Hydrodynamic Limit of the Boltzmann Equation toward Generic Riemann Solutions with Shocks

Abstract

We establish the hydrodynamic limit of the one-dimensional Boltzmann equation with hard-sphere collisions toward Riemann solutions of the compressible Euler system. The Riemann solutions covered by our result include generic superpositions of elementary waves: either two shock waves and a contact discontinuity, or a rarefaction wave, a contact discontinuity, and a shock wave. For suitably well-prepared initial data and sufficiently small wave strength, we prove that the corresponding Boltzmann solution exists globally in time and converges, as the Knudsen number vanishes, to the local Maxwellian associated with the Riemann solution in $L^2([0,T]\times\mathbb R_x\times\mathbb R^3_\xi)$ for any $T>0$. The proof combines the macro--micro decomposition with a kinetic adaptation of the $a$-contraction method, in which the propagation speeds of Boltzmann shocks are determined by Rakine-Hugoniot speed with dynamical modulation parameters. The resulting coercive control of the shock translation modes, together with layer analysis and the uniform bound of the Shifts, allows us to pass to the Knudsen limit in the full space-time domain. To the best of our knowledge, this is the first rigorous result on the hydrodynamic limit towards generic Riemann solutions containing shocks: either the shock--contact--shock case or the rarefaction--contact--shock case, in a global space-time energy norm without removing neighborhoods of either the initial time or the shock layers. In the special case of a single shock, the argument further yields a sharp quantitative description of the kinetic shock layer, up to the dynamically selected Shift.