Abstract
We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if s>0 is large then for every given A there exists initial data with a norm that is small in H s such that the H s norm of corresponding solution at some time exceeds A. The idea of the construction is quasilinear. We use a small perturbation of a stable shear flow. The shear flow can be shown to create small scales in the perturbation part of the flow. The control is lost once the nonlinear effects become too large.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)
Cordoba, D.: Nonexistence of simple hyperbolic blow up for the quasi-geostrophic equation. Ann. Math. 148, 1135–1152 (1998)
Denisov, S.: Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete Contin. Dyn. Syst., Ser. A 23, 755–764 (2009)
Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)
Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)
Nadirashvili, N.S.: Wandering solutions of the two-dimensional Euler equation. Funkc. Anal. Prilozh. 25, 70–71 (1991) (in Russian). Translation in Funct. Anal. Appl. 25 (1991), 220–221 (1992)
Ohkitani, K., Yamada, M.: Inviscid and inviscid-limit behavior of a SQG flow. Phys. Fluids 9, 876–882 (1997)
Pedlosky, J.: Geophysical Fluid Dynamics Springer, New York (1987)
Yudovich, V.I.: The loss of smoothness of the solutions of the Euler equation with time. Dinamika Sploshn. Sredy, Nestacionarnye Problemy Gidrodinamiki 16, 71–78 (1974) (in Russian)
Yudovich, V.I.: On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid. Chaos 10, 705–719 (2000)
Acknowledgements
Research of AK has been supported in part by the NSF-DMS grant 1104415. Research of FN has been partially supported by the NSF-DMS grant 0800243.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kiselev, A., Nazarov, F. (2012). A Simple Energy Pump for the Surface Quasi-geostrophic Equation. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-25361-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25360-7
Online ISBN: 978-3-642-25361-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)