A simple energy pump for the surface quasi-geostrophic equation

TL;DR

This paper demonstrates that solutions to the surface quasi-geostrophic equation can exhibit unbounded growth in high Sobolev norms, using a quasilinear construction with small perturbations of shear flows.

Contribution

It introduces a novel quasilinear method to show potential unbounded growth of high Sobolev norms for the surface quasi-geostrophic equation.

Findings

High Sobolev norms can grow arbitrarily large for certain initial data.

Small perturbations of shear flows can generate small scales leading to norm growth.

Nonlinear effects eventually dominate, causing loss of control over the solution.

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 is exist small in $H^s$ initial data such that the corresponding solution's $H^s$ norm exceeds $A$ at some time. 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.