Abstract
This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $\xi$. All of these equations are treated in the Fréchet space $L_{{\text {loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.
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