線性化波茲曼方程之巨觀變量的邊界奇異性

Abstract

在各種氣體模型中，氣體動力學適合用來模擬其靠近邊界的現象。在研究由邊界所造成的影響時，關於稀薄氣體在兩片平面夾層間的靜態問題是最簡化卻必要的。在本論文中，我們研究具截斷軟位勢的線性化波茲曼方程之邊界奇異性。我們對於其巨觀變量的微分建立了漸進展開式。對於硬球模式與硬位勢，類似的展開式已經發表在陳逸昆2013年的文章以及陳逸昆和夏俊雄2015年的文章中。本研究將把他們的結果推廣到軟位勢模型 -3/2 < γ < 0。但因為已知的解所坐落的函數空間對於速度變數的二次加權可積性之權重的特性，在軟位勢與硬位勢是有所不同，所以我們不能直接採用陳夏兩位的論證方式。為了要克服這個癥結點，我們針對該加權二次可積空間採用了一個與之前不同版本的平滑化估計，該估計是來自於高爾斯與珀波德1989年的文章。利用此一估計及陳夏2015年文章的想法，我們成功地建立了對於軟位勢模型之巨觀變量的邊界奇異性。

Among various models of gases, the kinetic theory is suitable for modelling boundary phenomena. In the studies of the effects of the boundary, problems related to the steady behavior of a rarefied gas bounded by a pair of planar walls are simplest yet indispensable. In this thesis, 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. 153(1), 93--118] and [Chen, I.-K., Hsia, C.-H.: SIAM J. Math. Anal. 47(6) 4332--4349 (2015)]. We extend their results to the case of soft potential -3/2< γ <0. Since the solution space from the known existence theory is equipped with a weighted L2 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 L2 space in [Golse, F., Poupaud, F.: Math. Methods Appl. Sci. 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 -3/2 < γ < 0.