Abstract
Let u( ¯x, t) be a weak solution of the Euler equation, governing the inviscid polytropic gas dynamics; in addition, u(¯ x, t) is assumed to respect the usual entropy conditions connected with the conservative Euler equations. We show that such entropy solutions of the gas dynamics equations satisfy a minimum entropy principle, namely, that the spatial minimum of their specific entropy, Ess inf ¯ x S( u(¯ x, t)), is an increasing function of time. This principle equally applies to discrete approximations of the Euler equations such as the Godunov-type and Lax—Friedrichs schemes. Our derivation of this minimum principle makes use of the fact that there is a family of generalized entropy functions connected with the conservative Euler equations.
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