In this lecture, we shall discuss functional-analytic methods for nonlinear diffusion equations. Nonlinear diffusion equations are introduced in order to describe diffusion phenomena which cannot be explained with a classical theory. There arise mathematical difficulties from degeneracy and singularity of leading terms of equations, which prevent us to apply classical methods such as Fourier analysis. Moreover, solutions exhibit characteristic properties and behaviors which look rather different from those of the normal diffusion equation. In particular, we essentially need to treat non-classical solutions. On the other hand, functional-analytic methods are known to effectively work for analysis of such nonlinear diffusion equations as well, and there have already been many researches in this direction. In this lecture, we shall start with fundamental issues such as well-posedness (existence and uniqueness of solutions and their continuous dependence on initial data) and comparison principle, and then discuss asymptotic behavior of solutions and some sort of stability issues (e.g., stability of equilibria).
We shall aim at helping audience to understand various problems on nonlinear parabolic equations and functional-analytic methods for them by explaining theories and recent developments on typical issues and their analysis for nonlinear diffusion equations.
Students are expected to understand
- derivations of nonlinear diffusion equations and related notions such as degeneracy and singularity.
- characteristics of solutions to nonlinear diffusion equations and difference from those of the classical diffusion equation
- proof of well-posedness of equations and comparison principles of solutions based on functional-analytic methods
- issues related to asymptotic behavior and profiles of solutions and stability of equilibria and asymptotic profiles
- analysis of the issues mentioned above based on functional-analytic methods
Nonlinear diffusion equation, degeneracy and singularity, functional-analytic method, asymptotic behavior of solutions, stability analysis
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||We shall discuss the following items: １．Nonlinear diffusion equations and their characteristics Derivation, interpretation through Fick's law, explicit solutions (ZKB solution) and their properties ２．Well-posedness (existence, uniqueness, continuous dependence of solutions on initial data and comparison principle Various approaches to prove well-posedness, nonlinear semigroup theory, energy methods, dynamical system ３．Long-time behavior and asymptotic profiles of solutions decay estimates of solutions for initial-boundary value problems, definition of asymptotic profiles and their existence and characterization ４．Stability of asymptotic profiles Notion of stability of asymptotic profiles, stability analysis for isolated (in a function space) asymptotic profiles ５．Łojasiewicz-Simon inequality and applications Example of non-isolated asymptotic profiles, convergence and stability analysis ６．Related topics||Details will be provided during each class session.|
J.L. Vázquez, The porous medium equation. Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.