×

zbMATH — the first resource for mathematics

Small-time existence of a strong solution of primitive equations for the ocean. (English) Zbl 1270.35326
The idea of weather forecast was given by V. Bjerknes in [Meteorol. Zs. 21, 1–7 (1904; JFM 35.0968.03)] and then the numerical weather forecast was executed by L. F. Richarson in [Weather prediction by numerical process. London: Cambridge Univ. Press (1922; JFM 48.0629.07)]. He gave a system of equations describing the motion of atmosphere very similar to the Navier-Stokes equations.
In this paper such primitive equations as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean are under consideration. The authors discuss in details the free boundary problem of the primitive equations for the ocean in three-dimensional strip with surface tension.They prove temporally local existence of the unique strong solution to the transformed problem in Sobolev-Slobodetskii spaces. For this purpose the authors use the so-called \(p\)-coordinates and a coordinate transformation similar to that in [J. T. Beale, Arch. Ration. Mech. Anal. 84, 307–352 (1984; Zbl 0545.76029)] – in such way they fix the time-dependent domain.

MSC:
35M10 PDEs of mixed type
35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
76D99 Incompressible viscous fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] P. Azerad and F. Guillen-Gonzalez, Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics, SIAM J. Math. Anal. 33 (2001), 847-859. · Zbl 0999.35072
[2] J. T. Beale, Large-time regularity of viscous surface waves, Arch. Rational Mech. Anal. 84 (1984), 307-352. · Zbl 0545.76029
[3] K. Bryan, A numerical method for the study of the circulation of the world ocean, J. Comp. Phys. 4 (1969), 347-376. · Zbl 0195.55504
[4] C. CAO and E. S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics, Ann. Math. 166 (2007), 245-267. · Zbl 1151.35074
[5] W. P. Crowly, A numerical model for viscous, free-surface, barotropic wind driven ocean circulations, J. Comp. Phys. 5 (1970), 139-168. · Zbl 0184.53705
[6] A. E. Gill, Atmosphere-Ocean Dynamics , Academic Press, 1982. · Zbl 0497.30003
[7] F. Guillen-Gonzalez and A. R. Bellido, A review on the improved regularity for the primitive equations, Banach Center 70 (2005), 85-103. · Zbl 1101.35359
[8] F. Guillen-Gonzalez, N. Masmoudi and M. A. Rodriguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations, Diff. and Integral Equations 14 (2001), 1381-1408. · Zbl 1161.76454
[9] C. Hu, Asymptotic analysis of the primitive equations under the small depth assumption, Nonlinear Anal. 61 (2005), 425-460. · Zbl 1081.35080
[10] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-linear Equations of Parabolic type , Translations of Mathematical Monographs, 23, American Mathematical Society, 1968. · Zbl 0174.15403
[11] J. L. Lions, R. Temam and S. Wang, On the equations of the large-scale-ocean, Nonlinearity 5 (1992), 1007-1053. · Zbl 0766.35039
[12] J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models, J. Math. Pures Appl. 74 (1995), 105-163. · Zbl 0866.76025
[13] J. L. Lions, R. Temam and S. Wang, Problemes a frontiere libre pour les modeles couples de l’ocean et de l’atmosphere, C. R. Acad. Sci. Paris 318 (1994), 1165-1171. · Zbl 0817.76010
[14] J. L. Lions, R. Temam and S. Wang, On mathematical probelms for the primitive equations of the ocean: the mesoscale midlatitude case, Nonlinear Anal. 40 (2000), 439-482. · Zbl 0978.76102
[15] J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAOI), Comput. Mech. Adv. 1 (1993), 5-54. · Zbl 0805.76011
[16] J. L. Lions, R. Temam and S. Wang, Numerical analysis of the coupled atmosphere and ocean models (CAOII), Comput. Mech. Adv. 1 (1993), 55-120. · Zbl 0805.76052
[17] J. L. Lions, R. Temam and S. Wang, Mathematical theory for the coupled atmosphere-ocean models (CAO III), J. Math. Pures Appl. 74 (1995), 105-163. · Zbl 0866.76025
[18] W. M. Pedlosky, Geophysical Fluid Dynamics , Springer-Verlag, 1979. · Zbl 0429.76001
[19] N. A. Phillips, Models for weather prediction, Annu. Rev. Fluid Mech. 2 (1970), 251-292.
[20] L. F. Richardson, Weather Prediction by Numerical Process , Cambridge University Press, 1922. · JFM 48.0629.07
[21] A. J. Semtner, A general circulation model for the world ocean, UCLA Dept. of Meteorology Tech. Rep. 8 (1974), 99-120.
[22] V. A. Solonnikov, Solvability of a problem of evolution of a viscous incompressible fluid bounded by a free surface in a finite time interval, St. Petersburg Math. J. 3 (1992), 189-220. · Zbl 0850.76132
[23] V. A. Solonnikov and A. Tani, Free boundary problem for a compressible flow with a surface tension, ed. M. Rassius, Constantin Carathéodory: An International Tribute, World Scientific Publishing, 1991, 1270-1303. · Zbl 0752.35096
[24] V. A. Solonnikov and A. Tani, Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid, The Navier-Stokes Equations 2-Thoery and Numerical Methods , eds. J. G. Heywood, K. Masuda, R. Rautmann, V. A. Solonnikov, Lecture Notes in Math. 1530 , Springer-Verlag, 1992, 30-55. · Zbl 0786.35106
[25] N. Tanaka and A. Tani, Large-time existence of surface waves of compressible viscous fluid, Proc. Japan Acad. 69 (1993), 230-233. · Zbl 0799.35180
[26] N. Tanaka and A. Tani, Large-time existence of compressible viscous and heat-conductive surface waves, Suurikaisekikennkyuusho Kokyuroku 914 (1995), 138-150.
[27] N. Tanaka and A. Tani, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal. 130 (1995), 303-314. · Zbl 0844.76025
[28] N. Tanaka and A. Tani, Surface waves for a compressible viscous fluid, J. Math. Fluid. Mech. 5 (2003), 303-363. · Zbl 1037.35061
[29] R. Temam and M. Ziane, Some mathematical problems in geophysical fluid dynamics, eds. S. J. Friedlander and D. Serre, Handbook of Mathematical Fluid Dynamics, North-Holland, vol III, 2004, 535-657. · Zbl 1222.35145
[30] W. M. Washington and C. L. Parkinson, An Introduction to Three-Dimensional Climate Modelling , Oxford University Press, 1986. · Zbl 0655.76003
[31] J. Wloka, Partielle Differentialgleichungen , B. G.Teubner, 1982. · Zbl 0482.35001
[32] M. Ziane, Regularity results for the stationary primitive equations of the atmosphere and the ocean, Nonlinear Anal. 28 (1997), 289-313. · Zbl 0863.35085
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.