Abstract:

The boundary singularity for stationary solutions of the linearized Boltzmann equation with cutoff soft potential in a slab is studied. An asymptotic formula for the gradient of the moments is established, which reveals the logarithmic singularity near the planar boundary. Similar results for cutoff hard-sphere and hard potential were proved in [Chen, I.-K.: J. Stat. Phys. \textbf{153}(1), 93--118] and [Chen, I.-K., Hsia, C.-H.: SIAM J. Math. Anal. \textbf{47}(6) 4332--4349 (2015)]. We extend their results to the case of soft potential Since the solution space from the known existence theory is equipped with a weighted integrability for the velocity variables that behaves differently from the solution space for hard potential case, we cannot apply their arguments directly. To overcome this crux, we employed a different version of smoothing property for weighted space in [Golse, F., Poupaud, F.: Math. Methods Appl. Sci. \textbf{11}(4), 483--502 (1989)] to carry out the idea of Chen and Hsia. We then successfully extend the boundary singularity result to the soft potential case