We construct degenerate quadratic models for the Euler equations which distinguish the stabilizing effect of an antisymmetry in the Lie structure of the Euler equations when this antisymmetry is accounted for versus when it is not. We derive a matrix, depending only on the mesh size, $N$, and the 2-wave strengths, whose powers propagate the 1- and 3-waves up to time $t$. We give sharp estimates for the magnitude of the largest eigenvalue of this matrix and conclude that the solution will decay for initial data of arbitrarily large total variation, of order the 4-th root of $N$ in the limit $N$ approaches infinity, when the antisymmetry is accounted for, and only for suffciently small total variation when it is not
"Spectral Analysis of a Model of the Compressible Euler Equations." Commun. Math. Sci. 5 (1) 205 - 232, March 2007.