Abstract
We consider an instationary generalized Stokes system with nonhomogeneous divergence data under a periodic condition in only some directions. The problem is set in the whole space, the half space or in (after an identification of the periodic directions with a torus) bounded domains with sufficiently regular boundary. We show unique solvability for all times in Muckenhoupt weighted Lebesgue spaces. The divergence condition is dealt with by analyzing the associated reduced Stokes system and in particular by showing maximal regularity of the partially periodic reduced Stokes operator.
References
Abels, H.: Reduced and generalized Stokes resolvent equations in asymptotically flat layers. I. Unique solvability. J. Math. Fluid Mech. 7(2), 201–222 (2005)
Abels, H.: Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discrete Contin. Dyn. Syst. Ser. S 3(2), 141–157 (2010)
Bothe, D., Prüss, J.: \(L_P\)-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007)
Bruhat, F.: Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes \(\wp \)-adiques. Bull. Soc. Math. Fr. 89, 43–75 (1961)
Burkholder, D.L.: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. I, II (Chicago, IL, 1981), Wadsworth Mathematics Series, pp. 270–286. Wadsworth, Belmont (1983)
Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41(4), 1027–1076 (1992)
de Rham, G.: Variétés différentiables. Formes, courants, formes harmoniques. Actualités Sci. Ind., no. 1222 = Publ. Inst. Math. Univ. Nancago III. Hermann et Cie, Paris (1955)
Denk, R., Nau, T.: Discrete Fourier multipliers and cylindrical boundary-value problems. Proc. R. Soc. Edinb. Sect. A 143(6), 1163–1183 (2013)
Farwig, R., Sohr, H.: Generalized resolvent estimates for the Stokes system in bounded and unbouded domains. J. Math. Soc. Jpn. 46(4), 607–643 (1994)
Fröhlich, A.: The Stokes operator in weighted \(L^q\)-spaces. I. Weighted estimates for the Stokes resolvent problem in a half space. J. Math. Fluid Mech. 5(2), 166–199 (2003)
Fröhlich, A.: The Stokes operator in weighted \(L^q\)-spaces. II. Weighted resolvent estimates and maximal \(L^p\)-regularity. Math. Ann. 339(2), 287–316 (2007)
García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics, volume 116 of North-Holland Mathematics Studies. North-Holland, Amsterdam (1985)
Iooss, G.: Théorie non linéaire de la stabilité des écoulements laminaires dans le cas de “l’échange des stabilités”. Arch. Ration. Mech. Anal. 40, 166–208 (1970/1971)
Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)
Kyed, M.: Maximal regularity of the time-periodic linearized Navier–Stokes system. J. Math. Fluid Mech. 16(3), 523–538 (2014)
Martínez Carracedo, C., Sanz Alix, M.: The Theory of Fractional Powers of Operators, volume 187 of North-Holland Mathematics Studies. North-Holland, Amsterdam (2001)
Nau, T.: The \(L^p\)-Helmholtz projection in finite cylinders. Czechoslovak Math. J. 65(1), 119–134 (2015)
Nekvinda, A.: Characterization of traces of the weighted Sobolev space \(W^{1, p}(\Omega, d^\epsilon _M)\) on \(M\). Czechoslov. Math. J. 43(4), 695–711 (1993)
Osborne, M.S.: On the Schwartz–Bruhat space and the Paley–Wiener theorem for locally compact Abelian groups. J. Funct. Anal. 19, 40–49 (1975)
Sauer, J.: An extrapolation theorem in non-Euclidean geometries and its application to partial differential equations. J. Elliptic Parabol. Equ. 1, 403–418 (2015)
Sauer, J.: Navier–Stokes Flow in Partially Periodic Domains. Sierke Verlag, Göttingen (2015)
Sauer, J.: Weighted resolvent estimates for the spatially periodic Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat. 61(2), 333–354 (2015)
Sauer, J.: Maximal regularity of the spatially periodic Stokes operator and application to nematic liquid crystal flows. Czechoslov. Math. J. 66(1), 41–55 (2016)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume of 43 Princeton Mathematical Series. Princeton University Press, Princeton (1993)
Weis, L.: A new approach to maximal \(L_p\)-regularity. In: Evolution Equations and Their Applications in Physical and Life Sciences (Bad Herrenalb, 1998), volume 215 of Lecture Notes in Pure and Applied Mathematics, pp. 195–214. Dekker, New York (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Giga
Jonas Sauer was partly supported by DFG and JSPS via the International Research Training Group 1529.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Sauer, J. Instationary Generalized Stokes Equations in Partially Periodic Domains. J. Math. Fluid Mech. 20, 289–327 (2018). https://doi.org/10.1007/s00021-017-0321-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-017-0321-x
Mathematics Subject Classification
- 35B10
- 35Q30
- 76D03
- 76D07
Keywords
- Generalized Stokes equations
- maximal regularity
- spatially periodic