Chapman-Enskog-Hilbert expansion for a class of solutions of the telegraph equation

The Chapman-Enskog-Hilbert expansion is a method for describing a gas in the "hydrodynamical stage," beginning from the Boltzmann equation. The present paper is devoted to the analog of this expansion for the problem ∂p/∂t + e∂p/∂x = p(-e) - p(+e), where e = ±1 and x ∈ R¹. Though the situation is vastly simpler than in the Boltzmann case, new and amusing mathematical phenomena are encountered. One studies solutions of ∂p/∂t + e∂p/∂x = ∈ ^-1[p(-e) - p(+e)] which are (formal) power series in ∈ (Hilbert solutions): such a solution solves ∂p/∂t = ∈^-1[(1 + ∈²∂²/∂x²)^1/2 - 1]p (hydrodynamical equation) and is completely determined by the initial value of p(-e) + p(+e) (Hilbert paradox). Also, every solution of the original problem comes very rapidly close to a Hilbert solution which is actually convergent (hydrodynamical stage).