## Tesi etd-01142015-154827 |

Tipo di tesi

Tesi di laurea magistrale

Autore

HIENTZSCH, LARS ERIC

URN

etd-01142015-154827

Titolo

Bilinear Estimates in Bourgain Spaces with Applications to the periodic KdV and Ostrovsky Equation

Struttura

MATEMATICA

Corso di studi

MATEMATICA

Commissione

**relatore**Prof. Georgiev, Vladimir

**controrelatore**Prof. Berselli, Luigi Carlo

Parole chiave

- bilinear estimates
- Bourgain spaces
- well-posedness
- low regularity
- periodic Korteweg-de Vries (KdV) equation
- periodic Ostrovsky equation
- Cauchy Problem
- nonlinear dispersive equation

Data inizio appello

30/01/2015;

Disponibilità

completa

Riassunto analitico

This work is devoted to the study of Cauchy Problems for nonlinear periodic evolution equations with low regularity initial data. Our concern are the local and global well-posedness for KdV and Ostrovsky equation.

We give a general introduction to the classic Fourier restriction method due to Bourgain for dispersive equations that reduces the well-posedness problem to multilinear estimates.

The former equation is known to be globally well-posed in the L^2-based Sobolev space H^s for s>=-1/2. We re-prove the local result without using Strichartz estimates following Kenig, Ponce, Vega.

The latter equation is a perturbation of the former. It was derived by L.A. Ostrovsky as model for nonlinear surface waves in the ocean in a rotating frame of reference. We develop a bilinear estimate in the respective Bourgain spaces. This allows us to implement an iterative method that leads to local well-posedness for the periodic Ostrovsky equation in H^s with s>=-1/2 provided that the initial data is of 0-mean and its H^s norm is sufficiently small.

We give a general introduction to the classic Fourier restriction method due to Bourgain for dispersive equations that reduces the well-posedness problem to multilinear estimates.

The former equation is known to be globally well-posed in the L^2-based Sobolev space H^s for s>=-1/2. We re-prove the local result without using Strichartz estimates following Kenig, Ponce, Vega.

The latter equation is a perturbation of the former. It was derived by L.A. Ostrovsky as model for nonlinear surface waves in the ocean in a rotating frame of reference. We develop a bilinear estimate in the respective Bourgain spaces. This allows us to implement an iterative method that leads to local well-posedness for the periodic Ostrovsky equation in H^s with s>=-1/2 provided that the initial data is of 0-mean and its H^s norm is sufficiently small.

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abstract...tzsch.pdf | 157.73 Kb |

thesisHientzsch.pdf | 1.07 Mb |

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