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This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of Boudin-Desvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

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SnTe is a good alternative to PbTe in the thermoelectric (TE) applications, in that it is a compound with no toxic element Pb. Besides, the compound SnTe has a relatively narrow bandgap (0.3~0.4 eV) and at the same time high Sn vacancy concentration (Sn_v). Accordingly, it gives a high carrier concentration (10²¹ cm^-3) at room temperature (RT), which is not favorable in thermoelectrics, thereby the regulation of both the electronic and phonon scattering mechanisms is strongly required. Up to date, there are many approaches to improve its TE performance. The typical examples are those involving the valence band convergence, nanostructuring, substitutional and interstitial defects, and lattice softening etc., which are all practical and effective to improve the TE performance of SnTe. However, we in this work take the entropy as an indicator to design the SnTe-based TE material with multicomponents and then optimize its TE performance. The detailed scheme involves the chemical composition design step by step. At first, SnTe alloys with 5% GaTe to form a solid solution Sn_0.95Ge_0.05Te, aiming to increase the solubility of the foreign species. The second step is to form another solid solution (Sn_0.95Ge_0.05Te)_0.95(Ag₂Se)_0.05 via the alloying Sn_0.95Ge_0.05Te with 5% Ag₂Se. The purpose of this step is to reduce the p-type carrier concentration of the system, as the species Ag₂Se is a typical n-type semiconductor. The last step is to form a series of solid solutions (Sn_{0.95-<i>x</i>}Ge_0.05Bi<i>_x</i>Te)_0.95(Ag₂Se)_0.05 by substituting different amount of Bi on Sn in (Sn_0.95Ge_0.05Te)_0.95(Ag₂Se)_0.05, to further enhance the configurational entropy (Δ<i>S</i>). Because of the above approaches, both the carrier concentration and thermal conductivity reduce while the highest TE figure of merit (ZT) increases from 0.22 for the pristine SnTe to~0.8 for the alloy (Sn_{0.95-<i>x</i>}Ge_0.05Bi<i>_x</i>Te)_0.95(Ag₂Se)_0.05 (<i>x</i>=0.075). This result proves that the entropy engineering is a practical way to improve the TE performance of SnTe and at the same time, it illustrates that it is very important to harmonize the entropy engineering with other electronic and phonon scattering mechanisms, in order to improve the TE performance of SnTe effectively.