Isentropic solutions of quasilinear equations of the first order

Conditions for the existence of non-trivial isentropic solutions of quasilinear conservation laws are found. Applications to the problem of the functional dependence between partial derivatives of a smooth function of two variables are presented. In particular, necessary conditions on a function $\varphi$ for the equation $\dfrac{\partial v}{\partial t} =\varphi\biggl(\dfrac{\partial v}{\partial x}\biggr)$ to have non-trivial $C^1$-smooth solutions are found.
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