×
Chueshov, Igor D.; Raugel, Geneviéve; Rekalo, Andrey M.

Interface boundary value problem for the Navier-Stokes equations in thin two-layer domains. (English) Zbl 1078.35084

J. Differ. Equations 208, No. 2, 449-493 (2005).
The authors consider the Navier-Stokes equations in a thin two-layer domain \(\Omega_\varepsilon= \Gamma\times (0,\varepsilon)\cup \Gamma\times(-\varepsilon, 0)\), where \(\Gamma= (0,l_1)\times (0,l_2)\subset \mathbb{R}^2\). On the interface \(\Gamma\times \{0\}\), conditions are imposed which essentially how that there is no interaction between the fluids in vertical direction. The authors prove the global existence of strong solutions for large initial data and external forces (the size of which is related to \(\varepsilon\)). The system can be seen as a simplified model for atmosphere-ocean interaction.
Reviewer: Klaus Deckelnick (Magdeburg)

Cited in 12 Documents

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems

Keywords:

Navier-Stokes equations; Thin domains; Interface conditions; Global strong solutions; Spectral decomposition; Attractors
Cite Review PDF
Full Text: DOI

References:

[1] Avrin, J. D., Large-eigenvalue global existence and regularity results for the Navier-Stokes equation, J. Differential Equations, 127, 2, 365-390 (1996) · Zbl 0863.35075
[2] I.D. Chueshov, A.M. Rekalo, Long-time dynamics of reaction-diffusion equations on thin two-layer domains, Proceedings of Equadiff, 2003.; I.D. Chueshov, A.M. Rekalo, Long-time dynamics of reaction-diffusion equations on thin two-layer domains, Proceedings of Equadiff, 2003. · Zbl 1108.35081
[3] I.D. Chueshov, A.M. Rekalo, Global attractor of contact parabolic problem on thin two-layer domain, Mat. Sb. 195(1) (2004) 103-128 (in Russian); (English translation in Sbornik: Mathematics 195(1) (2004)).; I.D. Chueshov, A.M. Rekalo, Global attractor of contact parabolic problem on thin two-layer domain, Mat. Sb. 195(1) (2004) 103-128 (in Russian); (English translation in Sbornik: Mathematics 195(1) (2004)). · Zbl 1114.35032
[4] Constantin, P.; Foiaş, C., Navier-Stokes Equations (1988), University of Chicago Press: University of Chicago Press Chicago · Zbl 0687.35071
[5] C. Foiaş, R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, Directions in Partial Differential Equations, Academic Press, New York, 1987, pp. 55-73.; C. Foiaş, R. Temam, The connection between the Navier-Stokes equations, dynamical systems, and turbulence theory, Directions in Partial Differential Equations, Academic Press, New York, 1987, pp. 55-73. · Zbl 0673.35084
[6] Hale, J. K.; Raugel, G., Reaction-diffusion equation on thin domains, J. Math. Pures Appl, 71, 33-95 (1992) · Zbl 0840.35044
[7] Hale, J. K.; Raugel, G., A damped hyperbolic equation on thin domains, Trans. Amer. Math. Soc, 329, 185-219 (1992) · Zbl 0761.35052
[8] J.K. Hale, G. Raugel, Partial differential equations on thin domains, Differential Equations and Mathematical Physics (Birmingham, AL, 1990), Academic Press, Boston, MA, 1992, pp. 63-97.; J.K. Hale, G. Raugel, Partial differential equations on thin domains, Differential Equations and Mathematical Physics (Birmingham, AL, 1990), Academic Press, Boston, MA, 1992, pp. 63-97. · Zbl 0785.35050
[9] Hu, C.; Temam, R.; Ziane, M., The primitive equations on the large scale ocean under the small depth hypothesis, Discrete Continuous Dynamical Systems, 9, 97-131 (2003) · Zbl 1048.35082
[10] Iftimie, D., The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France, 127, 473-517 (1999) · Zbl 0946.35059
[11] Iftimie, D.; Raugel, G., Some results on the Navier-Stokes equations in thin 3D domains, J. Differential Equations, 169, 281-331 (2001) · Zbl 0972.35085
[12] Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer New York · Zbl 0148.12601
[13] Lions, J.-L.; Magenes, E., Nonhomogeneous Boundary Value Problems and Applications (1972), Springer: Springer New York · Zbl 0227.35001
[14] Lions, J.-L.; Temam, R.; Wang, S., Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl, 74, 105-163 (1995) · Zbl 0866.76025
[15] Moise, I.; Temam, R.; Ziane, M., Asymptotic analysis of the Navier-Stokes equations in thin domains, Topol. Methods Nonlinear Anal, 10, 2, 249-282 (1997) · Zbl 0957.35108
[16] Montgomery-Smith, S., Global regularity of the Navier-Stokes equation on thin three dimensional domains with periodic boundary conditions, Electronic J. Differential Equations, 11, 1-19 (1999) · Zbl 0923.35120
[17] Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer: Springer New York · Zbl 0713.76005
[18] Raugel, G.; Sell, G., Navier-Stokes equations on thin domains. I: global attractors and global regularity of solutions, J. Amer. Math. Soc, 6, 503-568 (1993) · Zbl 0787.34039
[19] G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XI, Longman Sci. Tech., Harlow, 1994, pp. 205-247.; G. Raugel, G. Sell, Navier-Stokes equations on thin 3D domains. II. Global regularity of spatially periodic solutions, Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar, Vol. XI, Longman Sci. Tech., Harlow, 1994, pp. 205-247. · Zbl 0804.35106
[20] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II (1975), Academic Press: Academic Press New York · Zbl 0308.47002
[21] Rekalo, A., Asymptotic behaviour of solutions of nonlinear parabolic equations on thin two-layer domain, Nonlinear Anal, 52, 1393-1410 (2003) · Zbl 1026.35016
[22] Schlichting, H., Boundary Layer Theory (1979), McGraw Hill: McGraw Hill New York · Zbl 0434.76027
[23] Sell, G. R., Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics Differential Equations, 8, 1-33 (1996) · Zbl 0855.35100
[24] Temam, R., Navier-Stokes Equations, Revised Edition, Studies in Mathematics and its Applications 2 (1979), North-Holland: North-Holland Amsterdam · Zbl 0454.35073
[25] Temam, R., Some mathematical aspects of the GCMs, Progr. Probab, 49, 117-138 (2001) · Zbl 0986.86002
[26] Temam, R.; Ziane, M., Navier-Stokes equations in three-dimensional thin domains with various boundary conditions, Adv. in Differential Equations, 1, 499-546 (1996) · Zbl 0864.35083
[27] Temam, R.; Ziane, M., Navier-Stokes equations in thin spherical domains, Contemp. Math. AMS, 209, 281-314 (1997) · Zbl 0891.35119
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.