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Dispersion Phenomena in Partial Differential Equations

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

The Mechanism of Dispersion

Strichartz Estimates

The Nonlinear Wave Equation

The Nonlinear Schrödinger Equation

Future Directions

Bibliography

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Abbreviations

Notations:

Partial derivatives are written as u t or \({\partial_{t}u}\), \({\partial^{\alpha}= \partial_{x_{1}}^{\alpha_{1}}\cdot\partial_{x_{n}}^{\alpha_{n}}}\), the Fourier transform of a function is defined as

$$ \mathcal{F}f(\xi)=\widehat f(\xi)=\int \text{e}^{-ix \cdot\xi}f(x)\text{d} x $$

and we frequently use the mute constant notation \({A \lesssim B}\) to mean \({A\le C B}\) for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context).

Evolution equations:

Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t.

Cauchy problem:

A system of evolution equations, combined with a set of initial conditions at an initial time \({t=t_{0}}\). The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem.

Blow up:

In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the \({L^{\infty}}\) norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution.

Sobolev spaces:

we shall use two instances of Sobolev space: the space \({W^{k,1}}\) with norm

$$ \|u\|_{W^{k,1}}=\sum_{|\alpha|\le k}\|\partial^{\alpha}u\|_{L^{1}}$$

and the space \({H^{s}_{q}}\) with norm

$$ \|u\|_{H^{s}}=\left\|(1-\Delta)^{s/2}u\right\|_{L^{q}}\:. $$

Recall that this definition does not reduce to the preceding one when \({q=1}\). We shall also use the homogeneous space \({\dot H^{s}_{q}}\), with norm

$$ \left\|u\right\|_{\dot H^{s}}=\left\|(-\Delta)^{s/2}u\right\|_{L^{q}}\:. $$
Dispersive estimate:

a pointwise decay estimate of the form \({|u(t,x)|\le Ct^{-\alpha}}\), usually for the solution of a partial differential equation.

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy

    Piero D'Ancona

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  1. Piero D'Ancona
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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D'Ancona, P. (2012). Dispersion Phenomena in Partial Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_11

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