By Solomon Lefschetz
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Additional resources for Differential Equations: Geometric Theory, 2nd ed
Pel?. ) In the general case, the tensor P is not spherical. In the same way ~~q is not equal to zero. 44) for the pressure tensor P and the heat flux vector if. We are continually constructing explicit expressions for the Euler equations. To this end, we give an explicit expression to P • ~u + ~q . At the chosen order of magnitude one has: _> _> /a? 1? ^a\\f -_>^ + m / + 2 a\a\ - a\ \ __> _> H f\ p e u - p e u "TI + T = 7T^P U + - 4 7 - 4 2a\ af(a2 _ a 4TPeu -peu . 456) 46 Lecture Notes on the Discretization of the Boltzmann Equation and u + q a a a i( 2 ~ \) a\a\ + a%d\ 2a% pit.
We emphasize that all these results are very closed to those obtained by other treatments of the Boltzmann equation from qualitative and also quantitative points of view. 3 Some remarks on the boundary condition problem Let us consider the mathematical problem of the existence of solutions for the kinetic equations with boundary conditions. First we pay attention to the steady problem between two plates. 62) where y € [0, d] and where ak and bk are positive. All the quantities Vk are nonzero constants.
Independently of this paper, Cornille and Cercignani [14; 16] developed a class of planar semisymmetric models with 9, 11, 13, 15 velocities, and, by a suitable superposition, symmetric models with 17 and 25 velocities. In a subsequent paper, Cornille and Cercignani  constructed models with arbitrarily many velocities, for the values 2, 3, 4, 5 of the mass ratio. 5 Concluding remarks In this survey we have first reviewed the result of  showing that the procedure used to approximate the Boltzmann equation for a simple gas by DVM can be extended to mixtures.