Programme

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Core Documentation

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Type of delivery of the course

Frontal teaching.

Attendance

Attendance is strongly recommended.

Type of evaluation

Oral examination on the program of the course.

Programme

Fourier analysi, distribution theory, Sobolev spaces.
Applications to the study of linear and non linear dispersive partial differential equations

Core Documentation

J-M Bony - Cours d'Analyse - Théorie des distributions at analyse de Fourier- Editions de l'École polytecnique
F. Linares, G. Ponce - Introduction to Nonlinear dispersive equations - Springer

Type of delivery of the course

Lectures on theory and exercises will be held.

Attendance

Attendance is not compulsory but is nevertheless strongly recommended.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.

Programme

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Core Documentation

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Type of delivery of the course

Frontal teaching.

Attendance

Attendance is strongly recommended.

Type of evaluation

Oral examination on the program of the course.

Programme

Fourier analysi, distribution theory, Sobolev spaces.
Applications to the study of linear and non linear dispersive partial differential equations

Core Documentation

J-M Bony - Cours d'Analyse - Théorie des distributions at analyse de Fourier- Editions de l'École polytecnique
F. Linares, G. Ponce - Introduction to Nonlinear dispersive equations - Springer

Type of delivery of the course

Lectures on theory and exercises will be held.

Attendance

Attendance is not compulsory but is nevertheless strongly recommended.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.

Programme

1. Theory of distributions

Definition and convergence of distributions.
Derivative of a distribution.
Spaces of distributions.
Convolutions.
Schwartz space, tempered distributions.
Fourier transform of tempered distributions.
Sobolev spaces and pseudo-differential operators.

2. Application to Schrödinger equations

Linear Schrödinger equation, local and global smoothing effects.
Nonlinear Schrödinger equation, local theory in L^2, H^1, H^2.
Asymptotic results and singularity formation for NLS.

Core Documentation

Bony - Cours d'analyse, Théorie des distributions et analyse de Fourier
Ponce, Linares - Introduction to nonlinear dispersive equations

Type of delivery of the course

Frontal teaching.

Attendance

Attendance is strongly recommended.

Type of evaluation

Oral examination on the program of the course.

Programme

Fourier analysi, distribution theory, Sobolev spaces.
Applications to the study of linear and non linear dispersive partial differential equations

Core Documentation

J-M Bony - Cours d'Analyse - Théorie des distributions at analyse de Fourier- Editions de l'École polytecnique
F. Linares, G. Ponce - Introduction to Nonlinear dispersive equations - Springer

Type of delivery of the course

Lectures on theory and exercises will be held.

Attendance

Attendance is not compulsory but is nevertheless strongly recommended.

Type of evaluation

An oral test will also be held to evaluate the knowledge of the course topics.