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Systematic Integration by Parts

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Entropy Methods for Diffusive Partial Differential Equations

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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, Jüngel, and others; see, e.g., Bukal et al., Commun Math Sci, 9:353–382, 2011, [4], Jüngel and Matthes, Nonlinearity, 19:633–659, 2006, [12], Jüngel 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 Jüngel 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], Jüngel 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].

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  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria

    Ansgar Jüngel

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  1. Ansgar Jüngel
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Jüngel, A. (2016). Systematic Integration by Parts. In: Entropy Methods for Diffusive Partial Differential Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-34219-1_3

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