**Speaker(s): ** Jacek Jendrej (CNRS and Université Paris 13).

**Time: **October 25 - December 13, 2018

**Venue: ** Room 78201, Jingchunyuan 78, BICMR

**Time: 14:00-17:00, every Thursday, from October 25th, 2018 to December 13th, 2018**** **

The aim of the course is to present selected methods in the study of nonlinear wave equations, which are among the simplest nonlinear models considered in the field theory.

In the first part, we will study the Cauchy problem, that is existence, uniqueness and stability of solutions on short time intervals. To this end, we will introduce/recall some tools from harmonic analysis (the Littlewood-Paley theory) and prove the classical Strichartz estimates for the linear wave equation.

In the second part, we will focus on a particular model, namely the equivariant wave maps equation from the Minkowski plane R1+2, with values in a rotationally symmetric manifold (for instance the two-dimensional sphere S2). Our goal is to understand dynamics of solutions, which means answering questions like:

1. do solutions exist globally or can they break down in finite time?

2. if they exist globally, can we describe their asymptotic behaviour for large times?

3. if they break down, how do they look like as the time approaches the final time of existence?

**Content.**

Part 1.

1. General introduction

2. Fourier transform, Function spaces, Littlewood-Paley decomposition

3. Strichartz estimates for the wave equation

4. Local well-posedness for nonlinear wave equations

Part 2.

1. Harmonic maps and bubbling (after Christodoulou, Shatah, Struwe, Tahvildar-Zadeh)

2. Profile decomposition (after Bahouri and Gérard)

3. Refined threshold theorem (after Côte, Kenig, Lawrie and Schlag)

4. Description of threshold energy solutions (after J. and Lawrie)

Prerequisites. Undergraduate level courses in Real Analysis, ODEs and PDEs (including the Fourier transform in Rn).

**Lecture note**