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Simple shear flow in inelastic Maxwell models

The Boltzmann equation for inelastic Maxwell models is considered to determine the velocity moments through fourth degree in the simple shear flow state. First, the rheological properties (which are related to the second-degree velocity moments) are {\em exactly} evaluated in terms of the coefficient of restitution $α$ and the (reduced) shear rate $a^*$. For a given value of $α$, the above transport properties decrease with increasing shear rate. Moreover, as expected, the third-degree and the asymmetric fourth-degree moments vanish in the long time limit when they are scaled with the thermal speed. On the other hand, as in the case of elastic collisions, our results show that, for a given value of $α$, the scaled symmetric fourth-degree moments diverge in time for shear rates larger than a certain critical value $a_c^*(α)$ which decreases with increasing dissipation. The explicit shear-rate dependence of the fourth-degree moments below this critical value is also obtained.