In this note we analyze smooth solutions of a $p$-system of the \textit{mixed} type. Motivating example for this is a 2-components reduction of the Benney moments chain which appears to be connected to theory of integrable systems. We don't assume a-priory that the solutions in question are in the Hyperbolic region. Our main result states that the only smooth solutions of the system which are periodic in $x$ are necessarily constants. As for initial value problem we prove that if the initial data is strictly hyperbolic and periodic in $x$ then the solution can not extend to $[t_0;+\infty)$ and shocks are necessarily created.
Approximate solutions to the initial value problem for some compressible flows in presence of shocks and void regions
The initial value problem for the binormal flow with rough data
Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form
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