Abstract:
We study the half-space problem of the nonlinear Boltzmann equation, assigning the Dirichlet data for outgoing particles at the boundary and a Maxwellian as the far field. We will show that the solvability of the problem changes with the Mach number ℳ∞ of the far Maxwellian. If ℳ∞<−1, there exists a unique smooth solution connecting the Dirichlet data and the far Maxwellian for any Dirichlet data sufficiently close to the far Maxwellian. Otherwise, such a solution exists only for the Dirichlet data satisfying certain admissible conditions. The set of admissible Dirichlet data forms a smooth manifold of codimension 1 for the case −1<ℳ∞<0, 4 for 0<ℳ∞<1 and 5 for ℳ∞>1, respectively. We also show that the same is true for the linearized problem at the far Maxwellian, and the manifold is, then, a hyperplane. The proof is essentially based on the macro-micro or hydrodynamics-kinetic decomposition of solutions combined with an artificial damping term and a spatially exponential decay weight.
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Received: 20 April 2002 / Accepted: 4 December 2002 Published online: 21 March 2003
Communicated by H.-T. Yau
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Ukai, S., Yang, T. & Yu, SH. Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence. Commun. Math. Phys. 236, 373–393 (2003). https://doi.org/10.1007/s00220-003-0822-8
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DOI: https://doi.org/10.1007/s00220-003-0822-8