An alternative energy bound derivation for a generalized Hasegawa–Mima equation

We present an alternative derivation of the H1H1-boundedness of solutions to a generalized Hasegawa–Mima equation, first investigated by Grauer (1998) [2]. We apply a Lyapunov function technique similar to the one used for constructing energy bounds for the Kuramoto–Sivashinsky equation. Different from Grauer (1998) [2], who uses this technique in a Fourier space approach, we employ the physical space construction of the Lyapunov function, as developed in Bronski and Gambill (2006) [11]. Our approach has the advantage that it is more transparent in what concerns the estimates and the dominant terms that are being retained. A key tool of the present work, which replaces the algebraic manipulations on the Fourier coefficients from the other approach, is a Hardy–Rellich type inequality.