A Simple Energy Pump for the Surface Quasi-geostrophic Equation
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.
KeywordsEuler Equation Ekman Number Weak Rate Stable Shear Vorticity Form
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.
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