Fourier Analysis And Nonlinear Partial Differential Equations
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Fourier Analysis and Nonlinear Partial Differential Equations
Author: Hajer Bahouri
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-01-03
In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.
Fourier Series in Several Variables with Applications to Partial Differential Equations
Discussing many results and studies from the literature, this work illustrates the value of Fourier series methods in solving difficult nonlinear PDEs. Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory, presents the periodic Ca-theory of Calderon and Zygmund, and explores the extension of Fatou's famous work on antiderivatives and nontangential limits to higher dimensions. The importance of surface spherical harmonic functions is emphasized throughout.
Fourier Analysis and Partial Differential Equations
Author: Iorio Júnior Iorio Jr.
language: en
Publisher: Cambridge University Press
Release Date: 2001-03-15
This book was first published in 2001. It provides an introduction to Fourier analysis and partial differential equations and is intended to be used with courses for beginning graduate students. With minimal prerequisites the authors take the reader from fundamentals to research topics in the area of nonlinear evolution equations. The first part of the book consists of some very classical material, followed by a discussion of the theory of periodic distributions and the periodic Sobolev spaces. The authors then turn to the study of linear and nonlinear equations in the setting provided by periodic distributions. They assume only some familiarity with Banach and Hilbert spaces and the elementary properties of bounded linear operators. After presenting a fairly complete discussion of local and global well-posedness for the nonlinear Schrödinger and the Korteweg-de Vries equations, they turn their attention, in the two final chapters, to the non-periodic setting, concentrating on problems that do not occur in the periodic case.