Skip to main content
main-content

Über dieses Buch

The first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.​

Inhaltsverzeichnis

Frontmatter

Nonlinear Dispersive Equations

Frontmatter

Chapter 1. Introduction

Abstract
Nonlinearly interacting waves are often described by asymptotic equations. The derivation typically involves an ansatz for an approximate solution where higher order terms – the precise meaning of higher order term depends on the context and the relevant scales – are neglected. Often a Taylor expansion of a Fourier multiplier is part of that process.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 2. Stationary phase and dispersive estimates

Abstract
We begin with the evaluations of several integrals. Let m d be the d-dimensional Lebesgue measure and define
$$ I_d = \int_{{\mathbb{R}}^d } {e^{ - \left| x \right|^2 } dm^d (x)}. $$
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 3. Strichartz estimates and small data for the nonlinear Schrödinger equation

Abstract
We return to the linear Schrödinger equation
$$ i\partial _t u + \Delta u = 0 $$
and the unitary operators \( S(t):u(0) \to u(t). \)
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 4. Functions of bounded p-variation

Abstract
The study of p-variation of functions of one variable has a long history. Function of bounded p-variation have been studied by Wiener in [33]. The generalization of the Riemann–Stieltjes integral to functions of bounded p-variation against the derivative of a function of bounded q-variation, 1/p + 1/q > 1, is due to Young [34]. Much later Lyons developed his theory of rough paths [23] and [24], building on Young’s ideas, but going much further.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 5. Convolution of measures on hypersurfaces, bilinear estimates, and local smoothing

Abstract
The contents of this section developed in discussions with S. Herr, T. Schottdorf and J. Li. Related results have been proven by Foschi and Klainerman [7] and by Grünrock for the Airy equation [10] and the Kadomtsev–Petviashvili II equation [11]. The bilinear estimates for the Kadomtsev–Petviashvili equation have been influenced by the careful work of M. Hadac. Bilinear estimates are standard tools in dispersive equations.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 6. Well-posedness for nonlinear dispersive equations

Abstract
In this section we will study local and global well-posedness for a number of different equations where the techniques developed so far are relevant. The first example describes the interaction of three waves of different velocities. It is elementary and displays the role of adapted function spaces on an elementary level. The limitations of our current understanding become obvious as well: The result should remain true under small perturbations of the system, but I have no idea how to approach perturbed equations.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 7. Appendix A: Young’s inequality and interpolation

Abstract
Young’s inequality bounds convolutions in Lebesgue spaces. It is part of the statement that the integral exists for almost all arguments of the convolution. Let m d denote the d-dimensional Lebesgue measure.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 8. Appendix B: Bessel functions

Abstract
The Bessel functions are confluent hypergeometric functions. They are solutions to confluent hypergeometric differential equations. Here is a very brief introduction.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 9. Appendic C: The Fourier transform

Abstract
Let f be an integrable complex-valued function. We define its Fourier transform by
$$ \hat f(\xi)\, = \,\frac{1} {{\left( {2\pi } \right)^{{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2}} }}\,\int {e^{ - ix \cdot \xi } \,f(x)\,dm^d \,\left( x \right)} .$$
Herbert Koch, Daniel Tataru, Monica Vişan

Backmatter

Geometric Dispersive Evolutions

Frontmatter

Chapter 1. Introduction

Abstract
Among the nonlinear dispersive equations, a distinguished class is that of geometric evolutions. Unlike the models seen earlier where nonlinear interactions are added to an underlying linear dispersive flow, here the nonlinear structure arises from the curvature of the state space itself. Precisely, our geometric evolutions are obtained by applying the standard linear Lagrangian or Hamiltonian formalism to a state space consiting of maps into (curved) manifolds.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 2. Maps into manifolds

Abstract
Instead of working with real or complex-valued functions, our main objects of study here are evolutions whose state space, in the simplest setting, consists of maps from the Euclidean space \(\mathbb{R}^n\) into a Riemannian manifold (M,g). More generally, one can consider maps whose domains are also Riemannian manifolds.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 3. Geometric pde’s

Abstract
We first review the linear Laplace equation. For functions
$$\phi\;:\;\mathbb{R}^n\;\rightarrow\;\mathbb{R}$$
we define the Lagrangian
$$ L^e(\phi)\;=\;\frac{1} {2}\int_{\mathbb{R}^{n}} {\left| {\nabla _x \phi } \right|^2 \,dx} \, = \,\frac{1} {2}\int_{\mathbb{R}^{n}} {\partial _\alpha \phi } \cdot \partial _\alpha \phi \,dx ,$$
with the Einstein summation convention.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 4. Wave maps

Abstract
Here we outline the main difficulties encountered in the study of the small data problem, and describe the ideas needed to overcome these difficulties. For simplicity we confine ourselves to the most interesting case of dimension two. Some simplifications arise in higher dimension, but the principles remain the same.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 5. Schrödinger maps

Abstract
Here we consider Schrödinger maps \(\phi\;:\;\mathbb{R}\;\times\;\mathbb{R}^n\;\rightarrow\;\mathbb{S}^2,\;n\;\geq\;2\), and prove the small data result in Theorem 3.8. We recall that in n space dimensions the initial data belongs to the space \(\dot{H}^{\frac{n}{2}}\). To keep the notations simple we will confine the discussion to the energy critical case n = 2; this is also the most difficult case.
Herbert Koch, Daniel Tataru, Monica Vişan

Backmatter

Dispersive Equations

Frontmatter

Chapter 1. Notation

Abstract
Throughout this text, we will be regularly referring to the space-time norms
$$ \left\| u \right\|_{L_t^q L_x^r \left( {{\mathbb{R}} \times {\mathbb{R}}^d } \right)} \,:\, = \left( {\int_{{\mathbb{R}}} {\left[ {\int_{{\mathbb{R}}^d } {\left| {u\left( {t,x} \right)} \right|^r } dx} \right]} ^{\tfrac{q} {r}} dt} \right)^{\tfrac{1} {q}}, $$
with obvious changes if q or r are infinity.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 2. Dispersive and Strichartz estimates

Abstract
What all linear dispersive-type equations have in common is a dispersive-type estimate, which expresses the fact that wave packets spread out as time goes by. An expression of this on the Fourier side is the fact that different frequencies move with different speeds and/or in different directions. Below we will discuss several instances of this phenomenon.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 3. An inverse Strichartz inequality

Abstract
In this section, we develop tools that we will employ to prove a linear profile decomposition for the Schrödinger propagator for bounded sequences in \(\dot{H}^{1}(\mathbb{R}^{d})\) with d ≥ 3. Such a linear profile decomposition was first obtained by Keraani [18], relying on an improved Sobolev inequality proved by Gérard, Meyer, and Oru [16]. We should also note the influential precursor [1], which treated the wave equation. In these notes we present a different proof of the result in [18], which relies instead on an inverse Strichartz inequality.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 4. A linear profile decomposition

Abstract
In this section, we use the inverse Strichartz inequality Proposition 3.2 to derive a linear profile decomposition for the Schrödinger propagator.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 5. Stability theory for the energy-critical NLS

Abstract
In this section we develop a stability theory for the energy-critical NLS
$$ i\partial _t u\, = \, - \Delta u\, \pm \,\left| u \right|^{\frac{4} {{d - 2}}} u\,\,\,\,{\rm with}\,\,\,\,u\left( 0 \right)\, = \,u_0 \, \in \,\dot H_x^1. $$
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 6. A large data critical problem

Abstract
Throughout the remainder of these notes we restrict attention to the defocusing energy-critical NLS
$$ i\partial _t u\, + \,\Delta u\, = \,\,\left| u \right|^{\frac{4} {{d - 2}}} u\,\,\,\,{\rm with}\,\,\,\,u\left( 0 \right)\, = \,u_0 \, \in \,\dot H_x^1 . $$
For arguments and further references in the focusing case, see [22].
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 7. A Palais–Smale type condition

Abstract
In this section we prove a Palais–Smale condition for minimizing sequences of blowup solutions to the defocusing energy-critical NLS. It was already observed in [5, 13] that such minimizing sequences have good tightness and equicontinuity properties. This was taken to its ultimate conclusion by Keraani [19], who showed the existence and almost periodicity of minimal blowup solutions in the context of the mass-critical NLS. The proof of the Palais–Smale condition is the crux of this argument.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 8. Existence of minimal blowup solutions and their properties

Abstract
In this section we prove the existence of minimal counterexamples to Theorem 6.1 and we study some of their properties.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 9. Long-time Strichartz estimates and applications

Abstract
In this section, we prove a long-time Strichartz inequality for solutions to (8.15) as described in Theorem 8.10. This will then be used to rule out rapid frequency cascade solutions, namely, solutions which also satisfy
$$ \int_0^{T_{\max } } {N\left( t \right)^{ - 1} \,dt\,<\;\infty }. $$
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 10. Frequency-localized interaction Morawetz inequalities and applications

Abstract
Our goal in this section is to prove a frequency-localized interaction Morawetz inequality. This will then be used to preclude the existence of almost periodic solutions as in Theorem 8.10 for which \(\int_0^{T_{\max } } {N\left( t \right)^{ - 1} \,dt\,=\;\infty }\). These results appear in [40]; we review the proof below.
Herbert Koch, Daniel Tataru, Monica Vişan

Chapter 11. Appendix A: Background material

Abstract
Recall that by the Arzelà–Ascoli theorem, a family of continuous functions on a compact set \(K\;\subset\;\mathbb{R}^d\) is precompact in \(C^{0}(K)\)if and only if it is uniformly bounded and equicontinuous. The natural generalization to L p spaces is due to M.
Herbert Koch, Daniel Tataru, Monica Vişan

Backmatter

Weitere Informationen

Premium Partner

    Bildnachweise