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2012 | Book

The De Giorgi Method for Nonlocal Fluid Dynamics Read chapter Read first chapter

Nonlinear Partial Differential Equations

Authors: Luis A. Caffarelli, François Golse, Yan Guo, Carlos E. Kenig, Alexis Vasseur

Publisher: Springer Basel

Book Series : Advanced Courses in Mathematics - CRM Barcelona

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The book covers several topics of current interest in the field of nonlinear partial differential equations and their applications to the physics of continuous media and particle interactions. It treats the quasigeostrophic equation, integral diffusions, periodic Lorentz gas, Boltzmann equation, and critical dispersive nonlinear Schrödinger and wave equations. The book describes in a careful and expository manner several powerful methods from recent top research articles.

Table of Contents

Frontmatter
Chapter 1. The De Giorgi Method for Nonlocal Fluid Dynamics
Abstract
In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and analyticity of variational (“energy minimizing weak”) solutions to nonlinear elliptic variational problems. In so doing, he developed a very geometric, basic method to deduce boundedness and regularity of solutions to a priori very discontinuous problems. The essence of his method has found applications in homogenization, phase transition, inverse problems, etc.
Luis A. Caffarelli, Alexis Vasseur
Chapter 2. Recent Results on the Periodic Lorentz Gas
Abstract
The kinetic theory of gases was proposed by J. Clerk Maxwell [34, 35] and L. Boltzmann [5] in the second half of the XIXth century. Because the existence of atoms, on which kinetic theory rested, remained controversial for some time, it was not until many years later, in the XXth century, that the tools of kinetic theory became of common use in various branches of physics such as neutron transport, radiative transfer, plasma and semiconductor physics, etc.
François Golse
Chapter 3. The Boltzmann Equation in Bounded Domains
Abstract
Boundary effects play a crucial role in the dynamics of gases governed by the Boltzmann equation
Yan Guo
Chapter 4. The Concentration-Compactness Rigidity Method for Critical Dispersive and Wave Equations
Abstract
In these lectures I will describe a program (which I will call the concentrationcompactness/rigidity method) that Frank Merle and I have been developing to study critical evolution problems. The issues studied center around global wellposedness and scattering. The method applies to nonlinear dispersive and wave equations in both defocusing and focusing cases.
Carlos E. Kenig
Metadata
Title
Nonlinear Partial Differential Equations
Authors
Luis A. Caffarelli
François Golse
Yan Guo
Carlos E. Kenig
Alexis Vasseur
Copyright Year
2012
Publisher
Springer Basel
Electronic ISBN
978-3-0348-0191-1
Print ISBN
978-3-0348-0190-4
DOI
https://doi.org/10.1007/978-3-0348-0191-1

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