Abstract
The abstract theory of critical spaces developed in Prüss and Wilke (J Evol Equ, 2017. doi:10.1007/s00028-017-0382-6), Prüss et al. (Critical spaces for quasilinear parabolic evolution equations and applications, 2017) is applied to the Navier–Stokes equations in bounded domains with Navier boundary conditions as well as no-slip conditions. Our approach unifies, simplifies and extends existing work in the \(L_p\)–\(L_q\) setting, considerably. As an essential step, it is shown that the strong and weak Stokes operators with Navier conditions admit an \(\mathcal {H}^\infty \)-calculus with \(\mathcal {H}^\infty \)-angle 0, and the real and complex interpolation spaces of these operators are identified.
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Amann, H.: Linear and Quasilinear Parabolic Problems, volume 89 of Monographs in Mathematics. Birkhäuser Verlag, Basel (1995)
Amann, H.: On the strong solvability of the Navier–Stokes equations. J. Math. Fluid Mech. 2, 16–98 (2000)
Amann, H.: Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Nonlinear problems in mathematical physics and related topics, II. Int. Math. Ser. Kluwer/Plenum, New York, 1–28, (2002)
Bothe, D., Köhne, M., Prüss, J.: On a class of energy preserving boundary conditions for incompressible Newtonian flows. SIAM J. Math. Anal. 45(6), 3768–3822 (2013)
Bourgain, J., Pavlović, N.: Ill-posedness of the Navier–Stokes equations in a critical space in 3D. J. Funct. Anal. 255, 2233–2247 (2008)
Cannone, M.: On a generalization of a theorem of Kato on the Navier–Stokes equations. Rev. Mat. Iberoamericana 13, 515–541 (1997)
Denk, R., Dore, G., Hieber, M., Prüss, J., Venni, A.: New thoughts on old results of R. T. Seeley. Math. Ann. 328(4), 545–583 (2004)
Farwig, R., Giga, Y., Hsu, P.-Y.: Initial values for the Navier–Stokes equations in spaces with weights in time. Funkc. Ekvac. 59, 199–216 (2016)
Farwig, R., Giga, Y., Hsu, P.-Y.: On the continuity of the solutions to the Navier–Stokes equations with initial data in critical Besov spaces. Hokkaido Univ. Prepr. Ser. Math. 121, 1093 (2016)
Farwig, R., Rosteck, V.: Resolvent estimates for the Stokes system with Navier boundary conditions in general unbounded domains. Adv. Differ. Equ. 21(5–6), 401–428 (2016)
Farwig, R., Sohr, H.: Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations. Math. Ann. 345, 631–642 (2009)
Fujita, H., Kato, I.: On the non-stationary Navier–Stokes system. Rend. Sem. Mat. Univ. Padova 32, 243–260 (1962)
Fujita, H., Morimoto, H.: On fractional powers of the Stokes operator. Proc. Jpn. Acad. 46(10), 1141–1143 (1970)
Giga, Y.: The nonstationary Navier–Stokes equations with some first order boundary condition. Proc. Jpn. Acad. Ser. A Math. Sci. 58(3), 101–104 (1982)
Giga, M., Giga, Y., Saal, J.: Nonlinear Partial Differential Equations. Asymptotic Behavior of Solutions and Self-Similar Solutions, Volume 79 of Progess in Nonlinear Differential Equations and Their Applications. Birkhäuser Verlag, Basel (2010)
Giga, Y., Miyakawa, T.: Solutions in \(L_r\) of the Navier–Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)
Hieber, M., Saal, J.: The stokes equation in the \(L_p\)-setting: wellposedness and regularity properties. In: Giga, Y., Novotny, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Berlin (2017)
Kalton, N., Kunstmann, P., Weis, L.: Perturbation and interpolation theorems for the \(H^\infty \)-calculus with applications to differential operators. Math. Ann. 336(4), 747–801 (2006)
Koch, H., Tataru, D.: Well-posedness for the Navier–Stokes equations. Adv. Math. 157, 22–35 (2001)
Köhne, M., Prüss, J., Wilke, M.: On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces. J. Evol. Equ. 10(2), 443–463 (2010)
Kunstmann, P., Weis, L.: Erratum to: perturbation and interpolation theorems for the \(H^\infty \)-calculus with applications to differential operators. Math. Ann. 357, 801–804 (2013)
LeCrone, J., Prüss, J., Wilke, M.: On quasilinear parabolic evolution equations in weighted \(L_p\)-spaces II. J. Evol. Equ. 14(3), 509–533 (2014)
Mitrea, M., Monniaux, S.: On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds. Trans. Am. Math. Soc. 361(6), 3125–3157 (2009)
Miyakawa, T.: The \(L_p\) approach to the Navier–Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10(3), 517–537 (1980)
Prüss, J.: On the quasi-geostrophic equations on compact closed surfaces in \(\mathbb{R}^3\). J. Funct. Anal. 272, 2641–2658 (2017)
Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted \(L_p\)-spaces. Arch. Math. (Basel) 82(5), 415–431 (2004)
Prüss, J., Simonett, G.: Moving Interfaces and Quasilinear Parabolic Evolution Equations, Volume 105 of Monographs in Mathematics. Springer, Cham (2016)
Prüss, J., Simonett, G., Wilke, M.: Critical spaces for quasilinear parabolic evolution equations and applications. (2017) (submitted). arXiv:1708.08550
Prüss, J., Simonett, G., Zacher, R.: On convergence of solutions to equilibria for quasilinear parabolic problems. J. Differ. Equ. 246(10), 3902–3931 (2009)
Prüss, J., Wilke, M.: Addendum to the paper: on quasilinear evolution equations in weighted \({L}_p\)-spaces II. J. Evol. Equ. (2017). doi:10.1007/s00028-017-0382-6
Ri, M.-H., Zhang, P., Zhang, Z.: Global well-posedness for Navier–Stokes equations with small initial value in \(B_{n,\infty }^0(\Omega )\). J. Math. Fluid Mech. 18, 103–131 (2016)
Saal, J.: Stokes and Navier–Stokes equations with Robin boundary conditions in a half-space. J. Math. Fluid Mech. 8(2), 211–241 (2006)
Shibata, Y., Shimada, R.: On a generalized resolvent estimate for the Stokes system with Robin boundary condition. J. Math. Soc. Jpn. 59(2), 469–519 (2007)
Shimada, R.: On the \(L_p\)-\(L_q\) maximal regularity for Stokes equations with Robin boundary conditions in a bounded domain. Math. Methods Appl. Sci. 30, 257–289 (2007)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators North-Holland Mathematical Library. 18 North-Holland Publishing Co., Amsterdam (1978)
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Prüss, J., Wilke, M. On Critical Spaces for the Navier–Stokes Equations. J. Math. Fluid Mech. 20, 733–755 (2018). https://doi.org/10.1007/s00021-017-0342-5
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DOI: https://doi.org/10.1007/s00021-017-0342-5