Abstract: Poincaré integral inequality plays an inportant role in the nonlinear stability theorem of Arnold. In this paper, a best possible Poincaré integral inequality is obtained, which can be used to get a better estimation of disturbance bound for two-dimensional quasi-geostrophic flow and a better nonlinear stability criterion as well.

Key words: variational principle, Fourier analysis, nonlinear stability, Poincaré inequality, variational principle, Fourier analysis, nonlinear stability