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Kukavica, Igor; Temam, Roger; Vicol, Vlad C.; Ziane, Mohammed

Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain. (English) Zbl 1204.35129

J. Differ. Equations 250, No. 3, 1719-1746 (2011).
Summary: We address the question of well-posedness in spaces of analytic functions for the Cauchy problem for the hydrostatic incompressible Euler equations (inviscid primitive equations) on domains with boundary. By a suitable extension of the Cauchy-Kowalewski theorem we construct a locally in time, unique, real-analytic solution and give an explicit rate of decay of the radius of real-analyticity.

Cited in 42 Documents

MSC:

35Q31 Euler equations
35A10 Cauchy-Kovalevskaya theorems
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
86A05 Hydrology, hydrography, oceanography

Keywords:

hydrostatic Euler equations; non-viscous primitive equations; hydrostatic approximation; well-posedness; bounded domain; analyticity
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References:

[1] Asano, K., A note on the abstract Cauchy-Kowalewski theorem, Proc. Japan Acad. Ser. A, 64, 102-105 (1988) · Zbl 0666.35003
[2] Bardos, C.; Benachour, S., Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de \(R^n\), Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 4, 647-687 (1977) · Zbl 0366.35022
[3] Brenier, Y., Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12, 495-512 (1999) · Zbl 0984.35131
[4] Brenier, Y., Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math., 127, 585-595 (2003) · Zbl 1040.35068
[5] Grenier, E., On the derivation of homogeneous hydrostatic equations, M2AN Math. Model. Numer. Anal., 33, 5, 965-970 (1999) · Zbl 0947.76013
[6] Grenier, E., On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53, 9, 1067-1091 (2000) · Zbl 1048.35081
[7] Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1953), Dover Publications: Dover Publications New York · JFM 49.0725.04
[8] Kukavica, I.; Temam, R.; Vicol, V.; Ziane, M., Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, C. R. Acad. Sci. Paris Ser. I, 348, 639-645 (2010) · Zbl 1194.35247
[9] Kukavica, I.; Vicol, V., On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137, 669-677 (2009) · Zbl 1156.76010
[10] I. Kukavica, V. Vicol, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., in press.; I. Kukavica, V. Vicol, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., in press. · Zbl 1308.35192
[11] Lions, P.-L., Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models, Oxford Lecture Ser. Math. Appl., vol. 3 (1996), Oxford University Press: Oxford University Press New York · Zbl 0866.76002
[12] Lions, J.-L.; Magenes, E., Problèmes aux limites non homogènes et applications, vol. 3 (1968), S.A. Dunod: S.A. Dunod Paris · Zbl 0165.10801
[13] Lions, J.-L.; Temam, R.; Wang, S., New formulations of the primitive equations of atmosphere and applications, Nonlinearity, 5, 2, 237-288 (1992) · Zbl 0746.76019
[14] Lions, J.-L.; Temam, R.; Wang, S., On the equations of the large-scale ocean, Nonlinearity, 5, 5, 1007-1053 (1992) · Zbl 0766.35039
[15] Lions, J.-L.; Temam, R.; Wang, S., Models of the coupled atmosphere and ocean (CAO I), and Numerical Analysis of the coupled models of atmosphere and Ocean (CAO II), (Oden, J. T., Comput. Mech. Adv., vol. 1 (1993)), 1-54, and 55-119 · Zbl 0805.76011
[16] Levermore, C. D.; Oliver, M., Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133, 2, 321-339 (1997) · Zbl 0876.35090
[17] Lombardo, M. C.; Cannone, M.; Sammartino, M., Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35, 4, 987-1004 (2003) · Zbl 1053.76013
[18] Masuda, K., On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation, Proc. Japan Acad., 43, 827-832 (1967) · Zbl 0204.26901
[19] Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0713.76005
[20] Renardy, M., Ill-posedness of the hydrostatic Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 194, 877-886 (2009) · Zbl 1292.76011
[21] Rousseau, A.; Temam, R.; Tribbia, J., Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity, Discrete Contin. Dyn. Syst. Ser. A, 13, 5, 1257-1276 (2005) · Zbl 1092.35089
[22] Rousseau, A.; Temam, R.; Tribbia, J., The 3D primitive equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89, 297-319 (2008) · Zbl 1137.35057
[23] Oliger, J.; Sundström, A., Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35, 3, 419-446 (1978) · Zbl 0397.35067
[24] Sammartino, M.; Caflisch, R. E., Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192, 433-461 (1998) · Zbl 0913.35102
[25] Temam, R., On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20, 1, 32-43 (1975) · Zbl 0309.35061
[26] Temam, R.; Tribbia, J., Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60, 21, 2647-2660 (2003)
[27] Temam, R.; Ziane, M., Some mathematical problems in geophysical fluid dynamics, (Handbook of Mathematical Fluid Dynamics, vol. III (2004), North-Holland: North-Holland Amsterdam), 535-657 · Zbl 1222.35145
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