166, 2003.
Y. Giga. Domains of fractional powers of the Stokes operator in \(L_r\) spaces. Arch. Rational Mech. Anal. 89, 251–265 (1985).
T. Hytönen, J. van Neerven, M. Veraar, L. Weis, Analysis in Banach Sapces. Springer, 2016.
P. Kunstmann, L. Weis, Maximal \(L_p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \)-functional calculus. In: Springer LNM 1855 (M. Ianelli, R. Nagel, S. Piazzera, eds.), 65–311 (2004).
A. Noll, J. Saal, \(H^\infty \)-calculus for the Stokes operator on \(L_q\)-spaces. Math. Z. 244, 651–688 (2003).
J. Prüss, Maximal regularity for evolution equations in \(L_p\)-spaces, Conf. Semin. Mat. Univ. Bari 285, 1–39 (2003).
J. Prüss, G. Simonett, Maximal regularity for evolution equations in weighted \(L_p\)-spaces, Arch. Math. 82, 415–431 (2004).
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics 105, Birkhäuser, 2016.
J. Prüss, G. Simonett, M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications. J. Differential Equations 264, 2028–2074 (2018).
J. Prüss, M. Wilke, On critical spaces for the Navier–Stokes equations. J. Math. Fluid Mech. 20, 733–755 (2018).
J. Saal, Stokes and Navier–Stokes equations with Robin boundary conditions in a half space. J. Math. Fluid Mech. 8, 211–241 (2006).
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Dedicated to Professor Herbert Amann on the occasion of his 80th anniversary.
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Prüss, J. \(H^\infty \)-calculus for generalized Stokes operators. J. Evol. Equ. 18, 1543–1574 (2018). https://doi.org/10.1007/s00028-018-0466-y
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DOI: https://doi.org/10.1007/s00028-018-0466-y
Keywords
- Generalized Stokes operators
- Maximal \(L_p\)-regularity
- \({\mathcal H}^\infty \)-calculus
- Fractional powers