AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Nonlinear stability of Ekman boundary layers in rotating stratified fluid
About this Title
Hajime Koba, Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan; email: iti@ms.u-tokyo.ac.jp
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 228, Number 1073
ISBNs: 978-0-8218-9133-9 (print); 978-1-4704-1485-6 (online)
DOI: https://doi.org/10.1090/memo/1073
Published electronically: August 1, 2013
Keywords: Stability of Ekman boundary layers,
Ekman spiral,
asymptotic stability,
weak solutions,
strong solutions,
strong energy inequality,
strong energy equality,
uniqueness of weak solutions,
smoothness and regularity,
maximal $L^p$-regularity,
real interpolation theory,
perturbation theory,
Coriolis force,
Stratification effect,
Boussinesq system
MSC: Primary 35Q86, 76E20; Secondary 35B35, 35B40, 35B65, 76D03, 76D05
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Formulation and Main Results
- 3. Linearized Problem
- 4. Existence of Global Weak Solutions
- 5. Uniqueness of Weak Solutions
- 6. Nonlinear Stability
- 7. Smoothness of Weak Solutions
- 8. Some Extensions of the Theory
- A. Toolbox
Abstract
A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This booklet constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. We call such stationary solutions Ekman layers. This booklet shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, we discuss the uniqueness of weak solutions and compute the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. It is also shown that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI 10.1002/cpa.3160170104
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385
- Wolfgang Borchers and Tetsuro Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann. 282 (1988), no. 1, 139–155. MR 960838, DOI 10.1007/BF01457017
- Wolfgang Borchers and Tetsuro Miyakawa, On stability of exterior stationary Navier-Stokes flows, Acta Math. 174 (1995), no. 2, 311–382. MR 1351321, DOI 10.1007/BF02392469
- J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168. MR 727340, DOI 10.1007/BF02384306
- Donald L. Burkholder, Martingales and Fourier analysis in Banach spaces, Probability and analysis (Varenna, 1985) Lecture Notes in Math., vol. 1206, Springer, Berlin, 1986, pp. 61–108. MR 864712, DOI 10.1007/BFb0076300
- Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MR 0230022
- L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. MR 673830, DOI 10.1002/cpa.3160350604
- Jean-Yves Chemin, Benoît Desjardins, Isabelle Gallagher, and Emmanuel Grenier, Ekman boundary layers in rotating fluids, ESAIM Control Optim. Calc. Var. 8 (2002), 441–466. A tribute to J. L. Lions. MR 1932959, DOI 10.1051/cocv:2002037
- J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Mathematical geophysics, Oxford Lecture Series in Mathematics and its Applications, vol. 32, The Clarendon Press, Oxford University Press, Oxford, 2006. An introduction to rotating fluids and the Navier-Stokes equations. MR 2228849
- Robert Denk, Matthias Hieber, and Jan Prüss, $\scr R$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc. 166 (2003), no. 788, viii+114. MR 2006641, DOI 10.1090/memo/0788
- Wolfgang Desch, Matthias Hieber, and Jan Prüss, $L^p$-theory of the Stokes equation in a half space, J. Evol. Equ. 1 (2001), no. 1, 115–142. MR 1838323, DOI 10.1007/PL00001362
- B. Desjardins, E. Dormy, and E. Grenier, Stability of mixed Ekman-Hartmann boundary layers, Nonlinearity 12 (1999), no. 2, 181–199. MR 1677778, DOI 10.1088/0951-7715/12/2/001
- B. Desjardins and E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 1, 87–106 (English, with English and French summaries). MR 1958163, DOI 10.1016/S0294-1449(02)00009-4
- Lars Diening, Petteri Harjulehto, Peter Hästö, and Michael Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, Springer, Heidelberg, 2011. MR 2790542
- Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
- Reinhard Farwig, Hideo Kozono, and Hermann Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math. 195 (2005), 21–53. MR 2233684, DOI 10.1007/BF02588049
- Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188
- Daisuke Fujiwara and Hiroko Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 685–700. MR 492980
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162
- Mi-Ho Giga, Yoshikazu Giga, and Jürgen Saal, Nonlinear partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 79, Birkhäuser Boston, Ltd., Boston, MA, 2010. Asymptotic behavior of solutions and self-similar solutions. MR 2656972
- Yoshikazu Giga, Katsuya Inui, Alex Mahalov, Shin’ya Matsui, and Jürgen Saal, Rotating Navier-Stokes equations in $\Bbb R_+^3$ with initial data nondecreasing at infinity: the Ekman boundary layer problem, Arch. Ration. Mech. Anal. 186 (2007), no. 2, 177–224. MR 2342201, DOI 10.1007/s00205-007-0053-9
- Yoshikazu Giga and Hermann Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), no. 1, 72–94. MR 1138838, DOI 10.1016/0022-1236(91)90136-S
- Harvey Philip Greenspan, The theory of rotating fluids, Cambridge University Press, Cambridge-New York, 1980. Reprint of the 1968 original; Cambridge Monographs on Mechanics and Applied Mathematics. MR 639897
- E. Grenier and N. Masmoudi, Ekman layers of rotating fluids, the case of well prepared initial data, Comm. Partial Differential Equations 22 (1997), no. 5-6, 953–975. MR 1452174, DOI 10.1080/03605309708821290
- Matthias Hess, Matthias Hieber, Alex Mahalov, and Jürgen Saal, Nonlinear stability of Ekman boundary layers, Bull. Lond. Math. Soc. 42 (2010), no. 4, 691–706. MR 2669690, DOI 10.1112/blms/bdq029
- John G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math. 136 (1976), no. 1-2, 61–102. MR 425390, DOI 10.1007/BF02392043
- Matthias Hieber and Wilhelm Stannat, Stochastic stability of the Ekman spiral. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12, No. 1, 189–208 (2013).
- Tosio Kato and Hiroshi Fujita, On the nonstationary Navier-Stokes system, Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. MR 142928
- Hideo Kozono, Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations, J. Funct. Anal. 176 (2000), no. 2, 153–197. MR 1784412, DOI 10.1006/jfan.2000.3625
- Hideo Kozono and Takayoshi Ogawa, Global strong solution and its decay properties for the Navier-Stokes equations in three-dimensional domains with noncompact boundaries, Math. Z. 216 (1994), no. 1, 1–30. MR 1273463, DOI 10.1007/BF02572306
- Peer C. Kunstmann and Lutz Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311. MR 2108959, DOI 10.1007/978-3-540-44653-8_{2}
- Peer Christian Kunstmann and Lutz Weis, Perturbation theorems for maximal $L_p$-regularity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 2, 415–435. MR 1895717
- Fanghua Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257. MR 1488514, DOI 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A
- Alessandra Lunardi, Interpolation theory, 2nd ed., Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009. MR 2523200
- Nader Masmoudi, Ekman layers of rotating fluids: the case of general initial data, Comm. Pure Appl. Math. 53 (2000), no. 4, 432–483. MR 1733696, DOI 10.1002/(SICI)1097-0312(200004)53:4<432::AID-CPA2>3.3.CO;2-P
- Kyūya Masuda, Weak solutions of Navier-Stokes equations, Tohoku Math. J. (2) 36 (1984), no. 4, 623–646. MR 767409, DOI 10.2748/tmj/1178228767
- Marjorie McCracken, The resolvent problem for the Stokes equations on halfspace in $L_{p}$, SIAM J. Math. Anal. 12 (1981), no. 2, 201–228. MR 605431, DOI 10.1137/0512021
- Tetsuro Miyakawa and Hermann Sohr, On energy inequality, smoothness and large time behavior in $L^2$ for weak solutions of the Navier-Stokes equations in exterior domains, Math. Z. 199 (1988), no. 4, 455–478. MR 968313, DOI 10.1007/BF01161636
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- J. Pedlosky, Geophysical Fluid Dynamics. 2nd edition. Springer-Verlag, 1987.
- F. Rousset, Large mixed Ekman-Hartmann boundary layers in magnetohydrodynamics, Nonlinearity 17 (2004), no. 2, 503–518. MR 2039055, DOI 10.1088/0951-7715/17/2/008
- F. Rousset, Stability of large Ekman boundary layers in rotating fluids, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 213–245. MR 2058164, DOI 10.1007/s00205-003-0302-5
- F. Rousset, Stability of large amplitude Ekman-Hartmann boundary layers in MHD: the case of ill-prepared data, Comm. Math. Phys. 259 (2005), no. 1, 223–256. MR 2169974, DOI 10.1007/s00220-005-1371-0
- Vladimir Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), no. 2, 97–112. MR 510154
- James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 69–98. MR 0150444
- V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8 (1977), 467–529.
- Hermann Sohr, The Navier-Stokes equations, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2001. An elementary functional analytic approach; [2013 reprint of the 2001 original] [MR1928881]. MR 3013225
- Christian G. Simader and Hermann Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains, Mathematical problems relating to the Navier-Stokes equation, Ser. Adv. Math. Appl. Sci., vol. 11, World Sci. Publ., River Edge, NJ, 1992, pp. 1–35. MR 1190728, DOI 10.1142/9789814503594_{0}001
- Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
- Roger Temam, Navier Stokes Equations. Theory and numerical analysis. Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
- Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903
- Seiji Ukai, A solution formula for the Stokes equation in $\textbf {R}^n_+$, Comm. Pure Appl. Math. 40 (1987), no. 5, 611–621. MR 896770, DOI 10.1002/cpa.3160400506