Download Entropy Methods for Diffusive Partial Differential Equations by Ansgar Jüngel PDF

By Ansgar Jüngel

This publication offers various entropy tools for diffusive PDEs devised through many researchers during the earlier few many years, which permit us to appreciate the qualitative habit of suggestions to diffusive equations (and Markov diffusion processes). functions contain the large-time asymptotics of options, the derivation of convex Sobolev inequalities, the life and distinctiveness of vulnerable ideas, and the research of discrete and geometric constructions of the PDEs. the aim of the booklet is to supply readers an advent to chose entropy equipment that may be present in the examine literature. so as to spotlight the center options, the implications aren't acknowledged within the widest generality and lots of the arguments are just formal (in the feel that the practical environment isn't precise or adequate regularity is supposed). The textual content is additionally compatible for complicated grasp and PhD scholars and will function a textbook for precise classes and seminars.

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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. , Arch Ration Mech Anal 199:563–596, 2011, [21]. 1 Motivation Systematic integration by parts is not only a method to simplify the proofs. It may also be useful to strengthen the results. As a motivating example, let us consider the thin-film equation in the one-dimension torus, ∂t u = −(u β u x x x )x , t > 0, u(0) = u 0 ≥ 0 in T, β > 0.

4 Relaxation to Self-Similarity 31 However, the entropy is useful to study the intermediate asymptotics, namely the relaxation of u(t) to the self-similar solution U (x, t) = 1 exp (2π(2t + 1))d/2 − |x|2 , x ∈ Rd , t > 0. 15) The main result (with sharp decay rate O(t −1/2 )) is as follows. 3 (Relaxation to self-similarity) Let u 0 ∈ L 1 (Rd ) be nonnegative with unit mass, Rd |x|2 u 0 dx < ∞, and Rd u 0 log u 0 dx < ∞. 15), and H be the Boltzmann entropy. Then u(t) − U (t) L 1 (Rd ) √ 2H[u 0 ] , t > 0.

71–305. North-Holland, Amsterdam (2002) 46. : Optimal Transport Old and New. Springer, Berlin (2009) Chapter 3 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. , 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].

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