Entropy Methods for Diffusive Partial Differential Equations
This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.
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algorithm Anal assumptions asymptotics Bakry Bakry–Emery approach Boltzmann boundary integrals coefficients computation constant convergence convex function convex Sobolev inequality cross-diffusion systems decision problem defined diffusion equations diffusion matrix Dolbeault eigenvalues entropy density entropy methods entropy production entropy structure entropy variables estimates Exponential decay Fokker–Planck equation formulation global existence gradient flow Gronwall’s lemma Hſu initial datum Jüngel linear logarithmic Sobolev inequality Lyapunov functional Markov chains mass Math mathematical mathematical entropy Matthes Maxwell–Stefan nonlinear nonnegative numerical parabolic Partial Differential Equations Poincaré inequality polynomial population model porous-medium equation positive definite positive semi-definite proof of Theorem proved quantifier elimination Rd Rd Remark result right-hand side Runge–Kutta satisfied scheme Sect semigroup shift polynomials shows SIAM smooth ſº Springer stochastic symmetric Theorem 4.1 theory thermodynamics Toscani uniqueness usedX vanish weak solutions