Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 6 (2014), 1253-1315.

Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface

Anne-Laure Dalibard and Christophe Prange

Full-text: Open access

Abstract

This paper is devoted to the well-posedness of the stationary 3D Stokes–Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of Gérard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks to a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes–Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes–Coriolis system with data in the Kato space Huloc12.

Article information

Source
Anal. PDE, Volume 7, Number 6 (2014), 1253-1315.

Dates
Received: 23 April 2013
Revised: 28 January 2014
Accepted: 1 March 2014
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731583

Digital Object Identifier
doi:10.2140/apde.2014.7.1253

Mathematical Reviews number (MathSciNet)
MR3270164

Zentralblatt MATH identifier
1304.35535

Subjects
Primary: 35A22: Transform methods (e.g. integral transforms) 35C15: Integral representations of solutions 35S99: None of the above, but in this section 35A01: Existence problems: global existence, local existence, non-existence
Secondary: 76U05: Rotating fluids 35Q35: PDEs in connection with fluid mechanics 35Q86: PDEs in connection with geophysics

Keywords
Stokes–Coriolis system Ekman boundary layer rough boundaries Dirichlet to Neumann operator Saint-Venant estimate Kato spaces

Citation

Dalibard, Anne-Laure; Prange, Christophe. Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface. Anal. PDE 7 (2014), no. 6, 1253--1315. doi:10.2140/apde.2014.7.1253. https://projecteuclid.org/euclid.apde/1513731583


Export citation

References

  • T. Alazard, N. Burq, and C. Zuily, “Cauchy theory for the gravity water waves system with non localized initial data”, preprint, 2013.
  • H.-O. Bae and B. J. Jin, “Existence of strong mild solution of the Navier–Stokes equations in the half space with nondecaying initial data”, J. Korean Math. Soc. 49:1 (2012), 113–138.
  • A. Basson, “Homogeneous statistical solutions and local energy inequality for $3$D Navier–Stokes equations”, Comm. Math. Phys. 266:1 (2006), 17–35.
  • J. R. Cannon and G. H. Knightly, “A note on the Cauchy problem for the Navier–Stokes equations”, SIAM J. Appl. Math. 18 (1970), 641–644.
  • J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, “Ekman boundary layers in rotating fluids”, ESAIM Control Optim. Calc. Var. 8 (2002), 441–466.
  • J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Mathematical geophysics: an introduction to rotating fluids and the Navier–Stokes equations, Oxford Lecture Series in Mathematics and its Applications 32, Oxford University Press, 2006.
  • A.-L. Dalibard and D. Gérard-Varet, “Effective boundary condition at a rough surface starting from a slip condition”, J. Differential Equations 251:12 (2011), 3450–3487.
  • J. Droniou and C. Imbert, “Fractal first-order partial differential equations”, Arch. Ration. Mech. Anal. 182:2 (2006), 299–331.
  • V. W. Ekman, “On the influence of the earth's rotation on ocean-currents”, Ark. Mat. Astr. Fys. 2 (1905), Article No. 11.
  • S. Gala, “Quasi-geostrophic equations with initial data in Banach spaces of local measures”, Electron. J. Differential Equations (2005), Article No. 63.
  • G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations, I: Linearised steady problems, Springer Tracts in Natural Philosophy 38, Springer, New York, 1994.
  • I. Gallagher and F. Planchon, “On global infinite energy solutions to the Navier–Stokes equations in two dimensions”, Arch. Ration. Mech. Anal. 161:4 (2002), 307–337.
  • D. Gérard-Varet and N. Masmoudi, “Relevance of the slip condition for fluid flows near an irregular boundary”, Comm. Math. Phys. 295:1 (2010), 99–137.
  • Y. Giga and T. Miyakawa, “Navier–Stokes flow in $\R^3$ with measures as initial vorticity and Morrey spaces”, Comm. Partial Differential Equations 14:5 (1989), 577–618.
  • Y. Giga, K. Inui, and S. Matsui, “On the Cauchy problem for the Navier–Stokes equations with nondecaying initial data”, pp. 27–68 in Advances in fluid dynamics, edited by P. Maremonti, Quad. Mat. 4, Dept. Math., Seconda Univ. Napoli, Caserta, 1999.
  • Y. Giga, S. Matsui, and O. Sawada, “Global existence of two-dimensional Navier–Stokes flow with nondecaying initial velocity”, J. Math. Fluid Mech. 3:3 (2001), 302–315.
  • Y. Giga, K. Inui, A. Mahalov, and S. Matsui, “Navier–Stokes equations in a rotating frame in $\R^3$ with initial data nondecreasing at infinity”, Hokkaido Math. J. 35:2 (2006), 321–364.
  • Y. Giga, K. Inui, A. Mahalov, S. Matsui, and J. Saal, “Rotating Navier–Stokes equations in $\R_+^3$ with initial data nondecreasing at infinity: the Ekman boundary layer problem”, Arch. Ration. Mech. Anal. 186:2 (2007), 177–224.
  • Y. Giga, K. Inui, A. Mahalov, and J. Saal, “Uniform global solvability of the rotating Navier–Stokes equations for nondecaying initial data”, Indiana Univ. Math. J. 57:6 (2008), 2775–2791.
  • H. P. Greenspan, The theory of rotating fluids, Cambridge University Press, 1980.
  • T. Kato, “The Cauchy problem for quasi-linear symmetric hyperbolic systems”, Arch. Ration. Mech. Anal. 58:3 (1975), 181–205.
  • T. Kato, “Strong solutions of the Navier–Stokes equation in Morrey spaces”, Bol. Soc. Brasil. Mat. $($N.S.$)$ 22:2 (1992), 127–155.
  • P. Konieczny and T. Yoneda, “On dispersive effect of the Coriolis force for the stationary Navier–Stokes equations”, J. Differential Equations 250:10 (2011), 3859–3873.
  • O. A. Ladyženskaja and V. A. Solonnikov, “\cyr O nakhozhdenii resheniĭ kraevykh zadach dlya statsionarnykh uravneniĭ Stoksa i Nav'e–Stoksa, imeyushchikh neogranichennyĭ integral Dirikhle”, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. $($LOMI$)$ 96 (1980), 117–160. Translated as “Determination of the solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral” in J. Sov. Math. 21:5 (1983), 728–761.
  • P. G. Lemarié-Rieusset, “Solutions faibles d'énergie infinie pour les équations de Navier–Stokes dans $\R^3$”, C. R. Acad. Sci. Paris Sér. I Math. 328:12 (1999), 1133–1138.
  • P. G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, Research Notes in Mathematics 431, CRC, Boca Raton, FL, 2002.
  • Y. Maekawa and Y. Terasawa, “The Navier–Stokes equations with initial data in uniformly local $L^p$ spaces”, Differential Integral Equations 19:4 (2006), 369–400.
  • J. Pedlosky, Geophysical fluid dynamics, Springer, New York, 1987.
  • V. A. Solonnikov, “On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity”, J. Math. Sci. $($N.Y.$)$ 114:5 (2003), 1726–1740.
  • M. E. Taylor, “Analysis on Morrey spaces and applications to Navier–Stokes and other evolution equations”, Comm. Partial Differential Equations 17:9-10 (1992), 1407–1456.
  • T. Yoneda, “Global solvability of the Navier–Stokes equations in a rotating frame with spatially almost periodic data”, RIMS Kôkyûroku 1640 (2009), 104–115.