Stokes-Fourier and Acoustic Limits for the Boltzmann Equation: Convergence Proofs

We establish a Stokes-Fourier limit for the Boltzmann equation considered over any periodic spatial domain of dimension 2 or more. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of Stokes-Fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate L² initial data of the Stokes-Fourier equations. Both the motion and heat equations are recovered in the limit by controlling the fluxes and the local conservation defects of the DiPerna-Lions solutions with dissipation rate estimates. The scaling of the fluctuations with respect to Knudsen number is essentially optimal. The assumptions on the collision kernel are little more than those required for the DiPerna-Lions theory and that the viscosity and heat conduction are finite. For the acoustic limit, these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations. Both weak limits become strong when the initial fluctuations converge entropically to appropriate L² initial data.