| We investigate the stabilizing effect of the antisymmetry in the geometric interaction structure of the Euler equations of gas dynamics. We find the simplest model which distinguishes the stabilizing effect of the degenerate quadratic model associated with the Euler equations when the antisymmetry is accounted for versus when it is not. The results prove that, in the model, solutions with arbitrarily large total variation will eventually decay when the antisymmetry effect is accounted for, but that Glimm's result is sharp, (decay only occurs when the total variation is sufficiently small), for bounded solutions in the case when the antisymmetry effect is neglected. The model problem has a compactly supported solution that has an oscillating field of N + 1 contact discontinuites of constant strength, with arbitrary waves in the 1- and 3-fields. We derive a matrix whose powers propagate the 1- and 3-waves, (and hence the solution), up to time t. The problem of stability is thus transformed into an eigenvalue problem for this matrix. We give a sharp estimate for the magnitude of the largest eigenvalue of this matrix, and from this we conclude that the solution will decay for initial data of arbitrarily large total variation when the antisymmetry is accounted for, and only for sufficiently small total variation when it is not. In fact, the sharp result is that if the total variation is bounded by order N 1/4 as N → infinity, then the solution will decay when the antisymmetry is accounted for, but when it is not, the total variation must be smaller than a bounded function of N (the threshold for Glimm's result in this case) to get decay. |
