The entropy producing phenomenon is fundamental in statistical physics. It demonstrates the tendency of non-equilibrium states to equilibrium states as time evolves. In the mathematical study of kinetic theory, it is one of the key goals to describe this phenomenon quantitatively and rigorously. Logarithmic Sobolev inequalities and their nonlinear analogues have played an important role in this direction. 

In this talk, I will present results and questions about this functional inequality approach. I will start by introducing basic facts and some related questions/progresses about logarithmic Sobolev inequality. Then I will review results and questions about its nonlinear analogues. More precisely, I will focus on the following models: the McKean-Vlasov equation, the Landau equation and the Kac model.