Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain

Journal of Differential Equations

Volumn 250, Issue 3, 2011, Pages 1719-1746

  Kukavica, Igor ^a   Temam, Roger ^b   Vicol, Vlad C ^c   Ziane, Mohammed ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b INDIANA UNIVERSITY   (United States)

^c UNIVERSITY OF CHICAGO   (United States)

Author keywords

Analyticity; Bounded domain; Hydrostatic approximation; Hydrostatic Euler equations; Non viscous primitive equations; Well posedness

Indexed keywords

EID: 78349261159     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.07.032     Document Type: Article

References (27)

1. Asano K. A note on the abstract Cauchy-Kowalewski theorem. Proc. Japan Acad. Ser. A 1988, 64:102-105.
2. Bardos C., Benachour S. Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de Rn. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 1977, 4:647-687.
3. Brenier Y. Homogeneous hydrostatic flows with convex velocity profiles. Nonlinearity 1999, 12:495-512.
4. Brenier Y. Remarks on the derivation of the hydrostatic Euler equations. Bull. Sci. Math. 2003, 127:585-595.
5. Grenier E. On the derivation of homogeneous hydrostatic equations. M2AN Math. Model. Numer. Anal. 1999, 33(5):965-970.
6. Grenier E. On the nonlinear instability of Euler and Prandtl equations. Comm. Pure Appl. Math. 2000, 53(9):1067-1091.
7. Hadamard J. Lectures on Cauchy's Problem in Linear Partial Differential Equations 1953, Dover Publications, New York.
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 2010, 348:639-645.
9. Kukavica I., Vicol V. On the radius of analyticity of solutions to the three-dimensional Euler equations. Proc. Amer. Math. Soc. 2009, 137:669-677.
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.
11. Lions P.-L. Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models. Oxford Lecture Ser. Math. Appl. 1996, vol. 3. Oxford University Press, New York.
12. Lions J.-L., Magenes E. Problèmes aux limites non homogènes et applications, vol. 3 1968, S.A. Dunod, Paris.
13. Lions J.-L., Temam R., Wang S. New formulations of the primitive equations of atmosphere and applications. Nonlinearity 1992, 5(2):237-288.
14. Lions J.-L., Temam R., Wang S. On the equations of the large-scale ocean. Nonlinearity 1992, 5(5):1007-1053.
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). Comput. Mech. Adv. 1993, vol. 1:1-54. and 55-119. J.T. Oden (Ed.).
16. Levermore C.D., Oliver M. Analyticity of solutions for a generalized Euler equation. J. Differential Equations 1997, 133(2):321-339.
17. Lombardo M.C., Cannone M., Sammartino M. Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 2003, 35(4):987-1004.
18. Masuda K. On the analyticity and the unique continuation theorem for solutions of the Navier-Stokes equation. Proc. Japan Acad. 1967, 43:827-832.
19. Pedlosky J. Geophysical Fluid Dynamics 1987, Springer-Verlag, New York.
20. Renardy M. Ill-posedness of the hydrostatic Euler and Navier-Stokes equations. Arch. Ration. Mech. Anal. 2009, 194:877-886.
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 2005, 13(5):1257-1276.
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. 2008, 89:297-319.
23. Oliger J., Sundström A. Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 1978, 35(3):419-446.
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. 1998, 192:433-461.
25. Temam R. On the Euler equations of incompressible perfect fluids. J. Funct. Anal. 1975, 20(1):32-43.
26. Temam R., Tribbia J. Open boundary conditions for the primitive and Boussinesq equations. J. Atmospheric Sci. 2003, 60(21):2647-2660.
27. Temam R., Ziane M. Some mathematical problems in geophysical fluid dynamics. Handbook of Mathematical Fluid Dynamics, vol. III 2004, 535-657. North-Holland, Amsterdam.