Entropy Generation and Integral Inequalities

As a typical irreversible process, the heat conduction in rectangles results in entropy generation. As time tends to infinity, this entropy generation evolves into a finite value when the heat conduction comes from the initial temperature distribution, but into infinity whenever a positively-averaged heat source is involved. An application of the second law of thermodynamics to this process leads to nine integral inequalities which are important for studying heat-conduction equations and for uncovering some basic features of the total multiplicity and the Boltzmann entropy. The work correlates the second law of thermodynamics in thermodynamics and integral inequalities in mathematics, and inspires the future work in offering some fundamental insights into our future.