Nonlinear Partial Differential Equations

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

Verlag: Springer Basel

Buchreihe : Advanced Courses in Mathematics - CRM Barcelona

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

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

Titel: Nonlinear Partial Differential Equations
verfasst von: Luis A. Caffarelli
François Golse
Yan Guo
Carlos E. Kenig
Alexis Vasseur
Verlag: 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