Geodesic
Isentropic solutions of quasilinear equations of the first order
Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 727-752

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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. Bibliography: 13 titles. 
@article{SM_2006_197_5_a3,
     author = {M. V. Korobkov and E. Yu. Panov},
     title = {Isentropic solutions of quasilinear equations of the first order},
     journal = {Sbornik. Mathematics},
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     volume = {197},
     number = {5},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2006_197_5_a3/}
}
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M. V. Korobkov; E. Yu. Panov. Isentropic solutions of quasilinear equations of the first order. Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 727-752. http://geodesic.mathdoc.fr/item/SM_2006_197_5_a3/