For a one-dimensional BGK kinetic model, the Chapman-Enskog expansion is a local Taylor approximation to the exact spectral hydrodynamic mode and diverges for every nonzero wave number, while the spectral closure remains well defined.
Grad, Communications on pure and applied mathe- matics 2, 331 (1949)
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On the Relation of Exact Hydrodynamics to the Chapman-Enskog Series
For a one-dimensional BGK kinetic model, the Chapman-Enskog expansion is a local Taylor approximation to the exact spectral hydrodynamic mode and diverges for every nonzero wave number, while the spectral closure remains well defined.