Blowup of solutions of the hydrostatic Euler equations

Wong, Tak Kwong

doi:10.1090/S0002-9939-2014-12243-X

Blowup of solutions of the hydrostatic Euler equations
HTML articles powered by AMS MathViewer

by Tak Kwong Wong

Proc. Amer. Math. Soc. 143 (2015), 1119-1125

DOI: https://doi.org/10.1090/S0002-9939-2014-12243-X

Published electronically: November 24, 2014

PDF | Request permission

Abstract:

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.

References

Yann Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity 12 (1999), no. 3, 495–512. MR 1690189, DOI 10.1088/0951-7715/12/3/004
Yann Brenier, Remarks on the derivation of the hydrostatic Euler equations, Bull. Sci. Math. 127 (2003), no. 7, 585–595 (English, with English and French summaries). MR 2004720, DOI 10.1016/S0007-4497(03)00024-1
Yann Brenier, Generalized solutions and hydrostatic approximation of the Euler equations, Phys. D 237 (2008), no. 14-17, 1982–1988. MR 2449787, DOI 10.1016/j.physd.2008.02.026
Chongsheng Cao, Slim Ibrahim, Kenji Nakanishi, and Edriss S. Titi, Finite-time blowup for the inviscid primitive equations of oceanic and atomspheric dynamics, arXiv:1210.7337v1 [math.AP].
Weinan E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 2, 207–218. MR 1778702, DOI 10.1007/s101140000034
Weinan E and Bjorn Engquist, Blowup of solutions of the unsteady Prandtl’s equation, Comm. Pure Appl. Math. 50 (1997), no. 12, 1287–1293. MR 1476316, DOI 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4
Emmanuel Grenier, On the derivation of homogeneous hydrostatic equations, M2AN Math. Model. Numer. Anal. 33 (1999), no. 5, 965–970. MR 1726718, DOI 10.1051/m2an:1999128
Igor Kukavica, Roger Temam, Vlad C. Vicol, and Mohammed Ziane, Local existence and uniqueness for the hydrostatic Euler equations on a bounded domain, J. Differential Equations 250 (2011), no. 3, 1719–1746. MR 2737223, DOI 10.1016/j.jde.2010.07.032
Pierre-Louis Lions, Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models; Oxford Science Publications. MR 1422251
Nader Masmoudi and Tak Kwong Wong, On the $H^s$ theory of hydrostatic Euler equations, Arch. Ration. Mech. Anal. 204 (2012), no. 1, 231–271. MR 2898740, DOI 10.1007/s00205-011-0485-0
Tak Kwong Wong, On the wellposedness of boundary layer equations, Ph.D. thesis, New York University, 2010.