Abstract
We consider completely exceptional models for 3 × 3 systems of conservation laws having a given geometric interaction structure. We show that the model for the equations of gas dynamics has globally bounded solutions which do not decay, and conjecture that similar behavior occurs for the full Euler equations. Moreover, other 3 × 3 systems with nontrivial Lie algebra have exponentially growing modes. Thus a necessary condition for globally bounded solutions is that the geometric structure be the same as that of the gas dynamics equations, namely the system must be symmetric hyperbolic. We introduce a simple numerical scheme in which the approximations are exact weak solutions, so that there is no residual, and convergence follows easily. This allows us to explain most observed phenomena of periodic solutions.
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