Abstract
We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et al. (Indiana Univ Math J 45:479–510, 1996). This paper is an extension of Levermore et al. (Indiana Univ Math J 45:479–510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et al. (Indiana Univ Math J 45:479–510, 1996).
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References
Barbu V.: Nonlinear differential equations of monotone type in Banach spaces. Springer, New-York (2010)
Barbu V., Sritharan S.: Flow invariance preserving feedback controllers for the Navier–Stokes equations. J. Math. Anal. appl. 255, 281–307 (2001)
D. Bresch, B. Desjardins, G. Metivier, Recent mathematical results and open problems about shallow water equations, Anal. Sim. of Fluid Flows, Birkhauser (2007), 15–32.
D. Bresch, G. Metivier, Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for the degenerate equations, Nonlinearity 19 (3) (2006), 591–610.
Camassa R., Holm D.D., Levermore C.D.: Long-time shallow water equations with a varying bottom. J. Fluid Mech. 349, 173–189 (1997)
Chua S.K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41, 1027–1076 (1992)
Fabes E., Kening C., Serapioni R.: The local regularity of solutions of degenerate elliptic equations. Comm. P.D.E. 7, 77–116 (1982)
Farwig R., Sohr H.: Weighted L q−theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49, 251–288 (1997)
A. Frohlich, The Navier–Stokes equations with low-regularity data in weighted function spaces, Doktor der Naturwissenschaften dissertation, (2006).
A. Frohlich, Stokes-und Navier–Stokes-Gleichungen in gewichteten funktionenraumen, Shaker Verlag, Aachen, 2001.
Frohlich A.: The Stokes operator in weighted L q−spaces I: weighted estimates for the Stokes resolvent problem in a half space. J. Math.Fluid Mech. 5,–166199 (2003)
J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland, Amsterdam, 1985.
V. Gol’dshtein, A. Ukhlov, Weighted Sobolev spaces and embedding theorems, arXiv:math/0703725v4 [math.FA].
Haroske D.: Sobolev spaces with Muckenhoupt weights, singularities and inequalities. Georgian Mathematical Journal Vol. 15, 263–280 (2008)
J. Heinonen, T. Kilpelainen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, Oxford University Press, 1993.
T. Kilpelainen, Smooth approximation in weightd Sobolev spaces, Comment. Math. Univ. Carolin., 38 (1997).
Kilpelainen T.: Weighted Sobolev spaces and capacity. Annales Acad. Scient. Fennicae Vol. 19, 95–113 (1994)
Levermore C.D., Oliver M., Titi E.S.: Global well-posedeness for models of shallow water in a basin with a varying bottom. Indiana University Mathematics Journal Vol. 45, 479–510 (1996)
Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal functions. Trans. Amer. Math. Soc. 165, 207–226 (1972)
E. Stein, Harmonic analysis: real-variable methods, orthogonality and oszillatory integrals, Princeton Mathematical Series, 43,Princeton University Press, Princeton, N. J., 1993.
B.O. Turesson, Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, vol. 1736, Springer-Verlag, 2000.
Youdovitch V.I.: Non-stationary flow of an ideal incompressible liquid, Zh. Vychisl. Mat. i Mat. Fiz. 6, 1032–1066 (1963)
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Munteanu, I. Existence of solutions for models of shallow water in a basin with a degenerate varying bottom. J. Evol. Equ. 12, 393–412 (2012). https://doi.org/10.1007/s00028-012-0137-3
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DOI: https://doi.org/10.1007/s00028-012-0137-3