On the boundary layer equations with phase transition in the kinetic theory of gases (Q2662022)

In the present paper, the authors deal with the nonlinear half-space problem for the Boltzmann equation written in terms of the relative fluctuation of distribution function about the normalized Maxwellian \(M\) \[ \left\{ \begin{array}{cc} (\xi _i+u)\partial _xf_u+\mathcal{L}f_u, & \xi \in \mathbb{R}^3,x>0, \\ f_u(0,\xi )=f_b(\xi ), & \xi _i+u>0. \end{array} \right. \] The authors prove the existence of the curve \(C\) corresponding to solutions of the equations given in some neighborhood of the point \((1, 0, 1)\) converging as \(x\rightarrow \infty\) with exponential speed uniformly in \(u\). The authors provides a self-contained construction of the solution to the Nicolaenko-Thurber generalized eigenvalue problem near \(u = 0\). Then, the authors introduce the penalization method, and formulate the problem to be solved by a fixed point argument. The linearized penalized problem is studied. Also, the authors investigate the (weakly) nonlinear penalized problem by a fixed point argument. The authors give an alternative, possibly simpler proof of one of the results discussed in [\textit{T.-P. Liu} and \textit{S.-H. Yu}, Arch. Ration. Mech. Anal. 209, No. 3, 869--997 (2013; Zbl 1290.35181)].