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We study a class of first order quasilinear equations on bounded domains in the L ∞ framework. Using the "semi Kružkov entropy-flux pairs", we define a weak-entropy solution, state an existence and uniqueness result, and a maximum principle.

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To the memory of R. J. DiPerna. AbstractUsing relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna-Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier-Stokes limit with an additional mild weak compactness assumption. The continuous time Navier-Stokes limit is also discussed.