Systematic Integration by Parts

Abstract

The computation of the second time derivative of the entropy in Chap. 2 involves a number of smartly chosen integrations by parts. In this chapter, we show that these calculations can be made systematic to some extent. This technique was elaborated by Matthes, Bukal, Jungel, and others; see, e.g., Bukal et al., Commun Math Sci, 9:353–382, 2011, [4], Jungel and Matthes, Nonlinearity, 19:633–659, 2006, [12], Jungel and Matthes, SIAM J. Math Anal, 39:1996–2015, 2008, [13]. After motivating systematic integration by parts as a tool for entropy computations (Sect. 3.1), we detail the one-dimensional case (Sect. 3.2) and consider some multidimensional extensions (Sect. 3.3). Furthermore, the Bakry–Emery approach is reconsidered using systematic integration by parts. The presentation of Sect. 3.2 is close to Jungel and Matthes, Nonlinearity, 19:633–659, 2006, [12], the theorems in Sect. 3.3 are due to Bukal et al., Commun Math Sci, 9:353–382, 2011, [4], Jungel and Matthes, SIAM J. Math Anal, 39:1996–2015, 2008, [13], Laugesen, Commun Pure Appl Anal 4:613–634, 2005, [18], and Sect. 3.4 summarizes results from Matthes et al., Arch Ration Mech Anal 199:563–596, 2011, [21].