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2011 | Buch

Basic Analysis Kapitel lesen Erstes Kapitel lesen

Fourier Analysis and Nonlinear Partial Differential Equations

verfasst von: Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

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

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
1. Basic Analysis
Abstract
Chapter 1 is devoted to a self-contained elementary presentation of classical Fourier analysis results. Even though none of the results are new, some of the proofs that we present are not the standard ones and are likely to be useful in other contexts. We also pay attention to the construction of explicit examples which illustrate the optimality of some refined estimates.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
2. Littlewood–Paley Theory
Abstract
In Chapter 2 we give a detailed presentation on Littlewood-Paley decomposition and define homogeneous and nonhomogeneous Besov spaces. We should emphasize that we have replaced the usual definition of homogeneous spaces (which are quotient distribution spaces modulo polynomials) by something better adapted to the study of partial differential equations (indeed, dealing with distributions modulo polynomials is not appropriate in this context). We also establish technical results (commutator estimates and functional inequalities, in particular) which will be used in the following chapters.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
3. Transport and Transport-Diffusion Equations
Abstract
In Chapter 3, we give a very complete theory of strong solutions for transport and transport-diffusion equations. In particular, we provide a priori estimates which are the key to solving nonlinear systems coming from fluid mechanics.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
4. Quasilinear Symmetric Systems
Abstract
Chapter 4 is devoted to solving linear and quasilinear symmetric systems with data in Sobolev spaces. Blow-up criteria and results concerning the continuity of the flow map are also given. The case of data with critical regularity (in a Besov space) is also investigated.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
5. The Incompressible Navier–Stokes System
Abstract
In Chapter 5 we take advantage of the tools introduced in the previous chapters to establish most of the classical results concerning the well-posedness of the incompressible Navier–Stokes system for data with critical regularity.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
6. Anisotropic Viscosity
Abstract
In order to emphasize the robustness of the tools that have been introduced hitherto in this book, we present in Chapter 6 a nonlinear system of partial differential equations with degenerate parabolicity. In fact, we show that some of the classical results for the Navier–Stokes system may be extended to the case where there is no vertical diffusion. Most of the results of this chapter are based on the use of an anisotropic Littlewood–Paley decomposition.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
7. Euler System for Perfect Incompressible Fluids
Abstract
Chapter 7 is the natural continuation of the previous chapter: The diffusion term is removed, leading to the study of the Euler system for inviscid incompressible fluids. Here, we state local (in dimension d≥3) and global (in dimension two) well-posedness results for data in general Besov spaces. In particular, we study the case where the data belong to Besov spaces for which the embedding in the set of Lipschitz functions is critical. In the two-dimensional case, we also give results concerning the inviscid limit. We stress the case of data with (generalized) vortex patch structure.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
8. Strichartz Estimates and Applications to Semilinear Dispersive Equations
Abstract
Chapter 8 is devoted to Strichartz estimates for dispersive equations with a focus on Schrödinger and wave equations. After proving a dispersive inequality (i.e., decay in time of the L ∞ norm in space) for these equations, we present, in a self-contained way, the celebrated TT ⋆ argument based on a duality method and on bilinear estimates. Some examples of applications to semilinear Schrödinger and wave equations are given at the end of the chapter.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
9. Smoothing Effect in Quasilinear Wave Equations
Abstract
Chapter 9 is devoted to the study of a class of quasilinear wave equations which can be seen as a toy model for the Einstein equations. First, by taking advantage of energy methods in the spirit of those of Chapter 4, we establish local well-posedness for “smooth” initial data (i.e., for data in Sobolev spaces embedded in the set of Lipschitz functions). Next, we weaken our regularity assumptions by taking advantage of the dispersive nature of the wave equation. The key to that improvement is a quasilinear Strichartz estimate and a refinement of the paradifferential calculus. To prove the quasilinear Strichartz estimate, we use a microlocal decomposition of the time interval (i.e., a decomposition in some interval, the length of which depends on the size of the frequency) and geometrical optics.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
10. The Compressible Navier–Stokes System
Abstract
In Chapter 10 we present a more complicated system of partial differential equations coming from fluid mechanics, the so-called barotropic compressible Navier–Stokes equations. Those equations are of mixed hyperbolic-parabolic type. We show how we may take advantage of the results of Chapter 3 and the techniques introduced in Chapter 2 so as to obtain local (or global) unique solutions with critical regularity. The last part of this chapter is dedicated to the study of the low Mach number limit for this system. It is shown that under appropriate assumptions on the data, the limit solution satisfies the incompressible Navier–Stokes system studied in Chapter 5.
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
Backmatter
Metadaten
Titel
Fourier Analysis and Nonlinear Partial Differential Equations
verfasst von
Hajer Bahouri
Jean-Yves Chemin
Raphaël Danchin
Copyright-Jahr
2011
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-16830-7
Print ISBN
978-3-642-16829-1
DOI
https://doi.org/10.1007/978-3-642-16830-7